diff --git a/lean/OAI/Probability/SKValue/GroundState/GaussianTree.lean b/lean/OAI/Probability/SKValue/GroundState/GaussianTree.lean
index fc00b833ffc3a628b94a8f375bc767d94471aec9..8e59e8590dba012e264d9aa2ff4fae1b59675ef4 100644
--- a/lean/OAI/Probability/SKValue/GroundState/GaussianTree.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/GaussianTree.lean
@@ -1,149 +1,149 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.AffineComparison
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-structure TreeLevel where
-  branchExcess : ℕ
-  endpoint : ℝ
-  height : ℝ
-
-abbrev GaussianTree := List TreeLevel
-
-abbrev TreeLeaves : GaussianTree → Type
-  | [] => PUnit
-  | l::T => Fin (l.branchExcess+1) × TreeLeaves T
-
-abbrev TreeEdges : GaussianTree → Type
-  | [] => PEmpty
-  | l::T => Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T)
-
-instance treeLeavesFintype (T : GaussianTree) : Fintype (TreeLeaves T) := by
-  induction T with
-  | nil => exact inferInstanceAs (Fintype PUnit)
-  | cons l T ih => exact inferInstanceAs (Fintype (Fin (l.branchExcess+1) × TreeLeaves T))
-instance treeLeavesNonempty (T : GaussianTree) : Nonempty (TreeLeaves T) := by
-  induction T with
-  | nil => exact inferInstanceAs (Nonempty PUnit)
-  | cons l T ih => exact inferInstanceAs (Nonempty (Fin (l.branchExcess+1) × TreeLeaves T))
-instance treeEdgesFintype (T : GaussianTree) : Fintype (TreeEdges T) := by
-  induction T with
-  | nil => exact inferInstanceAs (Fintype PEmpty)
-  | cons l T ih => exact inferInstanceAs (Fintype (Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T)))
-
-noncomputable def treePathCoeff (c : ℝ → TreeLevel → ℝ) (q : ℝ) :
-    (T : GaussianTree) → TreeLeaves T → TreeEdges T → ℝ
-  | [],_,e => PEmpty.elim e
-  | l::T,α,e => match e with
-    | Sum.inl i => if α.1=i then c q l else 0
-    | Sum.inr (i,e) => if α.1=i then treePathCoeff c l.endpoint T α.2 e else 0
-
-noncomputable def treeOverlap (q : ℝ) : (T : GaussianTree) → TreeLeaves T → TreeLeaves T → ℝ
-  | [],_,_ => q
-  | l::T,α,β => if α.1=β.1 then treeOverlap l.endpoint T α.2 β.2 else q
-
-def ValidTreeGrid (q : ℝ) : GaussianTree → Prop
-  | [] => q=1
-  | l::T => q≤l.endpoint ∧ ValidTreeGrid l.endpoint T
-
-lemma treePath_inner_node (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
-    (α β : TreeLeaves (l::T)) :
-    (∑ e,treePathCoeff c q (l::T) α e*treePathCoeff c q (l::T) β e)=
-    if α.1=β.1 then (c q l)^2+∑ e,treePathCoeff c l.endpoint T α.2 e*treePathCoeff c l.endpoint T β.2 e else 0 := by
-  classical
-  change (∑ e : Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T),_)=_
-  rw [Fintype.sum_sum_type,Fintype.sum_prod_type]
-  simp only [treePathCoeff]
-  by_cases h : α.1=β.1
-  · simp only [h,ite_true]
-    simp [←h,ite_mul,mul_ite,sq]
-  · simp only [h,ite_false]
-    have he (i : Fin (l.branchExcess+1)) : ¬(α.1=i ∧ β.1=i) := by
-      rintro ⟨ha,hb⟩; exact h (ha.trans hb.symm)
-    simp_rw [ite_mul,zero_mul]
-    simp only [mul_ite,mul_zero]
-    have hzero (i : Fin (l.branchExcess+1)) (r s : ℝ) :
-        (if α.1=i then if β.1=i then r*s else 0 else 0)=0 := by
-      split_ifs with ha hb <;> try rfl
-      exact (he i ⟨ha,hb⟩).elim
-    simp only [hzero,Finset.sum_const_zero,add_zero]
-
-noncomputable def treeFieldVector (q : ℝ) (T : GaussianTree) : TreeLeaves T → TreeEdges T → ℝ :=
-  treePathCoeff (fun q l ↦ Real.sqrt (l.endpoint-q)) q T
-
-noncomputable def treeAuxVector (n : ℕ) (q : ℝ) (T : GaussianTree) : TreeLeaves T → TreeEdges T → ℝ :=
-  treePathCoeff (fun q l ↦ Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)) q T
-
-lemma treeField_inner (q : ℝ) (T : GaussianTree) (hT : ValidTreeGrid q T)
-    (α β : TreeLeaves T) :
-    (∑ e,treeFieldVector q T α e*treeFieldVector q T β e)=treeOverlap q T α β-q := by
-  induction T generalizing q with
-  | nil =>
-    simp only [treeFieldVector,treePathCoeff,treeOverlap,sub_self]
-    exact Finset.sum_eq_zero (by intro e he; exact PEmpty.elim e)
-  | cons l T ih =>
-    change q≤l.endpoint ∧ ValidTreeGrid l.endpoint T at hT
-    rw [treeFieldVector,treePath_inner_node]
-    dsimp [treeOverlap]
-    split_ifs with h
-    · rw [Real.sq_sqrt (sub_nonneg.mpr hT.1)]
-      have hh := ih l.endpoint hT.2 α.2 β.2
-      change (l.endpoint-q)+(∑ e,treeFieldVector l.endpoint T α.2 e*treeFieldVector l.endpoint T β.2 e)=_
-      rw [hh]; ring
-    · ring
-
-lemma treeAux_inner (n : ℕ) (q : ℝ) (hq : 0≤q) (T : GaussianTree) (hT : ValidTreeGrid q T)
-    (α β : TreeLeaves T) :
-    (∑ e,treeAuxVector n q T α e*treeAuxVector n q T β e)=
-      ((n : ℝ)/2)*((treeOverlap q T α β)^2-q^2) := by
-  induction T generalizing q with
-  | nil =>
-    simp only [treeAuxVector,treePathCoeff,treeOverlap,sub_self,mul_zero]
-    exact Finset.sum_eq_zero (by intro e he; exact PEmpty.elim e)
-  | cons l T ih =>
-    change q≤l.endpoint ∧ ValidTreeGrid l.endpoint T at hT
-    rw [treeAuxVector,treePath_inner_node]
-    dsimp [treeOverlap]
-    split_ifs with h
-    · rw [mul_pow,Real.sq_sqrt (by positivity),Real.sq_sqrt (by nlinarith [hT.1])]
-      have hh := ih l.endpoint (hq.trans hT.1) hT.2 α.2 β.2
-      change ((n : ℝ)/2)*(l.endpoint^2-q^2)+
-        (∑ e,treeAuxVector n l.endpoint T α.2 e*treeAuxVector n l.endpoint T β.2 e)=_
-      rw [hh]; ring
-    · ring
-
-lemma treeOverlap_self (q : ℝ) (T : GaussianTree) (hT : ValidTreeGrid q T) (α : TreeLeaves T) :
-    treeOverlap q T α α=1 := by
-  induction T generalizing q with
-  | nil => exact hT
-  | cons l T ih => simp only [treeOverlap,ite_true]; exact ih l.endpoint hT.2 α.2
-
-lemma treeField_unit (T : GaussianTree) (hT : ValidTreeGrid 0 T) (α : TreeLeaves T) :
-    (∑ e,(treeFieldVector 0 T α e)^2)=1 := by
-  simp only [sq]
-  rw [treeField_inner 0 T hT,treeOverlap_self 0 T hT]
-  norm_num
-
-lemma treeAux_distance (n : ℕ) (T : GaussianTree) (hT : ValidTreeGrid 0 T)
-    (α β : TreeLeaves T) :
-    (∑ e,(treeAuxVector n 0 T α e-treeAuxVector n 0 T β e)^2)=
-      (n : ℝ)*(1-(treeOverlap 0 T α β)^2) := by
-  rw [sq_distance_sum]
-  simp only [sq]
-  rw [treeAux_inner n 0 (by norm_num) T hT,
-    treeAux_inner n 0 (by norm_num) T hT,
-    treeAux_inner n 0 (by norm_num) T hT,
-    treeOverlap_self 0 T hT,treeOverlap_self 0 T hT]
-  ring
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.AffineComparison
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+structure TreeLevel where
+  branchExcess : ℕ
+  endpoint : ℝ
+  height : ℝ
+
+abbrev GaussianTree := List TreeLevel
+
+abbrev TreeLeaves : GaussianTree → Type
+  | [] => PUnit
+  | l::T => Fin (l.branchExcess+1) × TreeLeaves T
+
+abbrev TreeEdges : GaussianTree → Type
+  | [] => PEmpty
+  | l::T => Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T)
+
+instance treeLeavesFintype (T : GaussianTree) : Fintype (TreeLeaves T) := by
+  induction T with
+  | nil => exact inferInstanceAs (Fintype PUnit)
+  | cons l T ih => exact inferInstanceAs (Fintype (Fin (l.branchExcess+1) × TreeLeaves T))
+instance treeLeavesNonempty (T : GaussianTree) : Nonempty (TreeLeaves T) := by
+  induction T with
+  | nil => exact inferInstanceAs (Nonempty PUnit)
+  | cons l T ih => exact inferInstanceAs (Nonempty (Fin (l.branchExcess+1) × TreeLeaves T))
+instance treeEdgesFintype (T : GaussianTree) : Fintype (TreeEdges T) := by
+  induction T with
+  | nil => exact inferInstanceAs (Fintype PEmpty)
+  | cons l T ih => exact inferInstanceAs (Fintype (Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T)))
+
+noncomputable def treePathCoeff (c : ℝ → TreeLevel → ℝ) (q : ℝ) :
+    (T : GaussianTree) → TreeLeaves T → TreeEdges T → ℝ
+  | [],_,e => PEmpty.elim e
+  | l::T,α,e => match e with
+    | Sum.inl i => if α.1=i then c q l else 0
+    | Sum.inr (i,e) => if α.1=i then treePathCoeff c l.endpoint T α.2 e else 0
+
+noncomputable def treeOverlap (q : ℝ) : (T : GaussianTree) → TreeLeaves T → TreeLeaves T → ℝ
+  | [],_,_ => q
+  | l::T,α,β => if α.1=β.1 then treeOverlap l.endpoint T α.2 β.2 else q
+
+def ValidTreeGrid (q : ℝ) : GaussianTree → Prop
+  | [] => q=1
+  | l::T => q≤l.endpoint ∧ ValidTreeGrid l.endpoint T
+
+lemma treePath_inner_node (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
+    (α β : TreeLeaves (l::T)) :
+    (∑ e,treePathCoeff c q (l::T) α e*treePathCoeff c q (l::T) β e)=
+    if α.1=β.1 then (c q l)^2+∑ e,treePathCoeff c l.endpoint T α.2 e*treePathCoeff c l.endpoint T β.2 e else 0 := by
+  classical
+  change (∑ e : Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T),_)=_
+  rw [Fintype.sum_sum_type,Fintype.sum_prod_type]
+  simp only [treePathCoeff]
+  by_cases h : α.1=β.1
+  · simp only [h,ite_true]
+    simp [←h,ite_mul,mul_ite,sq]
+  · simp only [h,ite_false]
+    have he (i : Fin (l.branchExcess+1)) : ¬(α.1=i ∧ β.1=i) := by
+      rintro ⟨ha,hb⟩; exact h (ha.trans hb.symm)
+    simp_rw [ite_mul,zero_mul]
+    simp only [mul_ite,mul_zero]
+    have hzero (i : Fin (l.branchExcess+1)) (r s : ℝ) :
+        (if α.1=i then if β.1=i then r*s else 0 else 0)=0 := by
+      split_ifs with ha hb <;> try rfl
+      exact (he i ⟨ha,hb⟩).elim
+    simp only [hzero,Finset.sum_const_zero,add_zero]
+
+noncomputable def treeFieldVector (q : ℝ) (T : GaussianTree) : TreeLeaves T → TreeEdges T → ℝ :=
+  treePathCoeff (fun q l ↦ Real.sqrt (l.endpoint-q)) q T
+
+noncomputable def treeAuxVector (n : ℕ) (q : ℝ) (T : GaussianTree) : TreeLeaves T → TreeEdges T → ℝ :=
+  treePathCoeff (fun q l ↦ Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)) q T
+
+lemma treeField_inner (q : ℝ) (T : GaussianTree) (hT : ValidTreeGrid q T)
+    (α β : TreeLeaves T) :
+    (∑ e,treeFieldVector q T α e*treeFieldVector q T β e)=treeOverlap q T α β-q := by
+  induction T generalizing q with
+  | nil =>
+    simp only [treeFieldVector,treePathCoeff,treeOverlap,sub_self]
+    exact Finset.sum_eq_zero (by intro e he; exact PEmpty.elim e)
+  | cons l T ih =>
+    change q≤l.endpoint ∧ ValidTreeGrid l.endpoint T at hT
+    rw [treeFieldVector,treePath_inner_node]
+    dsimp [treeOverlap]
+    split_ifs with h
+    · rw [Real.sq_sqrt (sub_nonneg.mpr hT.1)]
+      have hh := ih l.endpoint hT.2 α.2 β.2
+      change (l.endpoint-q)+(∑ e,treeFieldVector l.endpoint T α.2 e*treeFieldVector l.endpoint T β.2 e)=_
+      rw [hh]; ring
+    · ring
+
+lemma treeAux_inner (n : ℕ) (q : ℝ) (hq : 0≤q) (T : GaussianTree) (hT : ValidTreeGrid q T)
+    (α β : TreeLeaves T) :
+    (∑ e,treeAuxVector n q T α e*treeAuxVector n q T β e)=
+      ((n : ℝ)/2)*((treeOverlap q T α β)^2-q^2) := by
+  induction T generalizing q with
+  | nil =>
+    simp only [treeAuxVector,treePathCoeff,treeOverlap,sub_self,mul_zero]
+    exact Finset.sum_eq_zero (by intro e he; exact PEmpty.elim e)
+  | cons l T ih =>
+    change q≤l.endpoint ∧ ValidTreeGrid l.endpoint T at hT
+    rw [treeAuxVector,treePath_inner_node]
+    dsimp [treeOverlap]
+    split_ifs with h
+    · rw [mul_pow,Real.sq_sqrt (by positivity),Real.sq_sqrt (by nlinarith [hT.1])]
+      have hh := ih l.endpoint (hq.trans hT.1) hT.2 α.2 β.2
+      change ((n : ℝ)/2)*(l.endpoint^2-q^2)+
+        (∑ e,treeAuxVector n l.endpoint T α.2 e*treeAuxVector n l.endpoint T β.2 e)=_
+      rw [hh]; ring
+    · ring
+
+lemma treeOverlap_self (q : ℝ) (T : GaussianTree) (hT : ValidTreeGrid q T) (α : TreeLeaves T) :
+    treeOverlap q T α α=1 := by
+  induction T generalizing q with
+  | nil => exact hT
+  | cons l T ih => simp only [treeOverlap,ite_true]; exact ih l.endpoint hT.2 α.2
+
+lemma treeField_unit (T : GaussianTree) (hT : ValidTreeGrid 0 T) (α : TreeLeaves T) :
+    (∑ e,(treeFieldVector 0 T α e)^2)=1 := by
+  simp only [sq]
+  rw [treeField_inner 0 T hT,treeOverlap_self 0 T hT]
+  norm_num
+
+lemma treeAux_distance (n : ℕ) (T : GaussianTree) (hT : ValidTreeGrid 0 T)
+    (α β : TreeLeaves T) :
+    (∑ e,(treeAuxVector n 0 T α e-treeAuxVector n 0 T β e)^2)=
+      (n : ℝ)*(1-(treeOverlap 0 T α β)^2) := by
+  rw [sq_distance_sum]
+  simp only [sq]
+  rw [treeAux_inner n 0 (by norm_num) T hT,
+    treeAux_inner n 0 (by norm_num) T hT,
+    treeAux_inner n 0 (by norm_num) T hT,
+    treeOverlap_self 0 T hT,treeOverlap_self 0 T hT]
+  ring
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/GramSchmidtLimit.lean b/lean/OAI/Probability/SKValue/GroundState/GramSchmidtLimit.lean
index 522e34d7ae349d6ce31075ac1e6282e16749f060..f067f3f49397f966a4f4341eefa2983691bbd43a 100644
--- a/lean/OAI/Probability/SKValue/GroundState/GramSchmidtLimit.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/GramSchmidtLimit.lean
@@ -1,182 +1,182 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.GaussianColumn
-
-namespace OAI
-
-section
-
-open Filter Set InnerProductSpace
-open scoped Topology BigOperators RealInnerProductSpace
-namespace SKValueG
-variable {E : ℕ → Type*} [∀ n, NormedAddCommGroup (E n)] [∀ n, InnerProductSpace ℝ (E n)]
-
-lemma inner_approx_tendsto {x x' y : (n : ℕ) → E n} {a b : ℝ}
-    (hx : Tendsto (fun n ↦ ‖x n-x' n‖) atTop (𝓝 0))
-    (hy : Tendsto (fun n ↦ ‖y n‖) atTop (𝓝 b))
-    (hxy : Tendsto (fun n ↦ ⟪x' n,y n⟫) atTop (𝓝 a)) :
-    Tendsto (fun n ↦ ⟪x n,y n⟫) atTop (𝓝 a) := by
-  have hdiff : Tendsto (fun n ↦ ⟪x n,y n⟫-⟪x' n,y n⟫) atTop (𝓝 0) := by
-    apply squeeze_zero_norm (fun n ↦ ?_) (by simpa using hx.mul hy)
-    rw [←inner_sub_left,Real.norm_eq_abs]
-    exact abs_real_inner_le_norm _ _
-  simpa only [sub_add_cancel,zero_add] using hdiff.add hxy
-
-omit [∀ n, InnerProductSpace ℝ (E n)] in
-lemma norm_approx_tendsto {x x' : (n : ℕ) → E n} {a : ℝ}
-    (hx : Tendsto (fun n ↦ ‖x n-x' n‖) atTop (𝓝 0))
-    (hx' : Tendsto (fun n ↦ ‖x' n‖) atTop (𝓝 a)) :
-    Tendsto (fun n ↦ ‖x n‖) atTop (𝓝 a) := by
-  have hd : Tendsto (fun n ↦ ‖x n‖-‖x' n‖) atTop (𝓝 0) := by
-    exact squeeze_zero_norm (fun n ↦ by simpa only [Real.norm_eq_abs] using abs_norm_sub_norm_le (x n) (x' n)) hx
-  simpa only [sub_add_cancel,zero_add] using hd.add hx'
-
-theorem gramSchmidt_approx (a : (n : ℕ) → ℕ → E n) (K : ℕ)
-    (hn : ∀ i≤K,Tendsto (fun n ↦ ‖a n i‖) atTop (𝓝 1))
-    (hi : ∀ i j,i<j → j≤K → Tendsto (fun n ↦ ⟪a n i,a n j⟫) atTop (𝓝 0))
-    (j : ℕ) (hj : j≤K) :
-    Tendsto (fun n ↦ ‖gramSchmidt ℝ (a n) j-a n j‖) atTop (𝓝 0) := by
-  induction j using Nat.strong_induction_on with
-  | h j ih =>
-    have hnorm (i : ℕ) (hij : i<j) :
-        Tendsto (fun n ↦ ‖gramSchmidt ℝ (a n) i‖) atTop (𝓝 1) :=
-      norm_approx_tendsto (ih i hij (hij.le.trans hj)) (hn i (hij.le.trans hj))
-    have hinner (i : ℕ) (hij : i<j) :
-        Tendsto (fun n ↦ ⟪gramSchmidt ℝ (a n) i,a n j⟫) atTop (𝓝 0) :=
-      inner_approx_tendsto (ih i hij (hij.le.trans hj)) (hn j hj) (hi i j hij hj)
-    have hterm (i : ℕ) (hij : i<j) :
-        Tendsto (fun n ↦ ‖(⟪gramSchmidt ℝ (a n) i,a n j⟫ /
-          ‖gramSchmidt ℝ (a n) i‖^2) • gramSchmidt ℝ (a n) i‖) atTop (𝓝 0) := by
-      simp only [norm_smul,Real.norm_eq_abs]
-      have h := ((hinner i hij).div ((hnorm i hij).pow 2) (by norm_num)).abs.mul (hnorm i hij)
-      simpa using h
-    have hs : Tendsto (fun n ↦ ∑ i∈Finset.range j,
-        ‖(⟪gramSchmidt ℝ (a n) i,a n j⟫/‖gramSchmidt ℝ (a n) i‖^2) •
-          gramSchmidt ℝ (a n) i‖) atTop (𝓝 0) := by
-      simpa only [Finset.sum_const_zero] using tendsto_finsetSum (Finset.range j)
-        (fun i hi ↦ hterm i (Finset.mem_range.mp hi))
-    apply squeeze_zero (fun n ↦ norm_nonneg _) (fun n ↦ ?_) hs
-    have he := gramSchmidt_def'' ℝ (a n) j
-    have he' : gramSchmidt ℝ (a n) j-a n j =
-        -(∑ i∈Finset.range j,(⟪gramSchmidt ℝ (a n) i,a n j⟫ /
-          ‖gramSchmidt ℝ (a n) i‖^2) • gramSchmidt ℝ (a n) i) := by
-      simp only [Nat.Iio_eq_range] at he
-      calc
-        _ = gramSchmidt ℝ (a n) j-(gramSchmidt ℝ (a n) j+_) := congrArg (fun x ↦ gramSchmidt ℝ (a n) j-x) he
-        _ = _ := by abel
-    rw [he',norm_neg]
-    exact norm_sum_le _ _
-
-theorem gramSchmidtNormed_approx (a : (n : ℕ) → ℕ → E n) (K : ℕ)
-    (hn : ∀ i≤K,Tendsto (fun n ↦ ‖a n i‖) atTop (𝓝 1))
-    (hi : ∀ i j,i<j → j≤K → Tendsto (fun n ↦ ⟪a n i,a n j⟫) atTop (𝓝 0))
-    (j : ℕ) (hj : j≤K) :
-    Tendsto (fun n ↦ ‖gramSchmidtNormed ℝ (a n) j-a n j‖) atTop (𝓝 0) := by
-  have hg := gramSchmidt_approx a K hn hi j hj
-  have hgn := norm_approx_tendsto hg (hn j hj)
-  have hscale : Tendsto (fun n ↦ |‖gramSchmidt ℝ (a n) j‖⁻¹-1| *
-      ‖gramSchmidt ℝ (a n) j‖) atTop (𝓝 0) := by
-    simpa using ((hgn.inv₀ (by norm_num)).sub_const 1).abs.mul hgn
-  apply squeeze_zero (fun n ↦ norm_nonneg _) (fun n ↦ ?_) (by simpa using hscale.add hg)
-  calc
-    _ ≤ ‖gramSchmidtNormed ℝ (a n) j-gramSchmidt ℝ (a n) j‖+
-        ‖gramSchmidt ℝ (a n) j-a n j‖ := by
-      calc
-        _ = ‖(gramSchmidtNormed ℝ (a n) j-gramSchmidt ℝ (a n) j)+(gramSchmidt ℝ (a n) j-a n j)‖ := by congr 1; abel
-        _ ≤ _ := norm_add_le _ _
-    _ = _ := by
-      congr 1
-      change ‖‖gramSchmidt ℝ (a n) j‖⁻¹ • gramSchmidt ℝ (a n) j-gramSchmidt ℝ (a n) j‖ = _
-      have he : ‖gramSchmidt ℝ (a n) j‖⁻¹ • gramSchmidt ℝ (a n) j-gramSchmidt ℝ (a n) j =
-          (‖gramSchmidt ℝ (a n) j‖⁻¹-1) • gramSchmidt ℝ (a n) j := by rw [sub_smul,one_smul]
-      rw [he,norm_smul,Real.norm_eq_abs]
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
-open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
-namespace SKValueG
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
-
-noncomputable def orthogonalRemainder (v : ℕ → E) (w : E) (j : ℕ) : E :=
-  w-∑ i∈Finset.range j,(⟪v i,w⟫ : ℝ) • v i
-
-lemma inner_orthogonalRemainder (v : ℕ → E) (w : E) (j : ℕ)
-    (ho : ∀ i<j,⟪v j,v i⟫=0) :
-    ⟪v j,orthogonalRemainder v w j⟫=⟪v j,w⟫ := by
-  simp only [orthogonalRemainder,inner_sub_right,inner_sum,real_inner_smul_right]
-  have hz : (∑ i∈Finset.range j,⟪v i,w⟫*⟪v j,v i⟫)=0 := by
-    apply Finset.sum_eq_zero
-    intro i hi
-    simp [ho i (Finset.mem_range.mp hi)]
-  rw [hz,sub_zero]
-
-lemma orthogonalRemainder_succ (v : ℕ → E) (w : E) (j : ℕ) :
-    orthogonalRemainder v w (j+1)=orthogonalRemainder v w j-⟪v j,w⟫ • v j := by
-  simp only [orthogonalRemainder,Finset.sum_range_succ]
-  abel
-
-lemma columnResidual_orthogonalRemainder {κ : Type*} [Fintype κ]
-    (v : ℕ → EuclideanSpace ℝ κ) (w : EuclideanSpace ℝ κ) (j : ℕ)
-    (ho : ∀ i<j,⟪v j,v i⟫=0) :
-    columnResidual (v j) (orthogonalRemainder v w j)=orthogonalRemainder v w (j+1) := by
-  rw [columnResidual,inner_orthogonalRemainder v w j ho,orthogonalRemainder_succ]
-
-lemma column_revealed_energy {κ : Type*} [Fintype κ]
-    (v : ℕ → EuclideanSpace ℝ κ) (w z : EuclideanSpace ℝ κ) (j : ℕ)
-    (ho : ∀ i<j,⟪v j,v i⟫=0) :
-    ⟪columnVector (v j) (orthogonalRemainder v w j),z⟫=
-      ⟪v j,w⟫*⟪orthogonalRemainder v w (j+1),z⟫+
-      (⟪v j,w⟫^2/Real.sqrt 2)*⟪v j,z⟫ := by
-  simp only [columnVector,inner_add_left,real_inner_smul_left]
-  rw [inner_orthogonalRemainder v w j ho,columnResidual_orthogonalRemainder v w j ho]
-
-section Limit
-variable {E' : ℕ → Type*} [∀ n,NormedAddCommGroup (E' n)] [∀ n,InnerProductSpace ℝ (E' n)]
-
-lemma orthogonalRemainder_inner_tendsto (v : (n : ℕ) → ℕ → E' n)
-    (w z : (n : ℕ) → E' n) (K : ℕ) (A : ℕ → ℝ) (B : ℝ)
-    (ha : ∀ j<K,Tendsto (fun n ↦ ⟪v n j,w n⟫) atTop (𝓝 (A j)))
-    (hz : ∀ j<K,Tendsto (fun n ↦ ⟪v n j,z n⟫) atTop (𝓝 0))
-    (hwz : Tendsto (fun n ↦ ⟪w n,z n⟫) atTop (𝓝 B)) :
-    Tendsto (fun n ↦ ⟪orthogonalRemainder (v n) (w n) K,z n⟫) atTop (𝓝 B) := by
-  have hs : Tendsto (fun n ↦ ∑ j∈Finset.range K,⟪v n j,w n⟫*⟪v n j,z n⟫) atTop (𝓝 0) := by
-    have hh := tendsto_finsetSum (s:=Finset.range K)
-      (fun j hj ↦ (ha j (Finset.mem_range.mp hj)).mul (hz j (Finset.mem_range.mp hj)))
-    simpa using hh
-  have ht := hwz.sub hs
-  simpa only [orthogonalRemainder,inner_sub_left,sum_inner,real_inner_smul_left,sub_zero] using ht
-
-theorem normalized_column_energy_tendsto (a : (n : ℕ) → ℕ → E' n)
-    (w z : (n : ℕ) → E' n) (K : ℕ) (A : ℕ → ℝ) (B : ℝ)
-    (hn : ∀ i≤K,Tendsto (fun n ↦ ‖a n i‖) atTop (𝓝 1))
-    (hi : ∀ i j,i<j → j≤K → Tendsto (fun n ↦ ⟪a n i,a n j⟫) atTop (𝓝 0))
-    (hwn : Tendsto (fun n ↦ ‖w n‖) atTop (𝓝 1))
-    (hzn : Tendsto (fun n ↦ ‖z n‖) atTop (𝓝 1))
-    (ha : ∀ j≤K,Tendsto (fun n ↦ ⟪a n j,w n⟫) atTop (𝓝 (A j)))
-    (hz : ∀ j≤K,Tendsto (fun n ↦ ⟪a n j,z n⟫) atTop (𝓝 0))
-    (hb : Tendsto (fun n ↦ ⟪w n,z n⟫) atTop (𝓝 B)) :
-    Tendsto (fun n ↦
-      ⟪gramSchmidtNormed ℝ (a n) K,w n⟫*
-        ⟪orthogonalRemainder (gramSchmidtNormed ℝ (a n)) (w n) (K+1),z n⟫+
-      (⟪gramSchmidtNormed ℝ (a n) K,w n⟫^2/Real.sqrt 2)*
-        ⟪gramSchmidtNormed ℝ (a n) K,z n⟫) atTop (𝓝 (A K*B)) := by
-  have hp (j : ℕ) (hj : j≤K) :
-      Tendsto (fun n ↦ ‖gramSchmidtNormed ℝ (a n) j-a n j‖) atTop (𝓝 0) :=
-    gramSchmidtNormed_approx a j (fun i h ↦ hn i (h.trans hj))
-      (fun i l hil hl ↦ hi i l hil (hl.trans hj)) j le_rfl
-  have ha' (j : ℕ) (hj : j≤K) := inner_approx_tendsto (hp j hj) hwn (ha j hj)
-  have hz' (j : ℕ) (hj : j≤K) := inner_approx_tendsto (hp j hj) hzn (hz j hj)
-  have hr := orthogonalRemainder_inner_tendsto (fun n ↦ gramSchmidtNormed ℝ (a n)) w z
-    (K+1) A B (fun j hj ↦ ha' j (by omega)) (fun j hj ↦ hz' j (by omega)) hb
-  simpa only [mul_zero,add_zero] using
-    ((ha' K le_rfl).mul hr).add (((ha' K le_rfl).pow 2 |>.div_const _).mul (hz' K le_rfl))
-end Limit
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.GaussianColumn
+
+namespace OAI
+
+section
+
+open Filter Set InnerProductSpace
+open scoped Topology BigOperators RealInnerProductSpace
+namespace SKValueG
+variable {E : ℕ → Type*} [∀ n, NormedAddCommGroup (E n)] [∀ n, InnerProductSpace ℝ (E n)]
+
+lemma inner_approx_tendsto {x x' y : (n : ℕ) → E n} {a b : ℝ}
+    (hx : Tendsto (fun n ↦ ‖x n-x' n‖) atTop (𝓝 0))
+    (hy : Tendsto (fun n ↦ ‖y n‖) atTop (𝓝 b))
+    (hxy : Tendsto (fun n ↦ ⟪x' n,y n⟫) atTop (𝓝 a)) :
+    Tendsto (fun n ↦ ⟪x n,y n⟫) atTop (𝓝 a) := by
+  have hdiff : Tendsto (fun n ↦ ⟪x n,y n⟫-⟪x' n,y n⟫) atTop (𝓝 0) := by
+    apply squeeze_zero_norm (fun n ↦ ?_) (by simpa using hx.mul hy)
+    rw [←inner_sub_left,Real.norm_eq_abs]
+    exact abs_real_inner_le_norm _ _
+  simpa only [sub_add_cancel,zero_add] using hdiff.add hxy
+
+omit [∀ n, InnerProductSpace ℝ (E n)] in
+lemma norm_approx_tendsto {x x' : (n : ℕ) → E n} {a : ℝ}
+    (hx : Tendsto (fun n ↦ ‖x n-x' n‖) atTop (𝓝 0))
+    (hx' : Tendsto (fun n ↦ ‖x' n‖) atTop (𝓝 a)) :
+    Tendsto (fun n ↦ ‖x n‖) atTop (𝓝 a) := by
+  have hd : Tendsto (fun n ↦ ‖x n‖-‖x' n‖) atTop (𝓝 0) := by
+    exact squeeze_zero_norm (fun n ↦ by simpa only [Real.norm_eq_abs] using abs_norm_sub_norm_le (x n) (x' n)) hx
+  simpa only [sub_add_cancel,zero_add] using hd.add hx'
+
+theorem gramSchmidt_approx (a : (n : ℕ) → ℕ → E n) (K : ℕ)
+    (hn : ∀ i≤K,Tendsto (fun n ↦ ‖a n i‖) atTop (𝓝 1))
+    (hi : ∀ i j,i<j → j≤K → Tendsto (fun n ↦ ⟪a n i,a n j⟫) atTop (𝓝 0))
+    (j : ℕ) (hj : j≤K) :
+    Tendsto (fun n ↦ ‖gramSchmidt ℝ (a n) j-a n j‖) atTop (𝓝 0) := by
+  induction j using Nat.strong_induction_on with
+  | h j ih =>
+    have hnorm (i : ℕ) (hij : i<j) :
+        Tendsto (fun n ↦ ‖gramSchmidt ℝ (a n) i‖) atTop (𝓝 1) :=
+      norm_approx_tendsto (ih i hij (hij.le.trans hj)) (hn i (hij.le.trans hj))
+    have hinner (i : ℕ) (hij : i<j) :
+        Tendsto (fun n ↦ ⟪gramSchmidt ℝ (a n) i,a n j⟫) atTop (𝓝 0) :=
+      inner_approx_tendsto (ih i hij (hij.le.trans hj)) (hn j hj) (hi i j hij hj)
+    have hterm (i : ℕ) (hij : i<j) :
+        Tendsto (fun n ↦ ‖(⟪gramSchmidt ℝ (a n) i,a n j⟫ /
+          ‖gramSchmidt ℝ (a n) i‖^2) • gramSchmidt ℝ (a n) i‖) atTop (𝓝 0) := by
+      simp only [norm_smul,Real.norm_eq_abs]
+      have h := ((hinner i hij).div ((hnorm i hij).pow 2) (by norm_num)).abs.mul (hnorm i hij)
+      simpa using h
+    have hs : Tendsto (fun n ↦ ∑ i∈Finset.range j,
+        ‖(⟪gramSchmidt ℝ (a n) i,a n j⟫/‖gramSchmidt ℝ (a n) i‖^2) •
+          gramSchmidt ℝ (a n) i‖) atTop (𝓝 0) := by
+      simpa only [Finset.sum_const_zero] using tendsto_finsetSum (Finset.range j)
+        (fun i hi ↦ hterm i (Finset.mem_range.mp hi))
+    apply squeeze_zero (fun n ↦ norm_nonneg _) (fun n ↦ ?_) hs
+    have he := gramSchmidt_def'' ℝ (a n) j
+    have he' : gramSchmidt ℝ (a n) j-a n j =
+        -(∑ i∈Finset.range j,(⟪gramSchmidt ℝ (a n) i,a n j⟫ /
+          ‖gramSchmidt ℝ (a n) i‖^2) • gramSchmidt ℝ (a n) i) := by
+      simp only [Nat.Iio_eq_range] at he
+      calc
+        _ = gramSchmidt ℝ (a n) j-(gramSchmidt ℝ (a n) j+_) := congrArg (fun x ↦ gramSchmidt ℝ (a n) j-x) he
+        _ = _ := by abel
+    rw [he',norm_neg]
+    exact norm_sum_le _ _
+
+theorem gramSchmidtNormed_approx (a : (n : ℕ) → ℕ → E n) (K : ℕ)
+    (hn : ∀ i≤K,Tendsto (fun n ↦ ‖a n i‖) atTop (𝓝 1))
+    (hi : ∀ i j,i<j → j≤K → Tendsto (fun n ↦ ⟪a n i,a n j⟫) atTop (𝓝 0))
+    (j : ℕ) (hj : j≤K) :
+    Tendsto (fun n ↦ ‖gramSchmidtNormed ℝ (a n) j-a n j‖) atTop (𝓝 0) := by
+  have hg := gramSchmidt_approx a K hn hi j hj
+  have hgn := norm_approx_tendsto hg (hn j hj)
+  have hscale : Tendsto (fun n ↦ |‖gramSchmidt ℝ (a n) j‖⁻¹-1| *
+      ‖gramSchmidt ℝ (a n) j‖) atTop (𝓝 0) := by
+    simpa using ((hgn.inv₀ (by norm_num)).sub_const 1).abs.mul hgn
+  apply squeeze_zero (fun n ↦ norm_nonneg _) (fun n ↦ ?_) (by simpa using hscale.add hg)
+  calc
+    _ ≤ ‖gramSchmidtNormed ℝ (a n) j-gramSchmidt ℝ (a n) j‖+
+        ‖gramSchmidt ℝ (a n) j-a n j‖ := by
+      calc
+        _ = ‖(gramSchmidtNormed ℝ (a n) j-gramSchmidt ℝ (a n) j)+(gramSchmidt ℝ (a n) j-a n j)‖ := by congr 1; abel
+        _ ≤ _ := norm_add_le _ _
+    _ = _ := by
+      congr 1
+      change ‖‖gramSchmidt ℝ (a n) j‖⁻¹ • gramSchmidt ℝ (a n) j-gramSchmidt ℝ (a n) j‖ = _
+      have he : ‖gramSchmidt ℝ (a n) j‖⁻¹ • gramSchmidt ℝ (a n) j-gramSchmidt ℝ (a n) j =
+          (‖gramSchmidt ℝ (a n) j‖⁻¹-1) • gramSchmidt ℝ (a n) j := by rw [sub_smul,one_smul]
+      rw [he,norm_smul,Real.norm_eq_abs]
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
+open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
+namespace SKValueG
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+noncomputable def orthogonalRemainder (v : ℕ → E) (w : E) (j : ℕ) : E :=
+  w-∑ i∈Finset.range j,(⟪v i,w⟫ : ℝ) • v i
+
+lemma inner_orthogonalRemainder (v : ℕ → E) (w : E) (j : ℕ)
+    (ho : ∀ i<j,⟪v j,v i⟫=0) :
+    ⟪v j,orthogonalRemainder v w j⟫=⟪v j,w⟫ := by
+  simp only [orthogonalRemainder,inner_sub_right,inner_sum,real_inner_smul_right]
+  have hz : (∑ i∈Finset.range j,⟪v i,w⟫*⟪v j,v i⟫)=0 := by
+    apply Finset.sum_eq_zero
+    intro i hi
+    simp [ho i (Finset.mem_range.mp hi)]
+  rw [hz,sub_zero]
+
+lemma orthogonalRemainder_succ (v : ℕ → E) (w : E) (j : ℕ) :
+    orthogonalRemainder v w (j+1)=orthogonalRemainder v w j-⟪v j,w⟫ • v j := by
+  simp only [orthogonalRemainder,Finset.sum_range_succ]
+  abel
+
+lemma columnResidual_orthogonalRemainder {κ : Type*} [Fintype κ]
+    (v : ℕ → EuclideanSpace ℝ κ) (w : EuclideanSpace ℝ κ) (j : ℕ)
+    (ho : ∀ i<j,⟪v j,v i⟫=0) :
+    columnResidual (v j) (orthogonalRemainder v w j)=orthogonalRemainder v w (j+1) := by
+  rw [columnResidual,inner_orthogonalRemainder v w j ho,orthogonalRemainder_succ]
+
+lemma column_revealed_energy {κ : Type*} [Fintype κ]
+    (v : ℕ → EuclideanSpace ℝ κ) (w z : EuclideanSpace ℝ κ) (j : ℕ)
+    (ho : ∀ i<j,⟪v j,v i⟫=0) :
+    ⟪columnVector (v j) (orthogonalRemainder v w j),z⟫=
+      ⟪v j,w⟫*⟪orthogonalRemainder v w (j+1),z⟫+
+      (⟪v j,w⟫^2/Real.sqrt 2)*⟪v j,z⟫ := by
+  simp only [columnVector,inner_add_left,real_inner_smul_left]
+  rw [inner_orthogonalRemainder v w j ho,columnResidual_orthogonalRemainder v w j ho]
+
+section Limit
+variable {E' : ℕ → Type*} [∀ n,NormedAddCommGroup (E' n)] [∀ n,InnerProductSpace ℝ (E' n)]
+
+lemma orthogonalRemainder_inner_tendsto (v : (n : ℕ) → ℕ → E' n)
+    (w z : (n : ℕ) → E' n) (K : ℕ) (A : ℕ → ℝ) (B : ℝ)
+    (ha : ∀ j<K,Tendsto (fun n ↦ ⟪v n j,w n⟫) atTop (𝓝 (A j)))
+    (hz : ∀ j<K,Tendsto (fun n ↦ ⟪v n j,z n⟫) atTop (𝓝 0))
+    (hwz : Tendsto (fun n ↦ ⟪w n,z n⟫) atTop (𝓝 B)) :
+    Tendsto (fun n ↦ ⟪orthogonalRemainder (v n) (w n) K,z n⟫) atTop (𝓝 B) := by
+  have hs : Tendsto (fun n ↦ ∑ j∈Finset.range K,⟪v n j,w n⟫*⟪v n j,z n⟫) atTop (𝓝 0) := by
+    have hh := tendsto_finsetSum (s:=Finset.range K)
+      (fun j hj ↦ (ha j (Finset.mem_range.mp hj)).mul (hz j (Finset.mem_range.mp hj)))
+    simpa using hh
+  have ht := hwz.sub hs
+  simpa only [orthogonalRemainder,inner_sub_left,sum_inner,real_inner_smul_left,sub_zero] using ht
+
+theorem normalized_column_energy_tendsto (a : (n : ℕ) → ℕ → E' n)
+    (w z : (n : ℕ) → E' n) (K : ℕ) (A : ℕ → ℝ) (B : ℝ)
+    (hn : ∀ i≤K,Tendsto (fun n ↦ ‖a n i‖) atTop (𝓝 1))
+    (hi : ∀ i j,i<j → j≤K → Tendsto (fun n ↦ ⟪a n i,a n j⟫) atTop (𝓝 0))
+    (hwn : Tendsto (fun n ↦ ‖w n‖) atTop (𝓝 1))
+    (hzn : Tendsto (fun n ↦ ‖z n‖) atTop (𝓝 1))
+    (ha : ∀ j≤K,Tendsto (fun n ↦ ⟪a n j,w n⟫) atTop (𝓝 (A j)))
+    (hz : ∀ j≤K,Tendsto (fun n ↦ ⟪a n j,z n⟫) atTop (𝓝 0))
+    (hb : Tendsto (fun n ↦ ⟪w n,z n⟫) atTop (𝓝 B)) :
+    Tendsto (fun n ↦
+      ⟪gramSchmidtNormed ℝ (a n) K,w n⟫*
+        ⟪orthogonalRemainder (gramSchmidtNormed ℝ (a n)) (w n) (K+1),z n⟫+
+      (⟪gramSchmidtNormed ℝ (a n) K,w n⟫^2/Real.sqrt 2)*
+        ⟪gramSchmidtNormed ℝ (a n) K,z n⟫) atTop (𝓝 (A K*B)) := by
+  have hp (j : ℕ) (hj : j≤K) :
+      Tendsto (fun n ↦ ‖gramSchmidtNormed ℝ (a n) j-a n j‖) atTop (𝓝 0) :=
+    gramSchmidtNormed_approx a j (fun i h ↦ hn i (h.trans hj))
+      (fun i l hil hl ↦ hi i l hil (hl.trans hj)) j le_rfl
+  have ha' (j : ℕ) (hj : j≤K) := inner_approx_tendsto (hp j hj) hwn (ha j hj)
+  have hz' (j : ℕ) (hj : j≤K) := inner_approx_tendsto (hp j hj) hzn (hz j hj)
+  have hr := orthogonalRemainder_inner_tendsto (fun n ↦ gramSchmidtNormed ℝ (a n)) w z
+    (K+1) A B (fun j hj ↦ ha' j (by omega)) (fun j hj ↦ hz' j (by omega)) hb
+  simpa only [mul_zero,add_zero] using
+    ((ha' K le_rfl).mul hr).add (((ha' K le_rfl).pow 2 |>.div_const _).mul (hz' K le_rfl))
+end Limit
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/GumbelMax.lean b/lean/OAI/Probability/SKValue/GroundState/GumbelMax.lean
index b3629617b0e83f56c67e3296daa231b240eb3bf9..bcb055804c486994ab3ad3816bccba983c9dc26a 100644
--- a/lean/OAI/Probability/SKValue/GroundState/GumbelMax.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/GumbelMax.lean
@@ -1,152 +1,152 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.RotatedCoefficients
-import OAI.Probability.SKValue.GroundState.ExponentialMoments
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology ENNReal NNReal BigOperators
-namespace SKValueG
-
-lemma finiteMaximum_le_iff {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (t : ℝ) :
-    finiteMaximum x ≤ t ↔ ∀ i, x i≤t := by
-  constructor
-  · intro h i; exact (le_finiteMaximum x i).trans h
-  · intro h; obtain ⟨i,hi⟩ := exists_finiteMaximum x; simpa only [←hi] using h i
-
-noncomputable def logSumExp {ι : Type*} [Fintype ι] (d : ι → ℝ) : ℝ :=
-  Real.log (∑ i, Real.exp (d i))
-
-lemma exp_sum_pos {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
-    0<∑ i, Real.exp (d i) := Finset.sum_pos (fun _ _ ↦ Real.exp_pos _) Finset.univ_nonempty
-
-lemma gumbel_shift_cdf (d t : ℝ) :
-    cdf (gumbelLaw.map (fun x ↦ x+d)) t=Real.exp (-Real.exp (-(t-d))) := by
-  have : IsProbabilityMeasure (gumbelLaw.map (fun x ↦ x+d)) := inferInstance
-  rw [cdf_eq_real,measureReal_def,Measure.map_apply (by fun_prop) measurableSet_Iic]
-  have he : (fun x : ℝ ↦ x+d) ⁻¹' Iic t=Iic (t-d) := by ext x; simp only [mem_preimage,mem_Iic]; constructor <;> intro h <;> linarith
-  rw [he,←measureReal_def,←cdf_eq_real,gumbel_cdf]
-
-lemma gumbel_max_cdf {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) (t : ℝ) :
-    cdf ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
-      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) t=
-      Real.exp (-Real.exp (-(t-logSumExp d))) := by
-  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
-    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
-  have : IsProbabilityMeasure ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
-      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) := inferInstance
-  rw [cdf_eq_real,measureReal_def,Measure.map_apply hm measurableSet_Iic]
-  have he : (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) ⁻¹' Iic t=
-      univ.pi (fun i ↦ Iic (t-d i)) := by
-    ext z
-    simp only [mem_preimage,mem_Iic,finiteMaximum_le_iff,mem_univ_pi]
-    exact forall_congr' (fun i ↦ by constructor <;> intro h <;> linarith)
-  rw [he,Measure.pi_pi,ENNReal.toReal_prod]
-  simp only [←measureReal_def,←cdf_eq_real,gumbel_cdf]
-  rw [←Real.exp_sum]
-  congr 1
-  simp only [logSumExp]
-  rw [show -(t-Real.log (∑ i, Real.exp (d i)))=-t+Real.log (∑ i, Real.exp (d i)) by ring,
-    Real.exp_add,Real.exp_log (exp_sum_pos d)]
-  simp_rw [show ∀ i, -(t-d i)=-t+d i by intro i; ring,Real.exp_add]
-  rw [Finset.sum_neg_distrib,←Finset.mul_sum]
-
-theorem gumbel_max_law {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
-    (Measure.pi (fun _ : ι ↦ gumbelLaw)).map
-      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))=
-      gumbelLaw.map (fun x ↦ x+logSumExp d) := by
-  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
-    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
-  have : IsProbabilityMeasure ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
-      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) := inferInstance
-  have : IsProbabilityMeasure (gumbelLaw.map (fun x ↦ x+logSumExp d)) := inferInstance
-  apply prob_measure_eq_of_cdf
-  intro t
-  rw [gumbel_max_cdf,gumbel_shift_cdf]
-
-lemma gumbel_max_integrable {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
-    Integrable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i))
-      (Measure.pi (fun _ : ι ↦ gumbelLaw)) := by
-  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
-    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
-  have h : Integrable (fun x : ℝ ↦ x) ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
-      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) := by
-    rw [gumbel_max_law,integrable_map_measure (by fun_prop) (by fun_prop)]
-    exact gumbel_integrable.add (integrable_const _)
-  exact (integrable_map_measure (by fun_prop) hm.aemeasurable).mp h
-
-lemma integral_gumbel_max {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
-    (∫ z, finiteMaximum (fun i ↦ d i+z i) ∂Measure.pi (fun _ : ι ↦ gumbelLaw))=
-      gumbelMean+logSumExp d := by
-  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
-    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
-  have h := congrArg (fun μ : Measure ℝ ↦ ∫ x, x ∂μ) (gumbel_max_law d)
-  rw [integral_map hm.aemeasurable (by fun_prop),integral_map (by fun_prop) (by fun_prop)] at h
-  rw [h,integral_add gumbel_integrable (integrable_const _)]
-  simp only [integral_const,probReal_univ,smul_eq_mul,one_mul,gumbelMean]
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology BigOperators ENNReal NNReal
-namespace SKValueG
-
-lemma log_tangent_bound {x m : ℝ} (hx : 0<x) (hm : 0 < m) :
-    Real.log x ≤ x/m-1+Real.log m := by
-  have h := Real.log_le_sub_one_of_pos (div_pos hx hm)
-  rw [Real.log_div hx.ne' hm.ne'] at h
-  linarith
-
-lemma log_mean_lower {ι : Type*} [Fintype ι] [Nonempty ι]
-    (x : ι → ℝ) (hx : ∀ i, 0<x i) :
-    (∑ i, Real.log (x i))/(Fintype.card ι : ℝ) ≤
-      Real.log ((∑ i, x i)/(Fintype.card ι : ℝ)) := by
-  let c : ℝ := Fintype.card ι
-  have hc : 0<c := by dsimp [c]; positivity
-  let m : ℝ := (∑ i, x i)/c
-  have hm : 0 < m := div_pos (Finset.sum_pos (fun i _ ↦ hx i) Finset.univ_nonempty) hc
-  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ ↦ log_tangent_bound (hx i) hm)
-  have hs : (∑ i, x i) ≠ 0 := (Finset.sum_pos (fun i _ ↦ hx i) Finset.univ_nonempty).ne'
-  have he : (∑ i, (x i/m-1+Real.log m))=c*Real.log m := by
-    rw [Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.sum_div]
-    simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    dsimp [m,c]
-    field_simp [hs]
-    ring
-  rw [he] at h
-  exact (div_le_iff₀ hc).mpr (by nlinarith)
-
-lemma integral_log_le_log_integral {Ω : Type*} [MeasurableSpace Ω]
-    {μ : Measure Ω} [IsProbabilityMeasure μ] {f : Ω → ℝ}
-    (hf : Integrable f μ) (hl : Integrable (fun ω ↦ Real.log (f ω)) μ)
-    (hp : ∀ᵐ ω ∂μ, 0<f ω) :
-    (∫ ω, Real.log (f ω) ∂μ) ≤ Real.log (∫ ω, f ω ∂μ) := by
-  have hm : 0<∫ ω, f ω ∂μ := integral_pos_iff_support_of_nonneg_ae (hp.mono (fun _ h ↦ h.le)) hf |>.mpr (by
-    have hs : Function.support f =ᵐ[μ] univ := hp.mono (fun ω hω ↦ by apply propext; change (f ω≠0) ↔ True; simp [hω.ne'])
-    rw [measure_congr hs]
-    simp)
-  have h := integral_mono_ae hl
-    (((hf.div_const (∫ ω, f ω ∂μ)).sub (integrable_const 1)).add
-      (integrable_const (Real.log (∫ ω, f ω ∂μ))))
-    (hp.mono (fun ω hω ↦ log_tangent_bound hω hm))
-  have he : (∫ ω, (f ω/(∫ ω, f ω ∂μ)-1)+Real.log (∫ ω, f ω ∂μ) ∂μ)=
-      Real.log (∫ ω, f ω ∂μ) := by
-    rw [integral_add (f := fun ω ↦ f ω/(∫ ω, f ω ∂μ)-1)
-      (g := fun _ ↦ Real.log (∫ ω, f ω ∂μ))
-      ((hf.div_const _).sub (integrable_const 1)) (integrable_const _),
-      integral_sub (f := fun ω ↦ f ω/(∫ ω, f ω ∂μ)) (g := fun _ ↦ (1 : ℝ))
-      (hf.div_const _) (integrable_const 1),integral_div]
-    simp [hm.ne']
-  exact h.trans_eq he
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.RotatedCoefficients
+import OAI.Probability.SKValue.GroundState.ExponentialMoments
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology ENNReal NNReal BigOperators
+namespace SKValueG
+
+lemma finiteMaximum_le_iff {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (t : ℝ) :
+    finiteMaximum x ≤ t ↔ ∀ i, x i≤t := by
+  constructor
+  · intro h i; exact (le_finiteMaximum x i).trans h
+  · intro h; obtain ⟨i,hi⟩ := exists_finiteMaximum x; simpa only [←hi] using h i
+
+noncomputable def logSumExp {ι : Type*} [Fintype ι] (d : ι → ℝ) : ℝ :=
+  Real.log (∑ i, Real.exp (d i))
+
+lemma exp_sum_pos {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
+    0<∑ i, Real.exp (d i) := Finset.sum_pos (fun _ _ ↦ Real.exp_pos _) Finset.univ_nonempty
+
+lemma gumbel_shift_cdf (d t : ℝ) :
+    cdf (gumbelLaw.map (fun x ↦ x+d)) t=Real.exp (-Real.exp (-(t-d))) := by
+  have : IsProbabilityMeasure (gumbelLaw.map (fun x ↦ x+d)) := inferInstance
+  rw [cdf_eq_real,measureReal_def,Measure.map_apply (by fun_prop) measurableSet_Iic]
+  have he : (fun x : ℝ ↦ x+d) ⁻¹' Iic t=Iic (t-d) := by ext x; simp only [mem_preimage,mem_Iic]; constructor <;> intro h <;> linarith
+  rw [he,←measureReal_def,←cdf_eq_real,gumbel_cdf]
+
+lemma gumbel_max_cdf {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) (t : ℝ) :
+    cdf ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
+      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) t=
+      Real.exp (-Real.exp (-(t-logSumExp d))) := by
+  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
+    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
+  have : IsProbabilityMeasure ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
+      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) := inferInstance
+  rw [cdf_eq_real,measureReal_def,Measure.map_apply hm measurableSet_Iic]
+  have he : (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) ⁻¹' Iic t=
+      univ.pi (fun i ↦ Iic (t-d i)) := by
+    ext z
+    simp only [mem_preimage,mem_Iic,finiteMaximum_le_iff,mem_univ_pi]
+    exact forall_congr' (fun i ↦ by constructor <;> intro h <;> linarith)
+  rw [he,Measure.pi_pi,ENNReal.toReal_prod]
+  simp only [←measureReal_def,←cdf_eq_real,gumbel_cdf]
+  rw [←Real.exp_sum]
+  congr 1
+  simp only [logSumExp]
+  rw [show -(t-Real.log (∑ i, Real.exp (d i)))=-t+Real.log (∑ i, Real.exp (d i)) by ring,
+    Real.exp_add,Real.exp_log (exp_sum_pos d)]
+  simp_rw [show ∀ i, -(t-d i)=-t+d i by intro i; ring,Real.exp_add]
+  rw [Finset.sum_neg_distrib,←Finset.mul_sum]
+
+theorem gumbel_max_law {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
+    (Measure.pi (fun _ : ι ↦ gumbelLaw)).map
+      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))=
+      gumbelLaw.map (fun x ↦ x+logSumExp d) := by
+  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
+    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
+  have : IsProbabilityMeasure ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
+      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) := inferInstance
+  have : IsProbabilityMeasure (gumbelLaw.map (fun x ↦ x+logSumExp d)) := inferInstance
+  apply prob_measure_eq_of_cdf
+  intro t
+  rw [gumbel_max_cdf,gumbel_shift_cdf]
+
+lemma gumbel_max_integrable {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
+    Integrable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i))
+      (Measure.pi (fun _ : ι ↦ gumbelLaw)) := by
+  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
+    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
+  have h : Integrable (fun x : ℝ ↦ x) ((Measure.pi (fun _ : ι ↦ gumbelLaw)).map
+      (fun z ↦ finiteMaximum (fun i ↦ d i+z i))) := by
+    rw [gumbel_max_law,integrable_map_measure (by fun_prop) (by fun_prop)]
+    exact gumbel_integrable.add (integrable_const _)
+  exact (integrable_map_measure (by fun_prop) hm.aemeasurable).mp h
+
+lemma integral_gumbel_max {ι : Type*} [Fintype ι] [Nonempty ι] (d : ι → ℝ) :
+    (∫ z, finiteMaximum (fun i ↦ d i+z i) ∂Measure.pi (fun _ : ι ↦ gumbelLaw))=
+      gumbelMean+logSumExp d := by
+  have hm : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ d i+z i)) :=
+    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
+  have h := congrArg (fun μ : Measure ℝ ↦ ∫ x, x ∂μ) (gumbel_max_law d)
+  rw [integral_map hm.aemeasurable (by fun_prop),integral_map (by fun_prop) (by fun_prop)] at h
+  rw [h,integral_add gumbel_integrable (integrable_const _)]
+  simp only [integral_const,probReal_univ,smul_eq_mul,one_mul,gumbelMean]
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology BigOperators ENNReal NNReal
+namespace SKValueG
+
+lemma log_tangent_bound {x m : ℝ} (hx : 0<x) (hm : 0 < m) :
+    Real.log x ≤ x/m-1+Real.log m := by
+  have h := Real.log_le_sub_one_of_pos (div_pos hx hm)
+  rw [Real.log_div hx.ne' hm.ne'] at h
+  linarith
+
+lemma log_mean_lower {ι : Type*} [Fintype ι] [Nonempty ι]
+    (x : ι → ℝ) (hx : ∀ i, 0<x i) :
+    (∑ i, Real.log (x i))/(Fintype.card ι : ℝ) ≤
+      Real.log ((∑ i, x i)/(Fintype.card ι : ℝ)) := by
+  let c : ℝ := Fintype.card ι
+  have hc : 0<c := by dsimp [c]; positivity
+  let m : ℝ := (∑ i, x i)/c
+  have hm : 0 < m := div_pos (Finset.sum_pos (fun i _ ↦ hx i) Finset.univ_nonempty) hc
+  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ ↦ log_tangent_bound (hx i) hm)
+  have hs : (∑ i, x i) ≠ 0 := (Finset.sum_pos (fun i _ ↦ hx i) Finset.univ_nonempty).ne'
+  have he : (∑ i, (x i/m-1+Real.log m))=c*Real.log m := by
+    rw [Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.sum_div]
+    simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    dsimp [m,c]
+    field_simp [hs]
+    ring
+  rw [he] at h
+  exact (div_le_iff₀ hc).mpr (by nlinarith)
+
+lemma integral_log_le_log_integral {Ω : Type*} [MeasurableSpace Ω]
+    {μ : Measure Ω} [IsProbabilityMeasure μ] {f : Ω → ℝ}
+    (hf : Integrable f μ) (hl : Integrable (fun ω ↦ Real.log (f ω)) μ)
+    (hp : ∀ᵐ ω ∂μ, 0<f ω) :
+    (∫ ω, Real.log (f ω) ∂μ) ≤ Real.log (∫ ω, f ω ∂μ) := by
+  have hm : 0<∫ ω, f ω ∂μ := integral_pos_iff_support_of_nonneg_ae (hp.mono (fun _ h ↦ h.le)) hf |>.mpr (by
+    have hs : Function.support f =ᵐ[μ] univ := hp.mono (fun ω hω ↦ by apply propext; change (f ω≠0) ↔ True; simp [hω.ne'])
+    rw [measure_congr hs]
+    simp)
+  have h := integral_mono_ae hl
+    (((hf.div_const (∫ ω, f ω ∂μ)).sub (integrable_const 1)).add
+      (integrable_const (Real.log (∫ ω, f ω ∂μ))))
+    (hp.mono (fun ω hω ↦ log_tangent_bound hω hm))
+  have he : (∫ ω, (f ω/(∫ ω, f ω ∂μ)-1)+Real.log (∫ ω, f ω ∂μ) ∂μ)=
+      Real.log (∫ ω, f ω ∂μ) := by
+    rw [integral_add (f := fun ω ↦ f ω/(∫ ω, f ω ∂μ)-1)
+      (g := fun _ ↦ Real.log (∫ ω, f ω ∂μ))
+      ((hf.div_const _).sub (integrable_const 1)) (integrable_const _),
+      integral_sub (f := fun ω ↦ f ω/(∫ ω, f ω ∂μ)) (g := fun _ ↦ (1 : ℝ))
+      (hf.div_const _) (integrable_const 1),integral_div]
+    simp [hm.ne']
+  exact h.trans_eq he
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/GumbelReplacement.lean b/lean/OAI/Probability/SKValue/GroundState/GumbelReplacement.lean
index 2f1c7a0ce66a469e81c9e929cc1ae5941fa1afc4..0497d9943b69064d7ca9e0b393cac6d34f518c8a 100644
--- a/lean/OAI/Probability/SKValue/GroundState/GumbelReplacement.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/GumbelReplacement.lean
@@ -1,132 +1,132 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.FiniteBranching
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology ENNReal NNReal BigOperators
-namespace SKValueG
-
-lemma lipschitz_integrable_comp {Ω : Type*} [MeasurableSpace Ω]
-    {μ : Measure Ω} [IsFiniteMeasure μ] {f : Ω → ℝ} (hi : Integrable f μ)
-    {ψ : ℝ → ℝ} {L : ℝ≥0} (hψ : LipschitzWith L ψ) : Integrable (fun ω ↦ ψ (f ω)) μ := by
-  apply ((hi.norm.const_mul (L : ℝ)).add (integrable_const ‖ψ 0‖)).mono'
-    (hψ.continuous.comp_aestronglyMeasurable hi.aestronglyMeasurable)
-  exact Eventually.of_forall (fun ω ↦ by
-    have h : ‖ψ (f ω)-ψ 0‖ ≤ (L : ℝ)*‖f ω‖ := by simpa only [sub_zero] using hψ.norm_sub_le (f ω) 0
-    have htri := norm_add_le (ψ (f ω)-ψ 0) (ψ 0)
-    simp only [sub_add_cancel] at htri
-    change ‖ψ (f ω)‖ ≤ (L : ℝ)*‖f ω‖+‖ψ 0‖
-    linarith)
-
-noncomputable def gumbelSmooth (m : ℝ) (ψ : ℝ → ℝ) (s : ℝ) : ℝ :=
-  ∫ g, ψ (g/m+s) ∂gumbelLaw
-
-lemma gumbelSmooth_lipschitz {m : ℝ} {ψ : ℝ → ℝ} {L : ℝ≥0}
-    (hψ : LipschitzWith L ψ) : LipschitzWith L (gumbelSmooth m ψ) := by
-  rw [lipschitzWith_iff_dist_le_mul]
-  intro x y
-  have hix := lipschitz_integrable_comp ((gumbel_integrable.div_const m).add (integrable_const x)) hψ
-  have hiy := lipschitz_integrable_comp ((gumbel_integrable.div_const m).add (integrable_const y)) hψ
-  change |(∫ g, ψ (g/m+x) ∂gumbelLaw)-(∫ g, ψ (g/m+y) ∂gumbelLaw)|≤L*|x-y|
-  rw [←integral_sub (f := fun g ↦ ψ (g/m+x)) (g := fun g ↦ ψ (g/m+y)) hix hiy]
-  have hbound (g : ℝ) : ‖ψ (g/m+x)-ψ (g/m+y)‖≤(L : ℝ)*|x-y| := by
-    simpa only [Real.norm_eq_abs,add_sub_add_left_eq_sub] using hψ.norm_sub_le (g/m+x) (g/m+y)
-  simpa using norm_integral_le_of_norm_le_const (μ := gumbelLaw) (Eventually.of_forall hbound)
-
-lemma integral_comp_diff_le {Ω : Type*} [MeasurableSpace Ω]
-    {μ : Measure Ω} [IsProbabilityMeasure μ] {s : Ω → ℝ} (hi : Integrable s μ)
-    {ψ : ℝ → ℝ} {L : ℝ≥0} (hψ : LipschitzWith L ψ) (c : ℝ) :
-    |(∫ ω,ψ (s ω) ∂μ)-ψ c|≤(L : ℝ)*(∫ ω,|s ω-c| ∂μ) := by
-  have his := lipschitz_integrable_comp hi hψ
-  have hie : Integrable (fun ω ↦ |s ω-c|) μ := (hi.sub (integrable_const c)).abs
-  calc
-    _ = |∫ ω,ψ (s ω)-ψ c ∂μ| := by rw [integral_sub his (integrable_const _)]; simp
-    _ ≤ ∫ ω,|ψ (s ω)-ψ c| ∂μ := by simpa only [Real.norm_eq_abs] using
-      norm_integral_le_integral_norm (fun ω ↦ ψ (s ω)-ψ c)
-    _ ≤ ∫ ω,(L : ℝ)*|s ω-c| ∂μ := integral_mono
-      (his.sub (integrable_const _)).abs (hie.const_mul _)
-      (fun ω ↦ by simpa only [Real.norm_eq_abs] using hψ.norm_sub_le (s ω) c)
-    _ = _ := integral_const_mul _ _
-
-lemma scaled_gumbel_test_integral {ι : Type*} [Fintype ι] [Nonempty ι]
-    (d : ι → ℝ) {m : ℝ} (hm : 0 < m) (ψ : ℝ → ℝ) (hψ : Measurable ψ) :
-    (∫ z,ψ (finiteMaximum (fun i ↦ d i+z i/m)-Real.log (Fintype.card ι : ℝ)/m)
-      ∂Measure.pi (fun _ : ι ↦ gumbelLaw))=
-      gumbelSmooth m ψ (Real.log ((∑ i,Real.exp (m*d i))/(Fintype.card ι : ℝ))/m) := by
-  have hmmax : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ m*d i+z i)) :=
-    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
-  have ht : Measurable (fun y : ℝ ↦ ψ (y/m-Real.log (Fintype.card ι : ℝ)/m)) := by fun_prop
-  have h := congrArg (fun μ : Measure ℝ ↦ ∫ y,ψ (y/m-Real.log (Fintype.card ι : ℝ)/m) ∂μ)
-    (gumbel_max_law (fun i ↦ m*d i))
-  rw [integral_map hmmax.aemeasurable ht.aestronglyMeasurable,
-    integral_map (by fun_prop) ht.aestronglyMeasurable] at h
-  have he (z : ι → ℝ) : finiteMaximum (fun i ↦ d i+z i/m)=
-      finiteMaximum (fun i ↦ m*d i+z i)/m := by
-    rw [←finiteMaximum_div_pos _ hm]
-    congr 1; ext i; field_simp
-  simp_rw [he]
-  rw [h]
-  apply integral_congr_ae
-  exact Eventually.of_forall (fun g ↦ by
-    apply congrArg ψ
-    rw [Real.log_div (exp_sum_pos _).ne' (by positivity)]
-    dsimp [logSumExp]
-    ring)
-
-theorem marked_branching_limit {Ω : Type*} [MeasurableSpace Ω]
-    {μ : Measure Ω} [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ)
-    {m : ℝ} (hm : 0 < m) (hi : Integrable (X 0) μ)
-    (he : Integrable (fun ω ↦ Real.exp (m*X 0 ω)) μ)
-    (hind : Pairwise (Function.onFun (fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) X))
-    (hid : ∀ i,IdentDistrib (X i) (X 0) μ μ)
-    {ψ : ℝ → ℝ} {L : ℝ≥0} (hψ : LipschitzWith L ψ) :
-    Tendsto (fun n ↦ ∫ ω,
-      (∫ z,ψ (finiteMaximum (fun i : Fin (n+1) ↦ X i ω+z i/m)-Real.log (n+1 : ℝ)/m)
-        ∂Measure.pi (fun _ : Fin (n+1) ↦ gumbelLaw)) ∂μ)
-      atTop (𝓝 (gumbelSmooth m ψ (Real.log (∫ ω,Real.exp (m*X 0 ω) ∂μ)/m))) := by
-  let Y := fun i ω ↦ Real.exp (m*X i ω)
-  have hf : Measurable (fun x : ℝ ↦ Real.exp (m*x)) := by fun_prop
-  have hl : Integrable (fun ω ↦ Real.log (Y 0 ω)) μ := by
-    simpa only [Y,Real.log_exp] using hi.const_mul m
-  have hiY (i) : Integrable (Y i) μ := ((hid i).comp hf).integrable_iff.mpr he
-  have hilogY (i) : Integrable (fun ω ↦ Real.log (Y i ω)) μ :=
-    (((hid i).comp hf).comp Real.measurable_log).integrable_iff.mpr hl
-  have hpY (i) : ∀ᵐ ω ∂μ,0<Y i ω := Eventually.of_forall (fun ω ↦ Real.exp_pos _)
-  have hlim := log_empiricalMean_L1 Y he hl hpY
-    (fun i j hij ↦ (hind hij).comp hf hf) (fun i ↦ (hid i).comp hf)
-  let c := Real.log (∫ ω,Y 0 ω ∂μ)/m
-  let s := fun n ω ↦ Real.log (empiricalMean Y n ω)/m
-  have his (n) : Integrable (s n) μ :=
-    (empiricalMean_log_integrable Y hiY hilogY hpY n).div_const m
-  have herr (n) : (∫ ω,|s n ω-c| ∂μ)=
-      (∫ ω,|Real.log (empiricalMean Y n ω)-Real.log (∫ ω,Y 0 ω ∂μ)| ∂μ)/m := by
-    rw [←integral_div]
-    apply integral_congr_ae
-    exact Eventually.of_forall (fun ω ↦ by dsimp [s,c]; rw [←sub_div,abs_div,abs_of_pos hm])
-  have ht : Tendsto (fun n ↦ ∫ ω,|s n ω-c| ∂μ) atTop (𝓝 0) := by
-    simpa only [herr,zero_div] using hlim.div_const m
-  have hg := gumbelSmooth_lipschitz (m := m) hψ
-  have hb (n) := integral_comp_diff_le (his n) hg c
-  have hx : Tendsto (fun n ↦ |(∫ ω,gumbelSmooth m ψ (s n ω) ∂μ)-gumbelSmooth m ψ c|)
-      atTop (𝓝 0) := squeeze_zero (fun n ↦ abs_nonneg _) hb (by simpa using ht.const_mul (L : ℝ))
-  have hconv : Tendsto (fun n ↦ ∫ ω,gumbelSmooth m ψ (s n ω) ∂μ) atTop
-      (𝓝 (gumbelSmooth m ψ c)) := tendsto_iff_norm_sub_tendsto_zero.mpr (by
-        simpa only [Real.norm_eq_abs] using hx)
-  have heq (n : ℕ) : (∫ ω,(∫ z,ψ (finiteMaximum (fun i : Fin (n+1) ↦ X i ω+z i/m)-
-      Real.log (n+1 : ℝ)/m) ∂Measure.pi (fun _ : Fin (n+1) ↦ gumbelLaw)) ∂μ)=
-      ∫ ω,gumbelSmooth m ψ (s n ω) ∂μ := by
-    apply integral_congr_ae
-    exact Eventually.of_forall (fun ω ↦ by
-      have hh := scaled_gumbel_test_integral (fun i : Fin (n+1) ↦ X i ω) hm ψ hψ.continuous.measurable
-      rw [Fin.sum_univ_eq_sum_range (fun i ↦ Real.exp (m*X i ω))] at hh
-      simpa only [Fintype.card_fin,Nat.cast_add,Nat.cast_one,empiricalMean,Y,s] using hh)
-  simpa only [heq,c,Y] using hconv
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.FiniteBranching
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology ENNReal NNReal BigOperators
+namespace SKValueG
+
+lemma lipschitz_integrable_comp {Ω : Type*} [MeasurableSpace Ω]
+    {μ : Measure Ω} [IsFiniteMeasure μ] {f : Ω → ℝ} (hi : Integrable f μ)
+    {ψ : ℝ → ℝ} {L : ℝ≥0} (hψ : LipschitzWith L ψ) : Integrable (fun ω ↦ ψ (f ω)) μ := by
+  apply ((hi.norm.const_mul (L : ℝ)).add (integrable_const ‖ψ 0‖)).mono'
+    (hψ.continuous.comp_aestronglyMeasurable hi.aestronglyMeasurable)
+  exact Eventually.of_forall (fun ω ↦ by
+    have h : ‖ψ (f ω)-ψ 0‖ ≤ (L : ℝ)*‖f ω‖ := by simpa only [sub_zero] using hψ.norm_sub_le (f ω) 0
+    have htri := norm_add_le (ψ (f ω)-ψ 0) (ψ 0)
+    simp only [sub_add_cancel] at htri
+    change ‖ψ (f ω)‖ ≤ (L : ℝ)*‖f ω‖+‖ψ 0‖
+    linarith)
+
+noncomputable def gumbelSmooth (m : ℝ) (ψ : ℝ → ℝ) (s : ℝ) : ℝ :=
+  ∫ g, ψ (g/m+s) ∂gumbelLaw
+
+lemma gumbelSmooth_lipschitz {m : ℝ} {ψ : ℝ → ℝ} {L : ℝ≥0}
+    (hψ : LipschitzWith L ψ) : LipschitzWith L (gumbelSmooth m ψ) := by
+  rw [lipschitzWith_iff_dist_le_mul]
+  intro x y
+  have hix := lipschitz_integrable_comp ((gumbel_integrable.div_const m).add (integrable_const x)) hψ
+  have hiy := lipschitz_integrable_comp ((gumbel_integrable.div_const m).add (integrable_const y)) hψ
+  change |(∫ g, ψ (g/m+x) ∂gumbelLaw)-(∫ g, ψ (g/m+y) ∂gumbelLaw)|≤L*|x-y|
+  rw [←integral_sub (f := fun g ↦ ψ (g/m+x)) (g := fun g ↦ ψ (g/m+y)) hix hiy]
+  have hbound (g : ℝ) : ‖ψ (g/m+x)-ψ (g/m+y)‖≤(L : ℝ)*|x-y| := by
+    simpa only [Real.norm_eq_abs,add_sub_add_left_eq_sub] using hψ.norm_sub_le (g/m+x) (g/m+y)
+  simpa using norm_integral_le_of_norm_le_const (μ := gumbelLaw) (Eventually.of_forall hbound)
+
+lemma integral_comp_diff_le {Ω : Type*} [MeasurableSpace Ω]
+    {μ : Measure Ω} [IsProbabilityMeasure μ] {s : Ω → ℝ} (hi : Integrable s μ)
+    {ψ : ℝ → ℝ} {L : ℝ≥0} (hψ : LipschitzWith L ψ) (c : ℝ) :
+    |(∫ ω,ψ (s ω) ∂μ)-ψ c|≤(L : ℝ)*(∫ ω,|s ω-c| ∂μ) := by
+  have his := lipschitz_integrable_comp hi hψ
+  have hie : Integrable (fun ω ↦ |s ω-c|) μ := (hi.sub (integrable_const c)).abs
+  calc
+    _ = |∫ ω,ψ (s ω)-ψ c ∂μ| := by rw [integral_sub his (integrable_const _)]; simp
+    _ ≤ ∫ ω,|ψ (s ω)-ψ c| ∂μ := by simpa only [Real.norm_eq_abs] using
+      norm_integral_le_integral_norm (fun ω ↦ ψ (s ω)-ψ c)
+    _ ≤ ∫ ω,(L : ℝ)*|s ω-c| ∂μ := integral_mono
+      (his.sub (integrable_const _)).abs (hie.const_mul _)
+      (fun ω ↦ by simpa only [Real.norm_eq_abs] using hψ.norm_sub_le (s ω) c)
+    _ = _ := integral_const_mul _ _
+
+lemma scaled_gumbel_test_integral {ι : Type*} [Fintype ι] [Nonempty ι]
+    (d : ι → ℝ) {m : ℝ} (hm : 0 < m) (ψ : ℝ → ℝ) (hψ : Measurable ψ) :
+    (∫ z,ψ (finiteMaximum (fun i ↦ d i+z i/m)-Real.log (Fintype.card ι : ℝ)/m)
+      ∂Measure.pi (fun _ : ι ↦ gumbelLaw))=
+      gumbelSmooth m ψ (Real.log ((∑ i,Real.exp (m*d i))/(Fintype.card ι : ℝ))/m) := by
+  have hmmax : Measurable (fun z : ι → ℝ ↦ finiteMaximum (fun i ↦ m*d i+z i)) :=
+    (continuous_finiteMaximum.comp (continuous_const.add continuous_id)).measurable
+  have ht : Measurable (fun y : ℝ ↦ ψ (y/m-Real.log (Fintype.card ι : ℝ)/m)) := by fun_prop
+  have h := congrArg (fun μ : Measure ℝ ↦ ∫ y,ψ (y/m-Real.log (Fintype.card ι : ℝ)/m) ∂μ)
+    (gumbel_max_law (fun i ↦ m*d i))
+  rw [integral_map hmmax.aemeasurable ht.aestronglyMeasurable,
+    integral_map (by fun_prop) ht.aestronglyMeasurable] at h
+  have he (z : ι → ℝ) : finiteMaximum (fun i ↦ d i+z i/m)=
+      finiteMaximum (fun i ↦ m*d i+z i)/m := by
+    rw [←finiteMaximum_div_pos _ hm]
+    congr 1; ext i; field_simp
+  simp_rw [he]
+  rw [h]
+  apply integral_congr_ae
+  exact Eventually.of_forall (fun g ↦ by
+    apply congrArg ψ
+    rw [Real.log_div (exp_sum_pos _).ne' (by positivity)]
+    dsimp [logSumExp]
+    ring)
+
+theorem marked_branching_limit {Ω : Type*} [MeasurableSpace Ω]
+    {μ : Measure Ω} [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ)
+    {m : ℝ} (hm : 0 < m) (hi : Integrable (X 0) μ)
+    (he : Integrable (fun ω ↦ Real.exp (m*X 0 ω)) μ)
+    (hind : Pairwise (Function.onFun (fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) X))
+    (hid : ∀ i,IdentDistrib (X i) (X 0) μ μ)
+    {ψ : ℝ → ℝ} {L : ℝ≥0} (hψ : LipschitzWith L ψ) :
+    Tendsto (fun n ↦ ∫ ω,
+      (∫ z,ψ (finiteMaximum (fun i : Fin (n+1) ↦ X i ω+z i/m)-Real.log (n+1 : ℝ)/m)
+        ∂Measure.pi (fun _ : Fin (n+1) ↦ gumbelLaw)) ∂μ)
+      atTop (𝓝 (gumbelSmooth m ψ (Real.log (∫ ω,Real.exp (m*X 0 ω) ∂μ)/m))) := by
+  let Y := fun i ω ↦ Real.exp (m*X i ω)
+  have hf : Measurable (fun x : ℝ ↦ Real.exp (m*x)) := by fun_prop
+  have hl : Integrable (fun ω ↦ Real.log (Y 0 ω)) μ := by
+    simpa only [Y,Real.log_exp] using hi.const_mul m
+  have hiY (i) : Integrable (Y i) μ := ((hid i).comp hf).integrable_iff.mpr he
+  have hilogY (i) : Integrable (fun ω ↦ Real.log (Y i ω)) μ :=
+    (((hid i).comp hf).comp Real.measurable_log).integrable_iff.mpr hl
+  have hpY (i) : ∀ᵐ ω ∂μ,0<Y i ω := Eventually.of_forall (fun ω ↦ Real.exp_pos _)
+  have hlim := log_empiricalMean_L1 Y he hl hpY
+    (fun i j hij ↦ (hind hij).comp hf hf) (fun i ↦ (hid i).comp hf)
+  let c := Real.log (∫ ω,Y 0 ω ∂μ)/m
+  let s := fun n ω ↦ Real.log (empiricalMean Y n ω)/m
+  have his (n) : Integrable (s n) μ :=
+    (empiricalMean_log_integrable Y hiY hilogY hpY n).div_const m
+  have herr (n) : (∫ ω,|s n ω-c| ∂μ)=
+      (∫ ω,|Real.log (empiricalMean Y n ω)-Real.log (∫ ω,Y 0 ω ∂μ)| ∂μ)/m := by
+    rw [←integral_div]
+    apply integral_congr_ae
+    exact Eventually.of_forall (fun ω ↦ by dsimp [s,c]; rw [←sub_div,abs_div,abs_of_pos hm])
+  have ht : Tendsto (fun n ↦ ∫ ω,|s n ω-c| ∂μ) atTop (𝓝 0) := by
+    simpa only [herr,zero_div] using hlim.div_const m
+  have hg := gumbelSmooth_lipschitz (m := m) hψ
+  have hb (n) := integral_comp_diff_le (his n) hg c
+  have hx : Tendsto (fun n ↦ |(∫ ω,gumbelSmooth m ψ (s n ω) ∂μ)-gumbelSmooth m ψ c|)
+      atTop (𝓝 0) := squeeze_zero (fun n ↦ abs_nonneg _) hb (by simpa using ht.const_mul (L : ℝ))
+  have hconv : Tendsto (fun n ↦ ∫ ω,gumbelSmooth m ψ (s n ω) ∂μ) atTop
+      (𝓝 (gumbelSmooth m ψ c)) := tendsto_iff_norm_sub_tendsto_zero.mpr (by
+        simpa only [Real.norm_eq_abs] using hx)
+  have heq (n : ℕ) : (∫ ω,(∫ z,ψ (finiteMaximum (fun i : Fin (n+1) ↦ X i ω+z i/m)-
+      Real.log (n+1 : ℝ)/m) ∂Measure.pi (fun _ : Fin (n+1) ↦ gumbelLaw)) ∂μ)=
+      ∫ ω,gumbelSmooth m ψ (s n ω) ∂μ := by
+    apply integral_congr_ae
+    exact Eventually.of_forall (fun ω ↦ by
+      have hh := scaled_gumbel_test_integral (fun i : Fin (n+1) ↦ X i ω) hm ψ hψ.continuous.measurable
+      rw [Fin.sum_univ_eq_sum_range (fun i ↦ Real.exp (m*X i ω))] at hh
+      simpa only [Fintype.card_fin,Nat.cast_add,Nat.cast_one,empiricalMean,Y,s] using hh)
+  simpa only [heq,c,Y] using hconv
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/LiteralLower.lean b/lean/OAI/Probability/SKValue/GroundState/LiteralLower.lean
index d13606ad315f954e9e2e03f962d962cd2a2e5a1f..037768a1fcb74753b7a33ea51ed6c5afaab273a5 100644
--- a/lean/OAI/Probability/SKValue/GroundState/LiteralLower.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/LiteralLower.lean
@@ -1,153 +1,153 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.LiteralNormalizer
-import OAI.Probability.SKValue.Control.Regularity
-import OAI.Probability.SKValue.GroundState.RoundingSpin
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-theorem shiftedCoefficientSum_le_groundState (W : BrownianSpace) (γ : OrderParameter)
-    {T : ℝ} (hT : 0≤T) (hT1 : T<1) {N : ℕ} (hN : 0<N)
-    (hpos : ∀ j<N,0<∫ z,(rawCoefficient W γ T N j z)^2 ∂gaussianProduct (Fin (N+1))) :
-    shiftedCoefficientSum W γ T N≤groundStateValue := by
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨hT,hT1⟩
-  rw [literal_shiftedCoefficientSum]
-  exact normalized_shifted_sum_le_groundState
-    (fun j : Fin N ↦ meshRaw T N γ.coeff (gradient W γ) j)
-    (meshTerminal T N γ.coeff (gradient W γ))
-    (fun j ↦ mesh_raw_measurable hT hG hN j.isLt.le)
-    (fun j ↦ mesh_raw_depends T j.isLt.le γ.coeff (gradient W γ))
-    (fun j ↦ mesh_raw_memLp hT hG hDb hN j.isLt.le)
-    (fun j ↦ hpos j j.isLt)
-    (mesh_terminal_measurable hT hG hN) (mesh_terminal_depends T N γ.coeff (gradient W γ))
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma finiteMaximum_prod_add {ι Λ : Type*} [Fintype ι] [Nonempty ι]
-    [Fintype Λ] [Nonempty Λ] (f : ι → ℝ) (g : Λ → ℝ) :
-    finiteMaximum (fun p : ι×Λ ↦ f p.1+g p.2)=finiteMaximum f+finiteMaximum g := by
-  apply le_antisymm
-  · obtain ⟨p,hp⟩ := exists_finiteMaximum (fun p : ι×Λ ↦ f p.1+g p.2)
-    rw [←hp]
-    exact add_le_add (le_finiteMaximum f p.1) (le_finiteMaximum g p.2)
-  · obtain ⟨i,hi⟩ := exists_finiteMaximum f
-    obtain ⟨a,ha⟩ := exists_finiteMaximum g
-    rw [←hi,←ha]
-    exact le_finiteMaximum (fun p : ι×Λ ↦ f p.1+g p.2) (i,a)
-
-lemma finiteMaximum_prod_curry {ι Λ : Type*} [Fintype ι] [Nonempty ι]
-    [Fintype Λ] [Nonempty Λ] (f : ι×Λ → ℝ) :
-    finiteMaximum f=finiteMaximum (fun a ↦ finiteMaximum (fun i ↦ f (i,a))) := by
-  apply le_antisymm
-  · obtain ⟨p,hp⟩ := exists_finiteMaximum f
-    rw [←hp]
-    exact (le_finiteMaximum (fun i ↦ f (i,p.2)) p.1).trans
-      (le_finiteMaximum (fun a ↦ finiteMaximum (fun i ↦ f (i,a))) p.2)
-  · obtain ⟨a,ha⟩ := exists_finiteMaximum (fun a ↦ finiteMaximum (fun i ↦ f (i,a)))
-    obtain ⟨i,hi⟩ := exists_finiteMaximum (fun i ↦ f (i,a))
-    rw [←ha,←hi]
-    exact le_finiteMaximum f (i,a)
-
-lemma spinMaximum_eq_sum_abs {ι : Type*} [Fintype ι] (x : ι → ℝ) :
-    finiteMaximum (fun σ : ι → Bool ↦ ∑ i, spin (σ i)*x i)=∑ i, |x i| := by
-  classical
-  apply le_antisymm
-  · obtain ⟨σ,hσ⟩ := exists_finiteMaximum (fun σ : ι → Bool ↦ ∑ i, spin (σ i)*x i)
-    rw [←hσ]
-    apply Finset.sum_le_sum
-    intro i _
-    calc
-      _ ≤ |spin (σ i)*x i| := le_abs_self _
-      _ = _ := by rw [abs_mul,abs_spin,one_mul]
-  · let σ : ι → Bool := fun i ↦ decide (0≤x i)
-    have he (i : ι) : spin (σ i)*x i=|x i| := by
-      by_cases h : 0≤x i
-      · simp [σ,spin,h,abs_of_nonneg h]
-      · simp [σ,spin,h,abs_of_neg (lt_of_not_ge h)]
-    simpa only [he] using le_finiteMaximum (fun σ : ι → Bool ↦ ∑ i, spin (σ i)*x i) σ
-
-noncomputable def leafField {Λ κ : Type*} [Fintype κ] {n : ℕ}
-    (v : Λ → κ → ℝ) (z : Fin n×κ → ℝ) (α : Λ) (i : Fin n) : ℝ :=
-  ∑ k, v α k*z (i,k)
-
-lemma linearProcess_leafField {Λ κ : Type*} [Fintype κ] (n : ℕ)
-    (v : Λ → κ → ℝ) (z : Fin n×κ → ℝ) (p : (Fin n → Bool)×Λ) :
-    linearProcess (leafFieldCoeff n v) z p=∑ i, spin (p.1 i)*leafField v z p.2 i := by
-  simp only [linearProcess,leafFieldCoeff,Fintype.sum_prod_type,leafField,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  apply Finset.sum_congr rfl
-  intro k _
-  ring
-
-lemma leafField_max {Λ κ : Type*} [Fintype Λ] [Nonempty Λ] [Fintype κ]
-    (n : ℕ) (v : Λ → κ → ℝ) (d : Λ → ℝ) (z : Fin n×κ → ℝ) :
-    finiteMaximum (fun p ↦ linearProcess (leafFieldCoeff n v) z p+d p.2)=
-      finiteMaximum (fun α ↦ (∑ i, |leafField v z α i|)+d α) := by
-  rw [finiteMaximum_prod_curry]
-  simp only [linearProcess_leafField,finiteMaximum_add_const,spinMaximum_eq_sum_abs]
-
-lemma expected_guerra_max {Λ κ : Type*} [Fintype Λ] [Nonempty Λ] [Fintype κ]
-    (n : ℕ) (v : Λ → κ → ℝ) (d : Λ → ℝ) :
-    (∫ z, finiteMaximum (fun p ↦ linearProcess (guerraCoeff n v) z p+d p.2)
-      ∂gaussianProduct ((Fin n×Fin n)⊕(κ×κ))) = expectedMaximum n+
-      ∫ z, finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z α+d α)
-        ∂gaussianProduct (κ×κ) := by
-  have he := measurePreserving_sumPiEquivProdPi_symm
-    (fun _ : (Fin n×Fin n)⊕(κ×κ) ↦ standardGaussian)
-  change (∫ z, finiteMaximum (fun p ↦ linearProcess (guerraCoeff n v) z p+d p.2)
-    ∂Measure.pi (fun _ : (Fin n×Fin n)⊕(κ×κ) ↦ standardGaussian)) = _
-  rw [←he.integral_comp' (fun z ↦ finiteMaximum (fun p ↦ linearProcess (guerraCoeff n v) z p+d p.2))]
-  have hlin (z : ((Fin n×Fin n) → ℝ)×((κ×κ) → ℝ)) :
-      (fun p ↦ linearProcess (guerraCoeff n v)
-        ((MeasurableEquiv.sumPiEquivProdPi (fun _ : (Fin n×Fin n)⊕(κ×κ) ↦ ℝ)).symm z) p+d p.2) =
-      (fun p ↦ linearProcess (hamCoeff n) z.1 p.1+(linearProcess (leafAuxCoeff n v) z.2 p.2+d p.2)) := by
-    funext p
-    simp [linearProcess,guerraCoeff,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm,add_assoc]
-  have hm (z : ((Fin n×Fin n) → ℝ)×((κ×κ) → ℝ)) :
-      finiteMaximum (fun p : (Fin n → Bool)×Λ ↦ linearProcess (hamCoeff n) z.1 p.1+
-        (linearProcess (leafAuxCoeff n v) z.2 p.2+d p.2)) =
-      finiteMaximum (linearProcess (hamCoeff n) z.1)+
-        finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z.2 α+d α) :=
-    finiteMaximum_prod_add (linearProcess (hamCoeff n) z.1)
-      (fun α ↦ linearProcess (leafAuxCoeff n v) z.2 α+d α)
-  simp only [hlin,hm]
-  change (∫ z, finiteMaximum (linearProcess (hamCoeff n) z.1)+
-    finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z.2 α+d α)
-      ∂(gaussianProduct (Fin n×Fin n)).prod (gaussianProduct (κ×κ))) = _
-  rw [integral_add ((finiteMaximum_integrable (hamCoeff n)).comp_fst _)
-    ((affineMaximum_integrable (leafAuxCoeff n v) d).comp_snd _),
-    integral_fun_fst (fun z ↦ finiteMaximum (linearProcess (hamCoeff n) z)),
-    integral_fun_snd (fun z ↦ finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z α+d α))]
-  simp [expectedMaximum]
-
-theorem finite_guerra_upper {Λ κ : Type*} [Fintype Λ] [Nonempty Λ] [Fintype κ]
-    {n : ℕ} (hn : 0<n) (v : Λ → κ → ℝ)
-    (hv : ∀ α, ∑ k, (v α k)^2=1) (d : Λ → ℝ) :
-    expectedMaximum n+
-      (∫ z, finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z α+d α)
-        ∂gaussianProduct (κ×κ)) ≤
-      ∫ z, finiteMaximum (fun α ↦ (∑ i, |leafField v z α i|)+d α)
-        ∂gaussianProduct (Fin n×κ) := by
-  have h := expected_affine_max_le (guerraCoeff n v) (leafFieldCoeff n v)
-    (fun p ↦ d p.2) (guerra_metric_le hn v hv)
-  simpa only [expected_guerra_max,leafField_max] using h
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.LiteralNormalizer
+import OAI.Probability.SKValue.Control.Regularity
+import OAI.Probability.SKValue.GroundState.RoundingSpin
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+theorem shiftedCoefficientSum_le_groundState (W : BrownianSpace) (γ : OrderParameter)
+    {T : ℝ} (hT : 0≤T) (hT1 : T<1) {N : ℕ} (hN : 0<N)
+    (hpos : ∀ j<N,0<∫ z,(rawCoefficient W γ T N j z)^2 ∂gaussianProduct (Fin (N+1))) :
+    shiftedCoefficientSum W γ T N≤groundStateValue := by
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨hT,hT1⟩
+  rw [literal_shiftedCoefficientSum]
+  exact normalized_shifted_sum_le_groundState
+    (fun j : Fin N ↦ meshRaw T N γ.coeff (gradient W γ) j)
+    (meshTerminal T N γ.coeff (gradient W γ))
+    (fun j ↦ mesh_raw_measurable hT hG hN j.isLt.le)
+    (fun j ↦ mesh_raw_depends T j.isLt.le γ.coeff (gradient W γ))
+    (fun j ↦ mesh_raw_memLp hT hG hDb hN j.isLt.le)
+    (fun j ↦ hpos j j.isLt)
+    (mesh_terminal_measurable hT hG hN) (mesh_terminal_depends T N γ.coeff (gradient W γ))
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma finiteMaximum_prod_add {ι Λ : Type*} [Fintype ι] [Nonempty ι]
+    [Fintype Λ] [Nonempty Λ] (f : ι → ℝ) (g : Λ → ℝ) :
+    finiteMaximum (fun p : ι×Λ ↦ f p.1+g p.2)=finiteMaximum f+finiteMaximum g := by
+  apply le_antisymm
+  · obtain ⟨p,hp⟩ := exists_finiteMaximum (fun p : ι×Λ ↦ f p.1+g p.2)
+    rw [←hp]
+    exact add_le_add (le_finiteMaximum f p.1) (le_finiteMaximum g p.2)
+  · obtain ⟨i,hi⟩ := exists_finiteMaximum f
+    obtain ⟨a,ha⟩ := exists_finiteMaximum g
+    rw [←hi,←ha]
+    exact le_finiteMaximum (fun p : ι×Λ ↦ f p.1+g p.2) (i,a)
+
+lemma finiteMaximum_prod_curry {ι Λ : Type*} [Fintype ι] [Nonempty ι]
+    [Fintype Λ] [Nonempty Λ] (f : ι×Λ → ℝ) :
+    finiteMaximum f=finiteMaximum (fun a ↦ finiteMaximum (fun i ↦ f (i,a))) := by
+  apply le_antisymm
+  · obtain ⟨p,hp⟩ := exists_finiteMaximum f
+    rw [←hp]
+    exact (le_finiteMaximum (fun i ↦ f (i,p.2)) p.1).trans
+      (le_finiteMaximum (fun a ↦ finiteMaximum (fun i ↦ f (i,a))) p.2)
+  · obtain ⟨a,ha⟩ := exists_finiteMaximum (fun a ↦ finiteMaximum (fun i ↦ f (i,a)))
+    obtain ⟨i,hi⟩ := exists_finiteMaximum (fun i ↦ f (i,a))
+    rw [←ha,←hi]
+    exact le_finiteMaximum f (i,a)
+
+lemma spinMaximum_eq_sum_abs {ι : Type*} [Fintype ι] (x : ι → ℝ) :
+    finiteMaximum (fun σ : ι → Bool ↦ ∑ i, spin (σ i)*x i)=∑ i, |x i| := by
+  classical
+  apply le_antisymm
+  · obtain ⟨σ,hσ⟩ := exists_finiteMaximum (fun σ : ι → Bool ↦ ∑ i, spin (σ i)*x i)
+    rw [←hσ]
+    apply Finset.sum_le_sum
+    intro i _
+    calc
+      _ ≤ |spin (σ i)*x i| := le_abs_self _
+      _ = _ := by rw [abs_mul,abs_spin,one_mul]
+  · let σ : ι → Bool := fun i ↦ decide (0≤x i)
+    have he (i : ι) : spin (σ i)*x i=|x i| := by
+      by_cases h : 0≤x i
+      · simp [σ,spin,h,abs_of_nonneg h]
+      · simp [σ,spin,h,abs_of_neg (lt_of_not_ge h)]
+    simpa only [he] using le_finiteMaximum (fun σ : ι → Bool ↦ ∑ i, spin (σ i)*x i) σ
+
+noncomputable def leafField {Λ κ : Type*} [Fintype κ] {n : ℕ}
+    (v : Λ → κ → ℝ) (z : Fin n×κ → ℝ) (α : Λ) (i : Fin n) : ℝ :=
+  ∑ k, v α k*z (i,k)
+
+lemma linearProcess_leafField {Λ κ : Type*} [Fintype κ] (n : ℕ)
+    (v : Λ → κ → ℝ) (z : Fin n×κ → ℝ) (p : (Fin n → Bool)×Λ) :
+    linearProcess (leafFieldCoeff n v) z p=∑ i, spin (p.1 i)*leafField v z p.2 i := by
+  simp only [linearProcess,leafFieldCoeff,Fintype.sum_prod_type,leafField,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro k _
+  ring
+
+lemma leafField_max {Λ κ : Type*} [Fintype Λ] [Nonempty Λ] [Fintype κ]
+    (n : ℕ) (v : Λ → κ → ℝ) (d : Λ → ℝ) (z : Fin n×κ → ℝ) :
+    finiteMaximum (fun p ↦ linearProcess (leafFieldCoeff n v) z p+d p.2)=
+      finiteMaximum (fun α ↦ (∑ i, |leafField v z α i|)+d α) := by
+  rw [finiteMaximum_prod_curry]
+  simp only [linearProcess_leafField,finiteMaximum_add_const,spinMaximum_eq_sum_abs]
+
+lemma expected_guerra_max {Λ κ : Type*} [Fintype Λ] [Nonempty Λ] [Fintype κ]
+    (n : ℕ) (v : Λ → κ → ℝ) (d : Λ → ℝ) :
+    (∫ z, finiteMaximum (fun p ↦ linearProcess (guerraCoeff n v) z p+d p.2)
+      ∂gaussianProduct ((Fin n×Fin n)⊕(κ×κ))) = expectedMaximum n+
+      ∫ z, finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z α+d α)
+        ∂gaussianProduct (κ×κ) := by
+  have he := measurePreserving_sumPiEquivProdPi_symm
+    (fun _ : (Fin n×Fin n)⊕(κ×κ) ↦ standardGaussian)
+  change (∫ z, finiteMaximum (fun p ↦ linearProcess (guerraCoeff n v) z p+d p.2)
+    ∂Measure.pi (fun _ : (Fin n×Fin n)⊕(κ×κ) ↦ standardGaussian)) = _
+  rw [←he.integral_comp' (fun z ↦ finiteMaximum (fun p ↦ linearProcess (guerraCoeff n v) z p+d p.2))]
+  have hlin (z : ((Fin n×Fin n) → ℝ)×((κ×κ) → ℝ)) :
+      (fun p ↦ linearProcess (guerraCoeff n v)
+        ((MeasurableEquiv.sumPiEquivProdPi (fun _ : (Fin n×Fin n)⊕(κ×κ) ↦ ℝ)).symm z) p+d p.2) =
+      (fun p ↦ linearProcess (hamCoeff n) z.1 p.1+(linearProcess (leafAuxCoeff n v) z.2 p.2+d p.2)) := by
+    funext p
+    simp [linearProcess,guerraCoeff,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm,add_assoc]
+  have hm (z : ((Fin n×Fin n) → ℝ)×((κ×κ) → ℝ)) :
+      finiteMaximum (fun p : (Fin n → Bool)×Λ ↦ linearProcess (hamCoeff n) z.1 p.1+
+        (linearProcess (leafAuxCoeff n v) z.2 p.2+d p.2)) =
+      finiteMaximum (linearProcess (hamCoeff n) z.1)+
+        finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z.2 α+d α) :=
+    finiteMaximum_prod_add (linearProcess (hamCoeff n) z.1)
+      (fun α ↦ linearProcess (leafAuxCoeff n v) z.2 α+d α)
+  simp only [hlin,hm]
+  change (∫ z, finiteMaximum (linearProcess (hamCoeff n) z.1)+
+    finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z.2 α+d α)
+      ∂(gaussianProduct (Fin n×Fin n)).prod (gaussianProduct (κ×κ))) = _
+  rw [integral_add ((finiteMaximum_integrable (hamCoeff n)).comp_fst _)
+    ((affineMaximum_integrable (leafAuxCoeff n v) d).comp_snd _),
+    integral_fun_fst (fun z ↦ finiteMaximum (linearProcess (hamCoeff n) z)),
+    integral_fun_snd (fun z ↦ finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z α+d α))]
+  simp [expectedMaximum]
+
+theorem finite_guerra_upper {Λ κ : Type*} [Fintype Λ] [Nonempty Λ] [Fintype κ]
+    {n : ℕ} (hn : 0<n) (v : Λ → κ → ℝ)
+    (hv : ∀ α, ∑ k, (v α k)^2=1) (d : Λ → ℝ) :
+    expectedMaximum n+
+      (∫ z, finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n v) z α+d α)
+        ∂gaussianProduct (κ×κ)) ≤
+      ∫ z, finiteMaximum (fun α ↦ (∑ i, |leafField v z α i|)+d α)
+        ∂gaussianProduct (Fin n×κ) := by
+  have h := expected_affine_max_le (guerraCoeff n v) (leafFieldCoeff n v)
+    (fun p ↦ d p.2) (guerra_metric_le hn v hv)
+  simpa only [expected_guerra_max,leafField_max] using h
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/LogComparison.lean b/lean/OAI/Probability/SKValue/GroundState/LogComparison.lean
index c6b6df111ef27cc54f10b452db24ee3a8079b844..88b6ee2f6f8302df827fe8215d6fc9dd5680f276 100644
--- a/lean/OAI/Probability/SKValue/GroundState/LogComparison.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/LogComparison.lean
@@ -1,165 +1,165 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.RotatedCoefficients
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma expected_logPartition_reindex {ι κ ν : Type*} [Fintype ι] [Fintype κ]
-    [Fintype ν] (e : κ ≃ ν) (a : ι → κ → ℝ) :
-    (∫ z, logPartition (linearProcess (fun i k ↦ a i (e.symm k)) z)
-      ∂gaussianProduct ν) = ∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ := by
-  simpa only [linearProcess_reindex] using gaussian_integral_reindex e
-    (fun z ↦ logPartition (linearProcess a z))
-
-lemma expected_logPartition_le_orthogonal {ι κ : Type*}
-    [Fintype ι] [Nonempty ι] [Fintype κ] (a b : ι → κ → ℝ)
-    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
-    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
-    (∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ) ≤
-      ∫ z, logPartition (linearProcess b z) ∂gaussianProduct κ := by
-  let e := Fintype.equivFin κ
-  have h := expected_logPartition_le_of_pair_distances
-    (fun i k ↦ a i (e.symm k)) (fun i k ↦ b i (e.symm k))
-    (fun i j ↦ by
-      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)*(b i k-b j k))]
-      exact horth i j)
-    (fun i j ↦ by
-      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)^2),
-        e.symm.sum_comp (fun k ↦ (b i k-b j k)^2)]
-      exact hdist i j)
-  simpa only [expected_logPartition_reindex] using h
-
-lemma expected_logPartition_embedLeft {ι κ ν : Type*} [Fintype ι] [Fintype κ] [Fintype ν]
-    (a : ι → κ → ℝ) :
-    (∫ z, logPartition (linearProcess (embedLeft (ν := ν) a) z) ∂gaussianProduct (κ ⊕ ν)) =
-      ∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ := by
-  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
-  change (∫ z, logPartition (linearProcess (embedLeft a) z)
-    ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
-  rw [← he.integral_comp' (fun z ↦ logPartition (linearProcess (embedLeft a) z))]
-  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
-      linearProcess (embedLeft a) ((MeasurableEquiv.sumPiEquivProdPi
-        (fun _ : κ ⊕ ν ↦ ℝ)).symm p) = linearProcess a p.1 := by
-    funext i
-    simp [linearProcess,embedLeft,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
-  simp only [hlin]
-  simpa [gaussianProduct] using integral_fun_fst (μ := gaussianProduct κ)
-    (ν := gaussianProduct ν) (fun z ↦ logPartition (linearProcess a z))
-
-lemma expected_logPartition_embedRight {ι κ ν : Type*} [Fintype ι] [Fintype κ] [Fintype ν]
-    (b : ι → ν → ℝ) :
-    (∫ z, logPartition (linearProcess (embedRight (κ := κ) b) z) ∂gaussianProduct (κ ⊕ ν)) =
-      ∫ z, logPartition (linearProcess b z) ∂gaussianProduct ν := by
-  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
-  change (∫ z, logPartition (linearProcess (embedRight b) z)
-    ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
-  rw [← he.integral_comp' (fun z ↦ logPartition (linearProcess (embedRight b) z))]
-  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
-      linearProcess (embedRight b) ((MeasurableEquiv.sumPiEquivProdPi
-        (fun _ : κ ⊕ ν ↦ ℝ)).symm p) = linearProcess b p.2 := by
-    funext i
-    simp [linearProcess,embedRight,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
-  simp only [hlin]
-  simpa [gaussianProduct] using integral_fun_snd (μ := gaussianProduct κ)
-    (ν := gaussianProduct ν) (fun z ↦ logPartition (linearProcess b z))
-
-lemma expected_logPartition_le {ι κ ν : Type*}
-    [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
-    (a : ι → κ → ℝ) (b : ι → ν → ℝ)
-    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
-    (∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ) ≤
-      ∫ z, logPartition (linearProcess b z) ∂gaussianProduct ν := by
-  have h := expected_logPartition_le_orthogonal
-    (embedLeft (ν := ν) a) (embedRight (κ := κ) b)
-    (fun i j ↦ by simp [embedLeft,embedRight,Fintype.sum_sum_type])
-    (fun i j ↦ by simpa only [Fintype.sum_sum_type,embedLeft,embedRight,
-      Sum.elim_inl,Sum.elim_inr,sub_self,zero_pow (by decide : 2 ≠ 0),
-      Finset.sum_const_zero,zero_add,add_zero] using hdist i j)
-  simpa only [expected_logPartition_embedLeft,expected_logPartition_embedRight] using h
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma finiteMaximum_mono {ι : Type*} [Fintype ι] [Nonempty ι] {x y : ι → ℝ}
-    (h : ∀ i, x i ≤ y i) : finiteMaximum x ≤ finiteMaximum y := by
-  obtain ⟨i,hi⟩ := exists_finiteMaximum x
-  rw [← hi]
-  exact (h i).trans (le_finiteMaximum y i)
-
-lemma finiteMaximum_add_const {ι : Type*} [Fintype ι] [Nonempty ι]
-    (x : ι → ℝ) (c : ℝ) : finiteMaximum (fun i ↦ x i+c)=finiteMaximum x+c := by
-  obtain ⟨i,hi⟩ := exists_finiteMaximum x
-  obtain ⟨j,hj⟩ := exists_finiteMaximum (fun i ↦ x i+c)
-  apply le_antisymm
-  · rw [← hj]
-    linarith [le_finiteMaximum x j]
-  · rw [← hi]
-    exact le_finiteMaximum (fun i ↦ x i+c) i
-
-lemma affineMaximum_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (a : ι → κ → ℝ) (d : ι → ℝ) :
-    Integrable (fun z ↦ finiteMaximum (fun i ↦ linearProcess a z i+d i)) (gaussianProduct κ) := by
-  apply (integrable_finsetSum Finset.univ (fun i _ ↦
-    ((linearProcess_integrable a i).add (integrable_const (d i))).abs)).mono'
-  · exact (continuous_finiteMaximum.comp ((continuous_linearProcess a).add continuous_const)).aestronglyMeasurable
-  · exact Eventually.of_forall (fun z ↦ by
-      simpa only [Real.norm_eq_abs,Pi.add_apply] using abs_finiteMaximum_le
-        (fun i ↦ linearProcess a z i + d i))
-
-lemma replicated_partition_bounds {ι : Type*} [Fintype ι] [Nonempty ι]
-    (x d : ι → ℝ) (β : ℝ) (hβ : 0≤β) (m : ι → ℕ)
-    (hl : ∀ i, Real.exp (β*d i) ≤ (m i : ℝ))
-    (hu : ∀ i, (m i : ℝ) ≤ 2*Real.exp (β*d i)) :
-    β*finiteMaximum (fun i ↦ x i+d i) ≤
-        logPartition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) ∧
-      logPartition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) ≤
-        β*finiteMaximum (fun i ↦ x i+d i)+Real.log (2*(Fintype.card ι : ℝ)) := by
-  have hp : partition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) =
-      ∑ i, (m i : ℝ)*Real.exp (β*x i) := by
-    simp [partition,Fintype.sum_sigma]
-  have hlo : Real.exp (β*finiteMaximum (fun i ↦ x i+d i)) ≤
-      partition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) := by
-    obtain ⟨i,hi⟩ := exists_finiteMaximum (fun i ↦ x i+d i)
-    rw [← hi,hp]
-    calc
-      _ = Real.exp (β*d i)*Real.exp (β*x i) := by rw [←Real.exp_add]; congr 1; ring
-      _ ≤ (m i : ℝ)*Real.exp (β*x i) := mul_le_mul_of_nonneg_right (hl i) (Real.exp_pos _).le
-      _ ≤ _ := Finset.single_le_sum (f := fun j ↦ (m j : ℝ)*Real.exp (β*x j)) (fun j _ ↦ by positivity) (Finset.mem_univ i)
-  have hup : partition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) ≤
-      2*(Fintype.card ι : ℝ)*Real.exp (β*finiteMaximum (fun i ↦ x i+d i)) := by
-    rw [hp]
-    calc
-      _ ≤ ∑ i : ι, 2*Real.exp (β*finiteMaximum (fun i ↦ x i+d i)) := by
-        apply Finset.sum_le_sum
-        intro i _
-        calc
-          _ ≤ 2*Real.exp (β*d i)*Real.exp (β*x i) :=
-            mul_le_mul_of_nonneg_right (hu i) (Real.exp_pos _).le
-          _ = 2*Real.exp (β*(x i+d i)) := by rw [mul_assoc,←Real.exp_add]; congr 2; ring
-          _ ≤ _ := mul_le_mul_of_nonneg_left
-            (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (le_finiteMaximum (fun i ↦ x i+d i) i) hβ)) (by norm_num)
-      _ = _ := by simp; ring
-  refine ⟨?_,?_⟩
-  · simpa only [Real.log_exp,logPartition] using Real.log_le_log (Real.exp_pos _) hlo
-  · have hc : 0<2*(Fintype.card ι : ℝ) := by positivity
-    have hh := Real.log_le_log ((Real.exp_pos _).trans_le hlo) hup
-    simpa only [logPartition,Real.log_mul hc.ne' (Real.exp_pos _).ne',
-      Real.log_exp,add_comm] using hh
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.RotatedCoefficients
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma expected_logPartition_reindex {ι κ ν : Type*} [Fintype ι] [Fintype κ]
+    [Fintype ν] (e : κ ≃ ν) (a : ι → κ → ℝ) :
+    (∫ z, logPartition (linearProcess (fun i k ↦ a i (e.symm k)) z)
+      ∂gaussianProduct ν) = ∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ := by
+  simpa only [linearProcess_reindex] using gaussian_integral_reindex e
+    (fun z ↦ logPartition (linearProcess a z))
+
+lemma expected_logPartition_le_orthogonal {ι κ : Type*}
+    [Fintype ι] [Nonempty ι] [Fintype κ] (a b : ι → κ → ℝ)
+    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
+    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
+    (∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ) ≤
+      ∫ z, logPartition (linearProcess b z) ∂gaussianProduct κ := by
+  let e := Fintype.equivFin κ
+  have h := expected_logPartition_le_of_pair_distances
+    (fun i k ↦ a i (e.symm k)) (fun i k ↦ b i (e.symm k))
+    (fun i j ↦ by
+      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)*(b i k-b j k))]
+      exact horth i j)
+    (fun i j ↦ by
+      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)^2),
+        e.symm.sum_comp (fun k ↦ (b i k-b j k)^2)]
+      exact hdist i j)
+  simpa only [expected_logPartition_reindex] using h
+
+lemma expected_logPartition_embedLeft {ι κ ν : Type*} [Fintype ι] [Fintype κ] [Fintype ν]
+    (a : ι → κ → ℝ) :
+    (∫ z, logPartition (linearProcess (embedLeft (ν := ν) a) z) ∂gaussianProduct (κ ⊕ ν)) =
+      ∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ := by
+  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
+  change (∫ z, logPartition (linearProcess (embedLeft a) z)
+    ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
+  rw [← he.integral_comp' (fun z ↦ logPartition (linearProcess (embedLeft a) z))]
+  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
+      linearProcess (embedLeft a) ((MeasurableEquiv.sumPiEquivProdPi
+        (fun _ : κ ⊕ ν ↦ ℝ)).symm p) = linearProcess a p.1 := by
+    funext i
+    simp [linearProcess,embedLeft,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
+  simp only [hlin]
+  simpa [gaussianProduct] using integral_fun_fst (μ := gaussianProduct κ)
+    (ν := gaussianProduct ν) (fun z ↦ logPartition (linearProcess a z))
+
+lemma expected_logPartition_embedRight {ι κ ν : Type*} [Fintype ι] [Fintype κ] [Fintype ν]
+    (b : ι → ν → ℝ) :
+    (∫ z, logPartition (linearProcess (embedRight (κ := κ) b) z) ∂gaussianProduct (κ ⊕ ν)) =
+      ∫ z, logPartition (linearProcess b z) ∂gaussianProduct ν := by
+  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
+  change (∫ z, logPartition (linearProcess (embedRight b) z)
+    ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
+  rw [← he.integral_comp' (fun z ↦ logPartition (linearProcess (embedRight b) z))]
+  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
+      linearProcess (embedRight b) ((MeasurableEquiv.sumPiEquivProdPi
+        (fun _ : κ ⊕ ν ↦ ℝ)).symm p) = linearProcess b p.2 := by
+    funext i
+    simp [linearProcess,embedRight,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
+  simp only [hlin]
+  simpa [gaussianProduct] using integral_fun_snd (μ := gaussianProduct κ)
+    (ν := gaussianProduct ν) (fun z ↦ logPartition (linearProcess b z))
+
+lemma expected_logPartition_le {ι κ ν : Type*}
+    [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
+    (a : ι → κ → ℝ) (b : ι → ν → ℝ)
+    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
+    (∫ z, logPartition (linearProcess a z) ∂gaussianProduct κ) ≤
+      ∫ z, logPartition (linearProcess b z) ∂gaussianProduct ν := by
+  have h := expected_logPartition_le_orthogonal
+    (embedLeft (ν := ν) a) (embedRight (κ := κ) b)
+    (fun i j ↦ by simp [embedLeft,embedRight,Fintype.sum_sum_type])
+    (fun i j ↦ by simpa only [Fintype.sum_sum_type,embedLeft,embedRight,
+      Sum.elim_inl,Sum.elim_inr,sub_self,zero_pow (by decide : 2 ≠ 0),
+      Finset.sum_const_zero,zero_add,add_zero] using hdist i j)
+  simpa only [expected_logPartition_embedLeft,expected_logPartition_embedRight] using h
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma finiteMaximum_mono {ι : Type*} [Fintype ι] [Nonempty ι] {x y : ι → ℝ}
+    (h : ∀ i, x i ≤ y i) : finiteMaximum x ≤ finiteMaximum y := by
+  obtain ⟨i,hi⟩ := exists_finiteMaximum x
+  rw [← hi]
+  exact (h i).trans (le_finiteMaximum y i)
+
+lemma finiteMaximum_add_const {ι : Type*} [Fintype ι] [Nonempty ι]
+    (x : ι → ℝ) (c : ℝ) : finiteMaximum (fun i ↦ x i+c)=finiteMaximum x+c := by
+  obtain ⟨i,hi⟩ := exists_finiteMaximum x
+  obtain ⟨j,hj⟩ := exists_finiteMaximum (fun i ↦ x i+c)
+  apply le_antisymm
+  · rw [← hj]
+    linarith [le_finiteMaximum x j]
+  · rw [← hi]
+    exact le_finiteMaximum (fun i ↦ x i+c) i
+
+lemma affineMaximum_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (a : ι → κ → ℝ) (d : ι → ℝ) :
+    Integrable (fun z ↦ finiteMaximum (fun i ↦ linearProcess a z i+d i)) (gaussianProduct κ) := by
+  apply (integrable_finsetSum Finset.univ (fun i _ ↦
+    ((linearProcess_integrable a i).add (integrable_const (d i))).abs)).mono'
+  · exact (continuous_finiteMaximum.comp ((continuous_linearProcess a).add continuous_const)).aestronglyMeasurable
+  · exact Eventually.of_forall (fun z ↦ by
+      simpa only [Real.norm_eq_abs,Pi.add_apply] using abs_finiteMaximum_le
+        (fun i ↦ linearProcess a z i + d i))
+
+lemma replicated_partition_bounds {ι : Type*} [Fintype ι] [Nonempty ι]
+    (x d : ι → ℝ) (β : ℝ) (hβ : 0≤β) (m : ι → ℕ)
+    (hl : ∀ i, Real.exp (β*d i) ≤ (m i : ℝ))
+    (hu : ∀ i, (m i : ℝ) ≤ 2*Real.exp (β*d i)) :
+    β*finiteMaximum (fun i ↦ x i+d i) ≤
+        logPartition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) ∧
+      logPartition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) ≤
+        β*finiteMaximum (fun i ↦ x i+d i)+Real.log (2*(Fintype.card ι : ℝ)) := by
+  have hp : partition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) =
+      ∑ i, (m i : ℝ)*Real.exp (β*x i) := by
+    simp [partition,Fintype.sum_sigma]
+  have hlo : Real.exp (β*finiteMaximum (fun i ↦ x i+d i)) ≤
+      partition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) := by
+    obtain ⟨i,hi⟩ := exists_finiteMaximum (fun i ↦ x i+d i)
+    rw [← hi,hp]
+    calc
+      _ = Real.exp (β*d i)*Real.exp (β*x i) := by rw [←Real.exp_add]; congr 1; ring
+      _ ≤ (m i : ℝ)*Real.exp (β*x i) := mul_le_mul_of_nonneg_right (hl i) (Real.exp_pos _).le
+      _ ≤ _ := Finset.single_le_sum (f := fun j ↦ (m j : ℝ)*Real.exp (β*x j)) (fun j _ ↦ by positivity) (Finset.mem_univ i)
+  have hup : partition (fun i : (Σ i, Fin (m i)) ↦ β*x i.1) ≤
+      2*(Fintype.card ι : ℝ)*Real.exp (β*finiteMaximum (fun i ↦ x i+d i)) := by
+    rw [hp]
+    calc
+      _ ≤ ∑ i : ι, 2*Real.exp (β*finiteMaximum (fun i ↦ x i+d i)) := by
+        apply Finset.sum_le_sum
+        intro i _
+        calc
+          _ ≤ 2*Real.exp (β*d i)*Real.exp (β*x i) :=
+            mul_le_mul_of_nonneg_right (hu i) (Real.exp_pos _).le
+          _ = 2*Real.exp (β*(x i+d i)) := by rw [mul_assoc,←Real.exp_add]; congr 2; ring
+          _ ≤ _ := mul_le_mul_of_nonneg_left
+            (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (le_finiteMaximum (fun i ↦ x i+d i) i) hβ)) (by norm_num)
+      _ = _ := by simp; ring
+  refine ⟨?_,?_⟩
+  · simpa only [Real.log_exp,logPartition] using Real.log_le_log (Real.exp_pos _) hlo
+  · have hc : 0<2*(Fintype.card ι : ℝ) := by positivity
+    have hh := Real.log_le_log ((Real.exp_pos _).trans_le hlo) hup
+    simpa only [logPartition,Real.log_mul hc.ne' (Real.exp_pos _).ne',
+      Real.log_exp,add_comm] using hh
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/MainConclusions.lean b/lean/OAI/Probability/SKValue/GroundState/MainConclusions.lean
index 83a1c042274c237c750b6035983194d32c3f2fde..9535baf8397288e63a67b20b1bfdc287fb23acac 100644
--- a/lean/OAI/Probability/SKValue/GroundState/MainConclusions.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/MainConclusions.lean
@@ -1,32 +1,32 @@
-import Mathlib
-import OAI.Probability.SKValue.Variation.MinimizerContact
-import OAI.Probability.SKValue.Variation.ContactAssembly
-import OAI.Probability.SKValue.Processes.DiffusionExistence
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal
-namespace SKValue
-
-theorem scalar_value_target (W:BrownianSpace) (γ:OrderParameter) (X:ℝ → W.Ω → ℝ)
-    (hmin:IsMinimizer W γ) (hX:IsDiffusion W γ X):ScalarValueConclusion W γ X :=
-  hX.scalar_value_of_contact (hmin.gradient_moment_contact hX)
-
-theorem finite_gaussian_target (W:BrownianSpace) (γ:OrderParameter) (X:ℝ → W.Ω → ℝ)
-    (T:ℝ) (hT:0<T) (hT1:T<1) (hmin:IsMinimizer W γ) (hX:IsDiffusion W γ X):
-    FiniteGaussianConclusion W γ X T :=
-  hX.finite_gaussian_of_contact (hmin.gradient_moment_contact hX) hT hT1
-lemma IsMinimizer.ground_state_parisi_of_diffusion {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
-    (hmin:IsMinimizer W γ) (hX:IsDiffusion W γ X):groundStateValue=parisi W γ :=
-  hX.groundState_eq_parisi_of_contact (hmin.gradient_moment_contact hX)
-
-theorem ground_state_parisi_target (W:BrownianSpace) (γ:OrderParameter)
-    (hmin:IsMinimizer W γ):groundStateValue=parisi W γ :=
-  hmin.ground_state_parisi_of_diffusion (diffusionProcess_isDiffusion W γ)
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Variation.MinimizerContact
+import OAI.Probability.SKValue.Variation.ContactAssembly
+import OAI.Probability.SKValue.Processes.DiffusionExistence
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal
+namespace SKValue
+
+theorem scalar_value_target (W:BrownianSpace) (γ:OrderParameter) (X:ℝ → W.Ω → ℝ)
+    (hmin:IsMinimizer W γ) (hX:IsDiffusion W γ X):ScalarValueConclusion W γ X :=
+  hX.scalar_value_of_contact (hmin.gradient_moment_contact hX)
+
+theorem finite_gaussian_target (W:BrownianSpace) (γ:OrderParameter) (X:ℝ → W.Ω → ℝ)
+    (T:ℝ) (hT:0<T) (hT1:T<1) (hmin:IsMinimizer W γ) (hX:IsDiffusion W γ X):
+    FiniteGaussianConclusion W γ X T :=
+  hX.finite_gaussian_of_contact (hmin.gradient_moment_contact hX) hT hT1
+lemma IsMinimizer.ground_state_parisi_of_diffusion {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
+    (hmin:IsMinimizer W γ) (hX:IsDiffusion W γ X):groundStateValue=parisi W γ :=
+  hX.groundState_eq_parisi_of_contact (hmin.gradient_moment_contact hX)
+
+theorem ground_state_parisi_target (W:BrownianSpace) (γ:OrderParameter)
+    (hmin:IsMinimizer W γ):groundStateValue=parisi W γ :=
+  hmin.ground_state_parisi_of_diffusion (diffusionProcess_isDiffusion W γ)
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/PredictableNormalizer.lean b/lean/OAI/Probability/SKValue/GroundState/PredictableNormalizer.lean
index fe46a0228aadc5f90a02109905147b5f8fa20bab..22f82427b32a69300f0d50decbf357d924038240 100644
--- a/lean/OAI/Probability/SKValue/GroundState/PredictableNormalizer.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/PredictableNormalizer.lean
@@ -1,294 +1,294 @@
-import Mathlib
-import OAI.Probability.SKValue.Model
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-namespace SKValue
-
-lemma independent_of_coordinate_not_in {ι β : Type*} [Fintype ι] [DecidableEq ι]
-    [MeasurableSpace β]
-    {μ : ι → Measure ℝ} [∀ i, IsProbabilityMeasure (μ i)]
-    (S : Finset ι) {j : ι} (hj : j ∉ S) {F : (ι → ℝ) → β}
-    (hF : Measurable F)
-    (hdep : ∀ x y, (∀ i ∈ S, x i=y i) → F x=F y) :
-    IndepFun F (fun x ↦ x j) (Measure.pi μ) := by
-  let fill : (S → ℝ) → ι → ℝ := fun x i ↦ if hi : i ∈ S then x ⟨i,hi⟩ else 0
-  have hfill : Measurable fill := by
-    apply Measurable.of_eval
-    intro i
-    by_cases hi : i ∈ S
-    · simpa only [fill, dite_eq_left hi] using (measurable_pi_apply (⟨i,hi⟩ : S))
-    · simp only [fill, dite_eq_right hi]
-      exact measurable_const
-  have hid : ∀ x : ι → ℝ, F (fill (fun i : S ↦ x i)) = F x := by
-    intro x
-    apply hdep
-    intro i hi
-    simp [fill,hi]
-  have hST : Disjoint S {j} := by simpa using hj
-  have hind : iIndepFun (fun i (x : ι → ℝ) ↦ x i) (Measure.pi μ) :=
-    iIndepFun_pi (fun i ↦ (measurable_id : Measurable (id : ℝ → ℝ)).aemeasurable)
-  have H := (hind.indepFun_finset S {j} hST (fun i ↦ measurable_pi_apply i)).comp
-    (hF.comp hfill) (measurable_pi_apply (⟨j,by simp⟩ : ({j} : Finset ι)))
-  simpa only [Function.comp_def, hid] using H
-
-lemma measurable_coordinate (N j : ℕ) : Measurable (coordinate N j) :=
-  measurable_pi_apply _
-
-lemma coordinate_eq_apply {N j : ℕ} (hj : j < N+1) (z : Fin (N+1) → ℝ) :
-    coordinate N j z = z ⟨j,hj⟩ := by
-  simp [coordinate, Nat.mod_eq_of_lt hj]
-
-lemma euler_depends_on_past (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (j : ℕ) (z z' : Fin (N+1) → ℝ)
-    (h : ∀ i < j, coordinate N i z=coordinate N i z') :
-    euler T N γ u z j=euler T N γ u z' j := by
-  induction j with
-  | zero => rfl
-  | succ j ih =>
-    have hp := ih (fun i hi ↦ h i (by omega))
-    simp only [euler, hp,h j (by omega)]
-
-lemma measurable_euler (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (hU : ∀ k < N, Measurable (u (meshTime T N k))) :
-    ∀ j ≤ N, Measurable (fun z ↦ euler T N γ u z j) := by
-  intro j hj
-  induction j with
-  | zero => exact measurable_const
-  | succ j ih =>
-    have hjN : j < N := by omega
-    have hy := ih (by omega)
-    have hz := measurable_coordinate N j
-    have hu := (hU j hjN).comp hy
-    simp only [euler]
-    fun_prop
-
-lemma euler_coefficient_independent {T : ℝ} {N j : ℕ} {γ : ℝ → ℝ}
-    {u a : ℝ → ℝ → ℝ} (hj : j < N+1)
-    (hU : ∀ k < N, Measurable (u (meshTime T N k)))
-    (hA : Measurable (a (meshTime T N j))) :
-    IndepFun (fun z ↦ a (meshTime T N j) (euler T N γ u z j))
-      (coordinate N j) (gaussianProduct (Fin (N+1))) := by
-  let S : Finset (Fin (N+1)) := Finset.univ.filter (fun i ↦ (i : ℕ)<j)
-  have hjS : (⟨j,hj⟩ : Fin (N+1)) ∉ S := by simp [S]
-  have hF : Measurable (fun z ↦ a (meshTime T N j) (euler T N γ u z j)) :=
-    hA.comp (measurable_euler T N γ u hU j (by omega))
-  have h := independent_of_coordinate_not_in (μ := fun _ : Fin (N+1) ↦ standardGaussian)
-    S hjS hF
-    (fun z z' heq ↦ by
-      apply congrArg (a (meshTime T N j))
-      apply euler_depends_on_past T N γ u j z z'
-      intro i hi
-      have hiN : i < N+1 := hi.trans hj
-      rw [coordinate_eq_apply hiN,coordinate_eq_apply hiN]
-      apply heq
-      simp [S,hi])
-  have heq : coordinate N j = fun z : Fin (N+1) → ℝ ↦ z ⟨j,hj⟩ :=
-    funext (coordinate_eq_apply hj)
-  rw [heq]
-  exact h
-
-variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-
-lemma gaussian_mean {Z : Ω → ℝ} (hZ : HasLaw Z (gaussianReal 0 1) μ) :
-    ∫ ω, Z ω ∂μ = 0 := by
-  rw [hZ.integral_eq, integral_id_gaussianReal]
-
-lemma gaussian_memLp {Z : Ω → ℝ} (hZ : HasLaw Z (gaussianReal 0 1) μ)
-    (p : ℝ≥0∞) (hp : p ≠ ∞) : MemLp Z p μ := by
-  have H : MemLp (id : ℝ → ℝ) p (μ.map Z) := by
-    rw [hZ.map_eq]
-    exact memLp_id_gaussianReal' p hp
-  simpa using (memLp_map_measure_iff H.aestronglyMeasurable hZ.aemeasurable).1 H
-
-lemma gaussian_second_moment {Z : Ω → ℝ} (hZ : HasLaw Z (gaussianReal 0 1) μ) :
-    ∫ ω, (Z ω)^2 ∂μ = 1 := by
-  have hvar := hZ.variance_eq
-  rw [variance_eq_integral hZ.aemeasurable, gaussian_mean hZ] at hvar
-  simpa using hvar
-
-noncomputable def predictableNormalizer (H : Ω → ℝ) (μ : Measure Ω) : ℝ :=
-  (Real.sqrt (∫ ω, (H ω)^2 ∂μ))⁻¹
-
-noncomputable def normalizedIncrement (H Z : Ω → ℝ) (μ : Measure Ω) : Ω → ℝ :=
-  fun ω ↦ predictableNormalizer H μ * H ω * Z ω
-
-lemma predictableNormalizer_pos {H : Ω → ℝ}
-    (hHpos : 0 < ∫ ω, (H ω)^2 ∂μ) :
-    0 < predictableNormalizer H μ := by
-  exact inv_pos.mpr (Real.sqrt_pos.mpr hHpos)
-
-lemma normalizedIncrement_mean {H Z : Ω → ℝ}
-    (hH : AEStronglyMeasurable H μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
-    (hind : IndepFun H Z μ) :
-    ∫ ω, normalizedIncrement H Z μ ω ∂μ = 0 := by
-  simp only [normalizedIncrement, mul_assoc]
-  rw [integral_const_mul]
-  have hi := hind.integral_mul_eq_mul_integral hH hZ.aemeasurable.aestronglyMeasurable
-  change (∫ ω, H ω * Z ω ∂μ) = _ at hi
-  rw [hi, gaussian_mean hZ, mul_zero, mul_zero]
-
-lemma normalizedIncrement_second_moment {H Z : Ω → ℝ}
-    (hH : AEStronglyMeasurable H μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
-    (hind : IndepFun H Z μ) (hHpos : 0 < ∫ ω, (H ω)^2 ∂μ) :
-    ∫ ω, (normalizedIncrement H Z μ ω)^2 ∂μ = 1 := by
-  have hind2 : IndepFun (fun ω ↦ (H ω)^2) (fun ω ↦ (Z ω)^2) μ :=
-    hind.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)
-  have hi := hind2.integral_mul_eq_mul_integral
-    (hH.fun_pow 2)
-    (hZ.aemeasurable.aestronglyMeasurable.fun_pow 2)
-  change (∫ ω, (H ω)^2 * (Z ω)^2 ∂μ) =
-    (∫ ω, (H ω)^2 ∂μ) * (∫ ω, (Z ω)^2 ∂μ) at hi
-  simp only [normalizedIncrement, mul_pow, mul_assoc]
-  rw [integral_const_mul, hi, gaussian_second_moment hZ, mul_one]
-  simp only [predictableNormalizer, inv_pow, Real.sq_sqrt (le_of_lt hHpos)]
-  exact inv_mul_cancel₀ (ne_of_gt hHpos)
-
-lemma normalizedIncrement_memLp_two {H Z : Ω → ℝ}
-    (hH : MemLp H 2 μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
-    (hind : IndepFun H Z μ) :
-    MemLp (normalizedIncrement H Z μ) 2 μ := by
-  apply (memLp_two_iff_integrable_sq
-    ((hH.aestronglyMeasurable.const_mul _).mul hZ.aemeasurable.aestronglyMeasurable)).2
-  have hi := (hind.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integrable_mul
-     hH.integrable_sq (gaussian_memLp hZ 2 (by norm_num)).integrable_sq
-  have hj := hi.const_mul (predictableNormalizer H μ ^ 2)
-  simpa only [normalizedIncrement, Pi.mul_apply, mul_pow, mul_assoc, Function.comp_def, id_eq] using hj
-
-lemma normalizedIncrement_gaussian_coefficient {H Z : Ω → ℝ}
-    (hH : AEStronglyMeasurable H μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
-    (hind : IndepFun H Z μ) :
-    ∫ ω, normalizedIncrement H Z μ ω * Z ω ∂μ =
-      predictableNormalizer H μ * ∫ ω, H ω ∂μ := by
-  have hind2 : IndepFun H (fun ω ↦ (Z ω)^2) μ :=
-    hind.comp measurable_id (measurable_id.pow_const 2)
-  have hi := hind2.integral_mul_eq_mul_integral hH
-    (hZ.aemeasurable.aestronglyMeasurable.fun_pow 2)
-  change (∫ ω, H ω * (Z ω)^2 ∂μ) =
-    (∫ ω, H ω ∂μ) * (∫ ω, (Z ω)^2 ∂μ) at hi
-  simp only [normalizedIncrement, mul_assoc, ← pow_two]
-  rw [integral_const_mul, hi, gaussian_second_moment hZ, mul_one]
-
-lemma predictable_product_mean_zero {H K Z : Ω → ℝ}
-    (hH : AEStronglyMeasurable H μ) (hK : AEStronglyMeasurable K μ)
-    (hZ : HasLaw Z (gaussianReal 0 1) μ)
-    (hind : IndepFun (fun ω ↦ H ω * K ω) Z μ) :
-    ∫ ω, K ω * normalizedIncrement H Z μ ω ∂μ = 0 := by
-  have hi := hind.integral_mul_eq_mul_integral (hH.mul hK)
-    hZ.aemeasurable.aestronglyMeasurable
-  change (∫ ω, (H ω * K ω) * Z ω ∂μ) = _ at hi
-  calc
-    _ = predictableNormalizer H μ * ∫ ω, (H ω * K ω) * Z ω ∂μ := by
-      rw [← integral_const_mul]
-      congr 1
-      ext ω
-      simp only [normalizedIncrement]
-      ring
-    _ = 0 := by rw [hi, gaussian_mean hZ, mul_zero, mul_zero]
-
-lemma coordinate_hasLaw (N j : ℕ) :
-    HasLaw (coordinate N j) (gaussianReal 0 1) (gaussianProduct (Fin (N+1))) :=
-  (measurePreserving_eval (fun _ : Fin (N+1) ↦ standardGaussian)
-    ⟨j % (N+1), Nat.mod_lt _ (Nat.succ_pos _)⟩).hasLaw
-
-def DependsBefore {N : ℕ} {β : Type*} (j : ℕ) (F : (Fin (N+1) → ℝ) → β) : Prop :=
-  ∀ z z', (∀ i : Fin (N+1), (i : ℕ)<j → z i=z' i) → F z=F z'
-
-lemma DependsBefore.independent {N j : ℕ} {β : Type*} [MeasurableSpace β] (hj : j<N+1)
-    {F : (Fin (N+1) → ℝ) → β} (hF : DependsBefore j F) (hm : Measurable F) :
-    IndepFun F (coordinate N j) (gaussianProduct (Fin (N+1))) := by
-  have H := independent_of_coordinate_not_in (μ := fun _ : Fin (N+1) ↦ standardGaussian)
-    (Finset.univ.filter (fun i : Fin (N+1) ↦ (i : ℕ)<j))
-    (j := ⟨j,hj⟩) (by simp) hm (fun z z' heq ↦ hF z z' (by simpa using heq))
-  have heq : coordinate N j = fun z : Fin (N+1) → ℝ ↦ z ⟨j,hj⟩ :=
-    funext (coordinate_eq_apply hj)
-  rw [heq]
-  exact H
-
-lemma DependsBefore.mono {N i j : ℕ} {β : Type*} {F : (Fin (N+1) → ℝ) → β}
-    (hF : DependsBefore i F) (hij : i≤j) : DependsBefore j F :=
-  fun z z' h ↦ hF z z' (fun k hk ↦ h k (hk.trans_le hij))
-
-lemma DependsBefore.prodMk {N j : ℕ} {β β' : Type*}
-    {F : (Fin (N+1) → ℝ) → β} {G : (Fin (N+1) → ℝ) → β'}
-    (hF : DependsBefore j F) (hG : DependsBefore j G) :
-    DependsBefore j (fun z ↦ (F z,G z)) := by
-  intro z z' h
-  exact Prod.ext (hF z z' h) (hG z z' h)
-
-lemma DependsBefore.mul {N j : ℕ} {F G : (Fin (N+1) → ℝ) → ℝ}
-    (hF : DependsBefore j F) (hG : DependsBefore j G) :
-    DependsBefore j (fun z ↦ F z * G z) := by
-  intro z z' h
-  dsimp only
-  rw [hF z z' h, hG z z' h]
-
-lemma eulerCoefficient_dependsBefore (T : ℝ) {N j : ℕ} (hj : j≤N)
-    (γ : ℝ → ℝ) (u a : ℝ → ℝ → ℝ) :
-    DependsBefore j (fun z ↦ a (meshTime T N j) (euler T N γ u z j)) := by
-  intro z z' h
-  apply congrArg (a (meshTime T N j))
-  apply euler_depends_on_past
-  intro i hi
-  have hiN : i<N+1 := by omega
-  rw [coordinate_eq_apply hiN, coordinate_eq_apply hiN]
-  exact h ⟨i,hiN⟩ hi
-
-lemma normalizedIncrement_dependsBefore {N j : ℕ} (hj : j<N+1)
-    {H : (Fin (N+1) → ℝ) → ℝ} (hH : DependsBefore j H) :
-    DependsBefore (j+1) (normalizedIncrement H (coordinate N j) (gaussianProduct (Fin (N+1)))) := by
-  intro z z' h
-  have heq := hH.mono (Nat.le_succ j) z z' h
-  have hz : coordinate N j z=coordinate N j z' := by
-    rw [coordinate_eq_apply hj, coordinate_eq_apply hj]
-    exact h ⟨j,hj⟩ (Nat.lt_succ_self j)
-  simp only [normalizedIncrement,heq,hz]
-
-lemma measurable_normalizedIncrement {N j : ℕ} {H : (Fin (N+1) → ℝ) → ℝ}
-    (hH : Measurable H) :
-    Measurable (normalizedIncrement H (coordinate N j) (gaussianProduct (Fin (N+1)))) :=
-  (hH.const_mul _).mul (measurable_coordinate N j)
-
-lemma normalizedIncrements_orthogonal {N i j : ℕ} (hij : i<j) (hj : j<N+1)
-    {H K : (Fin (N+1) → ℝ) → ℝ} (hHm : Measurable H) (hKm : Measurable K)
-    (hH : DependsBefore i H) (hK : DependsBefore j K) :
-    ∫ z, normalizedIncrement H (coordinate N i) (gaussianProduct (Fin (N+1))) z *
-      normalizedIncrement K (coordinate N j) (gaussianProduct (Fin (N+1))) z
-      ∂gaussianProduct (Fin (N+1)) = 0 := by
-  have hi : i<N+1 := hij.trans hj
-  have hprev := (normalizedIncrement_dependsBefore hi hH).mono (by omega : i+1≤j)
-  exact predictable_product_mean_zero hKm.aestronglyMeasurable
-    (measurable_normalizedIncrement hHm).aestronglyMeasurable (coordinate_hasLaw N j)
-    ((hK.mul hprev).independent hj
-      (hKm.mul (measurable_normalizedIncrement hHm)))
-
-lemma euler_normalized_increment_properties {T : ℝ} {N j : ℕ} {γ : ℝ → ℝ}
-    {u a : ℝ → ℝ → ℝ} (hj : j<N+1)
-    (hU : ∀ k<N, Measurable (u (meshTime T N k)))
-    (hA : Measurable (a (meshTime T N j)))
-    (hpos : 0 < ∫ z, (a (meshTime T N j) (euler T N γ u z j))^2
-      ∂gaussianProduct (Fin (N+1))) :
-    let H := fun z ↦ a (meshTime T N j) (euler T N γ u z j)
-    let μ := gaussianProduct (Fin (N+1))
-    0 < predictableNormalizer H μ ∧
-    (∫ z, normalizedIncrement H (coordinate N j) μ z ∂μ)=0 ∧
-    (∫ z, (normalizedIncrement H (coordinate N j) μ z)^2 ∂μ)=1 := by
-  dsimp only
-  have hm := hA.comp (measurable_euler T N γ u hU j (by omega))
-  have hind := euler_coefficient_independent (γ := γ) hj hU hA
-  exact ⟨predictableNormalizer_pos hpos,
-    normalizedIncrement_mean hm.aestronglyMeasurable (coordinate_hasLaw N j) hind,
-    normalizedIncrement_second_moment hm.aestronglyMeasurable (coordinate_hasLaw N j) hind hpos⟩
-
-end SKValue
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Model
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+namespace SKValue
+
+lemma independent_of_coordinate_not_in {ι β : Type*} [Fintype ι] [DecidableEq ι]
+    [MeasurableSpace β]
+    {μ : ι → Measure ℝ} [∀ i, IsProbabilityMeasure (μ i)]
+    (S : Finset ι) {j : ι} (hj : j ∉ S) {F : (ι → ℝ) → β}
+    (hF : Measurable F)
+    (hdep : ∀ x y, (∀ i ∈ S, x i=y i) → F x=F y) :
+    IndepFun F (fun x ↦ x j) (Measure.pi μ) := by
+  let fill : (S → ℝ) → ι → ℝ := fun x i ↦ if hi : i ∈ S then x ⟨i,hi⟩ else 0
+  have hfill : Measurable fill := by
+    apply Measurable.of_eval
+    intro i
+    by_cases hi : i ∈ S
+    · simpa only [fill, dite_eq_left hi] using (measurable_pi_apply (⟨i,hi⟩ : S))
+    · simp only [fill, dite_eq_right hi]
+      exact measurable_const
+  have hid : ∀ x : ι → ℝ, F (fill (fun i : S ↦ x i)) = F x := by
+    intro x
+    apply hdep
+    intro i hi
+    simp [fill,hi]
+  have hST : Disjoint S {j} := by simpa using hj
+  have hind : iIndepFun (fun i (x : ι → ℝ) ↦ x i) (Measure.pi μ) :=
+    iIndepFun_pi (fun i ↦ (measurable_id : Measurable (id : ℝ → ℝ)).aemeasurable)
+  have H := (hind.indepFun_finset S {j} hST (fun i ↦ measurable_pi_apply i)).comp
+    (hF.comp hfill) (measurable_pi_apply (⟨j,by simp⟩ : ({j} : Finset ι)))
+  simpa only [Function.comp_def, hid] using H
+
+lemma measurable_coordinate (N j : ℕ) : Measurable (coordinate N j) :=
+  measurable_pi_apply _
+
+lemma coordinate_eq_apply {N j : ℕ} (hj : j < N+1) (z : Fin (N+1) → ℝ) :
+    coordinate N j z = z ⟨j,hj⟩ := by
+  simp [coordinate, Nat.mod_eq_of_lt hj]
+
+lemma euler_depends_on_past (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (j : ℕ) (z z' : Fin (N+1) → ℝ)
+    (h : ∀ i < j, coordinate N i z=coordinate N i z') :
+    euler T N γ u z j=euler T N γ u z' j := by
+  induction j with
+  | zero => rfl
+  | succ j ih =>
+    have hp := ih (fun i hi ↦ h i (by omega))
+    simp only [euler, hp,h j (by omega)]
+
+lemma measurable_euler (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (hU : ∀ k < N, Measurable (u (meshTime T N k))) :
+    ∀ j ≤ N, Measurable (fun z ↦ euler T N γ u z j) := by
+  intro j hj
+  induction j with
+  | zero => exact measurable_const
+  | succ j ih =>
+    have hjN : j < N := by omega
+    have hy := ih (by omega)
+    have hz := measurable_coordinate N j
+    have hu := (hU j hjN).comp hy
+    simp only [euler]
+    fun_prop
+
+lemma euler_coefficient_independent {T : ℝ} {N j : ℕ} {γ : ℝ → ℝ}
+    {u a : ℝ → ℝ → ℝ} (hj : j < N+1)
+    (hU : ∀ k < N, Measurable (u (meshTime T N k)))
+    (hA : Measurable (a (meshTime T N j))) :
+    IndepFun (fun z ↦ a (meshTime T N j) (euler T N γ u z j))
+      (coordinate N j) (gaussianProduct (Fin (N+1))) := by
+  let S : Finset (Fin (N+1)) := Finset.univ.filter (fun i ↦ (i : ℕ)<j)
+  have hjS : (⟨j,hj⟩ : Fin (N+1)) ∉ S := by simp [S]
+  have hF : Measurable (fun z ↦ a (meshTime T N j) (euler T N γ u z j)) :=
+    hA.comp (measurable_euler T N γ u hU j (by omega))
+  have h := independent_of_coordinate_not_in (μ := fun _ : Fin (N+1) ↦ standardGaussian)
+    S hjS hF
+    (fun z z' heq ↦ by
+      apply congrArg (a (meshTime T N j))
+      apply euler_depends_on_past T N γ u j z z'
+      intro i hi
+      have hiN : i < N+1 := hi.trans hj
+      rw [coordinate_eq_apply hiN,coordinate_eq_apply hiN]
+      apply heq
+      simp [S,hi])
+  have heq : coordinate N j = fun z : Fin (N+1) → ℝ ↦ z ⟨j,hj⟩ :=
+    funext (coordinate_eq_apply hj)
+  rw [heq]
+  exact h
+
+variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+
+lemma gaussian_mean {Z : Ω → ℝ} (hZ : HasLaw Z (gaussianReal 0 1) μ) :
+    ∫ ω, Z ω ∂μ = 0 := by
+  rw [hZ.integral_eq, integral_id_gaussianReal]
+
+lemma gaussian_memLp {Z : Ω → ℝ} (hZ : HasLaw Z (gaussianReal 0 1) μ)
+    (p : ℝ≥0∞) (hp : p ≠ ∞) : MemLp Z p μ := by
+  have H : MemLp (id : ℝ → ℝ) p (μ.map Z) := by
+    rw [hZ.map_eq]
+    exact memLp_id_gaussianReal' p hp
+  simpa using (memLp_map_measure_iff H.aestronglyMeasurable hZ.aemeasurable).1 H
+
+lemma gaussian_second_moment {Z : Ω → ℝ} (hZ : HasLaw Z (gaussianReal 0 1) μ) :
+    ∫ ω, (Z ω)^2 ∂μ = 1 := by
+  have hvar := hZ.variance_eq
+  rw [variance_eq_integral hZ.aemeasurable, gaussian_mean hZ] at hvar
+  simpa using hvar
+
+noncomputable def predictableNormalizer (H : Ω → ℝ) (μ : Measure Ω) : ℝ :=
+  (Real.sqrt (∫ ω, (H ω)^2 ∂μ))⁻¹
+
+noncomputable def normalizedIncrement (H Z : Ω → ℝ) (μ : Measure Ω) : Ω → ℝ :=
+  fun ω ↦ predictableNormalizer H μ * H ω * Z ω
+
+lemma predictableNormalizer_pos {H : Ω → ℝ}
+    (hHpos : 0 < ∫ ω, (H ω)^2 ∂μ) :
+    0 < predictableNormalizer H μ := by
+  exact inv_pos.mpr (Real.sqrt_pos.mpr hHpos)
+
+lemma normalizedIncrement_mean {H Z : Ω → ℝ}
+    (hH : AEStronglyMeasurable H μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
+    (hind : IndepFun H Z μ) :
+    ∫ ω, normalizedIncrement H Z μ ω ∂μ = 0 := by
+  simp only [normalizedIncrement, mul_assoc]
+  rw [integral_const_mul]
+  have hi := hind.integral_mul_eq_mul_integral hH hZ.aemeasurable.aestronglyMeasurable
+  change (∫ ω, H ω * Z ω ∂μ) = _ at hi
+  rw [hi, gaussian_mean hZ, mul_zero, mul_zero]
+
+lemma normalizedIncrement_second_moment {H Z : Ω → ℝ}
+    (hH : AEStronglyMeasurable H μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
+    (hind : IndepFun H Z μ) (hHpos : 0 < ∫ ω, (H ω)^2 ∂μ) :
+    ∫ ω, (normalizedIncrement H Z μ ω)^2 ∂μ = 1 := by
+  have hind2 : IndepFun (fun ω ↦ (H ω)^2) (fun ω ↦ (Z ω)^2) μ :=
+    hind.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)
+  have hi := hind2.integral_mul_eq_mul_integral
+    (hH.fun_pow 2)
+    (hZ.aemeasurable.aestronglyMeasurable.fun_pow 2)
+  change (∫ ω, (H ω)^2 * (Z ω)^2 ∂μ) =
+    (∫ ω, (H ω)^2 ∂μ) * (∫ ω, (Z ω)^2 ∂μ) at hi
+  simp only [normalizedIncrement, mul_pow, mul_assoc]
+  rw [integral_const_mul, hi, gaussian_second_moment hZ, mul_one]
+  simp only [predictableNormalizer, inv_pow, Real.sq_sqrt (le_of_lt hHpos)]
+  exact inv_mul_cancel₀ (ne_of_gt hHpos)
+
+lemma normalizedIncrement_memLp_two {H Z : Ω → ℝ}
+    (hH : MemLp H 2 μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
+    (hind : IndepFun H Z μ) :
+    MemLp (normalizedIncrement H Z μ) 2 μ := by
+  apply (memLp_two_iff_integrable_sq
+    ((hH.aestronglyMeasurable.const_mul _).mul hZ.aemeasurable.aestronglyMeasurable)).2
+  have hi := (hind.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integrable_mul
+     hH.integrable_sq (gaussian_memLp hZ 2 (by norm_num)).integrable_sq
+  have hj := hi.const_mul (predictableNormalizer H μ ^ 2)
+  simpa only [normalizedIncrement, Pi.mul_apply, mul_pow, mul_assoc, Function.comp_def, id_eq] using hj
+
+lemma normalizedIncrement_gaussian_coefficient {H Z : Ω → ℝ}
+    (hH : AEStronglyMeasurable H μ) (hZ : HasLaw Z (gaussianReal 0 1) μ)
+    (hind : IndepFun H Z μ) :
+    ∫ ω, normalizedIncrement H Z μ ω * Z ω ∂μ =
+      predictableNormalizer H μ * ∫ ω, H ω ∂μ := by
+  have hind2 : IndepFun H (fun ω ↦ (Z ω)^2) μ :=
+    hind.comp measurable_id (measurable_id.pow_const 2)
+  have hi := hind2.integral_mul_eq_mul_integral hH
+    (hZ.aemeasurable.aestronglyMeasurable.fun_pow 2)
+  change (∫ ω, H ω * (Z ω)^2 ∂μ) =
+    (∫ ω, H ω ∂μ) * (∫ ω, (Z ω)^2 ∂μ) at hi
+  simp only [normalizedIncrement, mul_assoc, ← pow_two]
+  rw [integral_const_mul, hi, gaussian_second_moment hZ, mul_one]
+
+lemma predictable_product_mean_zero {H K Z : Ω → ℝ}
+    (hH : AEStronglyMeasurable H μ) (hK : AEStronglyMeasurable K μ)
+    (hZ : HasLaw Z (gaussianReal 0 1) μ)
+    (hind : IndepFun (fun ω ↦ H ω * K ω) Z μ) :
+    ∫ ω, K ω * normalizedIncrement H Z μ ω ∂μ = 0 := by
+  have hi := hind.integral_mul_eq_mul_integral (hH.mul hK)
+    hZ.aemeasurable.aestronglyMeasurable
+  change (∫ ω, (H ω * K ω) * Z ω ∂μ) = _ at hi
+  calc
+    _ = predictableNormalizer H μ * ∫ ω, (H ω * K ω) * Z ω ∂μ := by
+      rw [← integral_const_mul]
+      congr 1
+      ext ω
+      simp only [normalizedIncrement]
+      ring
+    _ = 0 := by rw [hi, gaussian_mean hZ, mul_zero, mul_zero]
+
+lemma coordinate_hasLaw (N j : ℕ) :
+    HasLaw (coordinate N j) (gaussianReal 0 1) (gaussianProduct (Fin (N+1))) :=
+  (measurePreserving_eval (fun _ : Fin (N+1) ↦ standardGaussian)
+    ⟨j % (N+1), Nat.mod_lt _ (Nat.succ_pos _)⟩).hasLaw
+
+def DependsBefore {N : ℕ} {β : Type*} (j : ℕ) (F : (Fin (N+1) → ℝ) → β) : Prop :=
+  ∀ z z', (∀ i : Fin (N+1), (i : ℕ)<j → z i=z' i) → F z=F z'
+
+lemma DependsBefore.independent {N j : ℕ} {β : Type*} [MeasurableSpace β] (hj : j<N+1)
+    {F : (Fin (N+1) → ℝ) → β} (hF : DependsBefore j F) (hm : Measurable F) :
+    IndepFun F (coordinate N j) (gaussianProduct (Fin (N+1))) := by
+  have H := independent_of_coordinate_not_in (μ := fun _ : Fin (N+1) ↦ standardGaussian)
+    (Finset.univ.filter (fun i : Fin (N+1) ↦ (i : ℕ)<j))
+    (j := ⟨j,hj⟩) (by simp) hm (fun z z' heq ↦ hF z z' (by simpa using heq))
+  have heq : coordinate N j = fun z : Fin (N+1) → ℝ ↦ z ⟨j,hj⟩ :=
+    funext (coordinate_eq_apply hj)
+  rw [heq]
+  exact H
+
+lemma DependsBefore.mono {N i j : ℕ} {β : Type*} {F : (Fin (N+1) → ℝ) → β}
+    (hF : DependsBefore i F) (hij : i≤j) : DependsBefore j F :=
+  fun z z' h ↦ hF z z' (fun k hk ↦ h k (hk.trans_le hij))
+
+lemma DependsBefore.prodMk {N j : ℕ} {β β' : Type*}
+    {F : (Fin (N+1) → ℝ) → β} {G : (Fin (N+1) → ℝ) → β'}
+    (hF : DependsBefore j F) (hG : DependsBefore j G) :
+    DependsBefore j (fun z ↦ (F z,G z)) := by
+  intro z z' h
+  exact Prod.ext (hF z z' h) (hG z z' h)
+
+lemma DependsBefore.mul {N j : ℕ} {F G : (Fin (N+1) → ℝ) → ℝ}
+    (hF : DependsBefore j F) (hG : DependsBefore j G) :
+    DependsBefore j (fun z ↦ F z * G z) := by
+  intro z z' h
+  dsimp only
+  rw [hF z z' h, hG z z' h]
+
+lemma eulerCoefficient_dependsBefore (T : ℝ) {N j : ℕ} (hj : j≤N)
+    (γ : ℝ → ℝ) (u a : ℝ → ℝ → ℝ) :
+    DependsBefore j (fun z ↦ a (meshTime T N j) (euler T N γ u z j)) := by
+  intro z z' h
+  apply congrArg (a (meshTime T N j))
+  apply euler_depends_on_past
+  intro i hi
+  have hiN : i<N+1 := by omega
+  rw [coordinate_eq_apply hiN, coordinate_eq_apply hiN]
+  exact h ⟨i,hiN⟩ hi
+
+lemma normalizedIncrement_dependsBefore {N j : ℕ} (hj : j<N+1)
+    {H : (Fin (N+1) → ℝ) → ℝ} (hH : DependsBefore j H) :
+    DependsBefore (j+1) (normalizedIncrement H (coordinate N j) (gaussianProduct (Fin (N+1)))) := by
+  intro z z' h
+  have heq := hH.mono (Nat.le_succ j) z z' h
+  have hz : coordinate N j z=coordinate N j z' := by
+    rw [coordinate_eq_apply hj, coordinate_eq_apply hj]
+    exact h ⟨j,hj⟩ (Nat.lt_succ_self j)
+  simp only [normalizedIncrement,heq,hz]
+
+lemma measurable_normalizedIncrement {N j : ℕ} {H : (Fin (N+1) → ℝ) → ℝ}
+    (hH : Measurable H) :
+    Measurable (normalizedIncrement H (coordinate N j) (gaussianProduct (Fin (N+1)))) :=
+  (hH.const_mul _).mul (measurable_coordinate N j)
+
+lemma normalizedIncrements_orthogonal {N i j : ℕ} (hij : i<j) (hj : j<N+1)
+    {H K : (Fin (N+1) → ℝ) → ℝ} (hHm : Measurable H) (hKm : Measurable K)
+    (hH : DependsBefore i H) (hK : DependsBefore j K) :
+    ∫ z, normalizedIncrement H (coordinate N i) (gaussianProduct (Fin (N+1))) z *
+      normalizedIncrement K (coordinate N j) (gaussianProduct (Fin (N+1))) z
+      ∂gaussianProduct (Fin (N+1)) = 0 := by
+  have hi : i<N+1 := hij.trans hj
+  have hprev := (normalizedIncrement_dependsBefore hi hH).mono (by omega : i+1≤j)
+  exact predictable_product_mean_zero hKm.aestronglyMeasurable
+    (measurable_normalizedIncrement hHm).aestronglyMeasurable (coordinate_hasLaw N j)
+    ((hK.mul hprev).independent hj
+      (hKm.mul (measurable_normalizedIncrement hHm)))
+
+lemma euler_normalized_increment_properties {T : ℝ} {N j : ℕ} {γ : ℝ → ℝ}
+    {u a : ℝ → ℝ → ℝ} (hj : j<N+1)
+    (hU : ∀ k<N, Measurable (u (meshTime T N k)))
+    (hA : Measurable (a (meshTime T N j)))
+    (hpos : 0 < ∫ z, (a (meshTime T N j) (euler T N γ u z j))^2
+      ∂gaussianProduct (Fin (N+1))) :
+    let H := fun z ↦ a (meshTime T N j) (euler T N γ u z j)
+    let μ := gaussianProduct (Fin (N+1))
+    0 < predictableNormalizer H μ ∧
+    (∫ z, normalizedIncrement H (coordinate N j) μ z ∂μ)=0 ∧
+    (∫ z, (normalizedIncrement H (coordinate N j) μ z)^2 ∂μ)=1 := by
+  dsimp only
+  have hm := hA.comp (measurable_euler T N γ u hU j (by omega))
+  have hind := euler_coefficient_independent (γ := γ) hj hU hA
+  exact ⟨predictableNormalizer_pos hpos,
+    normalizedIncrement_mean hm.aestronglyMeasurable (coordinate_hasLaw N j) hind,
+    normalizedIncrement_second_moment hm.aestronglyMeasurable (coordinate_hasLaw N j) hind hpos⟩
+
+end SKValue
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/QuadraticSK.lean b/lean/OAI/Probability/SKValue/GroundState/QuadraticSK.lean
index afed7f6bf784a68952ae3c922f459a93d3630bba..9c3b3686fb8ccbb6adc452a33f4ada0134d86151 100644
--- a/lean/OAI/Probability/SKValue/GroundState/QuadraticSK.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/QuadraticSK.lean
@@ -1,199 +1,199 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.AdaptiveColumns
-import OAI.Probability.SKValue.GroundState.GramSchmidtLimit
-import OAI.Probability.SKValue.GroundState.EmpiricalVectors
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
-open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
-namespace SKValueG
-
-noncomputable def spinVector (n : ℕ) (s : Fin n → Bool) : EuclideanSpace ℝ (Fin n) :=
-  WithLp.toLp 2 (fun i ↦ spin (s i))
-
-lemma spinVector_inner (n : ℕ) (s t : Fin n → Bool) :
-    ⟪spinVector n s,spinVector n t⟫=∑ i,spin (s i)*spin (t i) := euclidean_inner_sum _ _
-
-lemma spinVector_sqNorm (n : ℕ) (s : Fin n → Bool) : ‖spinVector n s‖^2=(n : ℝ) := by
-  rw [←real_inner_self_eq_norm_sq,spinVector_inner]
-  simp only [←sq,spin_sq,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma spinVector_norm (n : ℕ) (s : Fin n → Bool) : ‖spinVector n s‖=Real.sqrt (n : ℝ) := by
-  rw [←spinVector_sqNorm n s,Real.sqrt_sq (norm_nonneg _)]
-
-lemma finiteMaximum_mul_nonneg {ι : Type*} [Fintype ι] [Nonempty ι]
-    (f : ι → ℝ) {c : ℝ} (hc : 0≤c) :
-    finiteMaximum (fun i ↦ c*f i)=c*finiteMaximum f := by
-  obtain ⟨i,hi⟩ := exists_finiteMaximum f
-  apply le_antisymm
-  · obtain ⟨j,hj⟩ := exists_finiteMaximum (fun i ↦ c*f i)
-    rw [←hj]
-    exact mul_le_mul_of_nonneg_left (le_finiteMaximum f j) hc
-  · rw [←hi]
-    exact le_finiteMaximum (fun i ↦ c*f i) i
-
-lemma scaled_linearProcess {ι κ : Type*} [Fintype κ] (a : ι → κ → ℝ) (c : ℝ)
-    (z : κ → ℝ) (i : ι) : linearProcess (fun i k ↦ c*a i k) z i=c*linearProcess a z i := by
-  simp only [linearProcess,Finset.mul_sum,mul_assoc]
-
-lemma quadratic_spin_distance {n : ℕ} (hn : 0<n) (s t : Fin n → Bool) :
-    (∑ k,(quadraticCoeff (spinVector n s) k-quadraticCoeff (spinVector n t) k)^2)=
-      ∑ k,(Real.sqrt (n : ℝ)*hamCoeff n s k-Real.sqrt (n : ℝ)*hamCoeff n t k)^2 := by
-  have hs0 : (n : ℝ)≠0 := by exact_mod_cast hn.ne'
-  have hq (s : Fin n → Bool) :
-      (∑ k,(quadraticCoeff (spinVector n s) k)^2)=(n : ℝ)^2/2 := by
-    simp only [sq]
-    rw [quadraticCoeff_inner,real_inner_self_eq_norm_sq,spinVector_sqNorm]
-    ring
-  rw [sq_distance_sum,hq,hq,quadraticCoeff_inner,spinVector_inner]
-  have hs : (∑ k,(Real.sqrt (n : ℝ)*hamCoeff n s k-Real.sqrt (n : ℝ)*hamCoeff n t k)^2)=
-      (n : ℝ)*(∑ k,(hamCoeff n s k-hamCoeff n t k)^2) := by
-    simp only [←mul_sub,mul_pow,Real.sq_sqrt (Nat.cast_nonneg n),Finset.mul_sum]
-  rw [hs,hamCoeff_distance hn]
-  field_simp
-  ring
-
-theorem quadraticValue_spin {n : ℕ} (hn : 0<n) :
-    quadraticValue (spinVector n) (fun _ ↦ 0)=Real.sqrt (n : ℝ)*expectedMaximum n := by
-  have he : (∫ z,finiteMaximum (fun s ↦ linearProcess (fun s ↦ quadraticCoeff (spinVector n s)) z s+0)
-      ∂gaussianProduct (Fin n×Fin n))=
-      ∫ z,finiteMaximum (fun s ↦ linearProcess (fun s k ↦ Real.sqrt (n : ℝ)*hamCoeff n s k) z s+0)
-      ∂gaussianProduct (Fin n×Fin n) := by
-    apply le_antisymm
-    · exact expected_affine_max_le _ _ (fun _ ↦ 0) (fun s t ↦ (quadratic_spin_distance hn s t).le)
-    · exact expected_affine_max_le _ _ (fun _ ↦ 0) (fun s t ↦ (quadratic_spin_distance hn s t).ge)
-  unfold quadraticValue
-  rw [he]
-  simp only [add_zero,scaled_linearProcess,finiteMaximum_mul_nonneg _ (Real.sqrt_nonneg _)]
-  rw [integral_const_mul]
-  rfl
-
-theorem finite_adaptive_SK_lower {n : ℕ} (hn : 0<n)
-    (v : (j : ℕ) → ColumnHistory (Fin n) j → EuclideanSpace ℝ (Fin n))
-    (hv : ∀ j,Measurable (v j)) (hu : ∀ j h,v j h=0 ∨ ⟪v j h,v j h⟫=1)
-    (K : ℕ) (s : ColumnHistory (Fin n) K → (Fin n → Bool))
-    (hs : Measurable (fun h ↦ adaptiveOffset (spinVector n) v K h (s h))) :
-    (∫ h,adaptiveOffset (spinVector n) v K h (s h) ∂columnHistoryLaw (Fin n) K)/
-      ((n : ℝ)*Real.sqrt (n : ℝ))≤groundStateSequence n := by
-  have hh := adaptive_spin_lower (spinVector n) v hv hu (Real.sqrt (n : ℝ))
-    (fun s ↦ (spinVector_norm n s).le) K s hs
-  rw [quadraticValue_spin hn] at hh
-  have hn' : (0 : ℝ)<n := by exact_mod_cast hn
-  have pro : (0 : ℝ)<(n : ℝ)*Real.sqrt (n : ℝ) := mul_pos hn' (Real.sqrt_pos.mpr hn')
-  apply (div_le_div_of_nonneg_right hh pro.le).trans_eq
-  rw [groundStateSequence_eq]
-  field_simp
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
-open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
-namespace SKValueG
-
-lemma orthogonalRemainder_smul {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
-    (v : ℕ → E) (w : E) (j : ℕ) (c : ℝ) :
-    orthogonalRemainder v (c • w) j=c • orthogonalRemainder v w j := by
-  simp only [orthogonalRemainder,smul_sub,Finset.smul_sum,real_inner_smul_right,smul_smul]
-
-lemma columnVector_smul {κ : Type*} [Fintype κ]
-    (v w : EuclideanSpace ℝ κ) (c : ℝ) : columnVector v (c • w)=c^2 • columnVector v w := by
-  simp only [columnVector,columnResidual,real_inner_smul_right,smul_sub,smul_add,smul_smul]
-  module
-
-lemma empiricalVector_scale {Ω : Type*} (f : Ω → ℝ) (x : ℕ → Ω) {n : ℕ} (hn : 0<n) :
-    Real.sqrt (n : ℝ) • empiricalVector f x n=WithLp.toLp 2 (fun i : Fin n ↦ f (x i)) := by
-  rw [empiricalVector,smul_smul,mul_inv_cancel₀,one_smul]
-  exact Real.sqrt_ne_zero'.mpr (by exact_mod_cast hn)
-
-variable (K n : ℕ) (f : ℕ → (Fin (K+1) → ℝ) → ℝ)
-  (ha : ∀ l x y,(∀ i : Fin (K+1),i.val≤l → x i=y i) → f l x=f l y)
-  (x : ℕ → (Fin (K+1) → ℝ))
-
-include ha in
-lemma predictableResidual_rows {ι : Type*} [Fintype ι] [Nonempty ι]
-    (w : ι → EuclideanSpace ℝ (Fin n)) (j : ℕ) (s : ι) :
-    adaptiveResidual w (predictableDirection K n f) (j+1)
-      (rowHistory K n (fun i ↦ x i) (j+1)) s=
-    orthogonalRemainder (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n)) (w s) j := by
-  induction j with
-  | zero => simp [adaptiveResidual,predictableDirection,columnResidual,orthogonalRemainder]
-  | succ j ih =>
-    change columnResidual (predictableDirection K n f (j+1)
-      (rowHistory K n (fun i ↦ x i) (j+1)))
-      (adaptiveResidual w (predictableDirection K n f) (j+1)
-        (rowHistory K n (fun i ↦ x i) (j+1)) s)=_
-    rw [predictableDirection_rows K n f ha x j,ih]
-    apply columnResidual_orthogonalRemainder
-    intro i hi
-    exact gramSchmidtNormed_pairwise_orthogonal _ (Ne.symm (Nat.ne_of_lt hi))
-
-include ha in
-lemma predictableOffset_rows {ι : Type*} [Fintype ι] [Nonempty ι]
-    (w : ι → EuclideanSpace ℝ (Fin n)) (J : ℕ) (s : ι) :
-    adaptiveOffset w (predictableDirection K n f) (J+1)
-      (rowHistory K n (fun i ↦ x i) (J+1)) s=
-      ∑ j∈Finset.range J,
-        ⟪columnVector (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n) j)
-          (orthogonalRemainder (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n)) (w s) j),
-          WithLp.toLp 2 (rowColumn K n (fun i ↦ x i) (j+1))⟫ := by
-  induction J with
-  | zero => simp [adaptiveOffset,predictableDirection,columnVector,columnResidual,linearProcess]
-  | succ J ih =>
-    change adaptiveOffset w (predictableDirection K n f) (J+1)
-      (rowHistory K n (fun i ↦ x i) (J+1)) s+
-      linearProcess (fun i k ↦ columnVector
-        (predictableDirection K n f (J+1) (rowHistory K n (fun i ↦ x i) (J+1)))
-        (adaptiveResidual w (predictableDirection K n f) (J+1)
-          (rowHistory K n (fun i ↦ x i) (J+1)) i) k)
-        (rowColumn K n (fun i ↦ x i) (J+1)) s=_
-    rw [ih,Finset.sum_range_succ,predictableDirection_rows K n f ha x J]
-    congr 1
-    unfold linearProcess
-    dsimp only
-    rw [predictableResidual_rows K n f ha x w J s]
-    rw [euclidean_inner_sum]
-
-noncomputable def empiricalColumnEnergy (F : (Fin (K+1) → ℝ) → ℝ) (j : ℕ) : ℝ :=
-  ⟪columnVector (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n) j)
-      (orthogonalRemainder (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n))
-        (empiricalVector F x n) j),
-    empiricalVector (fun y ↦ if hj : j+1<K+1 then y ⟨j+1,hj⟩ else 0) x n⟫
-
-include ha in
-lemma normalizedOffset_rows (hn : 0<n) (F : (Fin (K+1) → ℝ) → Bool) :
-    adaptiveOffset (spinVector n) (predictableDirection K n f) (K+1)
-      (rowHistory K n (fun i ↦ x i) (K+1)) (fun i ↦ F (x i))/
-      ((n : ℝ)*Real.sqrt (n : ℝ))=
-      ∑ j∈Finset.range K,empiricalColumnEnergy K n f x (fun y ↦ spin (F y)) j := by
-  rw [predictableOffset_rows K n f ha x (spinVector n) K]
-  rw [Finset.sum_div]
-  apply Finset.sum_congr rfl
-  intro j hj
-  have hw : spinVector n (fun i ↦ F (x i))=
-      Real.sqrt (n : ℝ) • empiricalVector (fun y ↦ spin (F y)) x n :=
-    (empiricalVector_scale (fun y ↦ spin (F y)) x hn).symm
-  have hz : WithLp.toLp 2 (rowColumn K n (fun i ↦ x i) (j+1))=
-      Real.sqrt (n : ℝ) • empiricalVector
-        (fun y ↦ if hj : j+1<K+1 then y ⟨j+1,hj⟩ else 0) x n :=
-    by
-    change (WithLp.toLp 2 (fun i : Fin n ↦ if hj : j+1<K+1 then x i ⟨j+1,hj⟩ else 0)) = _
-    exact (empiricalVector_scale (fun y : Fin (K+1) → ℝ ↦ if hj : j+1<K+1 then y ⟨j+1,hj⟩ else 0) x hn).symm
-  rw [hw,hz,orthogonalRemainder_smul,columnVector_smul,real_inner_smul_left,real_inner_smul_right,
-    Real.sq_sqrt (Nat.cast_nonneg n)]
-  unfold empiricalColumnEnergy
-  have hn' : (n : ℝ)≠0 := by exact_mod_cast hn.ne'
-  have hs : Real.sqrt (n : ℝ)≠0 := Real.sqrt_ne_zero'.mpr (by exact_mod_cast hn)
-  field_simp
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.AdaptiveColumns
+import OAI.Probability.SKValue.GroundState.GramSchmidtLimit
+import OAI.Probability.SKValue.GroundState.EmpiricalVectors
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
+open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
+namespace SKValueG
+
+noncomputable def spinVector (n : ℕ) (s : Fin n → Bool) : EuclideanSpace ℝ (Fin n) :=
+  WithLp.toLp 2 (fun i ↦ spin (s i))
+
+lemma spinVector_inner (n : ℕ) (s t : Fin n → Bool) :
+    ⟪spinVector n s,spinVector n t⟫=∑ i,spin (s i)*spin (t i) := euclidean_inner_sum _ _
+
+lemma spinVector_sqNorm (n : ℕ) (s : Fin n → Bool) : ‖spinVector n s‖^2=(n : ℝ) := by
+  rw [←real_inner_self_eq_norm_sq,spinVector_inner]
+  simp only [←sq,spin_sq,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma spinVector_norm (n : ℕ) (s : Fin n → Bool) : ‖spinVector n s‖=Real.sqrt (n : ℝ) := by
+  rw [←spinVector_sqNorm n s,Real.sqrt_sq (norm_nonneg _)]
+
+lemma finiteMaximum_mul_nonneg {ι : Type*} [Fintype ι] [Nonempty ι]
+    (f : ι → ℝ) {c : ℝ} (hc : 0≤c) :
+    finiteMaximum (fun i ↦ c*f i)=c*finiteMaximum f := by
+  obtain ⟨i,hi⟩ := exists_finiteMaximum f
+  apply le_antisymm
+  · obtain ⟨j,hj⟩ := exists_finiteMaximum (fun i ↦ c*f i)
+    rw [←hj]
+    exact mul_le_mul_of_nonneg_left (le_finiteMaximum f j) hc
+  · rw [←hi]
+    exact le_finiteMaximum (fun i ↦ c*f i) i
+
+lemma scaled_linearProcess {ι κ : Type*} [Fintype κ] (a : ι → κ → ℝ) (c : ℝ)
+    (z : κ → ℝ) (i : ι) : linearProcess (fun i k ↦ c*a i k) z i=c*linearProcess a z i := by
+  simp only [linearProcess,Finset.mul_sum,mul_assoc]
+
+lemma quadratic_spin_distance {n : ℕ} (hn : 0<n) (s t : Fin n → Bool) :
+    (∑ k,(quadraticCoeff (spinVector n s) k-quadraticCoeff (spinVector n t) k)^2)=
+      ∑ k,(Real.sqrt (n : ℝ)*hamCoeff n s k-Real.sqrt (n : ℝ)*hamCoeff n t k)^2 := by
+  have hs0 : (n : ℝ)≠0 := by exact_mod_cast hn.ne'
+  have hq (s : Fin n → Bool) :
+      (∑ k,(quadraticCoeff (spinVector n s) k)^2)=(n : ℝ)^2/2 := by
+    simp only [sq]
+    rw [quadraticCoeff_inner,real_inner_self_eq_norm_sq,spinVector_sqNorm]
+    ring
+  rw [sq_distance_sum,hq,hq,quadraticCoeff_inner,spinVector_inner]
+  have hs : (∑ k,(Real.sqrt (n : ℝ)*hamCoeff n s k-Real.sqrt (n : ℝ)*hamCoeff n t k)^2)=
+      (n : ℝ)*(∑ k,(hamCoeff n s k-hamCoeff n t k)^2) := by
+    simp only [←mul_sub,mul_pow,Real.sq_sqrt (Nat.cast_nonneg n),Finset.mul_sum]
+  rw [hs,hamCoeff_distance hn]
+  field_simp
+  ring
+
+theorem quadraticValue_spin {n : ℕ} (hn : 0<n) :
+    quadraticValue (spinVector n) (fun _ ↦ 0)=Real.sqrt (n : ℝ)*expectedMaximum n := by
+  have he : (∫ z,finiteMaximum (fun s ↦ linearProcess (fun s ↦ quadraticCoeff (spinVector n s)) z s+0)
+      ∂gaussianProduct (Fin n×Fin n))=
+      ∫ z,finiteMaximum (fun s ↦ linearProcess (fun s k ↦ Real.sqrt (n : ℝ)*hamCoeff n s k) z s+0)
+      ∂gaussianProduct (Fin n×Fin n) := by
+    apply le_antisymm
+    · exact expected_affine_max_le _ _ (fun _ ↦ 0) (fun s t ↦ (quadratic_spin_distance hn s t).le)
+    · exact expected_affine_max_le _ _ (fun _ ↦ 0) (fun s t ↦ (quadratic_spin_distance hn s t).ge)
+  unfold quadraticValue
+  rw [he]
+  simp only [add_zero,scaled_linearProcess,finiteMaximum_mul_nonneg _ (Real.sqrt_nonneg _)]
+  rw [integral_const_mul]
+  rfl
+
+theorem finite_adaptive_SK_lower {n : ℕ} (hn : 0<n)
+    (v : (j : ℕ) → ColumnHistory (Fin n) j → EuclideanSpace ℝ (Fin n))
+    (hv : ∀ j,Measurable (v j)) (hu : ∀ j h,v j h=0 ∨ ⟪v j h,v j h⟫=1)
+    (K : ℕ) (s : ColumnHistory (Fin n) K → (Fin n → Bool))
+    (hs : Measurable (fun h ↦ adaptiveOffset (spinVector n) v K h (s h))) :
+    (∫ h,adaptiveOffset (spinVector n) v K h (s h) ∂columnHistoryLaw (Fin n) K)/
+      ((n : ℝ)*Real.sqrt (n : ℝ))≤groundStateSequence n := by
+  have hh := adaptive_spin_lower (spinVector n) v hv hu (Real.sqrt (n : ℝ))
+    (fun s ↦ (spinVector_norm n s).le) K s hs
+  rw [quadraticValue_spin hn] at hh
+  have hn' : (0 : ℝ)<n := by exact_mod_cast hn
+  have pro : (0 : ℝ)<(n : ℝ)*Real.sqrt (n : ℝ) := mul_pos hn' (Real.sqrt_pos.mpr hn')
+  apply (div_le_div_of_nonneg_right hh pro.le).trans_eq
+  rw [groundStateSequence_eq]
+  field_simp
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
+open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
+namespace SKValueG
+
+lemma orthogonalRemainder_smul {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    (v : ℕ → E) (w : E) (j : ℕ) (c : ℝ) :
+    orthogonalRemainder v (c • w) j=c • orthogonalRemainder v w j := by
+  simp only [orthogonalRemainder,smul_sub,Finset.smul_sum,real_inner_smul_right,smul_smul]
+
+lemma columnVector_smul {κ : Type*} [Fintype κ]
+    (v w : EuclideanSpace ℝ κ) (c : ℝ) : columnVector v (c • w)=c^2 • columnVector v w := by
+  simp only [columnVector,columnResidual,real_inner_smul_right,smul_sub,smul_add,smul_smul]
+  module
+
+lemma empiricalVector_scale {Ω : Type*} (f : Ω → ℝ) (x : ℕ → Ω) {n : ℕ} (hn : 0<n) :
+    Real.sqrt (n : ℝ) • empiricalVector f x n=WithLp.toLp 2 (fun i : Fin n ↦ f (x i)) := by
+  rw [empiricalVector,smul_smul,mul_inv_cancel₀,one_smul]
+  exact Real.sqrt_ne_zero'.mpr (by exact_mod_cast hn)
+
+variable (K n : ℕ) (f : ℕ → (Fin (K+1) → ℝ) → ℝ)
+  (ha : ∀ l x y,(∀ i : Fin (K+1),i.val≤l → x i=y i) → f l x=f l y)
+  (x : ℕ → (Fin (K+1) → ℝ))
+
+include ha in
+lemma predictableResidual_rows {ι : Type*} [Fintype ι] [Nonempty ι]
+    (w : ι → EuclideanSpace ℝ (Fin n)) (j : ℕ) (s : ι) :
+    adaptiveResidual w (predictableDirection K n f) (j+1)
+      (rowHistory K n (fun i ↦ x i) (j+1)) s=
+    orthogonalRemainder (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n)) (w s) j := by
+  induction j with
+  | zero => simp [adaptiveResidual,predictableDirection,columnResidual,orthogonalRemainder]
+  | succ j ih =>
+    change columnResidual (predictableDirection K n f (j+1)
+      (rowHistory K n (fun i ↦ x i) (j+1)))
+      (adaptiveResidual w (predictableDirection K n f) (j+1)
+        (rowHistory K n (fun i ↦ x i) (j+1)) s)=_
+    rw [predictableDirection_rows K n f ha x j,ih]
+    apply columnResidual_orthogonalRemainder
+    intro i hi
+    exact gramSchmidtNormed_pairwise_orthogonal _ (Ne.symm (Nat.ne_of_lt hi))
+
+include ha in
+lemma predictableOffset_rows {ι : Type*} [Fintype ι] [Nonempty ι]
+    (w : ι → EuclideanSpace ℝ (Fin n)) (J : ℕ) (s : ι) :
+    adaptiveOffset w (predictableDirection K n f) (J+1)
+      (rowHistory K n (fun i ↦ x i) (J+1)) s=
+      ∑ j∈Finset.range J,
+        ⟪columnVector (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n) j)
+          (orthogonalRemainder (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n)) (w s) j),
+          WithLp.toLp 2 (rowColumn K n (fun i ↦ x i) (j+1))⟫ := by
+  induction J with
+  | zero => simp [adaptiveOffset,predictableDirection,columnVector,columnResidual,linearProcess]
+  | succ J ih =>
+    change adaptiveOffset w (predictableDirection K n f) (J+1)
+      (rowHistory K n (fun i ↦ x i) (J+1)) s+
+      linearProcess (fun i k ↦ columnVector
+        (predictableDirection K n f (J+1) (rowHistory K n (fun i ↦ x i) (J+1)))
+        (adaptiveResidual w (predictableDirection K n f) (J+1)
+          (rowHistory K n (fun i ↦ x i) (J+1)) i) k)
+        (rowColumn K n (fun i ↦ x i) (J+1)) s=_
+    rw [ih,Finset.sum_range_succ,predictableDirection_rows K n f ha x J]
+    congr 1
+    unfold linearProcess
+    dsimp only
+    rw [predictableResidual_rows K n f ha x w J s]
+    rw [euclidean_inner_sum]
+
+noncomputable def empiricalColumnEnergy (F : (Fin (K+1) → ℝ) → ℝ) (j : ℕ) : ℝ :=
+  ⟪columnVector (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n) j)
+      (orthogonalRemainder (gramSchmidtNormed ℝ (fun l ↦ empiricalVector (f l) x n))
+        (empiricalVector F x n) j),
+    empiricalVector (fun y ↦ if hj : j+1<K+1 then y ⟨j+1,hj⟩ else 0) x n⟫
+
+include ha in
+lemma normalizedOffset_rows (hn : 0<n) (F : (Fin (K+1) → ℝ) → Bool) :
+    adaptiveOffset (spinVector n) (predictableDirection K n f) (K+1)
+      (rowHistory K n (fun i ↦ x i) (K+1)) (fun i ↦ F (x i))/
+      ((n : ℝ)*Real.sqrt (n : ℝ))=
+      ∑ j∈Finset.range K,empiricalColumnEnergy K n f x (fun y ↦ spin (F y)) j := by
+  rw [predictableOffset_rows K n f ha x (spinVector n) K]
+  rw [Finset.sum_div]
+  apply Finset.sum_congr rfl
+  intro j hj
+  have hw : spinVector n (fun i ↦ F (x i))=
+      Real.sqrt (n : ℝ) • empiricalVector (fun y ↦ spin (F y)) x n :=
+    (empiricalVector_scale (fun y ↦ spin (F y)) x hn).symm
+  have hz : WithLp.toLp 2 (rowColumn K n (fun i ↦ x i) (j+1))=
+      Real.sqrt (n : ℝ) • empiricalVector
+        (fun y ↦ if hj : j+1<K+1 then y ⟨j+1,hj⟩ else 0) x n :=
+    by
+    change (WithLp.toLp 2 (fun i : Fin n ↦ if hj : j+1<K+1 then x i ⟨j+1,hj⟩ else 0)) = _
+    exact (empiricalVector_scale (fun y : Fin (K+1) → ℝ ↦ if hj : j+1<K+1 then y ⟨j+1,hj⟩ else 0) x hn).symm
+  rw [hw,hz,orthogonalRemainder_smul,columnVector_smul,real_inner_smul_left,real_inner_smul_right,
+    Real.sq_sqrt (Nat.cast_nonneg n)]
+  unfold empiricalColumnEnergy
+  have hn' : (n : ℝ)≠0 := by exact_mod_cast hn.ne'
+  have hs : Real.sqrt (n : ℝ)≠0 := Real.sqrt_ne_zero'.mpr (by exact_mod_cast hn)
+  field_simp
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/RandomColumn.lean b/lean/OAI/Probability/SKValue/GroundState/RandomColumn.lean
index ac1e8011f7bfe4a16298a3148d343cdfdc29e252..2cb7c6af5363fadf2a69c62be37808e529bc63d2 100644
--- a/lean/OAI/Probability/SKValue/GroundState/RandomColumn.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/RandomColumn.lean
@@ -1,155 +1,155 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.ColumnBounds
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
-open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
-namespace SKValueG
-variable {Ω ι κ ν : Type*} [MeasurableSpace Ω] [Fintype ι] [Nonempty ι]
-  [Fintype κ] [Fintype ν] {μ : Measure Ω} [IsProbabilityMeasure μ]
-
-lemma measurable_finiteMaximum {f : Ω → ι → ℝ} (hf : ∀ i,Measurable (fun ω ↦ f ω i)) :
-    Measurable (fun ω ↦ finiteMaximum (f ω)) :=
-  continuous_finiteMaximum.measurable.comp (Measurable.of_eval hf)
-
-omit [IsProbabilityMeasure μ] in
-lemma integrable_randomMaximum [IsProbabilityMeasure μ] {f : Ω → ι → ℝ}
-    (hf : ∀ i,Integrable (fun ω ↦ f ω i) μ) :
-    Integrable (fun ω ↦ finiteMaximum (f ω)) μ := by
-  apply (integrable_finsetSum Finset.univ (fun i _ ↦ (hf i).abs)).mono'
-    (continuous_finiteMaximum.comp_aestronglyMeasurable
-      ((integrable_pi_iff.mpr hf).aestronglyMeasurable))
-  exact Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs] using abs_finiteMaximum_le (f ω))
-
-omit [Fintype ι] [Nonempty ι] in
-lemma random_linear_integrable [Fintype ι] [Nonempty ι] (a : Ω → ι → ν → ℝ)
-    (ha : ∀ i k,Measurable (fun ω ↦ a ω i k)) (C : ℝ)
-    (hC : ∀ ω i k,|a ω i k|≤C) (i : ι) :
-    Integrable (fun p : Ω×(ν → ℝ) ↦ linearProcess (a p.1) p.2 i)
-      (μ.prod (gaussianProduct ν)) := by
-  unfold linearProcess
-  apply integrable_finsetSum
-  intro k _
-  apply ((gaussian_coordinate_integrable k).abs.const_mul C |>.comp_snd μ).mono'
-    (((ha i k).comp measurable_fst).mul ((measurable_pi_apply k).comp measurable_snd)).aestronglyMeasurable
-  exact Eventually.of_forall (fun p ↦ by
-    change |a p.1 i k * p.2 k| ≤ C * |p.2 k|
-    rw [abs_mul]
-    exact mul_le_mul_of_nonneg_right (hC p.1 i k) (abs_nonneg _))
-
-lemma measurable_quadraticValue (w : Ω → ι → EuclideanSpace ℝ κ) (d : Ω → ι → ℝ)
-    (hw : ∀ i,Measurable (fun ω ↦ w ω i)) (hd : ∀ i,Measurable (fun ω ↦ d ω i)) :
-    Measurable (fun ω ↦ quadraticValue (w ω) (d ω)) := by
-  have hm : Measurable (fun p : Ω×((κ×κ) → ℝ) ↦
-      finiteMaximum (fun i ↦ linearProcess (fun i ↦ quadraticCoeff (w p.1 i)) p.2 i+d p.1 i)) := by
-    apply measurable_finiteMaximum
-    intro i
-    unfold linearProcess quadraticCoeff
-    fun_prop
-  exact hm.stronglyMeasurable.integral_prod_right.measurable
-
-lemma random_quadratic_integrable (w : Ω → ι → EuclideanSpace ℝ κ) (d : Ω → ι → ℝ)
-    (hw : ∀ i,Measurable (fun ω ↦ w ω i)) (hd : ∀ i,Integrable (fun ω ↦ d ω i) μ)
-    (R : ℝ) (hR : ∀ ω i,‖w ω i‖≤R) :
-    Integrable (fun ω ↦ quadraticValue (w ω) (d ω)) μ := by
-  have hi : Integrable (fun p : Ω×((κ×κ) → ℝ) ↦
-      finiteMaximum (fun i ↦ linearProcess (fun i ↦ quadraticCoeff (w p.1 i)) p.2 i+d p.1 i))
-      (μ.prod (gaussianProduct (κ×κ))) := by
-    apply integrable_randomMaximum
-    intro i
-    apply Integrable.add _ ((hd i).comp_fst _)
-    apply random_linear_integrable (fun ω i ↦ quadraticCoeff (w ω i)) _ (R^2) _ i
-    · intro i k; unfold quadraticCoeff; fun_prop
-    · intro ω i k
-      exact (quadraticCoeff_coord_bound _ _).trans (sq_le_sq₀ (norm_nonneg _) (le_trans (norm_nonneg _) (hR ω i)) |>.mpr (hR ω i))
-  exact hi.integral_prod_left
-
-lemma columnVector_coord_bound {v : EuclideanSpace ℝ κ} (hv : v=0 ∨ ⟪v,v⟫=1)
-    (w : EuclideanSpace ℝ κ) (k : κ) : |columnVector v w k|≤‖w‖^2 :=
-  (PiLp.norm_apply_le _ k).trans (columnVector_norm_le hv w)
-
-theorem random_column_identity (w : Ω → ι → EuclideanSpace ℝ κ) (d : Ω → ι → ℝ)
-    (v : Ω → EuclideanSpace ℝ κ) (hw : ∀ i,Measurable (fun ω ↦ w ω i))
-    (_hdm : ∀ i,Measurable (fun ω ↦ d ω i))
-    (hd : ∀ i,Integrable (fun ω ↦ d ω i) μ) (hv : Measurable v)
-    (hu : ∀ ω,v ω=0 ∨ ⟪v ω,v ω⟫=1) (R : ℝ) (hR : ∀ ω i,‖w ω i‖≤R) :
-    (∫ ω,quadraticValue (w ω) (d ω) ∂μ)=
-      ∫ p : Ω×(κ → ℝ),quadraticValue (fun i ↦ columnResidual (v p.1) (w p.1 i))
-        (fun i ↦ d p.1 i+linearProcess (fun i k ↦ columnVector (v p.1) (w p.1 i) k) p.2 i)
-        ∂μ.prod (gaussianProduct κ) := by
-  have hw' (i : ι) : Measurable (fun p : Ω×(κ → ℝ) ↦ columnResidual (v p.1) (w p.1 i)) :=
-    measurable_columnResidual (hv.comp measurable_fst) ((hw i).comp measurable_fst)
-  have hd' (i : ι) : Integrable (fun p : Ω×(κ → ℝ) ↦
-      d p.1 i+linearProcess (fun i k ↦ columnVector (v p.1) (w p.1 i) k) p.2 i)
-      (μ.prod (gaussianProduct κ)) := by
-    apply ((hd i).comp_fst _).add
-    apply random_linear_integrable (fun ω i k ↦ columnVector (v ω) (w ω i) k) _ (R^2) _ i
-    · intro i k
-      exact (PiLp.continuous_apply 2 (fun _ : κ ↦ ℝ) k).measurable.comp (measurable_columnVector hv (hw i))
-    · intro ω i k
-      exact (columnVector_coord_bound (hu ω) _ _).trans
-        ((sq_le_sq₀ (norm_nonneg _) ((norm_nonneg _).trans (hR ω i))).mpr (hR ω i))
-  have hi := random_quadratic_integrable _ _ hw' hd' R
-    (fun p i ↦ (columnResidual_norm_le (hu p.1) (w p.1 i)).trans (hR p.1 i))
-  rw [integral_prod _ hi]
-  apply integral_congr_ae
-  exact Eventually.of_forall (fun ω ↦ quadraticValue_column (w ω) (d ω) (v ω) (hu ω))
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
-open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
-namespace SKValueG
-variable (κ : Type*) [Fintype κ]
-
-abbrev ColumnHistory (j : ℕ) := Fin j → κ → ℝ
-noncomputable def columnHistoryLaw (j : ℕ) : Measure (ColumnHistory κ j) :=
-  Measure.pi (fun _ ↦ gaussianProduct κ)
-instance (j : ℕ) : IsProbabilityMeasure (columnHistoryLaw κ j) := by
-  unfold columnHistoryLaw; infer_instance
-
-def historyInit {j : ℕ} (h : ColumnHistory κ (j+1)) : ColumnHistory κ j := fun i ↦ h i.castSucc
-
-def historySplit (j : ℕ) : ColumnHistory κ (j+1) ≃ᵐ ColumnHistory κ j × (κ → ℝ) :=
-  (MeasurableEquiv.piFinSuccAbove (fun _ : Fin (j+1) ↦ κ → ℝ) (Fin.last j)).trans
-    MeasurableEquiv.prodComm
-
-omit [Fintype κ] in
-lemma historySplit_apply [Fintype κ] (j : ℕ) (h : ColumnHistory κ (j+1)) :
-    historySplit κ j h=(historyInit κ h,h (Fin.last j)) := by
-  simp [historySplit,MeasurableEquiv.piFinSuccAbove,Fin.insertNthEquiv]
-  rfl
-
-lemma historySplit_preserving (j : ℕ) :
-    MeasurePreserving (historySplit κ j) (columnHistoryLaw κ (j+1))
-      ((columnHistoryLaw κ j).prod (gaussianProduct κ)) := by
-  exact (Measure.measurePreserving_swap).comp
-    (measurePreserving_piFinSuccAbove (fun _ : Fin (j+1) ↦ gaussianProduct κ) (Fin.last j))
-
-omit [Fintype κ] in
-lemma historyInit_measurable [Fintype κ] (j : ℕ) : Measurable (historyInit κ (j:=j)) := by
-  unfold historyInit; fun_prop
-
-lemma history_integral_succ (j : ℕ) (f : ColumnHistory κ j × (κ → ℝ) → ℝ) :
-    (∫ h,f (historyInit κ h,h (Fin.last j)) ∂columnHistoryLaw κ (j+1))=
-      ∫ p,f p ∂(columnHistoryLaw κ j).prod (gaussianProduct κ) := by
-  simpa only [historySplit_apply] using (historySplit_preserving κ j).integral_comp' f
-
-lemma history_integrable_succ (j : ℕ) {f : ColumnHistory κ j × (κ → ℝ) → ℝ}
-    (hf : Integrable f ((columnHistoryLaw κ j).prod (gaussianProduct κ))) :
-    Integrable (fun h ↦ f (historyInit κ h,h (Fin.last j))) (columnHistoryLaw κ (j+1)) := by
-  simpa only [Function.comp_def,historySplit_apply] using (historySplit_preserving κ j).integrable_comp
-    hf.aestronglyMeasurable |>.mpr hf
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.ColumnBounds
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
+open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
+namespace SKValueG
+variable {Ω ι κ ν : Type*} [MeasurableSpace Ω] [Fintype ι] [Nonempty ι]
+  [Fintype κ] [Fintype ν] {μ : Measure Ω} [IsProbabilityMeasure μ]
+
+lemma measurable_finiteMaximum {f : Ω → ι → ℝ} (hf : ∀ i,Measurable (fun ω ↦ f ω i)) :
+    Measurable (fun ω ↦ finiteMaximum (f ω)) :=
+  continuous_finiteMaximum.measurable.comp (Measurable.of_eval hf)
+
+omit [IsProbabilityMeasure μ] in
+lemma integrable_randomMaximum [IsProbabilityMeasure μ] {f : Ω → ι → ℝ}
+    (hf : ∀ i,Integrable (fun ω ↦ f ω i) μ) :
+    Integrable (fun ω ↦ finiteMaximum (f ω)) μ := by
+  apply (integrable_finsetSum Finset.univ (fun i _ ↦ (hf i).abs)).mono'
+    (continuous_finiteMaximum.comp_aestronglyMeasurable
+      ((integrable_pi_iff.mpr hf).aestronglyMeasurable))
+  exact Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs] using abs_finiteMaximum_le (f ω))
+
+omit [Fintype ι] [Nonempty ι] in
+lemma random_linear_integrable [Fintype ι] [Nonempty ι] (a : Ω → ι → ν → ℝ)
+    (ha : ∀ i k,Measurable (fun ω ↦ a ω i k)) (C : ℝ)
+    (hC : ∀ ω i k,|a ω i k|≤C) (i : ι) :
+    Integrable (fun p : Ω×(ν → ℝ) ↦ linearProcess (a p.1) p.2 i)
+      (μ.prod (gaussianProduct ν)) := by
+  unfold linearProcess
+  apply integrable_finsetSum
+  intro k _
+  apply ((gaussian_coordinate_integrable k).abs.const_mul C |>.comp_snd μ).mono'
+    (((ha i k).comp measurable_fst).mul ((measurable_pi_apply k).comp measurable_snd)).aestronglyMeasurable
+  exact Eventually.of_forall (fun p ↦ by
+    change |a p.1 i k * p.2 k| ≤ C * |p.2 k|
+    rw [abs_mul]
+    exact mul_le_mul_of_nonneg_right (hC p.1 i k) (abs_nonneg _))
+
+lemma measurable_quadraticValue (w : Ω → ι → EuclideanSpace ℝ κ) (d : Ω → ι → ℝ)
+    (hw : ∀ i,Measurable (fun ω ↦ w ω i)) (hd : ∀ i,Measurable (fun ω ↦ d ω i)) :
+    Measurable (fun ω ↦ quadraticValue (w ω) (d ω)) := by
+  have hm : Measurable (fun p : Ω×((κ×κ) → ℝ) ↦
+      finiteMaximum (fun i ↦ linearProcess (fun i ↦ quadraticCoeff (w p.1 i)) p.2 i+d p.1 i)) := by
+    apply measurable_finiteMaximum
+    intro i
+    unfold linearProcess quadraticCoeff
+    fun_prop
+  exact hm.stronglyMeasurable.integral_prod_right.measurable
+
+lemma random_quadratic_integrable (w : Ω → ι → EuclideanSpace ℝ κ) (d : Ω → ι → ℝ)
+    (hw : ∀ i,Measurable (fun ω ↦ w ω i)) (hd : ∀ i,Integrable (fun ω ↦ d ω i) μ)
+    (R : ℝ) (hR : ∀ ω i,‖w ω i‖≤R) :
+    Integrable (fun ω ↦ quadraticValue (w ω) (d ω)) μ := by
+  have hi : Integrable (fun p : Ω×((κ×κ) → ℝ) ↦
+      finiteMaximum (fun i ↦ linearProcess (fun i ↦ quadraticCoeff (w p.1 i)) p.2 i+d p.1 i))
+      (μ.prod (gaussianProduct (κ×κ))) := by
+    apply integrable_randomMaximum
+    intro i
+    apply Integrable.add _ ((hd i).comp_fst _)
+    apply random_linear_integrable (fun ω i ↦ quadraticCoeff (w ω i)) _ (R^2) _ i
+    · intro i k; unfold quadraticCoeff; fun_prop
+    · intro ω i k
+      exact (quadraticCoeff_coord_bound _ _).trans (sq_le_sq₀ (norm_nonneg _) (le_trans (norm_nonneg _) (hR ω i)) |>.mpr (hR ω i))
+  exact hi.integral_prod_left
+
+lemma columnVector_coord_bound {v : EuclideanSpace ℝ κ} (hv : v=0 ∨ ⟪v,v⟫=1)
+    (w : EuclideanSpace ℝ κ) (k : κ) : |columnVector v w k|≤‖w‖^2 :=
+  (PiLp.norm_apply_le _ k).trans (columnVector_norm_le hv w)
+
+theorem random_column_identity (w : Ω → ι → EuclideanSpace ℝ κ) (d : Ω → ι → ℝ)
+    (v : Ω → EuclideanSpace ℝ κ) (hw : ∀ i,Measurable (fun ω ↦ w ω i))
+    (_hdm : ∀ i,Measurable (fun ω ↦ d ω i))
+    (hd : ∀ i,Integrable (fun ω ↦ d ω i) μ) (hv : Measurable v)
+    (hu : ∀ ω,v ω=0 ∨ ⟪v ω,v ω⟫=1) (R : ℝ) (hR : ∀ ω i,‖w ω i‖≤R) :
+    (∫ ω,quadraticValue (w ω) (d ω) ∂μ)=
+      ∫ p : Ω×(κ → ℝ),quadraticValue (fun i ↦ columnResidual (v p.1) (w p.1 i))
+        (fun i ↦ d p.1 i+linearProcess (fun i k ↦ columnVector (v p.1) (w p.1 i) k) p.2 i)
+        ∂μ.prod (gaussianProduct κ) := by
+  have hw' (i : ι) : Measurable (fun p : Ω×(κ → ℝ) ↦ columnResidual (v p.1) (w p.1 i)) :=
+    measurable_columnResidual (hv.comp measurable_fst) ((hw i).comp measurable_fst)
+  have hd' (i : ι) : Integrable (fun p : Ω×(κ → ℝ) ↦
+      d p.1 i+linearProcess (fun i k ↦ columnVector (v p.1) (w p.1 i) k) p.2 i)
+      (μ.prod (gaussianProduct κ)) := by
+    apply ((hd i).comp_fst _).add
+    apply random_linear_integrable (fun ω i k ↦ columnVector (v ω) (w ω i) k) _ (R^2) _ i
+    · intro i k
+      exact (PiLp.continuous_apply 2 (fun _ : κ ↦ ℝ) k).measurable.comp (measurable_columnVector hv (hw i))
+    · intro ω i k
+      exact (columnVector_coord_bound (hu ω) _ _).trans
+        ((sq_le_sq₀ (norm_nonneg _) ((norm_nonneg _).trans (hR ω i))).mpr (hR ω i))
+  have hi := random_quadratic_integrable _ _ hw' hd' R
+    (fun p i ↦ (columnResidual_norm_le (hu p.1) (w p.1 i)).trans (hR p.1 i))
+  rw [integral_prod _ hi]
+  apply integral_congr_ae
+  exact Eventually.of_forall (fun ω ↦ quadraticValue_column (w ω) (d ω) (v ω) (hu ω))
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set InnerProductSpace
+open scoped Topology NNReal ENNReal BigOperators RealInnerProductSpace
+namespace SKValueG
+variable (κ : Type*) [Fintype κ]
+
+abbrev ColumnHistory (j : ℕ) := Fin j → κ → ℝ
+noncomputable def columnHistoryLaw (j : ℕ) : Measure (ColumnHistory κ j) :=
+  Measure.pi (fun _ ↦ gaussianProduct κ)
+instance (j : ℕ) : IsProbabilityMeasure (columnHistoryLaw κ j) := by
+  unfold columnHistoryLaw; infer_instance
+
+def historyInit {j : ℕ} (h : ColumnHistory κ (j+1)) : ColumnHistory κ j := fun i ↦ h i.castSucc
+
+def historySplit (j : ℕ) : ColumnHistory κ (j+1) ≃ᵐ ColumnHistory κ j × (κ → ℝ) :=
+  (MeasurableEquiv.piFinSuccAbove (fun _ : Fin (j+1) ↦ κ → ℝ) (Fin.last j)).trans
+    MeasurableEquiv.prodComm
+
+omit [Fintype κ] in
+lemma historySplit_apply [Fintype κ] (j : ℕ) (h : ColumnHistory κ (j+1)) :
+    historySplit κ j h=(historyInit κ h,h (Fin.last j)) := by
+  simp [historySplit,MeasurableEquiv.piFinSuccAbove,Fin.insertNthEquiv]
+  rfl
+
+lemma historySplit_preserving (j : ℕ) :
+    MeasurePreserving (historySplit κ j) (columnHistoryLaw κ (j+1))
+      ((columnHistoryLaw κ j).prod (gaussianProduct κ)) := by
+  exact (Measure.measurePreserving_swap).comp
+    (measurePreserving_piFinSuccAbove (fun _ : Fin (j+1) ↦ gaussianProduct κ) (Fin.last j))
+
+omit [Fintype κ] in
+lemma historyInit_measurable [Fintype κ] (j : ℕ) : Measurable (historyInit κ (j:=j)) := by
+  unfold historyInit; fun_prop
+
+lemma history_integral_succ (j : ℕ) (f : ColumnHistory κ j × (κ → ℝ) → ℝ) :
+    (∫ h,f (historyInit κ h,h (Fin.last j)) ∂columnHistoryLaw κ (j+1))=
+      ∫ p,f p ∂(columnHistoryLaw κ j).prod (gaussianProduct κ) := by
+  simpa only [historySplit_apply] using (historySplit_preserving κ j).integral_comp' f
+
+lemma history_integrable_succ (j : ℕ) {f : ColumnHistory κ j × (κ → ℝ) → ℝ}
+    (hf : Integrable f ((columnHistoryLaw κ j).prod (gaussianProduct κ))) :
+    Integrable (fun h ↦ f (historyInit κ h,h (Fin.last j))) (columnHistoryLaw κ (j+1)) := by
+  simpa only [Function.comp_def,historySplit_apply] using (historySplit_preserving κ j).integrable_comp
+    hf.aestronglyMeasurable |>.mpr hf
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/RandomGuerra.lean b/lean/OAI/Probability/SKValue/GroundState/RandomGuerra.lean
index ec008d33fce253691dbe79ce79af0590287294c9..386a42b1a570314eb721cdf29d84d6350ae66002 100644
--- a/lean/OAI/Probability/SKValue/GroundState/RandomGuerra.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/RandomGuerra.lean
@@ -1,178 +1,178 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.FieldTreeLaw
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma scalar_pairing_law (T : GaussianTree) :
-    MeasurePreserving (fun p : (TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ fun e ↦ (p.1 e,p.2 e))
-      ((gaussianProduct (TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw)))
-      (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)) :=
-  (measurePreserving_arrowProdEquivProdArrow ℝ ℝ (TreeEdges T)
-    (fun _ ↦ standardGaussian) (fun _ ↦ gumbelLaw)).symm
-
-lemma field_pairing_law (n : ℕ) (T : GaussianTree) :
-    MeasurePreserving (fun p : (Fin n×TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ fun e ↦ ((fun i ↦ p.1 (i,e)),p.2 e))
-      ((gaussianProduct (Fin n×TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw)))
-      (Measure.pi (fun _ : TreeEdges T ↦ fieldNoiseLaw n)) := by
-  have hs := measurePreserving_piCongrLeft (fun _ : TreeEdges T×Fin n ↦ standardGaussian)
-    (Equiv.prodComm (Fin n) (TreeEdges T))
-  have hc := (measurePreserving_finite_curry (I := TreeEdges T) (E := Fin n) standardGaussian).comp hs
-  have hp := (measurePreserving_arrowProdEquivProdArrow (Fin n → ℝ) ℝ (TreeEdges T)
-    (fun _ ↦ gaussianProduct (Fin n)) (fun _ ↦ gumbelLaw)).symm
-  exact hp.comp (hc.prod (MeasurePreserving.id (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))))
-
-lemma scalarTreeEval_integrable (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
-    Integrable (scalarTreeEval c q T) (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)) :=
-  (scalarTreeEval_law c q T).integrable_comp aestronglyMeasurable_id |>.mpr (scalarTreeLaw_integrable c q T)
-
-lemma scalarTreeValue_integrable_prod (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
-    Integrable (fun p : (TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ scalarTreeValue c q T p.1 p.2)
-      ((gaussianProduct (TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))) := by
-  exact (scalar_pairing_law T).integrable_comp (continuous_scalarTreeEval c q T).aestronglyMeasurable
-    |>.mpr (scalarTreeEval_integrable c q T)
-
-lemma fieldTreeValue_integrable_prod (n : ℕ) (q : ℝ) (T : GaussianTree) (x : Fin n → ℝ) :
-    Integrable (fun p : (Fin n×TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ fieldTreeValue n q T x p.1 p.2)
-      ((gaussianProduct (Fin n×TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))) := by
-  have hc : Continuous (fieldTreeEval n q T x) :=
-    (continuous_fieldTreeEval n q T).comp (continuous_const.prodMk continuous_id)
-  exact (field_pairing_law n T).integrable_comp hc.aestronglyMeasurable
-    |>.mpr (fieldTreeEval_integrable n q T x)
-
-lemma scalarTreeValue_mean (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
-    (∫ g,∫ z,scalarTreeValue c q T z g ∂gaussianProduct (TreeEdges T)
-      ∂Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))=∫ y,y ∂scalarTreeLaw c q T := by
-  rw [←integral_prod_symm _ (scalarTreeValue_integrable_prod c q T)]
-  have hcomp := (scalarTreeEval_law c q T).comp (scalar_pairing_law T)
-  have he := hcomp.hasLaw.integral_comp aestronglyMeasurable_id
-  exact he
-
-lemma fieldTreeValue_mean (n : ℕ) (q : ℝ) (T : GaussianTree) (x : Fin n → ℝ) :
-    (∫ g,∫ z,fieldTreeValue n q T x z g ∂gaussianProduct (Fin n×TreeEdges T)
-      ∂Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))=
-      ∫ w,fieldTreeEval n q T x w ∂Measure.pi (fun _ : TreeEdges T ↦ fieldNoiseLaw n) := by
-  rw [←integral_prod_symm _ (fieldTreeValue_integrable_prod n q T x)]
-  have hc : Continuous (fieldTreeEval n q T x) :=
-    (continuous_fieldTreeEval n q T).comp (continuous_const.prodMk continuous_id)
-  exact (field_pairing_law n T).hasLaw.integral_comp hc.aestronglyMeasurable
-
-theorem randomized_tree_guerra_upper {n : ℕ} (hn : 0 < n) (T : GaussianTree) (hT : ValidTreeGrid 0 T) :
-    expectedMaximum n+(∫ y,y ∂scalarTreeLaw (fun q l ↦ Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)) 0 T) ≤
-      ∫ w,fieldTreeEval n 0 T (fun _ ↦ 0) w ∂Measure.pi (fun _ : TreeEdges T ↦ fieldNoiseLaw n) := by
-  let c : ℝ → TreeLevel → ℝ := fun q l ↦ Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)
-  have hiA := scalarTreeValue_integrable_prod c 0 T
-  have hiF := fieldTreeValue_integrable_prod n 0 T (fun _ ↦ 0)
-  have hg : ∀ g : TreeEdges T → ℝ, expectedMaximum n+
-      (∫ z,scalarTreeValue c 0 T z g ∂gaussianProduct (TreeEdges T)) ≤
-      ∫ z,fieldTreeValue n 0 T (fun _ ↦ 0) z g ∂gaussianProduct (Fin n×TreeEdges T) := by
-    intro g
-    simpa only [scalarTreeValue,fieldTreeValue,zero_add,treeAuxVector,c] using
-      finite_tree_guerra_upper hn T hT (treeOffset T g)
-  have hu := integral_mono ((integrable_const (expectedMaximum n)).add hiA.integral_prod_right)
-    hiF.integral_prod_right hg
-  simp only [Pi.add_apply] at hu
-  rw [integral_add (integrable_const _) hiA.integral_prod_right,integral_const,probReal_univ,one_smul,
-    scalarTreeValue_mean,fieldTreeValue_mean] at hu
-  exact hu
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma coleHopf_one_sided {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ) {m : ℝ} (hm : 0 < m)
-    (h x y : ℝ) : SKValue.coleHopf m h ψ x ≤ SKValue.coleHopf m h ψ y+|x-y| := by
-  have hiX := SKValue.lipschitz_exp_integrable hψ hm.le x (Real.sqrt h)
-  have hiY := SKValue.lipschitz_exp_integrable hψ hm.le y (Real.sqrt h)
-  have hpX := SKValue.lipschitz_exp_integral_pos hψ hm.le x (Real.sqrt h)
-  have hpY := SKValue.lipschitz_exp_integral_pos hψ hm.le y (Real.sqrt h)
-  have e : SKValue.standardGaussian=standardGaussian := rfl
-  rw [e] at hiX hiY hpX hpY
-  have hbound : (∫ z,Real.exp (m*ψ (x+Real.sqrt h*z)) ∂standardGaussian) ≤
-      Real.exp (m*|x-y|)*(∫ z,Real.exp (m*ψ (y+Real.sqrt h*z)) ∂standardGaussian) := by
-    rw [←integral_const_mul]
-    apply integral_mono hiX (hiY.const_mul _)
-    intro z
-    have hl := hψ.dist_le_mul (x+Real.sqrt h*z) (y+Real.sqrt h*z)
-    simp only [Real.dist_eq,NNReal.coe_one,one_mul] at hl
-    rw [show x+Real.sqrt h*z-(y+Real.sqrt h*z)=x-y by ring] at hl
-    dsimp only
-    rw [←Real.exp_add]
-    apply Real.exp_le_exp.mpr
-    have hh := le_abs_self (ψ (x+Real.sqrt h*z)-ψ (y+Real.sqrt h*z))
-    nlinarith
-  have hh := Real.log_le_log hpX hbound
-  rw [Real.log_mul (Real.exp_pos _).ne' hpY.ne',Real.log_exp] at hh
-  simp only [SKValue.coleHopf,ite_eq_right hm.ne',SKValue.heat,e]
-  apply (div_le_iff₀ hm).mpr
-  have he : (Real.log (∫ z,Real.exp (m*ψ (y+Real.sqrt h*z)) ∂standardGaussian)/m+|x-y|)*m=
-      m*|x-y|+Real.log (∫ z,Real.exp (m*ψ (y+Real.sqrt h*z)) ∂standardGaussian) := by field_simp; ring
-  rw [he]
-  exact hh
-
-lemma coleHopf_lipschitz {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ) {m : ℝ} (hm : 0 < m)
-    (h : ℝ) : LipschitzWith 1 (SKValue.coleHopf m h ψ) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro x y
-  simp only [NNReal.coe_one,one_mul,Real.dist_eq]
-  have hxy := coleHopf_one_sided hψ hm h x y
-  have hyx := coleHopf_one_sided hψ hm h y x
-  rw [abs_sub_comm y x] at hyx
-  exact abs_le.mpr ⟨by linarith,by linarith⟩
-
-noncomputable def fieldProfile (q : ℝ) : GaussianTree → ℝ → ℝ
-  | [] => abs
-  | l::T => SKValue.coleHopf l.height (l.endpoint-q) (fieldProfile l.endpoint T)
-
-def TreeScale (a : ℝ) : GaussianTree → Prop
-  | [] => 0 ≤ a
-  | l::T => 0 ≤ a ∧ a < l.height ∧ TreeScale l.height T
-
-lemma fieldProfile_lipschitz {T : GaussianTree} {a : ℝ} (h : TreeScale a T) (q : ℝ) :
-    LipschitzWith 1 (fieldProfile q T) := by
-  induction T generalizing a q with
-  | nil =>
-    apply LipschitzWith.of_dist_le_mul
-    intro x y
-    simpa only [fieldProfile,NNReal.coe_one,one_mul,Real.dist_eq] using abs_abs_sub_abs_le_abs_sub x y
-  | cons l T ih =>
-    obtain ⟨ha,ham,hT⟩ := h
-    exact coleHopf_lipschitz (ih hT _) (ha.trans_lt ham) _
-
-lemma gaussian_exp_sum_profile_integrable {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ)
-    {m : ℝ} (hm : 0 ≤ m) (n : ℕ) (x : Fin n → ℝ) (s : ℝ) :
-    Integrable (fun z : Fin n → ℝ ↦ Real.exp (m*∑ i,ψ (x i+s*z i))) (gaussianProduct (Fin n)) := by
-  have hi := Integrable.fintype_prod (fun i : Fin n ↦ SKValue.lipschitz_exp_integrable hψ hm (x i) s)
-  convert hi using 1
-  · funext z
-    simp only [Finset.mul_sum,Real.exp_sum]
-  · rfl
-
-lemma gaussian_exp_sum_coleHopf {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ)
-    {m : ℝ} (hm : 0 < m) (n : ℕ) (x : Fin n → ℝ) (h : ℝ) :
-    (∫ z : Fin n → ℝ,Real.exp (m*∑ i,ψ (x i+Real.sqrt h*z i)) ∂gaussianProduct (Fin n))=
-      Real.exp (m*∑ i,SKValue.coleHopf m h ψ (x i)) := by
-  simp only [Finset.mul_sum,Real.exp_sum]
-  rw [gaussianProduct,integral_fintype_prod_eq_prod (fun i : Fin n ↦ fun z : ℝ ↦ Real.exp (m*ψ (x i+Real.sqrt h*z)))]
-  apply Finset.prod_congr rfl
-  intro i _
-  simp only [SKValue.coleHopf,ite_eq_right hm.ne',SKValue.heat]
-  rw [mul_div_cancel₀ _ hm.ne',Real.exp_log (SKValue.lipschitz_exp_integral_pos hψ hm.le (x i) (Real.sqrt h))]
-  rfl
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.FieldTreeLaw
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma scalar_pairing_law (T : GaussianTree) :
+    MeasurePreserving (fun p : (TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ fun e ↦ (p.1 e,p.2 e))
+      ((gaussianProduct (TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw)))
+      (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)) :=
+  (measurePreserving_arrowProdEquivProdArrow ℝ ℝ (TreeEdges T)
+    (fun _ ↦ standardGaussian) (fun _ ↦ gumbelLaw)).symm
+
+lemma field_pairing_law (n : ℕ) (T : GaussianTree) :
+    MeasurePreserving (fun p : (Fin n×TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ fun e ↦ ((fun i ↦ p.1 (i,e)),p.2 e))
+      ((gaussianProduct (Fin n×TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw)))
+      (Measure.pi (fun _ : TreeEdges T ↦ fieldNoiseLaw n)) := by
+  have hs := measurePreserving_piCongrLeft (fun _ : TreeEdges T×Fin n ↦ standardGaussian)
+    (Equiv.prodComm (Fin n) (TreeEdges T))
+  have hc := (measurePreserving_finite_curry (I := TreeEdges T) (E := Fin n) standardGaussian).comp hs
+  have hp := (measurePreserving_arrowProdEquivProdArrow (Fin n → ℝ) ℝ (TreeEdges T)
+    (fun _ ↦ gaussianProduct (Fin n)) (fun _ ↦ gumbelLaw)).symm
+  exact hp.comp (hc.prod (MeasurePreserving.id (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))))
+
+lemma scalarTreeEval_integrable (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
+    Integrable (scalarTreeEval c q T) (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)) :=
+  (scalarTreeEval_law c q T).integrable_comp aestronglyMeasurable_id |>.mpr (scalarTreeLaw_integrable c q T)
+
+lemma scalarTreeValue_integrable_prod (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
+    Integrable (fun p : (TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ scalarTreeValue c q T p.1 p.2)
+      ((gaussianProduct (TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))) := by
+  exact (scalar_pairing_law T).integrable_comp (continuous_scalarTreeEval c q T).aestronglyMeasurable
+    |>.mpr (scalarTreeEval_integrable c q T)
+
+lemma fieldTreeValue_integrable_prod (n : ℕ) (q : ℝ) (T : GaussianTree) (x : Fin n → ℝ) :
+    Integrable (fun p : (Fin n×TreeEdges T → ℝ)×(TreeEdges T → ℝ) ↦ fieldTreeValue n q T x p.1 p.2)
+      ((gaussianProduct (Fin n×TreeEdges T)).prod (Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))) := by
+  have hc : Continuous (fieldTreeEval n q T x) :=
+    (continuous_fieldTreeEval n q T).comp (continuous_const.prodMk continuous_id)
+  exact (field_pairing_law n T).integrable_comp hc.aestronglyMeasurable
+    |>.mpr (fieldTreeEval_integrable n q T x)
+
+lemma scalarTreeValue_mean (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
+    (∫ g,∫ z,scalarTreeValue c q T z g ∂gaussianProduct (TreeEdges T)
+      ∂Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))=∫ y,y ∂scalarTreeLaw c q T := by
+  rw [←integral_prod_symm _ (scalarTreeValue_integrable_prod c q T)]
+  have hcomp := (scalarTreeEval_law c q T).comp (scalar_pairing_law T)
+  have he := hcomp.hasLaw.integral_comp aestronglyMeasurable_id
+  exact he
+
+lemma fieldTreeValue_mean (n : ℕ) (q : ℝ) (T : GaussianTree) (x : Fin n → ℝ) :
+    (∫ g,∫ z,fieldTreeValue n q T x z g ∂gaussianProduct (Fin n×TreeEdges T)
+      ∂Measure.pi (fun _ : TreeEdges T ↦ gumbelLaw))=
+      ∫ w,fieldTreeEval n q T x w ∂Measure.pi (fun _ : TreeEdges T ↦ fieldNoiseLaw n) := by
+  rw [←integral_prod_symm _ (fieldTreeValue_integrable_prod n q T x)]
+  have hc : Continuous (fieldTreeEval n q T x) :=
+    (continuous_fieldTreeEval n q T).comp (continuous_const.prodMk continuous_id)
+  exact (field_pairing_law n T).hasLaw.integral_comp hc.aestronglyMeasurable
+
+theorem randomized_tree_guerra_upper {n : ℕ} (hn : 0 < n) (T : GaussianTree) (hT : ValidTreeGrid 0 T) :
+    expectedMaximum n+(∫ y,y ∂scalarTreeLaw (fun q l ↦ Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)) 0 T) ≤
+      ∫ w,fieldTreeEval n 0 T (fun _ ↦ 0) w ∂Measure.pi (fun _ : TreeEdges T ↦ fieldNoiseLaw n) := by
+  let c : ℝ → TreeLevel → ℝ := fun q l ↦ Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)
+  have hiA := scalarTreeValue_integrable_prod c 0 T
+  have hiF := fieldTreeValue_integrable_prod n 0 T (fun _ ↦ 0)
+  have hg : ∀ g : TreeEdges T → ℝ, expectedMaximum n+
+      (∫ z,scalarTreeValue c 0 T z g ∂gaussianProduct (TreeEdges T)) ≤
+      ∫ z,fieldTreeValue n 0 T (fun _ ↦ 0) z g ∂gaussianProduct (Fin n×TreeEdges T) := by
+    intro g
+    simpa only [scalarTreeValue,fieldTreeValue,zero_add,treeAuxVector,c] using
+      finite_tree_guerra_upper hn T hT (treeOffset T g)
+  have hu := integral_mono ((integrable_const (expectedMaximum n)).add hiA.integral_prod_right)
+    hiF.integral_prod_right hg
+  simp only [Pi.add_apply] at hu
+  rw [integral_add (integrable_const _) hiA.integral_prod_right,integral_const,probReal_univ,one_smul,
+    scalarTreeValue_mean,fieldTreeValue_mean] at hu
+  exact hu
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma coleHopf_one_sided {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ) {m : ℝ} (hm : 0 < m)
+    (h x y : ℝ) : SKValue.coleHopf m h ψ x ≤ SKValue.coleHopf m h ψ y+|x-y| := by
+  have hiX := SKValue.lipschitz_exp_integrable hψ hm.le x (Real.sqrt h)
+  have hiY := SKValue.lipschitz_exp_integrable hψ hm.le y (Real.sqrt h)
+  have hpX := SKValue.lipschitz_exp_integral_pos hψ hm.le x (Real.sqrt h)
+  have hpY := SKValue.lipschitz_exp_integral_pos hψ hm.le y (Real.sqrt h)
+  have e : SKValue.standardGaussian=standardGaussian := rfl
+  rw [e] at hiX hiY hpX hpY
+  have hbound : (∫ z,Real.exp (m*ψ (x+Real.sqrt h*z)) ∂standardGaussian) ≤
+      Real.exp (m*|x-y|)*(∫ z,Real.exp (m*ψ (y+Real.sqrt h*z)) ∂standardGaussian) := by
+    rw [←integral_const_mul]
+    apply integral_mono hiX (hiY.const_mul _)
+    intro z
+    have hl := hψ.dist_le_mul (x+Real.sqrt h*z) (y+Real.sqrt h*z)
+    simp only [Real.dist_eq,NNReal.coe_one,one_mul] at hl
+    rw [show x+Real.sqrt h*z-(y+Real.sqrt h*z)=x-y by ring] at hl
+    dsimp only
+    rw [←Real.exp_add]
+    apply Real.exp_le_exp.mpr
+    have hh := le_abs_self (ψ (x+Real.sqrt h*z)-ψ (y+Real.sqrt h*z))
+    nlinarith
+  have hh := Real.log_le_log hpX hbound
+  rw [Real.log_mul (Real.exp_pos _).ne' hpY.ne',Real.log_exp] at hh
+  simp only [SKValue.coleHopf,ite_eq_right hm.ne',SKValue.heat,e]
+  apply (div_le_iff₀ hm).mpr
+  have he : (Real.log (∫ z,Real.exp (m*ψ (y+Real.sqrt h*z)) ∂standardGaussian)/m+|x-y|)*m=
+      m*|x-y|+Real.log (∫ z,Real.exp (m*ψ (y+Real.sqrt h*z)) ∂standardGaussian) := by field_simp; ring
+  rw [he]
+  exact hh
+
+lemma coleHopf_lipschitz {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ) {m : ℝ} (hm : 0 < m)
+    (h : ℝ) : LipschitzWith 1 (SKValue.coleHopf m h ψ) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro x y
+  simp only [NNReal.coe_one,one_mul,Real.dist_eq]
+  have hxy := coleHopf_one_sided hψ hm h x y
+  have hyx := coleHopf_one_sided hψ hm h y x
+  rw [abs_sub_comm y x] at hyx
+  exact abs_le.mpr ⟨by linarith,by linarith⟩
+
+noncomputable def fieldProfile (q : ℝ) : GaussianTree → ℝ → ℝ
+  | [] => abs
+  | l::T => SKValue.coleHopf l.height (l.endpoint-q) (fieldProfile l.endpoint T)
+
+def TreeScale (a : ℝ) : GaussianTree → Prop
+  | [] => 0 ≤ a
+  | l::T => 0 ≤ a ∧ a < l.height ∧ TreeScale l.height T
+
+lemma fieldProfile_lipschitz {T : GaussianTree} {a : ℝ} (h : TreeScale a T) (q : ℝ) :
+    LipschitzWith 1 (fieldProfile q T) := by
+  induction T generalizing a q with
+  | nil =>
+    apply LipschitzWith.of_dist_le_mul
+    intro x y
+    simpa only [fieldProfile,NNReal.coe_one,one_mul,Real.dist_eq] using abs_abs_sub_abs_le_abs_sub x y
+  | cons l T ih =>
+    obtain ⟨ha,ham,hT⟩ := h
+    exact coleHopf_lipschitz (ih hT _) (ha.trans_lt ham) _
+
+lemma gaussian_exp_sum_profile_integrable {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ)
+    {m : ℝ} (hm : 0 ≤ m) (n : ℕ) (x : Fin n → ℝ) (s : ℝ) :
+    Integrable (fun z : Fin n → ℝ ↦ Real.exp (m*∑ i,ψ (x i+s*z i))) (gaussianProduct (Fin n)) := by
+  have hi := Integrable.fintype_prod (fun i : Fin n ↦ SKValue.lipschitz_exp_integrable hψ hm (x i) s)
+  convert hi using 1
+  · funext z
+    simp only [Finset.mul_sum,Real.exp_sum]
+  · rfl
+
+lemma gaussian_exp_sum_coleHopf {ψ : ℝ → ℝ} (hψ : LipschitzWith 1 ψ)
+    {m : ℝ} (hm : 0 < m) (n : ℕ) (x : Fin n → ℝ) (h : ℝ) :
+    (∫ z : Fin n → ℝ,Real.exp (m*∑ i,ψ (x i+Real.sqrt h*z i)) ∂gaussianProduct (Fin n))=
+      Real.exp (m*∑ i,SKValue.coleHopf m h ψ (x i)) := by
+  simp only [Finset.mul_sum,Real.exp_sum]
+  rw [gaussianProduct,integral_fintype_prod_eq_prod (fun i : Fin n ↦ fun z : ℝ ↦ Real.exp (m*ψ (x i+Real.sqrt h*z)))]
+  apply Finset.prod_congr rfl
+  intro i _
+  simp only [SKValue.coleHopf,ite_eq_right hm.ne',SKValue.heat]
+  rw [mul_div_cancel₀ _ hm.ne',Real.exp_log (SKValue.lipschitz_exp_integral_pos hψ hm.le (x i) (Real.sqrt h))]
+  rfl
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/RotatedCoefficients.lean b/lean/OAI/Probability/SKValue/GroundState/RotatedCoefficients.lean
index f50e51965c7b383bd63a80a73fcf533de3a4a398..fb75473d9e4fd9f3ba97febbb9bcd44ed1a4092d 100644
--- a/lean/OAI/Probability/SKValue/GroundState/RotatedCoefficients.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/RotatedCoefficients.lean
@@ -1,396 +1,396 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.StandardGaussian
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma gaussian_gibbs_derivative_nonneg {ι : Type*} [Fintype ι] [Nonempty ι]
-    {n : ℕ} (c d : ι → Fin n → ℝ)
-    (h : ∀ i j, 0 ≤ ∑ k, (d i k-d j k)*(c i k-c j k)) :
-    0 ≤ ∫ z, ∑ i, gibbsWeight (linearProcess c z) i*linearProcess d z i ∂gaussianProduct (Fin n) := by
-  have hi : ∀ i, Integrable (fun z ↦ gibbsWeight (linearProcess c z) i*linearProcess d z i)
-      (gaussianProduct (Fin n)) := fun i ↦ gibbsWeight_mul_integrable c i (linearProcess_integrable d i)
-  have hi' : ∀ i, Integrable (fun z ↦ gibbsWeight (linearProcess c z) i *
-      ((∑ k, d i k*c i k)-∑ j, gibbsWeight (linearProcess c z) j*(∑ k, d i k*c j k)))
-      (gaussianProduct (Fin n)) := by
-    intro i
-    apply gibbsWeight_mul_integrable
-    exact (integrable_const _).sub (integrable_finsetSum _ (fun j _ ↦ (integrable_linear_gibbsWeight c j).mul_const _))
-  rw [integral_finsetSum _ (fun i _ ↦ hi i)]
-  simp_rw [gaussian_linear_gibbsWeight]
-  rw [← integral_finsetSum _ (fun i _ ↦ hi' i)]
-  apply integral_nonneg
-  intro z
-  change (0 : ℝ) ≤ ∑ i, gibbsWeight (linearProcess c z) i *
-    ((∑ k, d i k*c i k)-∑ j, gibbsWeight (linearProcess c z) j*(∑ k, d i k*c j k))
-  rw [weighted_covariance_pairs _ (fun i j ↦ ∑ k, d i k*c j k) (sum_gibbsWeight (linearProcess c z))]
-  apply mul_nonneg (by norm_num)
-  apply Finset.sum_nonneg
-  intro i _
-  apply Finset.sum_nonneg
-  intro j _
-  apply mul_nonneg (mul_nonneg (gibbsWeight_nonneg _ _) (gibbsWeight_nonneg _ _))
-  have he : (∑ k, d i k*c i k)+(∑ k, d j k*c j k)-
-      (∑ k, d i k*c j k)-(∑ k, d j k*c i k) = ∑ k, (d i k-d j k)*(c i k-c j k) := by
-    rw [← Finset.sum_add_distrib,← Finset.sum_sub_distrib,← Finset.sum_sub_distrib]
-    apply Finset.sum_congr rfl; intro k _; ring
-  rw [he]
-  exact h i j
-
-noncomputable def rotatedCoefficients {ι κ : Type*} (a b : ι → κ → ℝ) (t : ℝ) (i : ι) (k : κ) : ℝ :=
-  Real.cos t*a i k + Real.sin t*b i k
-
-noncomputable def rotationVelocity {ι κ : Type*} (a b : ι → κ → ℝ) (t : ℝ) (i : ι) (k : κ) : ℝ :=
-  -Real.sin t*a i k + Real.cos t*b i k
-
-lemma hasDerivAt_rotatedProcess {ι κ : Type*} [Fintype κ] (a b : ι → κ → ℝ)
-    (t : ℝ) (z : κ → ℝ) (i : ι) :
-    HasDerivAt (fun s ↦ linearProcess (rotatedCoefficients a b s) z i)
-      (linearProcess (rotationVelocity a b t) z i) t := by
-  exact HasDerivAt.fun_sum (u := Finset.univ) (fun k _ ↦
-    (((Real.hasDerivAt_cos t).mul_const (a i k)).add ((Real.hasDerivAt_sin t).mul_const (b i k))).mul_const (z k))
-
-lemma abs_rotationVelocity_le {ι κ : Type*} (a b : ι → κ → ℝ) (t : ℝ) (i : ι) (k : κ) :
-    |rotationVelocity a b t i k| ≤ |a i k|+|b i k| := by
-  calc
-    _ ≤ |-Real.sin t*a i k| + |Real.cos t*b i k| := abs_add_le _ _
-    _ = |Real.sin t| *|a i k| + |Real.cos t| *|b i k| := by rw [abs_mul,abs_neg,abs_mul]
-    _ ≤ 1*|a i k| + 1*|b i k| := add_le_add
-      (mul_le_mul_of_nonneg_right (Real.abs_sin_le_one t) (abs_nonneg _))
-      (mul_le_mul_of_nonneg_right (Real.abs_cos_le_one t) (abs_nonneg _))
-    _ = _ := by ring
-
-lemma rotation_derivative_bound {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (a b : ι → κ → ℝ) (t : ℝ) (z : κ → ℝ) :
-    |∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
-        linearProcess (rotationVelocity a b t) z i| ≤
-      ∑ i, ∑ k, (|a i k|+|b i k|)*|z k| := by
-  apply abs_gibbsMean_le
-  intro i
-  calc
-    _ ≤ ∑ k, |rotationVelocity a b t i k*z k| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ k, (|a i k|+|b i k|)*|z k| := by
-      apply Finset.sum_le_sum
-      intro k _
-      rw [abs_mul]
-      exact mul_le_mul_of_nonneg_right (abs_rotationVelocity_le _ _ _ _ _) (abs_nonneg _)
-    _ ≤ _ := Finset.single_le_sum (f := fun j ↦ ∑ k, (|a j k|+|b j k|)*|z k|) (fun j _ ↦ Finset.sum_nonneg (fun k _ ↦ mul_nonneg (add_nonneg (abs_nonneg _) (abs_nonneg _)) (abs_nonneg _))) (Finset.mem_univ i)
-
-lemma hasDerivAt_expected_rotated_logPartition {ι : Type*} [Fintype ι] [Nonempty ι]
-    {n : ℕ} (a b : ι → Fin n → ℝ) (t : ℝ) :
-    HasDerivAt (fun s ↦ ∫ z, logPartition (linearProcess (rotatedCoefficients a b s) z) ∂gaussianProduct (Fin n))
-      (∫ z, ∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
-        linearProcess (rotationVelocity a b t) z i ∂gaussianProduct (Fin n)) t := by
-  have hb : Integrable (fun z : Fin n → ℝ ↦ ∑ i, ∑ k, (|a i k|+|b i k|)*|z k|)
-      (gaussianProduct (Fin n)) :=
-    integrable_finsetSum _ (fun i _ ↦ integrable_finsetSum _ (fun k _ ↦
-      (gaussian_coordinate_integrable k).abs.const_mul _))
-  have hmc : Continuous (fun z : Fin n → ℝ ↦ ∑ i,
-      gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
-      linearProcess (rotationVelocity a b t) z i) := by
-    apply continuous_finsetSum
-    intro i _
-    apply (continuous_linear_gibbsWeight _ _).mul
-    exact (continuous_apply i).comp (continuous_linearProcess _)
-  exact (hasDerivAt_integral_of_dominated_loc_of_deriv_le (s := Set.univ) (μ := gaussianProduct (Fin n))
-    (F := fun s z ↦ logPartition (linearProcess (rotatedCoefficients a b s) z))
-    (F' := fun s z ↦ ∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b s) z) i *
-      linearProcess (rotationVelocity a b s) z i)
-    (bound := fun z ↦ ∑ i, ∑ k, (|a i k|+|b i k|)*|z k|)
-    (Filter.univ_mem)
-    (Eventually.of_forall (fun s ↦ (continuous_logPartition.comp (continuous_linearProcess _)).aestronglyMeasurable))
-    (logPartition_integrable _) hmc.aestronglyMeasurable
-    (Eventually.of_forall (fun z s _ ↦ by simpa only [Real.norm_eq_abs] using rotation_derivative_bound a b s z)) hb
-    (Eventually.of_forall (fun z s _ ↦ hasDerivAt_logPartition (hasDerivAt_rotatedProcess a b s z)))).2
-
-lemma rotation_pair_covariance {ι κ : Type*} [Fintype κ] (a b : ι → κ → ℝ) (t : ℝ) (i j : ι) :
-    (∑ k, (rotationVelocity a b t i k-rotationVelocity a b t j k)*
-      (rotatedCoefficients a b t i k-rotatedCoefficients a b t j k)) =
-    Real.sin t*Real.cos t*((∑ k, (b i k-b j k)^2)-(∑ k, (a i k-a j k)^2)) +
-      ((Real.cos t)^2-(Real.sin t)^2)*∑ k, (a i k-a j k)*(b i k-b j k) := by
-  simp only [rotationVelocity,rotatedCoefficients,← Finset.sum_sub_distrib,Finset.mul_sum,← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro k _
-  ring
-
-lemma expected_logPartition_le_of_pair_distances {ι : Type*} [Fintype ι] [Nonempty ι]
-    {n : ℕ} (a b : ι → Fin n → ℝ)
-    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
-    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
-    (∫ z, logPartition (linearProcess a z) ∂gaussianProduct (Fin n)) ≤
-    ∫ z, logPartition (linearProcess b z) ∂gaussianProduct (Fin n) := by
-  let F := fun t ↦ ∫ z, logPartition (linearProcess (rotatedCoefficients a b t) z) ∂gaussianProduct (Fin n)
-  have hd : ∀ t, HasDerivAt F (∫ z, ∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
-      linearProcess (rotationVelocity a b t) z i ∂gaussianProduct (Fin n)) t :=
-    hasDerivAt_expected_rotated_logPartition a b
-  have hc : Continuous F := continuous_iff_continuousAt.mpr (fun t ↦ (hd t).continuousAt)
-  have hm : MonotoneOn F (Icc (0 : ℝ) (Real.pi/2)) := by
-    apply monotoneOn_of_deriv_nonneg (convex_Icc _ _) hc.continuousOn
-      (fun t _ ↦ (hd t).differentiableAt.differentiableWithinAt)
-    intro t ht
-    have ht' : t ∈ Icc (0 : ℝ) (Real.pi/2) := interior_subset ht
-    rw [(hd t).deriv]
-    apply gaussian_gibbs_derivative_nonneg
-    intro i j
-    rw [rotation_pair_covariance,horth,mul_zero,add_zero]
-    exact mul_nonneg (mul_nonneg
-      (Real.sin_nonneg_of_nonneg_of_le_pi ht'.1 (ht'.2.trans (by linarith [Real.pi_pos])))
-      (Real.cos_nonneg_of_mem_Icc ⟨by linarith [Real.pi_pos, ht'.1],ht'.2⟩)) (sub_nonneg.mpr (hdist i j))
-  have hπ : (0 : ℝ) ≤ Real.pi/2 := by positivity
-  have hh := hm ⟨le_rfl,hπ⟩ ⟨hπ,le_rfl⟩ hπ
-  have ha0 : rotatedCoefficients a b 0 = a := by funext i k; simp [rotatedCoefficients]
-  have hb0 : rotatedCoefficients a b (Real.pi/2) = b := by funext i k; simp [rotatedCoefficients]
-  simpa only [F,ha0,hb0] using hh
-
-noncomputable def finiteMaximum {ι : Type*} (x : ι → ℝ) : ℝ := sSup (Set.range x)
-
-lemma le_finiteMaximum {ι : Type*} [Fintype ι] (x : ι → ℝ) (i : ι) : x i ≤ finiteMaximum x :=
-  le_csSup (Set.finite_range x).bddAbove (Set.mem_range_self i)
-
-lemma exists_finiteMaximum {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
-    ∃ i, x i = finiteMaximum x := (Set.range_nonempty x).csSup_mem (Set.finite_range x)
-
-lemma continuous_finiteMaximum {ι : Type*} [Fintype ι] [Nonempty ι] :
-    Continuous (@finiteMaximum ι) := by
-  have he (x : ι → ℝ) : finiteMaximum x = Finset.univ.sup' Finset.univ_nonempty x := by
-    rw [Finset.sup'_eq_csSup_image]
-    simp only [Finset.coe_univ,Set.image_univ,finiteMaximum]
-  simp_rw [funext he]
-  exact Continuous.finset_sup'_apply Finset.univ_nonempty (fun i _ ↦ continuous_apply i)
-
-lemma abs_finiteMaximum_le {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
-    |finiteMaximum x| ≤ ∑ i, |x i| := by
-  obtain ⟨i,hi⟩ := exists_finiteMaximum x
-  rw [← hi]
-  exact Finset.single_le_sum (fun j _ ↦ abs_nonneg (x j)) (Finset.mem_univ i)
-
-lemma finiteMaximum_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (c : ι → κ → ℝ) : Integrable (fun z ↦ finiteMaximum (linearProcess c z)) (gaussianProduct κ) := by
-  apply (integrable_finsetSum Finset.univ (fun i _ ↦ (linearProcess_integrable c i).abs)).mono'
-    ((continuous_finiteMaximum.comp (continuous_linearProcess c)).aestronglyMeasurable)
-  exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_def] using abs_finiteMaximum_le (linearProcess c z))
-
-lemma scaled_max_le_logPartition {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ)
-    (β : ℝ) : β*finiteMaximum x ≤ logPartition (fun i ↦ β*x i) := by
-  obtain ⟨i,hi⟩ := exists_finiteMaximum x
-  rw [← hi]
-  exact le_logPartition (fun i ↦ β*x i) i
-
-lemma logPartition_le_scaled_max {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ)
-    {β : ℝ} (hβ : 0 ≤ β) : logPartition (fun i ↦ β*x i) ≤ β*finiteMaximum x+Real.log (Fintype.card ι) := by
-  exact (logPartition_le _ (fun i ↦ mul_le_mul_of_nonneg_left (le_finiteMaximum x i) hβ)).trans_eq (add_comm _ _)
-
-lemma linearProcess_scaled {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) (β : ℝ) (z : κ → ℝ) :
-    linearProcess (fun i k ↦ β*c i k) z = fun i ↦ β*linearProcess c z i := by
-  funext i
-  simp only [linearProcess,Finset.mul_sum,mul_assoc]
-
-lemma expected_max_le_of_pair_distances_orthogonal {ι : Type*} [Fintype ι] [Nonempty ι]
-    {n : ℕ} (a b : ι → Fin n → ℝ)
-    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
-    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
-    (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) ≤
-    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n) := by
-  have hb (β : ℝ) (hβ : 0 ≤ β) :
-      β*(∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) ≤
-      β*(∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n)) + Real.log (Fintype.card ι) := by
-    have hiA := logPartition_integrable (fun i k ↦ β*a i k)
-    have hiB := logPartition_integrable (fun i k ↦ β*b i k)
-    have hcomp := expected_logPartition_le_of_pair_distances
-      (fun i k ↦ β*a i k) (fun i k ↦ β*b i k)
-      (fun i j ↦ by
-        calc
-          (∑ k, (β*a i k-β*a j k)*(β*b i k-β*b j k)) =
-              β^2*(∑ k, (a i k-a j k)*(b i k-b j k)) := by
-            rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro k _; ring
-          _ = 0 := by rw [horth,mul_zero])
-      (fun i j ↦ by
-        have he (c : ι → Fin n → ℝ) : (∑ k, (β*c i k-β*c j k)^2) = β^2*∑ k, (c i k-c j k)^2 := by
-          rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro k _; ring
-        rw [he,he]
-        exact mul_le_mul_of_nonneg_left (hdist i j) (sq_nonneg β))
-    calc
-      _ = ∫ z, β*finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n) := by rw [integral_const_mul]
-      _ ≤ ∫ z, logPartition (linearProcess (fun i k ↦ β*a i k) z) ∂gaussianProduct (Fin n) := by
-        apply integral_mono ((finiteMaximum_integrable a).const_mul _) hiA
-        intro z
-        change β*finiteMaximum (linearProcess a z) ≤ logPartition (linearProcess (fun i k ↦ β*a i k) z)
-        rw [linearProcess_scaled]
-        exact scaled_max_le_logPartition _ _
-      _ ≤ _ := hcomp
-      _ ≤ ∫ z, β*finiteMaximum (linearProcess b z)+Real.log (Fintype.card ι) ∂gaussianProduct (Fin n) := by
-        apply integral_mono hiB (((finiteMaximum_integrable b).const_mul _).add (integrable_const _))
-        intro z
-        change logPartition (linearProcess (fun i k ↦ β*b i k) z) ≤ β*finiteMaximum (linearProcess b z)+Real.log (Fintype.card ι)
-        rw [linearProcess_scaled]
-        exact logPartition_le_scaled_max _ hβ
-      _ = _ := by rw [integral_add ((finiteMaximum_integrable b).const_mul _) (integrable_const _),integral_const_mul,integral_const]; simp
-  by_contra hh
-  have hΔ : 0 < (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) -
-      (∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n)) := sub_pos.mpr (lt_of_not_ge hh)
-  let Δ := (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) -
-      (∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n))
-  let β := (|Real.log (Fintype.card ι)|+1)/Δ
-  have hβ : 0 < β := div_pos (by positivity) hΔ
-  have hh' := hb β hβ.le
-  have he : β*Δ = |Real.log (Fintype.card ι)|+1 := div_mul_cancel₀ _ hΔ.ne'
-  have hlog := le_abs_self (Real.log (Fintype.card ι))
-  dsimp [Δ] at he
-  nlinarith
-
-lemma gaussian_integral_reindex {κ ν : Type*} [Fintype κ] [Fintype ν] (e : κ ≃ ν)
-    (f : (κ → ℝ) → ℝ) :
-    (∫ z : ν → ℝ, f (fun k ↦ z (e k)) ∂gaussianProduct ν) = ∫ z, f z ∂gaussianProduct κ := by
-  have he := measurePreserving_piCongrLeft (α := fun _ : κ ↦ ℝ)
-    (fun _ : κ ↦ standardGaussian) e.symm
-  have hf (z : ν → ℝ) : (MeasurableEquiv.piCongrLeft (π := fun _ : κ ↦ ℝ) e.symm) z = fun k ↦ z (e k) := by
-    funext k
-    simp [MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply]
-  simpa only [gaussianProduct,hf] using he.integral_comp' f
-
-lemma linearProcess_reindex {ι κ ν : Type*} [Fintype κ] [Fintype ν]
-    (c : ι → κ → ℝ) (e : κ ≃ ν) (z : ν → ℝ) :
-    linearProcess c (fun k ↦ z (e k)) = linearProcess (fun i l ↦ c i (e.symm l)) z := by
-  funext i
-  simpa only [linearProcess, Equiv.symm_apply_apply] using e.sum_comp (fun l ↦ c i (e.symm l)*z l)
-
-lemma expected_max_reindex {ι κ ν : Type*} [Fintype ι] [Fintype κ] [Fintype ν]
-    (c : ι → κ → ℝ) (e : κ ≃ ν) :
-    (∫ z, finiteMaximum (linearProcess (fun i l ↦ c i (e.symm l)) z) ∂gaussianProduct ν) =
-    ∫ z, finiteMaximum (linearProcess c z) ∂gaussianProduct κ := by
-  simpa only [linearProcess_reindex] using gaussian_integral_reindex e (fun z ↦ finiteMaximum (linearProcess c z))
-
-lemma expected_max_le_of_pair_distances_orthogonal_fintype {ι κ : Type*}
-    [Fintype ι] [Nonempty ι] [Fintype κ] (a b : ι → κ → ℝ)
-    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
-    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
-    (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct κ) ≤
-    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct κ := by
-  let e := Fintype.equivFin κ
-  have hh := expected_max_le_of_pair_distances_orthogonal
-    (fun i k ↦ a i (e.symm k)) (fun i k ↦ b i (e.symm k))
-    (fun i j ↦ by
-      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)*(b i k-b j k))]
-      exact horth i j)
-    (fun i j ↦ by
-      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)^2), e.symm.sum_comp (fun k ↦ (b i k-b j k)^2)]
-      exact hdist i j)
-  simpa only [expected_max_reindex] using hh
-
-noncomputable def embedLeft {ι κ ν : Type*} (a : ι → κ → ℝ) (i : ι) : κ ⊕ ν → ℝ :=
-  Sum.elim (a i) (fun _ ↦ 0)
-noncomputable def embedRight {ι κ ν : Type*} (b : ι → ν → ℝ) (i : ι) : κ ⊕ ν → ℝ :=
-  Sum.elim (fun _ ↦ 0) (b i)
-
-lemma expected_max_embedLeft {ι κ ν : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
-    (a : ι → κ → ℝ) :
-    (∫ z, finiteMaximum (linearProcess (embedLeft (ν := ν) a) z) ∂gaussianProduct (κ ⊕ ν)) =
-    ∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct κ := by
-  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
-  change (∫ z, finiteMaximum (linearProcess (embedLeft a) z) ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
-  rw [← he.integral_comp' (fun z ↦ finiteMaximum (linearProcess (embedLeft a) z))]
-  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
-      linearProcess (embedLeft a) ((MeasurableEquiv.sumPiEquivProdPi (fun _ : κ ⊕ ν ↦ ℝ)).symm p) =
-      linearProcess a p.1 := by
-    funext i
-    simp [linearProcess,embedLeft,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
-  simp only [hlin]
-  simpa [gaussianProduct] using integral_fun_fst (μ := gaussianProduct κ) (ν := gaussianProduct ν) (fun z ↦ finiteMaximum (linearProcess a z))
-
-lemma expected_max_embedRight {ι κ ν : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
-    (b : ι → ν → ℝ) :
-    (∫ z, finiteMaximum (linearProcess (embedRight (κ := κ) b) z) ∂gaussianProduct (κ ⊕ ν)) =
-    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct ν := by
-  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
-  change (∫ z, finiteMaximum (linearProcess (embedRight b) z) ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
-  rw [← he.integral_comp' (fun z ↦ finiteMaximum (linearProcess (embedRight b) z))]
-  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
-      linearProcess (embedRight b) ((MeasurableEquiv.sumPiEquivProdPi (fun _ : κ ⊕ ν ↦ ℝ)).symm p) =
-      linearProcess b p.2 := by
-    funext i
-    simp [linearProcess,embedRight,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
-  simp only [hlin]
-  simpa [gaussianProduct] using integral_fun_snd (μ := gaussianProduct κ) (ν := gaussianProduct ν) (fun z ↦ finiteMaximum (linearProcess b z))
-
-lemma expected_max_le_of_pair_distances {ι κ ν : Type*}
-    [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
-    (a : ι → κ → ℝ) (b : ι → ν → ℝ)
-    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
-    (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct κ) ≤
-    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct ν := by
-  have h := expected_max_le_of_pair_distances_orthogonal_fintype
-    (embedLeft (ν := ν) a) (embedRight (κ := κ) b)
-    (fun i j ↦ by simp [embedLeft,embedRight,Fintype.sum_sum_type])
-    (fun i j ↦ by simpa only [Fintype.sum_sum_type,embedLeft,embedRight,
-      Sum.elim_inl,Sum.elim_inr,sub_self,zero_pow (by decide : 2 ≠ 0),Finset.sum_const_zero,zero_add,add_zero] using hdist i j)
-  simpa only [expected_max_embedLeft,expected_max_embedRight] using h
-
-noncomputable def spin (b : Bool) : ℝ := if b then 1 else -1
-noncomputable def hamiltonian (n : ℕ) (J : (Fin n × Fin n) → ℝ)
-    (σ : Fin n → Bool) : ℝ :=
-  (Real.sqrt (n : ℝ))⁻¹ *
-    ∑ i : Fin n, ∑ j : Fin n, if i < j then J (i,j) * spin (σ i) * spin (σ j) else 0
-noncomputable def groundStateSequence (n : ℕ) : ℝ :=
-  (n : ℝ)⁻¹ * ∫ J, sSup (range (hamiltonian n J)) ∂gaussianProduct (Fin n × Fin n)
-noncomputable def groundStateValue : ℝ := limUnder atTop groundStateSequence
-
-noncomputable def hamCoeff (n : ℕ) (σ : Fin n → Bool) (k : Fin n × Fin n) : ℝ :=
-  if k.1 < k.2 then (Real.sqrt (n : ℝ))⁻¹ * spin (σ k.1) * spin (σ k.2) else 0
-noncomputable def expectedMaximum (n : ℕ) : ℝ :=
-  ∫ J, finiteMaximum (linearProcess (hamCoeff n) J) ∂gaussianProduct (Fin n × Fin n)
-
-lemma spin_sq (b : Bool) : spin b ^ 2 = 1 := by cases b <;> norm_num [spin]
-lemma abs_spin (b : Bool) : |spin b| = 1 := by cases b <;> norm_num [spin]
-
-lemma linearProcess_hamCoeff (n : ℕ) (J : (Fin n × Fin n) → ℝ) :
-    linearProcess (hamCoeff n) J = hamiltonian n J := by
-  funext σ
-  simp only [linearProcess,hamCoeff,hamiltonian,Fintype.sum_prod_type,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  apply Finset.sum_congr rfl
-  intro j _
-  split_ifs <;> ring
-
-lemma groundStateSequence_eq (n : ℕ) : groundStateSequence n = (n : ℝ)⁻¹ * expectedMaximum n := by
-  simp only [groundStateSequence,expectedMaximum,linearProcess_hamCoeff,finiteMaximum]
-
-lemma twice_upper_sum {ι : Type*} [Fintype ι] [LinearOrder ι] (f : ι → ι → ℝ)
-    (hsymm : ∀ i j, f i j = f j i) (hdiag : ∀ i, f i i = 0) :
-    2*(∑ i, ∑ j, if i < j then f i j else 0) = ∑ i, ∑ j, f i j := by
-  have hs : (∑ i, ∑ j, if j < i then f i j else 0) = ∑ i, ∑ j, if i < j then f i j else 0 := by
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl
-    intro i _
-    apply Finset.sum_congr rfl
-    intro j _
-    rw [hsymm j i]
-  have hsplit (i j : ι) : f i j = (if i < j then f i j else 0)+(if j < i then f i j else 0) := by
-    rcases lt_trichotomy i j with h | h | h
-    · simp [h,not_lt_of_ge h.le]
-    · subst j; simp [hdiag]
-    · simp [h,not_lt_of_ge h.le]
-  conv_rhs => arg 2; ext i; arg 2; ext j; rw [hsplit i j]
-  simp only [Finset.sum_add_distrib,hs]
-  ring
-
-lemma spin_pair_difference {dimension : ℕ} (σ τ : Fin dimension → Bool) (i j : Fin dimension) :
-    (spin (σ i)*spin (σ j)-spin (τ i)*spin (τ j))^2 =
-      2-2*(spin (σ i)*spin (τ i))*(spin (σ j)*spin (τ j)) := by
-  have h1 := spin_sq (σ i)
-  have h2 := spin_sq (σ j)
-  have h3 := spin_sq (τ i)
-  have h4 := spin_sq (τ j)
-  calc
-    _ = (spin (σ i))^2*(spin (σ j))^2+(spin (τ i))^2*(spin (τ j))^2 -
-      2*(spin (σ i)*spin (τ i))*(spin (σ j)*spin (τ j)) := by ring
-    _ = _ := by rw [h1,h2,h3,h4]; ring
-
-end SKValueG
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.StandardGaussian
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma gaussian_gibbs_derivative_nonneg {ι : Type*} [Fintype ι] [Nonempty ι]
+    {n : ℕ} (c d : ι → Fin n → ℝ)
+    (h : ∀ i j, 0 ≤ ∑ k, (d i k-d j k)*(c i k-c j k)) :
+    0 ≤ ∫ z, ∑ i, gibbsWeight (linearProcess c z) i*linearProcess d z i ∂gaussianProduct (Fin n) := by
+  have hi : ∀ i, Integrable (fun z ↦ gibbsWeight (linearProcess c z) i*linearProcess d z i)
+      (gaussianProduct (Fin n)) := fun i ↦ gibbsWeight_mul_integrable c i (linearProcess_integrable d i)
+  have hi' : ∀ i, Integrable (fun z ↦ gibbsWeight (linearProcess c z) i *
+      ((∑ k, d i k*c i k)-∑ j, gibbsWeight (linearProcess c z) j*(∑ k, d i k*c j k)))
+      (gaussianProduct (Fin n)) := by
+    intro i
+    apply gibbsWeight_mul_integrable
+    exact (integrable_const _).sub (integrable_finsetSum _ (fun j _ ↦ (integrable_linear_gibbsWeight c j).mul_const _))
+  rw [integral_finsetSum _ (fun i _ ↦ hi i)]
+  simp_rw [gaussian_linear_gibbsWeight]
+  rw [← integral_finsetSum _ (fun i _ ↦ hi' i)]
+  apply integral_nonneg
+  intro z
+  change (0 : ℝ) ≤ ∑ i, gibbsWeight (linearProcess c z) i *
+    ((∑ k, d i k*c i k)-∑ j, gibbsWeight (linearProcess c z) j*(∑ k, d i k*c j k))
+  rw [weighted_covariance_pairs _ (fun i j ↦ ∑ k, d i k*c j k) (sum_gibbsWeight (linearProcess c z))]
+  apply mul_nonneg (by norm_num)
+  apply Finset.sum_nonneg
+  intro i _
+  apply Finset.sum_nonneg
+  intro j _
+  apply mul_nonneg (mul_nonneg (gibbsWeight_nonneg _ _) (gibbsWeight_nonneg _ _))
+  have he : (∑ k, d i k*c i k)+(∑ k, d j k*c j k)-
+      (∑ k, d i k*c j k)-(∑ k, d j k*c i k) = ∑ k, (d i k-d j k)*(c i k-c j k) := by
+    rw [← Finset.sum_add_distrib,← Finset.sum_sub_distrib,← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl; intro k _; ring
+  rw [he]
+  exact h i j
+
+noncomputable def rotatedCoefficients {ι κ : Type*} (a b : ι → κ → ℝ) (t : ℝ) (i : ι) (k : κ) : ℝ :=
+  Real.cos t*a i k + Real.sin t*b i k
+
+noncomputable def rotationVelocity {ι κ : Type*} (a b : ι → κ → ℝ) (t : ℝ) (i : ι) (k : κ) : ℝ :=
+  -Real.sin t*a i k + Real.cos t*b i k
+
+lemma hasDerivAt_rotatedProcess {ι κ : Type*} [Fintype κ] (a b : ι → κ → ℝ)
+    (t : ℝ) (z : κ → ℝ) (i : ι) :
+    HasDerivAt (fun s ↦ linearProcess (rotatedCoefficients a b s) z i)
+      (linearProcess (rotationVelocity a b t) z i) t := by
+  exact HasDerivAt.fun_sum (u := Finset.univ) (fun k _ ↦
+    (((Real.hasDerivAt_cos t).mul_const (a i k)).add ((Real.hasDerivAt_sin t).mul_const (b i k))).mul_const (z k))
+
+lemma abs_rotationVelocity_le {ι κ : Type*} (a b : ι → κ → ℝ) (t : ℝ) (i : ι) (k : κ) :
+    |rotationVelocity a b t i k| ≤ |a i k|+|b i k| := by
+  calc
+    _ ≤ |-Real.sin t*a i k| + |Real.cos t*b i k| := abs_add_le _ _
+    _ = |Real.sin t| *|a i k| + |Real.cos t| *|b i k| := by rw [abs_mul,abs_neg,abs_mul]
+    _ ≤ 1*|a i k| + 1*|b i k| := add_le_add
+      (mul_le_mul_of_nonneg_right (Real.abs_sin_le_one t) (abs_nonneg _))
+      (mul_le_mul_of_nonneg_right (Real.abs_cos_le_one t) (abs_nonneg _))
+    _ = _ := by ring
+
+lemma rotation_derivative_bound {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (a b : ι → κ → ℝ) (t : ℝ) (z : κ → ℝ) :
+    |∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
+        linearProcess (rotationVelocity a b t) z i| ≤
+      ∑ i, ∑ k, (|a i k|+|b i k|)*|z k| := by
+  apply abs_gibbsMean_le
+  intro i
+  calc
+    _ ≤ ∑ k, |rotationVelocity a b t i k*z k| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ k, (|a i k|+|b i k|)*|z k| := by
+      apply Finset.sum_le_sum
+      intro k _
+      rw [abs_mul]
+      exact mul_le_mul_of_nonneg_right (abs_rotationVelocity_le _ _ _ _ _) (abs_nonneg _)
+    _ ≤ _ := Finset.single_le_sum (f := fun j ↦ ∑ k, (|a j k|+|b j k|)*|z k|) (fun j _ ↦ Finset.sum_nonneg (fun k _ ↦ mul_nonneg (add_nonneg (abs_nonneg _) (abs_nonneg _)) (abs_nonneg _))) (Finset.mem_univ i)
+
+lemma hasDerivAt_expected_rotated_logPartition {ι : Type*} [Fintype ι] [Nonempty ι]
+    {n : ℕ} (a b : ι → Fin n → ℝ) (t : ℝ) :
+    HasDerivAt (fun s ↦ ∫ z, logPartition (linearProcess (rotatedCoefficients a b s) z) ∂gaussianProduct (Fin n))
+      (∫ z, ∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
+        linearProcess (rotationVelocity a b t) z i ∂gaussianProduct (Fin n)) t := by
+  have hb : Integrable (fun z : Fin n → ℝ ↦ ∑ i, ∑ k, (|a i k|+|b i k|)*|z k|)
+      (gaussianProduct (Fin n)) :=
+    integrable_finsetSum _ (fun i _ ↦ integrable_finsetSum _ (fun k _ ↦
+      (gaussian_coordinate_integrable k).abs.const_mul _))
+  have hmc : Continuous (fun z : Fin n → ℝ ↦ ∑ i,
+      gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
+      linearProcess (rotationVelocity a b t) z i) := by
+    apply continuous_finsetSum
+    intro i _
+    apply (continuous_linear_gibbsWeight _ _).mul
+    exact (continuous_apply i).comp (continuous_linearProcess _)
+  exact (hasDerivAt_integral_of_dominated_loc_of_deriv_le (s := Set.univ) (μ := gaussianProduct (Fin n))
+    (F := fun s z ↦ logPartition (linearProcess (rotatedCoefficients a b s) z))
+    (F' := fun s z ↦ ∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b s) z) i *
+      linearProcess (rotationVelocity a b s) z i)
+    (bound := fun z ↦ ∑ i, ∑ k, (|a i k|+|b i k|)*|z k|)
+    (Filter.univ_mem)
+    (Eventually.of_forall (fun s ↦ (continuous_logPartition.comp (continuous_linearProcess _)).aestronglyMeasurable))
+    (logPartition_integrable _) hmc.aestronglyMeasurable
+    (Eventually.of_forall (fun z s _ ↦ by simpa only [Real.norm_eq_abs] using rotation_derivative_bound a b s z)) hb
+    (Eventually.of_forall (fun z s _ ↦ hasDerivAt_logPartition (hasDerivAt_rotatedProcess a b s z)))).2
+
+lemma rotation_pair_covariance {ι κ : Type*} [Fintype κ] (a b : ι → κ → ℝ) (t : ℝ) (i j : ι) :
+    (∑ k, (rotationVelocity a b t i k-rotationVelocity a b t j k)*
+      (rotatedCoefficients a b t i k-rotatedCoefficients a b t j k)) =
+    Real.sin t*Real.cos t*((∑ k, (b i k-b j k)^2)-(∑ k, (a i k-a j k)^2)) +
+      ((Real.cos t)^2-(Real.sin t)^2)*∑ k, (a i k-a j k)*(b i k-b j k) := by
+  simp only [rotationVelocity,rotatedCoefficients,← Finset.sum_sub_distrib,Finset.mul_sum,← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro k _
+  ring
+
+lemma expected_logPartition_le_of_pair_distances {ι : Type*} [Fintype ι] [Nonempty ι]
+    {n : ℕ} (a b : ι → Fin n → ℝ)
+    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
+    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
+    (∫ z, logPartition (linearProcess a z) ∂gaussianProduct (Fin n)) ≤
+    ∫ z, logPartition (linearProcess b z) ∂gaussianProduct (Fin n) := by
+  let F := fun t ↦ ∫ z, logPartition (linearProcess (rotatedCoefficients a b t) z) ∂gaussianProduct (Fin n)
+  have hd : ∀ t, HasDerivAt F (∫ z, ∑ i, gibbsWeight (linearProcess (rotatedCoefficients a b t) z) i *
+      linearProcess (rotationVelocity a b t) z i ∂gaussianProduct (Fin n)) t :=
+    hasDerivAt_expected_rotated_logPartition a b
+  have hc : Continuous F := continuous_iff_continuousAt.mpr (fun t ↦ (hd t).continuousAt)
+  have hm : MonotoneOn F (Icc (0 : ℝ) (Real.pi/2)) := by
+    apply monotoneOn_of_deriv_nonneg (convex_Icc _ _) hc.continuousOn
+      (fun t _ ↦ (hd t).differentiableAt.differentiableWithinAt)
+    intro t ht
+    have ht' : t ∈ Icc (0 : ℝ) (Real.pi/2) := interior_subset ht
+    rw [(hd t).deriv]
+    apply gaussian_gibbs_derivative_nonneg
+    intro i j
+    rw [rotation_pair_covariance,horth,mul_zero,add_zero]
+    exact mul_nonneg (mul_nonneg
+      (Real.sin_nonneg_of_nonneg_of_le_pi ht'.1 (ht'.2.trans (by linarith [Real.pi_pos])))
+      (Real.cos_nonneg_of_mem_Icc ⟨by linarith [Real.pi_pos, ht'.1],ht'.2⟩)) (sub_nonneg.mpr (hdist i j))
+  have hπ : (0 : ℝ) ≤ Real.pi/2 := by positivity
+  have hh := hm ⟨le_rfl,hπ⟩ ⟨hπ,le_rfl⟩ hπ
+  have ha0 : rotatedCoefficients a b 0 = a := by funext i k; simp [rotatedCoefficients]
+  have hb0 : rotatedCoefficients a b (Real.pi/2) = b := by funext i k; simp [rotatedCoefficients]
+  simpa only [F,ha0,hb0] using hh
+
+noncomputable def finiteMaximum {ι : Type*} (x : ι → ℝ) : ℝ := sSup (Set.range x)
+
+lemma le_finiteMaximum {ι : Type*} [Fintype ι] (x : ι → ℝ) (i : ι) : x i ≤ finiteMaximum x :=
+  le_csSup (Set.finite_range x).bddAbove (Set.mem_range_self i)
+
+lemma exists_finiteMaximum {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
+    ∃ i, x i = finiteMaximum x := (Set.range_nonempty x).csSup_mem (Set.finite_range x)
+
+lemma continuous_finiteMaximum {ι : Type*} [Fintype ι] [Nonempty ι] :
+    Continuous (@finiteMaximum ι) := by
+  have he (x : ι → ℝ) : finiteMaximum x = Finset.univ.sup' Finset.univ_nonempty x := by
+    rw [Finset.sup'_eq_csSup_image]
+    simp only [Finset.coe_univ,Set.image_univ,finiteMaximum]
+  simp_rw [funext he]
+  exact Continuous.finset_sup'_apply Finset.univ_nonempty (fun i _ ↦ continuous_apply i)
+
+lemma abs_finiteMaximum_le {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
+    |finiteMaximum x| ≤ ∑ i, |x i| := by
+  obtain ⟨i,hi⟩ := exists_finiteMaximum x
+  rw [← hi]
+  exact Finset.single_le_sum (fun j _ ↦ abs_nonneg (x j)) (Finset.mem_univ i)
+
+lemma finiteMaximum_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (c : ι → κ → ℝ) : Integrable (fun z ↦ finiteMaximum (linearProcess c z)) (gaussianProduct κ) := by
+  apply (integrable_finsetSum Finset.univ (fun i _ ↦ (linearProcess_integrable c i).abs)).mono'
+    ((continuous_finiteMaximum.comp (continuous_linearProcess c)).aestronglyMeasurable)
+  exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_def] using abs_finiteMaximum_le (linearProcess c z))
+
+lemma scaled_max_le_logPartition {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ)
+    (β : ℝ) : β*finiteMaximum x ≤ logPartition (fun i ↦ β*x i) := by
+  obtain ⟨i,hi⟩ := exists_finiteMaximum x
+  rw [← hi]
+  exact le_logPartition (fun i ↦ β*x i) i
+
+lemma logPartition_le_scaled_max {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ)
+    {β : ℝ} (hβ : 0 ≤ β) : logPartition (fun i ↦ β*x i) ≤ β*finiteMaximum x+Real.log (Fintype.card ι) := by
+  exact (logPartition_le _ (fun i ↦ mul_le_mul_of_nonneg_left (le_finiteMaximum x i) hβ)).trans_eq (add_comm _ _)
+
+lemma linearProcess_scaled {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) (β : ℝ) (z : κ → ℝ) :
+    linearProcess (fun i k ↦ β*c i k) z = fun i ↦ β*linearProcess c z i := by
+  funext i
+  simp only [linearProcess,Finset.mul_sum,mul_assoc]
+
+lemma expected_max_le_of_pair_distances_orthogonal {ι : Type*} [Fintype ι] [Nonempty ι]
+    {n : ℕ} (a b : ι → Fin n → ℝ)
+    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
+    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
+    (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) ≤
+    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n) := by
+  have hb (β : ℝ) (hβ : 0 ≤ β) :
+      β*(∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) ≤
+      β*(∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n)) + Real.log (Fintype.card ι) := by
+    have hiA := logPartition_integrable (fun i k ↦ β*a i k)
+    have hiB := logPartition_integrable (fun i k ↦ β*b i k)
+    have hcomp := expected_logPartition_le_of_pair_distances
+      (fun i k ↦ β*a i k) (fun i k ↦ β*b i k)
+      (fun i j ↦ by
+        calc
+          (∑ k, (β*a i k-β*a j k)*(β*b i k-β*b j k)) =
+              β^2*(∑ k, (a i k-a j k)*(b i k-b j k)) := by
+            rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro k _; ring
+          _ = 0 := by rw [horth,mul_zero])
+      (fun i j ↦ by
+        have he (c : ι → Fin n → ℝ) : (∑ k, (β*c i k-β*c j k)^2) = β^2*∑ k, (c i k-c j k)^2 := by
+          rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro k _; ring
+        rw [he,he]
+        exact mul_le_mul_of_nonneg_left (hdist i j) (sq_nonneg β))
+    calc
+      _ = ∫ z, β*finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n) := by rw [integral_const_mul]
+      _ ≤ ∫ z, logPartition (linearProcess (fun i k ↦ β*a i k) z) ∂gaussianProduct (Fin n) := by
+        apply integral_mono ((finiteMaximum_integrable a).const_mul _) hiA
+        intro z
+        change β*finiteMaximum (linearProcess a z) ≤ logPartition (linearProcess (fun i k ↦ β*a i k) z)
+        rw [linearProcess_scaled]
+        exact scaled_max_le_logPartition _ _
+      _ ≤ _ := hcomp
+      _ ≤ ∫ z, β*finiteMaximum (linearProcess b z)+Real.log (Fintype.card ι) ∂gaussianProduct (Fin n) := by
+        apply integral_mono hiB (((finiteMaximum_integrable b).const_mul _).add (integrable_const _))
+        intro z
+        change logPartition (linearProcess (fun i k ↦ β*b i k) z) ≤ β*finiteMaximum (linearProcess b z)+Real.log (Fintype.card ι)
+        rw [linearProcess_scaled]
+        exact logPartition_le_scaled_max _ hβ
+      _ = _ := by rw [integral_add ((finiteMaximum_integrable b).const_mul _) (integrable_const _),integral_const_mul,integral_const]; simp
+  by_contra hh
+  have hΔ : 0 < (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) -
+      (∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n)) := sub_pos.mpr (lt_of_not_ge hh)
+  let Δ := (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct (Fin n)) -
+      (∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct (Fin n))
+  let β := (|Real.log (Fintype.card ι)|+1)/Δ
+  have hβ : 0 < β := div_pos (by positivity) hΔ
+  have hh' := hb β hβ.le
+  have he : β*Δ = |Real.log (Fintype.card ι)|+1 := div_mul_cancel₀ _ hΔ.ne'
+  have hlog := le_abs_self (Real.log (Fintype.card ι))
+  dsimp [Δ] at he
+  nlinarith
+
+lemma gaussian_integral_reindex {κ ν : Type*} [Fintype κ] [Fintype ν] (e : κ ≃ ν)
+    (f : (κ → ℝ) → ℝ) :
+    (∫ z : ν → ℝ, f (fun k ↦ z (e k)) ∂gaussianProduct ν) = ∫ z, f z ∂gaussianProduct κ := by
+  have he := measurePreserving_piCongrLeft (α := fun _ : κ ↦ ℝ)
+    (fun _ : κ ↦ standardGaussian) e.symm
+  have hf (z : ν → ℝ) : (MeasurableEquiv.piCongrLeft (π := fun _ : κ ↦ ℝ) e.symm) z = fun k ↦ z (e k) := by
+    funext k
+    simp [MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply]
+  simpa only [gaussianProduct,hf] using he.integral_comp' f
+
+lemma linearProcess_reindex {ι κ ν : Type*} [Fintype κ] [Fintype ν]
+    (c : ι → κ → ℝ) (e : κ ≃ ν) (z : ν → ℝ) :
+    linearProcess c (fun k ↦ z (e k)) = linearProcess (fun i l ↦ c i (e.symm l)) z := by
+  funext i
+  simpa only [linearProcess, Equiv.symm_apply_apply] using e.sum_comp (fun l ↦ c i (e.symm l)*z l)
+
+lemma expected_max_reindex {ι κ ν : Type*} [Fintype ι] [Fintype κ] [Fintype ν]
+    (c : ι → κ → ℝ) (e : κ ≃ ν) :
+    (∫ z, finiteMaximum (linearProcess (fun i l ↦ c i (e.symm l)) z) ∂gaussianProduct ν) =
+    ∫ z, finiteMaximum (linearProcess c z) ∂gaussianProduct κ := by
+  simpa only [linearProcess_reindex] using gaussian_integral_reindex e (fun z ↦ finiteMaximum (linearProcess c z))
+
+lemma expected_max_le_of_pair_distances_orthogonal_fintype {ι κ : Type*}
+    [Fintype ι] [Nonempty ι] [Fintype κ] (a b : ι → κ → ℝ)
+    (horth : ∀ i j, (∑ k, (a i k-a j k)*(b i k-b j k)) = 0)
+    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
+    (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct κ) ≤
+    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct κ := by
+  let e := Fintype.equivFin κ
+  have hh := expected_max_le_of_pair_distances_orthogonal
+    (fun i k ↦ a i (e.symm k)) (fun i k ↦ b i (e.symm k))
+    (fun i j ↦ by
+      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)*(b i k-b j k))]
+      exact horth i j)
+    (fun i j ↦ by
+      rw [e.symm.sum_comp (fun k ↦ (a i k-a j k)^2), e.symm.sum_comp (fun k ↦ (b i k-b j k)^2)]
+      exact hdist i j)
+  simpa only [expected_max_reindex] using hh
+
+noncomputable def embedLeft {ι κ ν : Type*} (a : ι → κ → ℝ) (i : ι) : κ ⊕ ν → ℝ :=
+  Sum.elim (a i) (fun _ ↦ 0)
+noncomputable def embedRight {ι κ ν : Type*} (b : ι → ν → ℝ) (i : ι) : κ ⊕ ν → ℝ :=
+  Sum.elim (fun _ ↦ 0) (b i)
+
+lemma expected_max_embedLeft {ι κ ν : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
+    (a : ι → κ → ℝ) :
+    (∫ z, finiteMaximum (linearProcess (embedLeft (ν := ν) a) z) ∂gaussianProduct (κ ⊕ ν)) =
+    ∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct κ := by
+  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
+  change (∫ z, finiteMaximum (linearProcess (embedLeft a) z) ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
+  rw [← he.integral_comp' (fun z ↦ finiteMaximum (linearProcess (embedLeft a) z))]
+  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
+      linearProcess (embedLeft a) ((MeasurableEquiv.sumPiEquivProdPi (fun _ : κ ⊕ ν ↦ ℝ)).symm p) =
+      linearProcess a p.1 := by
+    funext i
+    simp [linearProcess,embedLeft,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
+  simp only [hlin]
+  simpa [gaussianProduct] using integral_fun_fst (μ := gaussianProduct κ) (ν := gaussianProduct ν) (fun z ↦ finiteMaximum (linearProcess a z))
+
+lemma expected_max_embedRight {ι κ ν : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
+    (b : ι → ν → ℝ) :
+    (∫ z, finiteMaximum (linearProcess (embedRight (κ := κ) b) z) ∂gaussianProduct (κ ⊕ ν)) =
+    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct ν := by
+  have he := measurePreserving_sumPiEquivProdPi_symm (fun _ : κ ⊕ ν ↦ standardGaussian)
+  change (∫ z, finiteMaximum (linearProcess (embedRight b) z) ∂Measure.pi (fun _ : κ ⊕ ν ↦ standardGaussian)) = _
+  rw [← he.integral_comp' (fun z ↦ finiteMaximum (linearProcess (embedRight b) z))]
+  have hlin (p : (κ → ℝ) × (ν → ℝ)) :
+      linearProcess (embedRight b) ((MeasurableEquiv.sumPiEquivProdPi (fun _ : κ ⊕ ν ↦ ℝ)).symm p) =
+      linearProcess b p.2 := by
+    funext i
+    simp [linearProcess,embedRight,Fintype.sum_sum_type,MeasurableEquiv.coe_sumPiEquivProdPi_symm]
+  simp only [hlin]
+  simpa [gaussianProduct] using integral_fun_snd (μ := gaussianProduct κ) (ν := gaussianProduct ν) (fun z ↦ finiteMaximum (linearProcess b z))
+
+lemma expected_max_le_of_pair_distances {ι κ ν : Type*}
+    [Fintype ι] [Nonempty ι] [Fintype κ] [Fintype ν]
+    (a : ι → κ → ℝ) (b : ι → ν → ℝ)
+    (hdist : ∀ i j, (∑ k, (a i k-a j k)^2) ≤ ∑ k, (b i k-b j k)^2) :
+    (∫ z, finiteMaximum (linearProcess a z) ∂gaussianProduct κ) ≤
+    ∫ z, finiteMaximum (linearProcess b z) ∂gaussianProduct ν := by
+  have h := expected_max_le_of_pair_distances_orthogonal_fintype
+    (embedLeft (ν := ν) a) (embedRight (κ := κ) b)
+    (fun i j ↦ by simp [embedLeft,embedRight,Fintype.sum_sum_type])
+    (fun i j ↦ by simpa only [Fintype.sum_sum_type,embedLeft,embedRight,
+      Sum.elim_inl,Sum.elim_inr,sub_self,zero_pow (by decide : 2 ≠ 0),Finset.sum_const_zero,zero_add,add_zero] using hdist i j)
+  simpa only [expected_max_embedLeft,expected_max_embedRight] using h
+
+noncomputable def spin (b : Bool) : ℝ := if b then 1 else -1
+noncomputable def hamiltonian (n : ℕ) (J : (Fin n × Fin n) → ℝ)
+    (σ : Fin n → Bool) : ℝ :=
+  (Real.sqrt (n : ℝ))⁻¹ *
+    ∑ i : Fin n, ∑ j : Fin n, if i < j then J (i,j) * spin (σ i) * spin (σ j) else 0
+noncomputable def groundStateSequence (n : ℕ) : ℝ :=
+  (n : ℝ)⁻¹ * ∫ J, sSup (range (hamiltonian n J)) ∂gaussianProduct (Fin n × Fin n)
+noncomputable def groundStateValue : ℝ := limUnder atTop groundStateSequence
+
+noncomputable def hamCoeff (n : ℕ) (σ : Fin n → Bool) (k : Fin n × Fin n) : ℝ :=
+  if k.1 < k.2 then (Real.sqrt (n : ℝ))⁻¹ * spin (σ k.1) * spin (σ k.2) else 0
+noncomputable def expectedMaximum (n : ℕ) : ℝ :=
+  ∫ J, finiteMaximum (linearProcess (hamCoeff n) J) ∂gaussianProduct (Fin n × Fin n)
+
+lemma spin_sq (b : Bool) : spin b ^ 2 = 1 := by cases b <;> norm_num [spin]
+lemma abs_spin (b : Bool) : |spin b| = 1 := by cases b <;> norm_num [spin]
+
+lemma linearProcess_hamCoeff (n : ℕ) (J : (Fin n × Fin n) → ℝ) :
+    linearProcess (hamCoeff n) J = hamiltonian n J := by
+  funext σ
+  simp only [linearProcess,hamCoeff,hamiltonian,Fintype.sum_prod_type,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  split_ifs <;> ring
+
+lemma groundStateSequence_eq (n : ℕ) : groundStateSequence n = (n : ℝ)⁻¹ * expectedMaximum n := by
+  simp only [groundStateSequence,expectedMaximum,linearProcess_hamCoeff,finiteMaximum]
+
+lemma twice_upper_sum {ι : Type*} [Fintype ι] [LinearOrder ι] (f : ι → ι → ℝ)
+    (hsymm : ∀ i j, f i j = f j i) (hdiag : ∀ i, f i i = 0) :
+    2*(∑ i, ∑ j, if i < j then f i j else 0) = ∑ i, ∑ j, f i j := by
+  have hs : (∑ i, ∑ j, if j < i then f i j else 0) = ∑ i, ∑ j, if i < j then f i j else 0 := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    rw [hsymm j i]
+  have hsplit (i j : ι) : f i j = (if i < j then f i j else 0)+(if j < i then f i j else 0) := by
+    rcases lt_trichotomy i j with h | h | h
+    · simp [h,not_lt_of_ge h.le]
+    · subst j; simp [hdiag]
+    · simp [h,not_lt_of_ge h.le]
+  conv_rhs => arg 2; ext i; arg 2; ext j; rw [hsplit i j]
+  simp only [Finset.sum_add_distrib,hs]
+  ring
+
+lemma spin_pair_difference {dimension : ℕ} (σ τ : Fin dimension → Bool) (i j : Fin dimension) :
+    (spin (σ i)*spin (σ j)-spin (τ i)*spin (τ j))^2 =
+      2-2*(spin (σ i)*spin (τ i))*(spin (σ j)*spin (τ j)) := by
+  have h1 := spin_sq (σ i)
+  have h2 := spin_sq (σ j)
+  have h3 := spin_sq (τ i)
+  have h4 := spin_sq (τ j)
+  calc
+    _ = (spin (σ i))^2*(spin (σ j))^2+(spin (τ i))^2*(spin (τ j))^2 -
+      2*(spin (σ i)*spin (τ i))*(spin (σ j)*spin (τ j)) := by ring
+    _ = _ := by rw [h1,h2,h3,h4]; ring
+
+end SKValueG
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/RoundingSpin.lean b/lean/OAI/Probability/SKValue/GroundState/RoundingSpin.lean
index bf83e17f300a6fca73e8e5488b99b56c05a24966..ff365f78614d027529343c78b8b9cf5969308563 100644
--- a/lean/OAI/Probability/SKValue/GroundState/RoundingSpin.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/RoundingSpin.lean
@@ -1,126 +1,126 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.ConstantFatou
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def boolSign (x : ℝ) : Bool := if 0≤x then true else false
-
-lemma measurable_boolSign : Measurable boolSign :=
-  Measurable.ite measurableSet_Ici measurable_const measurable_const
-
-lemma spin_boolSign_eq {x : ℝ} (hx : x≠0) : SKValueG.spin (boolSign x)=Real.sign x := by
-  rcases lt_or_gt_of_ne hx with h | h
-  · simp [boolSign,SKValueG.spin,not_le.mpr h,Real.sign_of_neg h]
-  · simp [boolSign,SKValueG.spin,h.le,Real.sign_of_pos h]
-
-lemma gaussian_rounding_nonzero (a : ℝ) :
-    ∀ᵐ z ∂standardGaussian,a+2*cdf standardGaussian z-1≠0 := by
-  have hh : ∀ᵐ z ∂standardGaussian,cdf standardGaussian z≠(1-a)/2 :=
-    (standardGaussian_cdf_hasLaw_uniform.ae_iff (measurableSet_setOfPred.mp ((measurableSet_eq_fun measurable_id measurable_const).compl))).mpr
-      ((volume.restrict (Ioc (0 : ℝ) 1)).ae_ne ((1-a)/2))
-  filter_upwards [hh] with z hz
-  intro h
-  apply hz
-  linarith
-
-lemma independent_rounding_nonzero {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    [IsProbabilityMeasure μ] {U Z : Ω → ℝ}
-    (hU : Measurable U) (hZ : HasLaw Z standardGaussian μ) (hind : IndepFun U Z μ) :
-    ∀ᵐ ω ∂μ,U ω+2*cdf standardGaussian (Z ω)-1≠0 := by
-  have hLaw : HasLaw U (μ.map U) μ := ⟨hU.aemeasurable,rfl⟩
-  have hJoint := hind.hasLaw_prod hLaw hZ
-  have hm : Measurable (fun p : ℝ×ℝ ↦ p.1+2*cdf standardGaussian p.2-1) :=
-    (measurable_fst.add (((monotone_cdf _).measurable.comp measurable_snd).const_mul 2)).sub_const 1
-  apply (hJoint.ae_iff (measurableSet_setOfPred.mp ((measurableSet_eq_fun hm measurable_const).compl))).mpr
-  apply (Measure.ae_prod_iff_ae_ae ((measurableSet_eq_fun hm measurable_const).compl)).mpr
-  exact Eventually.of_forall gaussian_rounding_nonzero
-
-lemma finite_rounding_signed_spin {N : ℕ} {U : (Fin (N+1) → ℝ) → ℝ}
-    (hU : Measurable U) (hUp : DependsBefore N U) :
-    (fun z ↦ SKValueG.spin (boolSign (U z+2*cdf standardGaussian (coordinate N N z)-1)))
-      =ᵐ[gaussianProduct (Fin (N+1))]
-    (fun z ↦ Real.sign (U z+2*cdf standardGaussian (coordinate N N z)-1)) := by
-  have := gaussianProduct_probability (Fin (N+1))
-  exact (independent_rounding_nonzero hU (coordinate_hasLaw N N)
-    (hUp.independent (by omega) hU)).mono (fun _ ↦ spin_boolSign_eq)
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-theorem normalized_shifted_sum_le_groundState {N : ℕ}
-    (H : Fin N → (Fin (N+1) → ℝ) → ℝ) (U : (Fin (N+1) → ℝ) → ℝ)
-    (hm : ∀ j,Measurable (H j)) (hp : ∀ j : Fin N,DependsBefore j (H j))
-    (hL : ∀ j,MemLp (H j) 2 (gaussianProduct (Fin (N+1))))
-    (hpos : ∀ j,0<∫ z,(H j z)^2 ∂gaussianProduct (Fin (N+1)))
-    (hU : Measurable U) (hUp : DependsBefore N U) :
-    (∑ j : Fin N,(∫ z,finiteRounded U z*
-      normalizedIncrement (H j) (coordinate N j) (gaussianProduct (Fin (N+1))) z
-        ∂gaussianProduct (Fin (N+1)))*
-      (∫ z,finiteRounded U z*coordinate N (j+1) z ∂gaussianProduct (Fin (N+1))))≤groundStateValue := by
-  let f := fun j : ℕ ↦ if hj : j<N then
-    normalizedIncrement (H ⟨j,hj⟩) (coordinate N j) (gaussianProduct (Fin (N+1))) else fun _ ↦ 0
-  let F := fun z ↦ boolSign (U z+2*cdf standardGaussian (coordinate N N z)-1)
-  have hfm (j : ℕ) : Measurable (f j) := by
-    dsimp only [f]
-    split_ifs with hj
-    · exact measurable_normalizedIncrement (hm ⟨j,hj⟩)
-    · exact measurable_const
-  have hfa (l : ℕ) (x y : Fin (N+1) → ℝ)
-      (he : ∀ i : Fin (N+1),i.val≤l → x i=y i) : f l x=f l y := by
-    dsimp only [f]
-    split_ifs with hl
-    · apply normalizedIncrement_dependsBefore (by omega) (hp ⟨l,hl⟩)
-      intro i hi
-      change i.val<l+1 at hi
-      exact he i (by omega)
-    · rfl
-  have hfL (j : ℕ) (hj : j<N) : MemLp (f j) 2 (SKValueG.gaussianProduct (Fin (N+1))) := by
-    simp only [f,dite_eq_left hj]
-    exact normalizedIncrement_memLp_two (hL ⟨j,hj⟩) (coordinate_hasLaw N j)
-      ((hp ⟨j,hj⟩).independent (by omega) (hm ⟨j,hj⟩))
-  have hfu (j : ℕ) (hj : j<N) : (∫ z,(f j z)^2 ∂SKValueG.gaussianProduct (Fin (N+1)))=1 := by
-    simp only [f,dite_eq_left hj]
-    exact normalizedIncrement_second_moment (hm ⟨j,hj⟩).aestronglyMeasurable (coordinate_hasLaw N j)
-      ((hp ⟨j,hj⟩).independent (by omega) (hm ⟨j,hj⟩)) (hpos ⟨j,hj⟩)
-  have hfo (i j : ℕ) (hij : i<j) (hj : j<N) :
-      (∫ z,f i z*f j z ∂SKValueG.gaussianProduct (Fin (N+1)))=0 := by
-    have hi : i<N := hij.trans hj
-    simp only [f,dite_eq_left hi,dite_eq_left hj]
-    exact normalizedIncrements_orthogonal hij (by omega) (hm ⟨i,hi⟩) (hm ⟨j,hj⟩)
-      (hp ⟨i,hi⟩) (hp ⟨j,hj⟩)
-  have hFm : Measurable F := measurable_boolSign.comp
-    ((hU.add (((monotone_cdf _).measurable.comp (measurable_coordinate N N)).const_mul 2)).sub_const 1)
-  have he : (fun z ↦ SKValueG.spin (F z))=ᵐ[gaussianProduct (Fin (N+1))] finiteRounded U :=
-    finite_rounding_signed_spin hU hUp
-  have hb := SKValueG.predictable_coefficients_le_groundState N f F hfm hfa hfL hfu hfo hFm
-  change _≤SKValueG.groundStateValue
-  convert hb using 1
-  apply Finset.sum_congr rfl
-  intro j hj
-  congr 1
-  · apply integral_congr_ae
-    filter_upwards [he] with z hz
-    simp only [f,dite_eq_left j.isLt,hz]
-    exact mul_comm _ _
-  · apply integral_congr_ae
-    filter_upwards [he] with z hz
-    rw [hz,coordinate_eq_apply (by omega)]
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.ConstantFatou
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def boolSign (x : ℝ) : Bool := if 0≤x then true else false
+
+lemma measurable_boolSign : Measurable boolSign :=
+  Measurable.ite measurableSet_Ici measurable_const measurable_const
+
+lemma spin_boolSign_eq {x : ℝ} (hx : x≠0) : SKValueG.spin (boolSign x)=Real.sign x := by
+  rcases lt_or_gt_of_ne hx with h | h
+  · simp [boolSign,SKValueG.spin,not_le.mpr h,Real.sign_of_neg h]
+  · simp [boolSign,SKValueG.spin,h.le,Real.sign_of_pos h]
+
+lemma gaussian_rounding_nonzero (a : ℝ) :
+    ∀ᵐ z ∂standardGaussian,a+2*cdf standardGaussian z-1≠0 := by
+  have hh : ∀ᵐ z ∂standardGaussian,cdf standardGaussian z≠(1-a)/2 :=
+    (standardGaussian_cdf_hasLaw_uniform.ae_iff (measurableSet_setOfPred.mp ((measurableSet_eq_fun measurable_id measurable_const).compl))).mpr
+      ((volume.restrict (Ioc (0 : ℝ) 1)).ae_ne ((1-a)/2))
+  filter_upwards [hh] with z hz
+  intro h
+  apply hz
+  linarith
+
+lemma independent_rounding_nonzero {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    [IsProbabilityMeasure μ] {U Z : Ω → ℝ}
+    (hU : Measurable U) (hZ : HasLaw Z standardGaussian μ) (hind : IndepFun U Z μ) :
+    ∀ᵐ ω ∂μ,U ω+2*cdf standardGaussian (Z ω)-1≠0 := by
+  have hLaw : HasLaw U (μ.map U) μ := ⟨hU.aemeasurable,rfl⟩
+  have hJoint := hind.hasLaw_prod hLaw hZ
+  have hm : Measurable (fun p : ℝ×ℝ ↦ p.1+2*cdf standardGaussian p.2-1) :=
+    (measurable_fst.add (((monotone_cdf _).measurable.comp measurable_snd).const_mul 2)).sub_const 1
+  apply (hJoint.ae_iff (measurableSet_setOfPred.mp ((measurableSet_eq_fun hm measurable_const).compl))).mpr
+  apply (Measure.ae_prod_iff_ae_ae ((measurableSet_eq_fun hm measurable_const).compl)).mpr
+  exact Eventually.of_forall gaussian_rounding_nonzero
+
+lemma finite_rounding_signed_spin {N : ℕ} {U : (Fin (N+1) → ℝ) → ℝ}
+    (hU : Measurable U) (hUp : DependsBefore N U) :
+    (fun z ↦ SKValueG.spin (boolSign (U z+2*cdf standardGaussian (coordinate N N z)-1)))
+      =ᵐ[gaussianProduct (Fin (N+1))]
+    (fun z ↦ Real.sign (U z+2*cdf standardGaussian (coordinate N N z)-1)) := by
+  have := gaussianProduct_probability (Fin (N+1))
+  exact (independent_rounding_nonzero hU (coordinate_hasLaw N N)
+    (hUp.independent (by omega) hU)).mono (fun _ ↦ spin_boolSign_eq)
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+theorem normalized_shifted_sum_le_groundState {N : ℕ}
+    (H : Fin N → (Fin (N+1) → ℝ) → ℝ) (U : (Fin (N+1) → ℝ) → ℝ)
+    (hm : ∀ j,Measurable (H j)) (hp : ∀ j : Fin N,DependsBefore j (H j))
+    (hL : ∀ j,MemLp (H j) 2 (gaussianProduct (Fin (N+1))))
+    (hpos : ∀ j,0<∫ z,(H j z)^2 ∂gaussianProduct (Fin (N+1)))
+    (hU : Measurable U) (hUp : DependsBefore N U) :
+    (∑ j : Fin N,(∫ z,finiteRounded U z*
+      normalizedIncrement (H j) (coordinate N j) (gaussianProduct (Fin (N+1))) z
+        ∂gaussianProduct (Fin (N+1)))*
+      (∫ z,finiteRounded U z*coordinate N (j+1) z ∂gaussianProduct (Fin (N+1))))≤groundStateValue := by
+  let f := fun j : ℕ ↦ if hj : j<N then
+    normalizedIncrement (H ⟨j,hj⟩) (coordinate N j) (gaussianProduct (Fin (N+1))) else fun _ ↦ 0
+  let F := fun z ↦ boolSign (U z+2*cdf standardGaussian (coordinate N N z)-1)
+  have hfm (j : ℕ) : Measurable (f j) := by
+    dsimp only [f]
+    split_ifs with hj
+    · exact measurable_normalizedIncrement (hm ⟨j,hj⟩)
+    · exact measurable_const
+  have hfa (l : ℕ) (x y : Fin (N+1) → ℝ)
+      (he : ∀ i : Fin (N+1),i.val≤l → x i=y i) : f l x=f l y := by
+    dsimp only [f]
+    split_ifs with hl
+    · apply normalizedIncrement_dependsBefore (by omega) (hp ⟨l,hl⟩)
+      intro i hi
+      change i.val<l+1 at hi
+      exact he i (by omega)
+    · rfl
+  have hfL (j : ℕ) (hj : j<N) : MemLp (f j) 2 (SKValueG.gaussianProduct (Fin (N+1))) := by
+    simp only [f,dite_eq_left hj]
+    exact normalizedIncrement_memLp_two (hL ⟨j,hj⟩) (coordinate_hasLaw N j)
+      ((hp ⟨j,hj⟩).independent (by omega) (hm ⟨j,hj⟩))
+  have hfu (j : ℕ) (hj : j<N) : (∫ z,(f j z)^2 ∂SKValueG.gaussianProduct (Fin (N+1)))=1 := by
+    simp only [f,dite_eq_left hj]
+    exact normalizedIncrement_second_moment (hm ⟨j,hj⟩).aestronglyMeasurable (coordinate_hasLaw N j)
+      ((hp ⟨j,hj⟩).independent (by omega) (hm ⟨j,hj⟩)) (hpos ⟨j,hj⟩)
+  have hfo (i j : ℕ) (hij : i<j) (hj : j<N) :
+      (∫ z,f i z*f j z ∂SKValueG.gaussianProduct (Fin (N+1)))=0 := by
+    have hi : i<N := hij.trans hj
+    simp only [f,dite_eq_left hi,dite_eq_left hj]
+    exact normalizedIncrements_orthogonal hij (by omega) (hm ⟨i,hi⟩) (hm ⟨j,hj⟩)
+      (hp ⟨i,hi⟩) (hp ⟨j,hj⟩)
+  have hFm : Measurable F := measurable_boolSign.comp
+    ((hU.add (((monotone_cdf _).measurable.comp (measurable_coordinate N N)).const_mul 2)).sub_const 1)
+  have he : (fun z ↦ SKValueG.spin (F z))=ᵐ[gaussianProduct (Fin (N+1))] finiteRounded U :=
+    finite_rounding_signed_spin hU hUp
+  have hb := SKValueG.predictable_coefficients_le_groundState N f F hfm hfa hfL hfu hfo hFm
+  change _≤SKValueG.groundStateValue
+  convert hb using 1
+  apply Finset.sum_congr rfl
+  intro j hj
+  congr 1
+  · apply integral_congr_ae
+    filter_upwards [he] with z hz
+    simp only [f,dite_eq_left j.isLt,hz]
+    exact mul_comm _ _
+  · apply integral_congr_ae
+    filter_upwards [he] with z hz
+    rw [hz,coordinate_eq_apply (by omega)]
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/StandardGaussian.lean b/lean/OAI/Probability/SKValue/GroundState/StandardGaussian.lean
index 0249749dcea94abf598b5239635e911b86783577..b935607b96478dcbada6765c1fd5687edb1e6571 100644
--- a/lean/OAI/Probability/SKValue/GroundState/StandardGaussian.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/StandardGaussian.lean
@@ -1,384 +1,384 @@
-import Mathlib
-import OAI.Probability.SKValue.Evolution.BoundedSmooth
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-noncomputable def standardGaussian : Measure ℝ := gaussianReal 0 1
-noncomputable def gaussianProduct (ι : Type*) [Fintype ι] : Measure (ι → ℝ) :=
-  Measure.pi (fun _ : ι ↦ standardGaussian)
-
-instance : IsProbabilityMeasure standardGaussian := by unfold standardGaussian; infer_instance
-instance {ι : Type*} [Fintype ι] : IsProbabilityMeasure (gaussianProduct ι) := by
-  unfold gaussianProduct; infer_instance
-
-lemma density_hasDerivAt (x : ℝ) :
-    HasDerivAt (gaussianPDFReal 0 1) (-x*gaussianPDFReal 0 1 x) x := by
-  have h1 : HasDerivAt (fun y : ℝ ↦ -y^2/2) (-x) x := by
-    have hh := ((hasDerivAt_id x).pow 2).neg.div_const (2 : ℝ)
-    simpa only [Pi.pow_def,Pi.neg_def,id_eq,pow_one,show (2 : ℕ)-1=1 from rfl,
-      Nat.cast_ofNat,mul_one,neg_div,mul_div_cancel_left₀ _ (by norm_num : (2 : ℝ) ≠ 0)] using hh
-  have hd := h1.exp.const_mul ((Real.sqrt (2*Real.pi))⁻¹)
-  convert hd using 1 <;> try rfl
-  · funext y
-    simp only [gaussianPDFReal,NNReal.coe_one,mul_one,sub_zero]
-  · simp only [gaussianPDFReal,NNReal.coe_one,mul_one,sub_zero]
-    ring
-
-lemma density_mul_integrable {f : ℝ → ℝ} (hf : Integrable f standardGaussian) :
-    Integrable (fun x ↦ gaussianPDFReal 0 1 x*f x) := by
-  rw [standardGaussian,gaussianReal_of_var_ne_zero _ (by norm_num)] at hf
-  have hh := (integrable_withDensity_iff_integrable_smul'
-    (measurable_gaussianPDF 0 1) (Eventually.of_forall (fun x ↦ gaussianPDF_lt_top (μ := 0) (v := 1) (x := x)))).mp hf
-  simpa only [toReal_gaussianPDF,smul_eq_mul] using hh
-
-lemma standardGaussian_integrable_id : Integrable (fun x : ℝ ↦ x) standardGaussian := by
-  have hh := (memLp_id_gaussianReal' (μ := (0 : ℝ)) (v := (1 : ℝ≥0)) (1 : ℝ≥0∞) (by norm_num))
-  exact hh.integrable (by norm_num)
-
-lemma gaussian_stein_bounded {f f' : ℝ → ℝ} {C D : ℝ}
-    (hf : ∀ x, HasDerivAt f (f' x) x) (hm : AEStronglyMeasurable f' standardGaussian)
-    (hC : ∀ x, |f x|≤C) (hD : ∀ x, |f' x|≤D) :
-    (∫ x, x*f x ∂standardGaussian) = ∫ x, f' x ∂standardGaussian := by
-  have hfm : Continuous f := continuous_iff_continuousAt.mpr (fun x ↦ (hf x).continuousAt)
-  have hfi : Integrable f standardGaussian := Integrable.of_bound hfm.aestronglyMeasurable C
-    (Eventually.of_forall (fun x ↦ (Real.norm_eq_abs _).trans_le (hC x)))
-  have hf'i : Integrable f' standardGaussian := Integrable.of_bound hm D
-    (Eventually.of_forall (fun x ↦ (Real.norm_eq_abs _).trans_le (hD x)))
-  have hxfi : Integrable (fun x ↦ x*f x) standardGaussian :=
-    standardGaussian_integrable_id.mul_bdd hfm.aestronglyMeasurable
-      (Eventually.of_forall (fun x ↦ (Real.norm_eq_abs _).trans_le (hC x)))
-  have hi1 : Integrable (fun x ↦ f x*(-x*gaussianPDFReal 0 1 x)) := by
-    convert (density_mul_integrable hxfi).neg using 1
-    ext x
-    simp only [Pi.neg_apply]
-    ring
-  have hi2 : Integrable (fun x ↦ f' x*gaussianPDFReal 0 1 x) := by
-    simpa only [mul_comm] using density_mul_integrable hf'i
-  have hi3 : Integrable (fun x ↦ f x*gaussianPDFReal 0 1 x) := by
-    simpa only [mul_comm] using density_mul_integrable hfi
-  have hibp := integral_mul_deriv_eq_deriv_mul_of_integrable
-    (u := f) (v := gaussianPDFReal 0 1) (u' := f') (v' := fun x ↦ -x*gaussianPDFReal 0 1 x)
-    (fun x _ ↦ hf x) (fun x _ ↦ density_hasDerivAt x) hi1 hi2 hi3
-  have hb : (∫ x, f x*(-x*gaussianPDFReal 0 1 x)) =
-      -(∫ x, gaussianPDFReal 0 1 x*(x*f x)) := by
-    rw [← integral_neg]
-    apply integral_congr_ae
-    exact Eventually.of_forall (fun x ↦ by ring)
-  rw [hb] at hibp
-  have he : (∫ x, gaussianPDFReal 0 1 x*(x*f x)) =
-      ∫ x, gaussianPDFReal 0 1 x*f' x := by
-    exact neg_injective hibp |>.trans (by congr 1; funext x; ring)
-  simpa only [standardGaussian,integral_gaussianReal_eq_integral_smul (by norm_num : (1 : ℝ≥0) ≠ 0),
-    smul_eq_mul] using he
-
-lemma gaussian_coordinate_integrable {ι : Type*} [Fintype ι] (i : ι) :
-    Integrable (fun z : ι → ℝ ↦ z i) (gaussianProduct ι) := by
-  exact ((measurePreserving_eval (fun _ : ι ↦ standardGaussian) i).integrable_comp
-    standardGaussian_integrable_id.aestronglyMeasurable).mpr standardGaussian_integrable_id
-
-lemma gaussian_pi_stein_bounded {n : ℕ} (i : Fin n) {f f' : (Fin n → ℝ) → ℝ} {C D : ℝ}
-    (hf : Continuous f) (hf' : Continuous f')
-    (hd : ∀ z, HasDerivAt (fun x ↦ f (Function.update z i x)) (f' z) (z i))
-    (hC : ∀ z, |f z| ≤ C) (hD : ∀ z, |f' z| ≤ D) :
-    (∫ z, z i*f z ∂gaussianProduct (Fin n)) = ∫ z, f' z ∂gaussianProduct (Fin n) := by
-  cases n with
-  | zero => exact Fin.elim0 i
-  | succ n =>
-    let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) ↦ ℝ) i
-    have he : MeasurePreserving e.symm (standardGaussian.prod (gaussianProduct (Fin n)))
-        (gaussianProduct (Fin (n+1))) :=
-      (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ standardGaussian) i).symm
-    have hi : Integrable (fun z : Fin (n+1) → ℝ ↦ z i*f z) (gaussianProduct (Fin (n+1))) :=
-      (gaussian_coordinate_integrable i).mul_bdd hf.aestronglyMeasurable
-        (Eventually.of_forall (fun z ↦ (Real.norm_eq_abs _).trans_le (hC z)))
-    have hi' : Integrable f' (gaussianProduct (Fin (n+1))) :=
-      Integrable.of_bound hf'.aestronglyMeasurable D
-        (Eventually.of_forall (fun z ↦ (Real.norm_eq_abs _).trans_le (hD z)))
-    have hip : Integrable (fun p ↦ e.symm p i * f (e.symm p))
-        (standardGaussian.prod (gaussianProduct (Fin n))) := (he.integrable_comp hi.aestronglyMeasurable).mpr hi
-    have hi'p : Integrable (fun p ↦ f' (e.symm p))
-        (standardGaussian.prod (gaussianProduct (Fin n))) := (he.integrable_comp hi'.aestronglyMeasurable).mpr hi'
-    rw [← he.integral_comp' (fun z ↦ z i*f z), ← he.integral_comp' f',
-      integral_prod _ hip, integral_prod _ hi'p,
-      integral_integral_swap (f := fun x y ↦ e.symm (x,y) i * f (e.symm (x,y))) hip,
-      integral_integral_swap (f := fun x y ↦ f' (e.symm (x,y))) hi'p]
-    apply integral_congr_ae
-    apply Eventually.of_forall
-    intro y
-    simp only [e, MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv,Equiv.coe_fn_mk,Fin.insertNth_apply_same]
-    change (∫ x, x*f (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y) ∂standardGaussian) =
-      ∫ x, f' (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y) ∂standardGaussian
-    have hmeas : Measurable (fun x : ℝ ↦ i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y) := by
-      exact e.symm.measurable.comp (measurable_id.prodMk measurable_const)
-    apply gaussian_stein_bounded (f' := fun x ↦ f' (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y))
-    · intro x
-      simpa only [Fin.update_insertNth,Fin.insertNth_apply_same] using hd (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y)
-    · exact (hf'.measurable.comp hmeas).aestronglyMeasurable
-    · exact fun x ↦ hC _
-    · exact fun x ↦ hD _
-
-noncomputable def partition {ι : Type*} [Fintype ι] (x : ι → ℝ) : ℝ :=
-  ∑ i, Real.exp (x i)
-
-noncomputable def gibbsWeight {ι : Type*} [Fintype ι] (x : ι → ℝ) (i : ι) : ℝ :=
-  Real.exp (x i) / partition x
-
-noncomputable def logPartition {ι : Type*} [Fintype ι] (x : ι → ℝ) : ℝ :=
-  Real.log (partition x)
-
-lemma partition_pos {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) : 0 < partition x := by
-  exact Finset.sum_pos (fun i _ ↦ Real.exp_pos _) Finset.univ_nonempty
-
-lemma gibbsWeight_nonneg {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
-    0 ≤ gibbsWeight x i := le_of_lt (div_pos (Real.exp_pos _) (partition_pos x))
-
-lemma sum_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
-    ∑ i, gibbsWeight x i = 1 := by
-  simp only [gibbsWeight,← Finset.sum_div]
-  change partition x / partition x = 1
-  exact div_self (ne_of_gt (partition_pos x))
-
-lemma gibbsWeight_le_one {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
-    gibbsWeight x i ≤ 1 := by
-  calc
-    _ ≤ ∑ j, gibbsWeight x j := Finset.single_le_sum (fun j _ ↦ gibbsWeight_nonneg x j) (Finset.mem_univ i)
-    _ = 1 := sum_gibbsWeight x
-
-lemma continuous_partition {ι : Type*} [Fintype ι] : Continuous (@partition ι _) := by
-  unfold partition
-  fun_prop
-
-lemma continuous_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι] (i : ι) :
-    Continuous (fun x : ι → ℝ ↦ gibbsWeight x i) := by
-  exact (Real.continuous_exp.comp (continuous_apply i)).div continuous_partition (fun x ↦ ne_of_gt (partition_pos x))
-
-lemma hasDerivAt_partition {ι : Type*} [Fintype ι] {x : ℝ → ι → ℝ} {v : ι → ℝ} {t : ℝ}
-    (h : ∀ i, HasDerivAt (fun s ↦ x s i) (v i) t) :
-    HasDerivAt (fun s ↦ partition (x s)) (∑ i, Real.exp (x t i)*v i) t := by
-  simpa only [partition,Finset.sum_apply] using (HasDerivAt.fun_sum (u := Finset.univ) (fun i _ ↦ (h i).exp))
-
-lemma hasDerivAt_logPartition {ι : Type*} [Fintype ι] [Nonempty ι]
-    {x : ℝ → ι → ℝ} {v : ι → ℝ} {t : ℝ}
-    (h : ∀ i, HasDerivAt (fun s ↦ x s i) (v i) t) :
-    HasDerivAt (fun s ↦ logPartition (x s)) (∑ i, gibbsWeight (x t) i*v i) t := by
-  convert (hasDerivAt_partition h).log (ne_of_gt (partition_pos _)) using 1 <;> try rfl
-  simp only [gibbsWeight,Finset.sum_div,mul_div_assoc]
-  apply Finset.sum_congr rfl
-  intro i _
-  ring
-
-lemma hasDerivAt_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι]
-    {x : ℝ → ι → ℝ} {v : ι → ℝ} {t : ℝ}
-    (h : ∀ i, HasDerivAt (fun s ↦ x s i) (v i) t) (i : ι) :
-    HasDerivAt (fun s ↦ gibbsWeight (x s) i)
-      (gibbsWeight (x t) i*(v i-∑ j, gibbsWeight (x t) j*v j)) t := by
-  convert (h i).exp.div (hasDerivAt_partition h) (ne_of_gt (partition_pos _)) using 1 <;> try rfl
-  simp only [gibbsWeight]
-  rw [show (∑ j, Real.exp (x t j)/partition (x t)*v j) =
-      (∑ j, Real.exp (x t j)*v j)/partition (x t) by
-    rw [Finset.sum_div]; apply Finset.sum_congr rfl; intro j _; ring]
-  field_simp [ne_of_gt (partition_pos (x t))]
-
-lemma abs_gibbsMean_le {ι : Type*} [Fintype ι] [Nonempty ι] (x v : ι → ℝ) {C : ℝ}
-    (hC : ∀ i, |v i| ≤ C) : |∑ i, gibbsWeight x i*v i| ≤ C := by
-  calc
-    _ ≤ ∑ i, |gibbsWeight x i*v i| := Finset.abs_sum_le_sum_abs _ _
-    _ = ∑ i, gibbsWeight x i*|v i| := by simp only [abs_mul,abs_of_nonneg (gibbsWeight_nonneg _ _)]
-    _ ≤ ∑ i, gibbsWeight x i*C := Finset.sum_le_sum (fun i _ ↦ mul_le_mul_of_nonneg_left (hC i) (gibbsWeight_nonneg _ _))
-    _ = C := by rw [← Finset.sum_mul,sum_gibbsWeight,one_mul]
-
-noncomputable def linearProcess {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) (z : κ → ℝ) (i : ι) : ℝ :=
-  ∑ k, c i k*z k
-
-lemma continuous_linearProcess {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) :
-    Continuous (linearProcess c) := by
-  unfold linearProcess
-  fun_prop
-
-lemma hasDerivAt_linearProcess_update {ι κ : Type*} [Fintype κ] [DecidableEq κ]
-    (c : ι → κ → ℝ) (z : κ → ℝ) (i : ι) (k : κ) :
-    HasDerivAt (fun x ↦ linearProcess c (Function.update z k x) i) (c i k) (z k) := by
-  have h : ∀ j, HasDerivAt (fun x ↦ c i j * Function.update z k x j)
-      (if j=k then c i k else 0) (z k) := by
-    intro j
-    by_cases hj : j=k
-    · subst j
-      simpa only [Function.update_self,ite_true,mul_one,id_eq] using (hasDerivAt_id (z k)).const_mul (c i k)
-    · simpa only [Function.update_of_ne hj,ite_eq_right hj] using hasDerivAt_const (z k) (c i j*z j)
-  simpa only [linearProcess,Finset.sum_apply,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
-    using HasDerivAt.fun_sum (u := Finset.univ) (fun j _ ↦ h j)
-
-lemma continuous_linear_gibbsWeight {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (c : ι → κ → ℝ) (i : ι) : Continuous (fun z ↦ gibbsWeight (linearProcess c z) i) :=
-  (continuous_gibbsWeight i).comp (continuous_linearProcess c)
-
-lemma integrable_linear_gibbsWeight {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (c : ι → κ → ℝ) (i : ι) : Integrable (fun z ↦ gibbsWeight (linearProcess c z) i) (gaussianProduct κ) := by
-  apply Integrable.of_bound (continuous_linear_gibbsWeight c i).aestronglyMeasurable 1
-  exact Eventually.of_forall (fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (gibbsWeight_nonneg _ _)]; exact gibbsWeight_le_one _ _)
-
-lemma coordinate_gibbsWeight_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (c : ι → κ → ℝ) (i : ι) (k : κ) :
-    Integrable (fun z ↦ z k*gibbsWeight (linearProcess c z) i) (gaussianProduct κ) := by
-  apply (gaussian_coordinate_integrable k).mul_bdd (continuous_linear_gibbsWeight c i).aestronglyMeasurable
-  exact Eventually.of_forall (fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (gibbsWeight_nonneg _ _)]; exact gibbsWeight_le_one _ _)
-
-lemma gaussian_coordinate_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι]
-    {n : ℕ} (c : ι → Fin n → ℝ) (i : ι) (k : Fin n) :
-    (∫ z, z k*gibbsWeight (linearProcess c z) i ∂gaussianProduct (Fin n)) =
-    ∫ z, gibbsWeight (linearProcess c z) i *
-      (c i k - ∑ j, gibbsWeight (linearProcess c z) j*c j k) ∂gaussianProduct (Fin n) := by
-  apply gaussian_pi_stein_bounded k (C := 1) (D := 2*∑ j, |c j k|)
-  · exact continuous_linear_gibbsWeight c i
-  · apply Continuous.mul (continuous_linear_gibbsWeight c i)
-    apply continuous_const.sub
-    exact continuous_finsetSum _ (fun j _ ↦ (continuous_linear_gibbsWeight c j).mul continuous_const)
-  · intro z
-    simpa only [Function.update_eq_self] using
-      hasDerivAt_gibbsWeight (fun j ↦ hasDerivAt_linearProcess_update c z j k) i
-  · intro z
-    rw [abs_of_nonneg (gibbsWeight_nonneg _ _)]
-    exact gibbsWeight_le_one _ _
-  · intro z
-    have hsum : ∀ j, |c j k| ≤ ∑ l, |c l k| := fun j ↦
-      Finset.single_le_sum (f := fun l ↦ |c l k|) (fun l _ ↦ abs_nonneg _) (Finset.mem_univ j)
-    calc
-      _ = gibbsWeight (linearProcess c z) i * |c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k| := by
-        rw [abs_mul,abs_of_nonneg (gibbsWeight_nonneg _ _)]
-      _ ≤ |c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k| :=
-        mul_le_of_le_one_left (abs_nonneg _) (gibbsWeight_le_one _ _)
-      _ ≤ |c i k| + |∑ j, gibbsWeight (linearProcess c z) j*c j k| := abs_sub _ _
-      _ ≤ (∑ j, |c j k|) + (∑ j, |c j k|) := add_le_add (hsum i) (abs_gibbsMean_le _ _ hsum)
-      _ = _ := by ring
-
-lemma le_logPartition {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
-    x i ≤ logPartition x := by
-  apply (Real.le_log_iff_exp_le (partition_pos x)).mpr
-  exact Finset.single_le_sum (fun j _ ↦ (Real.exp_pos _).le) (Finset.mem_univ i)
-
-lemma logPartition_le {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) {C : ℝ}
-    (h : ∀ i, x i ≤ C) : logPartition x ≤ Real.log (Fintype.card ι) + C := by
-  have hc : (0 : ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos
-  calc
-    logPartition x ≤ Real.log ((Fintype.card ι : ℝ)*Real.exp C) := by
-      apply Real.log_le_log (partition_pos x)
-      calc
-        partition x ≤ ∑ _ : ι, Real.exp C := Finset.sum_le_sum (fun i _ ↦ Real.exp_le_exp.mpr (h i))
-        _ = _ := by simp
-    _ = _ := by rw [Real.log_mul (ne_of_gt hc) (Real.exp_ne_zero _),Real.log_exp]
-
-lemma abs_logPartition_le {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
-    |logPartition x| ≤ |Real.log (Fintype.card ι)| + ∑ i, |x i| := by
-  have hsum : ∀ i, |x i| ≤ ∑ j, |x j| := fun i ↦
-    Finset.single_le_sum (f := fun j ↦ |x j|) (fun j _ ↦ abs_nonneg _) (Finset.mem_univ i)
-  have hu := logPartition_le x (fun i ↦ (le_abs_self _).trans (hsum i))
-  obtain ⟨i⟩ := ‹Nonempty ι›
-  have hl := le_logPartition x i
-  have hx := neg_abs_le (x i)
-  have hxi := hsum i
-  rw [abs_le]
-  constructor <;> linarith [le_abs_self (Real.log (Fintype.card ι)),abs_nonneg (Real.log (Fintype.card ι))]
-
-lemma continuous_logPartition {ι : Type*} [Fintype ι] [Nonempty ι] :
-    Continuous (@logPartition ι _) :=
-  continuous_partition.log (fun x ↦ ne_of_gt (partition_pos x))
-
-lemma linearProcess_integrable {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) (i : ι) :
-    Integrable (fun z ↦ linearProcess c z i) (gaussianProduct κ) :=
-  integrable_finsetSum _ (fun k _ ↦ (gaussian_coordinate_integrable k).const_mul (c i k))
-
-lemma logPartition_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (c : ι → κ → ℝ) : Integrable (fun z ↦ logPartition (linearProcess c z)) (gaussianProduct κ) := by
-  have hb := (integrable_const (|Real.log (Fintype.card ι)| : ℝ)).add
-    (integrable_finsetSum Finset.univ (fun i _ ↦ (linearProcess_integrable c i).abs))
-  apply hb.mono' ((continuous_logPartition.comp (continuous_linearProcess c)).aestronglyMeasurable)
-  exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_def,Pi.add_apply] using abs_logPartition_le (linearProcess c z))
-
-lemma gibbsWeight_mul_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
-    (c : ι → κ → ℝ) (i : ι) {f : (κ → ℝ) → ℝ} (hi : Integrable f (gaussianProduct κ)) :
-    Integrable (fun z ↦ gibbsWeight (linearProcess c z) i*f z) (gaussianProduct κ) := by
-  simpa only [mul_comm] using hi.mul_bdd (continuous_linear_gibbsWeight c i).aestronglyMeasurable
-    (Eventually.of_forall (fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (gibbsWeight_nonneg _ _)]; exact gibbsWeight_le_one _ _))
-
-lemma gaussian_linear_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι]
-    {n : ℕ} (c d : ι → Fin n → ℝ) (i : ι) :
-    (∫ z, gibbsWeight (linearProcess c z) i * linearProcess d z i ∂gaussianProduct (Fin n)) =
-    ∫ z, gibbsWeight (linearProcess c z) i *
-      ((∑ k, d i k*c i k) - ∑ j, gibbsWeight (linearProcess c z) j*(∑ k, d i k*c j k))
-        ∂gaussianProduct (Fin n) := by
-  have hid : ∀ k, Integrable (fun z ↦ d i k * (z k*gibbsWeight (linearProcess c z) i))
-      (gaussianProduct (Fin n)) := fun k ↦ (coordinate_gibbsWeight_integrable c i k).const_mul _
-  have hic : ∀ k, Integrable (fun z ↦ d i k * (gibbsWeight (linearProcess c z) i *
-      (c i k - ∑ j, gibbsWeight (linearProcess c z) j*c j k))) (gaussianProduct (Fin n)) := by
-    intro k
-    apply Integrable.const_mul
-    apply gibbsWeight_mul_integrable
-    exact (integrable_const _).sub (integrable_finsetSum Finset.univ
-      (fun j _ ↦ (integrable_linear_gibbsWeight c j).mul_const _))
-  calc
-    _ = ∫ z, ∑ k, d i k*(z k*gibbsWeight (linearProcess c z) i) ∂gaussianProduct (Fin n) := by
-      apply integral_congr_ae
-      exact Eventually.of_forall (fun z ↦ by simp only [linearProcess,Finset.mul_sum]; apply Finset.sum_congr rfl; intro k _; ring)
-    _ = ∑ k, d i k * (∫ z, z k*gibbsWeight (linearProcess c z) i ∂gaussianProduct (Fin n)) := by
-      rw [integral_finsetSum Finset.univ (fun k _ ↦ hid k)]
-      simp only [integral_const_mul]
-    _ = ∑ k, d i k * (∫ z, gibbsWeight (linearProcess c z) i *
-        (c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k) ∂gaussianProduct (Fin n)) := by
-      simp only [gaussian_coordinate_gibbsWeight]
-    _ = ∫ z, ∑ k, d i k * (gibbsWeight (linearProcess c z) i *
-        (c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k)) ∂gaussianProduct (Fin n) := by
-      rw [integral_finsetSum Finset.univ (fun k _ ↦ hic k)]
-      simp only [integral_const_mul]
-    _ = _ := by
-      apply integral_congr_ae
-      apply Eventually.of_forall
-      intro z
-      simp only [mul_sub,Finset.sum_sub_distrib,Finset.mul_sum]
-      congr 1
-      · apply Finset.sum_congr rfl; intro k _; ring
-      · rw [Finset.sum_comm]
-        apply Finset.sum_congr rfl; intro j _
-        apply Finset.sum_congr rfl; intro k _; ring
-
-lemma weighted_covariance_pairs {ι : Type*} [Fintype ι] (p : ι → ℝ) (D : ι → ι → ℝ)
-    (hp : ∑ i, p i = 1) :
-    ∑ i, p i*(D i i-∑ j, p j*D i j) =
-    (1/2 : ℝ)*∑ i, ∑ j, p i*p j*(D i i+D j j-D i j-D j i) := by
-  have h1 : (∑ i, ∑ j, p i*p j*D i i) = ∑ i, p i*D i i := by
-    apply Finset.sum_congr rfl
-    intro i _
-    calc
-      _ = p i*D i i*∑ j, p j := by rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j _; ring
-      _ = _ := by rw [hp,mul_one]
-  have h2 : (∑ i, ∑ j, p i*p j*D j j) = ∑ j, p j*D j j := by
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl
-    intro j _
-    calc
-      _ = (∑ i, p i)*(p j*D j j) := by rw [Finset.sum_mul]; apply Finset.sum_congr rfl; intro i _; ring
-      _ = _ := by rw [hp,one_mul]
-  have h3 : (∑ i, ∑ j, p i*p j*D j i) = ∑ i, ∑ j, p i*p j*D i j := by
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl; intro i _
-    apply Finset.sum_congr rfl; intro j _; ring
-  have hl : (∑ i, p i*(D i i-∑ j, p j*D i j)) =
-      (∑ i, p i*D i i) - ∑ i, ∑ j, p i*p j*D i j := by
-    simp only [mul_sub,Finset.sum_sub_distrib,Finset.mul_sum,mul_assoc]
-  have hr : (∑ i, ∑ j, p i*p j*(D i i+D j j-D i j-D j i)) =
-      2*(∑ i, p i*D i i)-2*(∑ i, ∑ j, p i*p j*D i j) := by
-    simp only [mul_sub,mul_add,Finset.sum_sub_distrib,Finset.sum_add_distrib]
-    rw [h1,h2,h3]
-    ring
-  rw [hl,hr]
-  ring
-
-end SKValueG
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Evolution.BoundedSmooth
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+noncomputable def standardGaussian : Measure ℝ := gaussianReal 0 1
+noncomputable def gaussianProduct (ι : Type*) [Fintype ι] : Measure (ι → ℝ) :=
+  Measure.pi (fun _ : ι ↦ standardGaussian)
+
+instance : IsProbabilityMeasure standardGaussian := by unfold standardGaussian; infer_instance
+instance {ι : Type*} [Fintype ι] : IsProbabilityMeasure (gaussianProduct ι) := by
+  unfold gaussianProduct; infer_instance
+
+lemma density_hasDerivAt (x : ℝ) :
+    HasDerivAt (gaussianPDFReal 0 1) (-x*gaussianPDFReal 0 1 x) x := by
+  have h1 : HasDerivAt (fun y : ℝ ↦ -y^2/2) (-x) x := by
+    have hh := ((hasDerivAt_id x).pow 2).neg.div_const (2 : ℝ)
+    simpa only [Pi.pow_def,Pi.neg_def,id_eq,pow_one,show (2 : ℕ)-1=1 from rfl,
+      Nat.cast_ofNat,mul_one,neg_div,mul_div_cancel_left₀ _ (by norm_num : (2 : ℝ) ≠ 0)] using hh
+  have hd := h1.exp.const_mul ((Real.sqrt (2*Real.pi))⁻¹)
+  convert hd using 1 <;> try rfl
+  · funext y
+    simp only [gaussianPDFReal,NNReal.coe_one,mul_one,sub_zero]
+  · simp only [gaussianPDFReal,NNReal.coe_one,mul_one,sub_zero]
+    ring
+
+lemma density_mul_integrable {f : ℝ → ℝ} (hf : Integrable f standardGaussian) :
+    Integrable (fun x ↦ gaussianPDFReal 0 1 x*f x) := by
+  rw [standardGaussian,gaussianReal_of_var_ne_zero _ (by norm_num)] at hf
+  have hh := (integrable_withDensity_iff_integrable_smul'
+    (measurable_gaussianPDF 0 1) (Eventually.of_forall (fun x ↦ gaussianPDF_lt_top (μ := 0) (v := 1) (x := x)))).mp hf
+  simpa only [toReal_gaussianPDF,smul_eq_mul] using hh
+
+lemma standardGaussian_integrable_id : Integrable (fun x : ℝ ↦ x) standardGaussian := by
+  have hh := (memLp_id_gaussianReal' (μ := (0 : ℝ)) (v := (1 : ℝ≥0)) (1 : ℝ≥0∞) (by norm_num))
+  exact hh.integrable (by norm_num)
+
+lemma gaussian_stein_bounded {f f' : ℝ → ℝ} {C D : ℝ}
+    (hf : ∀ x, HasDerivAt f (f' x) x) (hm : AEStronglyMeasurable f' standardGaussian)
+    (hC : ∀ x, |f x|≤C) (hD : ∀ x, |f' x|≤D) :
+    (∫ x, x*f x ∂standardGaussian) = ∫ x, f' x ∂standardGaussian := by
+  have hfm : Continuous f := continuous_iff_continuousAt.mpr (fun x ↦ (hf x).continuousAt)
+  have hfi : Integrable f standardGaussian := Integrable.of_bound hfm.aestronglyMeasurable C
+    (Eventually.of_forall (fun x ↦ (Real.norm_eq_abs _).trans_le (hC x)))
+  have hf'i : Integrable f' standardGaussian := Integrable.of_bound hm D
+    (Eventually.of_forall (fun x ↦ (Real.norm_eq_abs _).trans_le (hD x)))
+  have hxfi : Integrable (fun x ↦ x*f x) standardGaussian :=
+    standardGaussian_integrable_id.mul_bdd hfm.aestronglyMeasurable
+      (Eventually.of_forall (fun x ↦ (Real.norm_eq_abs _).trans_le (hC x)))
+  have hi1 : Integrable (fun x ↦ f x*(-x*gaussianPDFReal 0 1 x)) := by
+    convert (density_mul_integrable hxfi).neg using 1
+    ext x
+    simp only [Pi.neg_apply]
+    ring
+  have hi2 : Integrable (fun x ↦ f' x*gaussianPDFReal 0 1 x) := by
+    simpa only [mul_comm] using density_mul_integrable hf'i
+  have hi3 : Integrable (fun x ↦ f x*gaussianPDFReal 0 1 x) := by
+    simpa only [mul_comm] using density_mul_integrable hfi
+  have hibp := integral_mul_deriv_eq_deriv_mul_of_integrable
+    (u := f) (v := gaussianPDFReal 0 1) (u' := f') (v' := fun x ↦ -x*gaussianPDFReal 0 1 x)
+    (fun x _ ↦ hf x) (fun x _ ↦ density_hasDerivAt x) hi1 hi2 hi3
+  have hb : (∫ x, f x*(-x*gaussianPDFReal 0 1 x)) =
+      -(∫ x, gaussianPDFReal 0 1 x*(x*f x)) := by
+    rw [← integral_neg]
+    apply integral_congr_ae
+    exact Eventually.of_forall (fun x ↦ by ring)
+  rw [hb] at hibp
+  have he : (∫ x, gaussianPDFReal 0 1 x*(x*f x)) =
+      ∫ x, gaussianPDFReal 0 1 x*f' x := by
+    exact neg_injective hibp |>.trans (by congr 1; funext x; ring)
+  simpa only [standardGaussian,integral_gaussianReal_eq_integral_smul (by norm_num : (1 : ℝ≥0) ≠ 0),
+    smul_eq_mul] using he
+
+lemma gaussian_coordinate_integrable {ι : Type*} [Fintype ι] (i : ι) :
+    Integrable (fun z : ι → ℝ ↦ z i) (gaussianProduct ι) := by
+  exact ((measurePreserving_eval (fun _ : ι ↦ standardGaussian) i).integrable_comp
+    standardGaussian_integrable_id.aestronglyMeasurable).mpr standardGaussian_integrable_id
+
+lemma gaussian_pi_stein_bounded {n : ℕ} (i : Fin n) {f f' : (Fin n → ℝ) → ℝ} {C D : ℝ}
+    (hf : Continuous f) (hf' : Continuous f')
+    (hd : ∀ z, HasDerivAt (fun x ↦ f (Function.update z i x)) (f' z) (z i))
+    (hC : ∀ z, |f z| ≤ C) (hD : ∀ z, |f' z| ≤ D) :
+    (∫ z, z i*f z ∂gaussianProduct (Fin n)) = ∫ z, f' z ∂gaussianProduct (Fin n) := by
+  cases n with
+  | zero => exact Fin.elim0 i
+  | succ n =>
+    let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) ↦ ℝ) i
+    have he : MeasurePreserving e.symm (standardGaussian.prod (gaussianProduct (Fin n)))
+        (gaussianProduct (Fin (n+1))) :=
+      (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ standardGaussian) i).symm
+    have hi : Integrable (fun z : Fin (n+1) → ℝ ↦ z i*f z) (gaussianProduct (Fin (n+1))) :=
+      (gaussian_coordinate_integrable i).mul_bdd hf.aestronglyMeasurable
+        (Eventually.of_forall (fun z ↦ (Real.norm_eq_abs _).trans_le (hC z)))
+    have hi' : Integrable f' (gaussianProduct (Fin (n+1))) :=
+      Integrable.of_bound hf'.aestronglyMeasurable D
+        (Eventually.of_forall (fun z ↦ (Real.norm_eq_abs _).trans_le (hD z)))
+    have hip : Integrable (fun p ↦ e.symm p i * f (e.symm p))
+        (standardGaussian.prod (gaussianProduct (Fin n))) := (he.integrable_comp hi.aestronglyMeasurable).mpr hi
+    have hi'p : Integrable (fun p ↦ f' (e.symm p))
+        (standardGaussian.prod (gaussianProduct (Fin n))) := (he.integrable_comp hi'.aestronglyMeasurable).mpr hi'
+    rw [← he.integral_comp' (fun z ↦ z i*f z), ← he.integral_comp' f',
+      integral_prod _ hip, integral_prod _ hi'p,
+      integral_integral_swap (f := fun x y ↦ e.symm (x,y) i * f (e.symm (x,y))) hip,
+      integral_integral_swap (f := fun x y ↦ f' (e.symm (x,y))) hi'p]
+    apply integral_congr_ae
+    apply Eventually.of_forall
+    intro y
+    simp only [e, MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv,Equiv.coe_fn_mk,Fin.insertNth_apply_same]
+    change (∫ x, x*f (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y) ∂standardGaussian) =
+      ∫ x, f' (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y) ∂standardGaussian
+    have hmeas : Measurable (fun x : ℝ ↦ i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y) := by
+      exact e.symm.measurable.comp (measurable_id.prodMk measurable_const)
+    apply gaussian_stein_bounded (f' := fun x ↦ f' (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y))
+    · intro x
+      simpa only [Fin.update_insertNth,Fin.insertNth_apply_same] using hd (i.insertNth (α := fun _ : Fin (n+1) ↦ ℝ) x y)
+    · exact (hf'.measurable.comp hmeas).aestronglyMeasurable
+    · exact fun x ↦ hC _
+    · exact fun x ↦ hD _
+
+noncomputable def partition {ι : Type*} [Fintype ι] (x : ι → ℝ) : ℝ :=
+  ∑ i, Real.exp (x i)
+
+noncomputable def gibbsWeight {ι : Type*} [Fintype ι] (x : ι → ℝ) (i : ι) : ℝ :=
+  Real.exp (x i) / partition x
+
+noncomputable def logPartition {ι : Type*} [Fintype ι] (x : ι → ℝ) : ℝ :=
+  Real.log (partition x)
+
+lemma partition_pos {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) : 0 < partition x := by
+  exact Finset.sum_pos (fun i _ ↦ Real.exp_pos _) Finset.univ_nonempty
+
+lemma gibbsWeight_nonneg {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
+    0 ≤ gibbsWeight x i := le_of_lt (div_pos (Real.exp_pos _) (partition_pos x))
+
+lemma sum_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
+    ∑ i, gibbsWeight x i = 1 := by
+  simp only [gibbsWeight,← Finset.sum_div]
+  change partition x / partition x = 1
+  exact div_self (ne_of_gt (partition_pos x))
+
+lemma gibbsWeight_le_one {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
+    gibbsWeight x i ≤ 1 := by
+  calc
+    _ ≤ ∑ j, gibbsWeight x j := Finset.single_le_sum (fun j _ ↦ gibbsWeight_nonneg x j) (Finset.mem_univ i)
+    _ = 1 := sum_gibbsWeight x
+
+lemma continuous_partition {ι : Type*} [Fintype ι] : Continuous (@partition ι _) := by
+  unfold partition
+  fun_prop
+
+lemma continuous_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι] (i : ι) :
+    Continuous (fun x : ι → ℝ ↦ gibbsWeight x i) := by
+  exact (Real.continuous_exp.comp (continuous_apply i)).div continuous_partition (fun x ↦ ne_of_gt (partition_pos x))
+
+lemma hasDerivAt_partition {ι : Type*} [Fintype ι] {x : ℝ → ι → ℝ} {v : ι → ℝ} {t : ℝ}
+    (h : ∀ i, HasDerivAt (fun s ↦ x s i) (v i) t) :
+    HasDerivAt (fun s ↦ partition (x s)) (∑ i, Real.exp (x t i)*v i) t := by
+  simpa only [partition,Finset.sum_apply] using (HasDerivAt.fun_sum (u := Finset.univ) (fun i _ ↦ (h i).exp))
+
+lemma hasDerivAt_logPartition {ι : Type*} [Fintype ι] [Nonempty ι]
+    {x : ℝ → ι → ℝ} {v : ι → ℝ} {t : ℝ}
+    (h : ∀ i, HasDerivAt (fun s ↦ x s i) (v i) t) :
+    HasDerivAt (fun s ↦ logPartition (x s)) (∑ i, gibbsWeight (x t) i*v i) t := by
+  convert (hasDerivAt_partition h).log (ne_of_gt (partition_pos _)) using 1 <;> try rfl
+  simp only [gibbsWeight,Finset.sum_div,mul_div_assoc]
+  apply Finset.sum_congr rfl
+  intro i _
+  ring
+
+lemma hasDerivAt_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι]
+    {x : ℝ → ι → ℝ} {v : ι → ℝ} {t : ℝ}
+    (h : ∀ i, HasDerivAt (fun s ↦ x s i) (v i) t) (i : ι) :
+    HasDerivAt (fun s ↦ gibbsWeight (x s) i)
+      (gibbsWeight (x t) i*(v i-∑ j, gibbsWeight (x t) j*v j)) t := by
+  convert (h i).exp.div (hasDerivAt_partition h) (ne_of_gt (partition_pos _)) using 1 <;> try rfl
+  simp only [gibbsWeight]
+  rw [show (∑ j, Real.exp (x t j)/partition (x t)*v j) =
+      (∑ j, Real.exp (x t j)*v j)/partition (x t) by
+    rw [Finset.sum_div]; apply Finset.sum_congr rfl; intro j _; ring]
+  field_simp [ne_of_gt (partition_pos (x t))]
+
+lemma abs_gibbsMean_le {ι : Type*} [Fintype ι] [Nonempty ι] (x v : ι → ℝ) {C : ℝ}
+    (hC : ∀ i, |v i| ≤ C) : |∑ i, gibbsWeight x i*v i| ≤ C := by
+  calc
+    _ ≤ ∑ i, |gibbsWeight x i*v i| := Finset.abs_sum_le_sum_abs _ _
+    _ = ∑ i, gibbsWeight x i*|v i| := by simp only [abs_mul,abs_of_nonneg (gibbsWeight_nonneg _ _)]
+    _ ≤ ∑ i, gibbsWeight x i*C := Finset.sum_le_sum (fun i _ ↦ mul_le_mul_of_nonneg_left (hC i) (gibbsWeight_nonneg _ _))
+    _ = C := by rw [← Finset.sum_mul,sum_gibbsWeight,one_mul]
+
+noncomputable def linearProcess {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) (z : κ → ℝ) (i : ι) : ℝ :=
+  ∑ k, c i k*z k
+
+lemma continuous_linearProcess {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) :
+    Continuous (linearProcess c) := by
+  unfold linearProcess
+  fun_prop
+
+lemma hasDerivAt_linearProcess_update {ι κ : Type*} [Fintype κ] [DecidableEq κ]
+    (c : ι → κ → ℝ) (z : κ → ℝ) (i : ι) (k : κ) :
+    HasDerivAt (fun x ↦ linearProcess c (Function.update z k x) i) (c i k) (z k) := by
+  have h : ∀ j, HasDerivAt (fun x ↦ c i j * Function.update z k x j)
+      (if j=k then c i k else 0) (z k) := by
+    intro j
+    by_cases hj : j=k
+    · subst j
+      simpa only [Function.update_self,ite_true,mul_one,id_eq] using (hasDerivAt_id (z k)).const_mul (c i k)
+    · simpa only [Function.update_of_ne hj,ite_eq_right hj] using hasDerivAt_const (z k) (c i j*z j)
+  simpa only [linearProcess,Finset.sum_apply,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+    using HasDerivAt.fun_sum (u := Finset.univ) (fun j _ ↦ h j)
+
+lemma continuous_linear_gibbsWeight {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (c : ι → κ → ℝ) (i : ι) : Continuous (fun z ↦ gibbsWeight (linearProcess c z) i) :=
+  (continuous_gibbsWeight i).comp (continuous_linearProcess c)
+
+lemma integrable_linear_gibbsWeight {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (c : ι → κ → ℝ) (i : ι) : Integrable (fun z ↦ gibbsWeight (linearProcess c z) i) (gaussianProduct κ) := by
+  apply Integrable.of_bound (continuous_linear_gibbsWeight c i).aestronglyMeasurable 1
+  exact Eventually.of_forall (fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (gibbsWeight_nonneg _ _)]; exact gibbsWeight_le_one _ _)
+
+lemma coordinate_gibbsWeight_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (c : ι → κ → ℝ) (i : ι) (k : κ) :
+    Integrable (fun z ↦ z k*gibbsWeight (linearProcess c z) i) (gaussianProduct κ) := by
+  apply (gaussian_coordinate_integrable k).mul_bdd (continuous_linear_gibbsWeight c i).aestronglyMeasurable
+  exact Eventually.of_forall (fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (gibbsWeight_nonneg _ _)]; exact gibbsWeight_le_one _ _)
+
+lemma gaussian_coordinate_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι]
+    {n : ℕ} (c : ι → Fin n → ℝ) (i : ι) (k : Fin n) :
+    (∫ z, z k*gibbsWeight (linearProcess c z) i ∂gaussianProduct (Fin n)) =
+    ∫ z, gibbsWeight (linearProcess c z) i *
+      (c i k - ∑ j, gibbsWeight (linearProcess c z) j*c j k) ∂gaussianProduct (Fin n) := by
+  apply gaussian_pi_stein_bounded k (C := 1) (D := 2*∑ j, |c j k|)
+  · exact continuous_linear_gibbsWeight c i
+  · apply Continuous.mul (continuous_linear_gibbsWeight c i)
+    apply continuous_const.sub
+    exact continuous_finsetSum _ (fun j _ ↦ (continuous_linear_gibbsWeight c j).mul continuous_const)
+  · intro z
+    simpa only [Function.update_eq_self] using
+      hasDerivAt_gibbsWeight (fun j ↦ hasDerivAt_linearProcess_update c z j k) i
+  · intro z
+    rw [abs_of_nonneg (gibbsWeight_nonneg _ _)]
+    exact gibbsWeight_le_one _ _
+  · intro z
+    have hsum : ∀ j, |c j k| ≤ ∑ l, |c l k| := fun j ↦
+      Finset.single_le_sum (f := fun l ↦ |c l k|) (fun l _ ↦ abs_nonneg _) (Finset.mem_univ j)
+    calc
+      _ = gibbsWeight (linearProcess c z) i * |c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k| := by
+        rw [abs_mul,abs_of_nonneg (gibbsWeight_nonneg _ _)]
+      _ ≤ |c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k| :=
+        mul_le_of_le_one_left (abs_nonneg _) (gibbsWeight_le_one _ _)
+      _ ≤ |c i k| + |∑ j, gibbsWeight (linearProcess c z) j*c j k| := abs_sub _ _
+      _ ≤ (∑ j, |c j k|) + (∑ j, |c j k|) := add_le_add (hsum i) (abs_gibbsMean_le _ _ hsum)
+      _ = _ := by ring
+
+lemma le_logPartition {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
+    x i ≤ logPartition x := by
+  apply (Real.le_log_iff_exp_le (partition_pos x)).mpr
+  exact Finset.single_le_sum (fun j _ ↦ (Real.exp_pos _).le) (Finset.mem_univ i)
+
+lemma logPartition_le {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) {C : ℝ}
+    (h : ∀ i, x i ≤ C) : logPartition x ≤ Real.log (Fintype.card ι) + C := by
+  have hc : (0 : ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos
+  calc
+    logPartition x ≤ Real.log ((Fintype.card ι : ℝ)*Real.exp C) := by
+      apply Real.log_le_log (partition_pos x)
+      calc
+        partition x ≤ ∑ _ : ι, Real.exp C := Finset.sum_le_sum (fun i _ ↦ Real.exp_le_exp.mpr (h i))
+        _ = _ := by simp
+    _ = _ := by rw [Real.log_mul (ne_of_gt hc) (Real.exp_ne_zero _),Real.log_exp]
+
+lemma abs_logPartition_le {ι : Type*} [Fintype ι] [Nonempty ι] (x : ι → ℝ) :
+    |logPartition x| ≤ |Real.log (Fintype.card ι)| + ∑ i, |x i| := by
+  have hsum : ∀ i, |x i| ≤ ∑ j, |x j| := fun i ↦
+    Finset.single_le_sum (f := fun j ↦ |x j|) (fun j _ ↦ abs_nonneg _) (Finset.mem_univ i)
+  have hu := logPartition_le x (fun i ↦ (le_abs_self _).trans (hsum i))
+  obtain ⟨i⟩ := ‹Nonempty ι›
+  have hl := le_logPartition x i
+  have hx := neg_abs_le (x i)
+  have hxi := hsum i
+  rw [abs_le]
+  constructor <;> linarith [le_abs_self (Real.log (Fintype.card ι)),abs_nonneg (Real.log (Fintype.card ι))]
+
+lemma continuous_logPartition {ι : Type*} [Fintype ι] [Nonempty ι] :
+    Continuous (@logPartition ι _) :=
+  continuous_partition.log (fun x ↦ ne_of_gt (partition_pos x))
+
+lemma linearProcess_integrable {ι κ : Type*} [Fintype κ] (c : ι → κ → ℝ) (i : ι) :
+    Integrable (fun z ↦ linearProcess c z i) (gaussianProduct κ) :=
+  integrable_finsetSum _ (fun k _ ↦ (gaussian_coordinate_integrable k).const_mul (c i k))
+
+lemma logPartition_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (c : ι → κ → ℝ) : Integrable (fun z ↦ logPartition (linearProcess c z)) (gaussianProduct κ) := by
+  have hb := (integrable_const (|Real.log (Fintype.card ι)| : ℝ)).add
+    (integrable_finsetSum Finset.univ (fun i _ ↦ (linearProcess_integrable c i).abs))
+  apply hb.mono' ((continuous_logPartition.comp (continuous_linearProcess c)).aestronglyMeasurable)
+  exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_def,Pi.add_apply] using abs_logPartition_le (linearProcess c z))
+
+lemma gibbsWeight_mul_integrable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ]
+    (c : ι → κ → ℝ) (i : ι) {f : (κ → ℝ) → ℝ} (hi : Integrable f (gaussianProduct κ)) :
+    Integrable (fun z ↦ gibbsWeight (linearProcess c z) i*f z) (gaussianProduct κ) := by
+  simpa only [mul_comm] using hi.mul_bdd (continuous_linear_gibbsWeight c i).aestronglyMeasurable
+    (Eventually.of_forall (fun z ↦ by rw [Real.norm_eq_abs,abs_of_nonneg (gibbsWeight_nonneg _ _)]; exact gibbsWeight_le_one _ _))
+
+lemma gaussian_linear_gibbsWeight {ι : Type*} [Fintype ι] [Nonempty ι]
+    {n : ℕ} (c d : ι → Fin n → ℝ) (i : ι) :
+    (∫ z, gibbsWeight (linearProcess c z) i * linearProcess d z i ∂gaussianProduct (Fin n)) =
+    ∫ z, gibbsWeight (linearProcess c z) i *
+      ((∑ k, d i k*c i k) - ∑ j, gibbsWeight (linearProcess c z) j*(∑ k, d i k*c j k))
+        ∂gaussianProduct (Fin n) := by
+  have hid : ∀ k, Integrable (fun z ↦ d i k * (z k*gibbsWeight (linearProcess c z) i))
+      (gaussianProduct (Fin n)) := fun k ↦ (coordinate_gibbsWeight_integrable c i k).const_mul _
+  have hic : ∀ k, Integrable (fun z ↦ d i k * (gibbsWeight (linearProcess c z) i *
+      (c i k - ∑ j, gibbsWeight (linearProcess c z) j*c j k))) (gaussianProduct (Fin n)) := by
+    intro k
+    apply Integrable.const_mul
+    apply gibbsWeight_mul_integrable
+    exact (integrable_const _).sub (integrable_finsetSum Finset.univ
+      (fun j _ ↦ (integrable_linear_gibbsWeight c j).mul_const _))
+  calc
+    _ = ∫ z, ∑ k, d i k*(z k*gibbsWeight (linearProcess c z) i) ∂gaussianProduct (Fin n) := by
+      apply integral_congr_ae
+      exact Eventually.of_forall (fun z ↦ by simp only [linearProcess,Finset.mul_sum]; apply Finset.sum_congr rfl; intro k _; ring)
+    _ = ∑ k, d i k * (∫ z, z k*gibbsWeight (linearProcess c z) i ∂gaussianProduct (Fin n)) := by
+      rw [integral_finsetSum Finset.univ (fun k _ ↦ hid k)]
+      simp only [integral_const_mul]
+    _ = ∑ k, d i k * (∫ z, gibbsWeight (linearProcess c z) i *
+        (c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k) ∂gaussianProduct (Fin n)) := by
+      simp only [gaussian_coordinate_gibbsWeight]
+    _ = ∫ z, ∑ k, d i k * (gibbsWeight (linearProcess c z) i *
+        (c i k-∑ j, gibbsWeight (linearProcess c z) j*c j k)) ∂gaussianProduct (Fin n) := by
+      rw [integral_finsetSum Finset.univ (fun k _ ↦ hic k)]
+      simp only [integral_const_mul]
+    _ = _ := by
+      apply integral_congr_ae
+      apply Eventually.of_forall
+      intro z
+      simp only [mul_sub,Finset.sum_sub_distrib,Finset.mul_sum]
+      congr 1
+      · apply Finset.sum_congr rfl; intro k _; ring
+      · rw [Finset.sum_comm]
+        apply Finset.sum_congr rfl; intro j _
+        apply Finset.sum_congr rfl; intro k _; ring
+
+lemma weighted_covariance_pairs {ι : Type*} [Fintype ι] (p : ι → ℝ) (D : ι → ι → ℝ)
+    (hp : ∑ i, p i = 1) :
+    ∑ i, p i*(D i i-∑ j, p j*D i j) =
+    (1/2 : ℝ)*∑ i, ∑ j, p i*p j*(D i i+D j j-D i j-D j i) := by
+  have h1 : (∑ i, ∑ j, p i*p j*D i i) = ∑ i, p i*D i i := by
+    apply Finset.sum_congr rfl
+    intro i _
+    calc
+      _ = p i*D i i*∑ j, p j := by rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j _; ring
+      _ = _ := by rw [hp,mul_one]
+  have h2 : (∑ i, ∑ j, p i*p j*D j j) = ∑ j, p j*D j j := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro j _
+    calc
+      _ = (∑ i, p i)*(p j*D j j) := by rw [Finset.sum_mul]; apply Finset.sum_congr rfl; intro i _; ring
+      _ = _ := by rw [hp,one_mul]
+  have h3 : (∑ i, ∑ j, p i*p j*D j i) = ∑ i, ∑ j, p i*p j*D i j := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl; intro i _
+    apply Finset.sum_congr rfl; intro j _; ring
+  have hl : (∑ i, p i*(D i i-∑ j, p j*D i j)) =
+      (∑ i, p i*D i i) - ∑ i, ∑ j, p i*p j*D i j := by
+    simp only [mul_sub,Finset.sum_sub_distrib,Finset.mul_sum,mul_assoc]
+  have hr : (∑ i, ∑ j, p i*p j*(D i i+D j j-D i j-D j i)) =
+      2*(∑ i, p i*D i i)-2*(∑ i, ∑ j, p i*p j*D i j) := by
+    simp only [mul_sub,mul_add,Finset.sum_sub_distrib,Finset.sum_add_distrib]
+    rw [h1,h2,h3]
+    ring
+  rw [hl,hr]
+  ring
+
+end SKValueG
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/TreeGuerra.lean b/lean/OAI/Probability/SKValue/GroundState/TreeGuerra.lean
index 537b13bf1c493b3710a9e6ad6630c329178108de..e895210217e73c154d2aeae255495d9e5716fbf1 100644
--- a/lean/OAI/Probability/SKValue/GroundState/TreeGuerra.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/TreeGuerra.lean
@@ -1,161 +1,161 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.LiteralLower
-import OAI.Probability.SKValue.GroundState.GaussianTree
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma treeAux_tensor_distance (n : ℕ) (T : GaussianTree) (hT : ValidTreeGrid 0 T)
-    (α β : TreeLeaves T) :
-    (∑ e,(leafAuxCoeff n (treeFieldVector 0 T) α e-leafAuxCoeff n (treeFieldVector 0 T) β e)^2)=
-      ∑ e,(treeAuxVector n 0 T α e-treeAuxVector n 0 T β e)^2 := by
-  rw [sq_distance_sum,leafAux_sq n _ (treeField_unit T hT),
-    leafAux_sq n _ (treeField_unit T hT),leafAux_inner,treeField_inner 0 T hT,
-    treeAux_distance n T hT]
-  ring
-
-lemma expected_treeAux_eq_tensor (n : ℕ) (T : GaussianTree) (hT : ValidTreeGrid 0 T)
-    (d : TreeLeaves T → ℝ) :
-    (∫ z,finiteMaximum (fun α ↦ linearProcess (treeAuxVector n 0 T) z α+d α)
-      ∂gaussianProduct (TreeEdges T))=
-    ∫ z,finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n (treeFieldVector 0 T)) z α+d α)
-      ∂gaussianProduct (TreeEdges T × TreeEdges T) := by
-  apply le_antisymm
-  · exact expected_affine_max_le _ _ d (fun α β ↦ (treeAux_tensor_distance n T hT α β).symm.le)
-  · exact expected_affine_max_le _ _ d (fun α β ↦ (treeAux_tensor_distance n T hT α β).le)
-
-theorem finite_tree_guerra_upper {n : ℕ} (hn : 0 < n)
-    (T : GaussianTree) (hT : ValidTreeGrid 0 T) (d : TreeLeaves T → ℝ) :
-    expectedMaximum n+
-      (∫ z,finiteMaximum (fun α ↦ linearProcess (treeAuxVector n 0 T) z α+d α)
-        ∂gaussianProduct (TreeEdges T))≤
-      ∫ z,finiteMaximum (fun α ↦ (∑ i,|leafField (treeFieldVector 0 T) z α i|)+d α)
-        ∂gaussianProduct (Fin n × TreeEdges T) := by
-  rw [expected_treeAux_eq_tensor n T hT d]
-  exact finite_guerra_upper hn (treeFieldVector 0 T) (treeField_unit T hT) d
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma finiteMaximum_prod_curry_left {ι κ : Type*} [Fintype ι] [Nonempty ι]
-    [Fintype κ] [Nonempty κ] (f : ι×κ → ℝ) :
-    finiteMaximum f=finiteMaximum (fun i ↦ finiteMaximum (fun j ↦ f (i,j))) := by
-  apply le_antisymm
-  · obtain ⟨p,hp⟩ := exists_finiteMaximum f
-    rw [←hp]
-    exact (le_finiteMaximum (fun j ↦ f (p.1,j)) p.2).trans
-      (le_finiteMaximum (fun i ↦ finiteMaximum (fun j ↦ f (i,j))) p.1)
-  · obtain ⟨i,hi⟩ := exists_finiteMaximum (fun i ↦ finiteMaximum (fun j ↦ f (i,j)))
-    obtain ⟨j,hj⟩ := exists_finiteMaximum (fun j ↦ f (i,j))
-    rw [←hi,←hj]
-    exact le_finiteMaximum f (i,j)
-
-lemma treeLinear_step (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
-    (z : TreeEdges (l::T) → ℝ) (α : TreeLeaves (l::T)) :
-    linearProcess (treePathCoeff c q (l::T)) z α=
-      c q l*z (Sum.inl α.1)+linearProcess (treePathCoeff c l.endpoint T)
-        (fun e ↦ z (Sum.inr (α.1,e))) α.2 := by
-  classical
-  change (∑ e : Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T),_)=_
-  rw [Fintype.sum_sum_type,Fintype.sum_prod_type]
-  simp only [treePathCoeff,ite_mul,zero_mul]
-  simp [linearProcess]
-
-noncomputable def treeCenter : GaussianTree → ℝ
-  | [] => 0
-  | l::T => Real.log (l.branchExcess+1 : ℝ)/l.height+treeCenter T
-
-noncomputable def treeOffset (T : GaussianTree) (g : TreeEdges T → ℝ) (α : TreeLeaves T) : ℝ :=
-  linearProcess (treePathCoeff (fun _ l ↦ 1/l.height) 0 T) g α-treeCenter T
-
-lemma treePathCoeff_const_q (c : TreeLevel → ℝ) (q r : ℝ) (T : GaussianTree) :
-    treePathCoeff (fun _ l ↦ c l) q T=treePathCoeff (fun _ l ↦ c l) r T := by
-  induction T with
-  | nil => rfl
-  | cons l T ih => funext α e; cases e <;> rfl
-
-lemma treeOffset_step (l : TreeLevel) (T : GaussianTree)
-    (g : TreeEdges (l::T) → ℝ) (α : TreeLeaves (l::T)) :
-    treeOffset (l::T) g α=(g (Sum.inl α.1)-Real.log (l.branchExcess+1 : ℝ))/l.height+
-      treeOffset T (fun e ↦ g (Sum.inr (α.1,e))) α.2 := by
-  rw [treeOffset,treeLinear_step,treeCenter]
-  rw [treePathCoeff_const_q (fun l ↦ 1/l.height) l.endpoint 0 T]
-  simp only [treeOffset]
-  ring
-
-noncomputable def scalarTreeValue (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree)
-    (z g : TreeEdges T → ℝ) : ℝ :=
-  finiteMaximum (fun α ↦ linearProcess (treePathCoeff c q T) z α+treeOffset T g α)
-
-lemma scalarTreeValue_step (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
-    (z g : TreeEdges (l::T) → ℝ) :
-    scalarTreeValue c q (l::T) z g=finiteMaximum (fun i : Fin (l.branchExcess+1) ↦
-      c q l*z (Sum.inl i)+scalarTreeValue c l.endpoint T
-        (fun e ↦ z (Sum.inr (i,e))) (fun e ↦ g (Sum.inr (i,e)))+
-      (g (Sum.inl i)-Real.log (l.branchExcess+1 : ℝ))/l.height) := by
-  unfold scalarTreeValue
-  change finiteMaximum (fun α : Fin (l.branchExcess+1) × TreeLeaves T ↦ _)=_
-  rw [finiteMaximum_prod_curry_left]
-  congr 1; funext i
-  conv_lhs =>
-    arg 1
-    ext α
-    rw [treeLinear_step c q l T z (i,α),treeOffset_step l T g (i,α)]
-  have he (a b d e : ℝ) : a+b+(d+e)=(b+e)+(a+d) := by ring
-  conv_lhs =>
-    arg 1
-    ext α
-    rw [he]
-  dsimp only
-  rw [finiteMaximum_add_const]
-  ring
-
-noncomputable def fieldTreeValue (n : ℕ) (q : ℝ) (T : GaussianTree) (x : Fin n → ℝ)
-    (z : Fin n × TreeEdges T → ℝ) (g : TreeEdges T → ℝ) : ℝ :=
-  finiteMaximum (fun α ↦ (∑ i,|x i+leafField (treeFieldVector q T) z α i|)+treeOffset T g α)
-
-lemma fieldTreeValue_step (n : ℕ) (q : ℝ) (l : TreeLevel) (T : GaussianTree) (x : Fin n → ℝ)
-    (z : Fin n × TreeEdges (l::T) → ℝ) (g : TreeEdges (l::T) → ℝ) :
-    fieldTreeValue n q (l::T) x z g=finiteMaximum (fun j : Fin (l.branchExcess+1) ↦
-      fieldTreeValue n l.endpoint T (fun i ↦ x i+Real.sqrt (l.endpoint-q)*z (i,Sum.inl j))
-        (fun p ↦ z (p.1,Sum.inr (j,p.2))) (fun e ↦ g (Sum.inr (j,e)))+
-      (g (Sum.inl j)-Real.log (l.branchExcess+1 : ℝ))/l.height) := by
-  have hh (α : TreeLeaves (l::T)) (i : Fin n) :
-      leafField (treeFieldVector q (l::T)) z α i=
-      Real.sqrt (l.endpoint-q)*z (i,Sum.inl α.1)+
-        leafField (treeFieldVector l.endpoint T) (fun p ↦ z (p.1,Sum.inr (α.1,p.2))) α.2 i := by
-    exact treeLinear_step (fun q l ↦ Real.sqrt (l.endpoint-q)) q l T (fun e ↦ z (i,e)) α
-  unfold fieldTreeValue
-  change finiteMaximum (fun α : Fin (l.branchExcess+1) × TreeLeaves T ↦ _)=_
-  rw [finiteMaximum_prod_curry_left]
-  congr 1; funext j
-  conv_lhs =>
-    arg 1
-    ext α
-    rw [treeOffset_step l T g (j,α)]
-    simp only [hh (j,α),←add_assoc]
-  dsimp only
-  have he (a b d : ℝ) : a+b+d=(a+d)+b := by ring
-  conv_lhs =>
-    arg 1
-    ext α
-    rw [he]
-  rw [finiteMaximum_add_const]
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.LiteralLower
+import OAI.Probability.SKValue.GroundState.GaussianTree
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma treeAux_tensor_distance (n : ℕ) (T : GaussianTree) (hT : ValidTreeGrid 0 T)
+    (α β : TreeLeaves T) :
+    (∑ e,(leafAuxCoeff n (treeFieldVector 0 T) α e-leafAuxCoeff n (treeFieldVector 0 T) β e)^2)=
+      ∑ e,(treeAuxVector n 0 T α e-treeAuxVector n 0 T β e)^2 := by
+  rw [sq_distance_sum,leafAux_sq n _ (treeField_unit T hT),
+    leafAux_sq n _ (treeField_unit T hT),leafAux_inner,treeField_inner 0 T hT,
+    treeAux_distance n T hT]
+  ring
+
+lemma expected_treeAux_eq_tensor (n : ℕ) (T : GaussianTree) (hT : ValidTreeGrid 0 T)
+    (d : TreeLeaves T → ℝ) :
+    (∫ z,finiteMaximum (fun α ↦ linearProcess (treeAuxVector n 0 T) z α+d α)
+      ∂gaussianProduct (TreeEdges T))=
+    ∫ z,finiteMaximum (fun α ↦ linearProcess (leafAuxCoeff n (treeFieldVector 0 T)) z α+d α)
+      ∂gaussianProduct (TreeEdges T × TreeEdges T) := by
+  apply le_antisymm
+  · exact expected_affine_max_le _ _ d (fun α β ↦ (treeAux_tensor_distance n T hT α β).symm.le)
+  · exact expected_affine_max_le _ _ d (fun α β ↦ (treeAux_tensor_distance n T hT α β).le)
+
+theorem finite_tree_guerra_upper {n : ℕ} (hn : 0 < n)
+    (T : GaussianTree) (hT : ValidTreeGrid 0 T) (d : TreeLeaves T → ℝ) :
+    expectedMaximum n+
+      (∫ z,finiteMaximum (fun α ↦ linearProcess (treeAuxVector n 0 T) z α+d α)
+        ∂gaussianProduct (TreeEdges T))≤
+      ∫ z,finiteMaximum (fun α ↦ (∑ i,|leafField (treeFieldVector 0 T) z α i|)+d α)
+        ∂gaussianProduct (Fin n × TreeEdges T) := by
+  rw [expected_treeAux_eq_tensor n T hT d]
+  exact finite_guerra_upper hn (treeFieldVector 0 T) (treeField_unit T hT) d
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma finiteMaximum_prod_curry_left {ι κ : Type*} [Fintype ι] [Nonempty ι]
+    [Fintype κ] [Nonempty κ] (f : ι×κ → ℝ) :
+    finiteMaximum f=finiteMaximum (fun i ↦ finiteMaximum (fun j ↦ f (i,j))) := by
+  apply le_antisymm
+  · obtain ⟨p,hp⟩ := exists_finiteMaximum f
+    rw [←hp]
+    exact (le_finiteMaximum (fun j ↦ f (p.1,j)) p.2).trans
+      (le_finiteMaximum (fun i ↦ finiteMaximum (fun j ↦ f (i,j))) p.1)
+  · obtain ⟨i,hi⟩ := exists_finiteMaximum (fun i ↦ finiteMaximum (fun j ↦ f (i,j)))
+    obtain ⟨j,hj⟩ := exists_finiteMaximum (fun j ↦ f (i,j))
+    rw [←hi,←hj]
+    exact le_finiteMaximum f (i,j)
+
+lemma treeLinear_step (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
+    (z : TreeEdges (l::T) → ℝ) (α : TreeLeaves (l::T)) :
+    linearProcess (treePathCoeff c q (l::T)) z α=
+      c q l*z (Sum.inl α.1)+linearProcess (treePathCoeff c l.endpoint T)
+        (fun e ↦ z (Sum.inr (α.1,e))) α.2 := by
+  classical
+  change (∑ e : Fin (l.branchExcess+1) ⊕ (Fin (l.branchExcess+1) × TreeEdges T),_)=_
+  rw [Fintype.sum_sum_type,Fintype.sum_prod_type]
+  simp only [treePathCoeff,ite_mul,zero_mul]
+  simp [linearProcess]
+
+noncomputable def treeCenter : GaussianTree → ℝ
+  | [] => 0
+  | l::T => Real.log (l.branchExcess+1 : ℝ)/l.height+treeCenter T
+
+noncomputable def treeOffset (T : GaussianTree) (g : TreeEdges T → ℝ) (α : TreeLeaves T) : ℝ :=
+  linearProcess (treePathCoeff (fun _ l ↦ 1/l.height) 0 T) g α-treeCenter T
+
+lemma treePathCoeff_const_q (c : TreeLevel → ℝ) (q r : ℝ) (T : GaussianTree) :
+    treePathCoeff (fun _ l ↦ c l) q T=treePathCoeff (fun _ l ↦ c l) r T := by
+  induction T with
+  | nil => rfl
+  | cons l T ih => funext α e; cases e <;> rfl
+
+lemma treeOffset_step (l : TreeLevel) (T : GaussianTree)
+    (g : TreeEdges (l::T) → ℝ) (α : TreeLeaves (l::T)) :
+    treeOffset (l::T) g α=(g (Sum.inl α.1)-Real.log (l.branchExcess+1 : ℝ))/l.height+
+      treeOffset T (fun e ↦ g (Sum.inr (α.1,e))) α.2 := by
+  rw [treeOffset,treeLinear_step,treeCenter]
+  rw [treePathCoeff_const_q (fun l ↦ 1/l.height) l.endpoint 0 T]
+  simp only [treeOffset]
+  ring
+
+noncomputable def scalarTreeValue (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree)
+    (z g : TreeEdges T → ℝ) : ℝ :=
+  finiteMaximum (fun α ↦ linearProcess (treePathCoeff c q T) z α+treeOffset T g α)
+
+lemma scalarTreeValue_step (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
+    (z g : TreeEdges (l::T) → ℝ) :
+    scalarTreeValue c q (l::T) z g=finiteMaximum (fun i : Fin (l.branchExcess+1) ↦
+      c q l*z (Sum.inl i)+scalarTreeValue c l.endpoint T
+        (fun e ↦ z (Sum.inr (i,e))) (fun e ↦ g (Sum.inr (i,e)))+
+      (g (Sum.inl i)-Real.log (l.branchExcess+1 : ℝ))/l.height) := by
+  unfold scalarTreeValue
+  change finiteMaximum (fun α : Fin (l.branchExcess+1) × TreeLeaves T ↦ _)=_
+  rw [finiteMaximum_prod_curry_left]
+  congr 1; funext i
+  conv_lhs =>
+    arg 1
+    ext α
+    rw [treeLinear_step c q l T z (i,α),treeOffset_step l T g (i,α)]
+  have he (a b d e : ℝ) : a+b+(d+e)=(b+e)+(a+d) := by ring
+  conv_lhs =>
+    arg 1
+    ext α
+    rw [he]
+  dsimp only
+  rw [finiteMaximum_add_const]
+  ring
+
+noncomputable def fieldTreeValue (n : ℕ) (q : ℝ) (T : GaussianTree) (x : Fin n → ℝ)
+    (z : Fin n × TreeEdges T → ℝ) (g : TreeEdges T → ℝ) : ℝ :=
+  finiteMaximum (fun α ↦ (∑ i,|x i+leafField (treeFieldVector q T) z α i|)+treeOffset T g α)
+
+lemma fieldTreeValue_step (n : ℕ) (q : ℝ) (l : TreeLevel) (T : GaussianTree) (x : Fin n → ℝ)
+    (z : Fin n × TreeEdges (l::T) → ℝ) (g : TreeEdges (l::T) → ℝ) :
+    fieldTreeValue n q (l::T) x z g=finiteMaximum (fun j : Fin (l.branchExcess+1) ↦
+      fieldTreeValue n l.endpoint T (fun i ↦ x i+Real.sqrt (l.endpoint-q)*z (i,Sum.inl j))
+        (fun p ↦ z (p.1,Sum.inr (j,p.2))) (fun e ↦ g (Sum.inr (j,e)))+
+      (g (Sum.inl j)-Real.log (l.branchExcess+1 : ℝ))/l.height) := by
+  have hh (α : TreeLeaves (l::T)) (i : Fin n) :
+      leafField (treeFieldVector q (l::T)) z α i=
+      Real.sqrt (l.endpoint-q)*z (i,Sum.inl α.1)+
+        leafField (treeFieldVector l.endpoint T) (fun p ↦ z (p.1,Sum.inr (α.1,p.2))) α.2 i := by
+    exact treeLinear_step (fun q l ↦ Real.sqrt (l.endpoint-q)) q l T (fun e ↦ z (i,e)) α
+  unfold fieldTreeValue
+  change finiteMaximum (fun α : Fin (l.branchExcess+1) × TreeLeaves T ↦ _)=_
+  rw [finiteMaximum_prod_curry_left]
+  congr 1; funext j
+  conv_lhs =>
+    arg 1
+    ext α
+    rw [treeOffset_step l T g (j,α)]
+    simp only [hh (j,α),←add_assoc]
+  dsimp only
+  have he (a b d : ℝ) : a+b+d=(a+d)+b := by ring
+  conv_lhs =>
+    arg 1
+    ext α
+    rw [he]
+  rw [finiteMaximum_add_const]
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/TreeProductLaw.lean b/lean/OAI/Probability/SKValue/GroundState/TreeProductLaw.lean
index 307df6d51b7da077aff0eef2747d018f3139b68d..21605575098cc58c9845364960458348b1200550 100644
--- a/lean/OAI/Probability/SKValue/GroundState/TreeProductLaw.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/TreeProductLaw.lean
@@ -1,204 +1,204 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.TreeGuerra
-import OAI.Probability.SKValue.GroundState.BranchLaw
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma measurePreserving_finite_curry {I E A : Type*} [Fintype I] [Fintype E]
-    [MeasurableSpace A] (μ : Measure A) [IsProbabilityMeasure μ] :
-    MeasurePreserving (fun z : I×E → A ↦ fun i e ↦ z (i,e))
-      (Measure.pi (fun _ ↦ μ)) (Measure.pi (fun _ : I ↦ Measure.pi (fun _ : E ↦ μ))) := by
-  refine ⟨by fun_prop,?_⟩
-  have h := Measure.infinitePi_map_curry (fun (_ : I) (_ : E) ↦ μ)
-  simp only [Measure.infinitePi_eq_pi,MeasurableEquiv.coe_curry] at h
-  convert h using 1
-  rfl
-
-lemma measurePreserving_nodeBlocks {I E A : Type*} [Fintype I] [Fintype E]
-    [MeasurableSpace A] (μ : Measure A) [IsProbabilityMeasure μ] :
-    MeasurePreserving (fun z : I⊕(I×E) → A ↦ fun i ↦ (z (Sum.inl i),fun e ↦ z (Sum.inr (i,e))))
-      (Measure.pi (fun _ ↦ μ))
-      (Measure.pi (fun _ : I ↦ μ.prod (Measure.pi (fun _ : E ↦ μ)))) := by
-  have hsum := measurePreserving_sumPiEquivProdPi (fun _ : I⊕(I×E) ↦ μ)
-  have hc := (MeasurePreserving.id (Measure.pi (fun _ : I ↦ μ))).prod
-    (measurePreserving_finite_curry (I := I) (E := E) μ)
-  have hp := (measurePreserving_arrowProdEquivProdArrow A (E → A) I
-    (fun _ ↦ μ) (fun _ ↦ Measure.pi (fun _ : E ↦ μ))).symm
-  exact hp.comp (hc.comp hsum)
-
-lemma measurePreserving_midSwap {A B C : Type*} [MeasurableSpace A]
-    [MeasurableSpace B] [MeasurableSpace C] (μ : Measure A) (ν : Measure B) (σ : Measure C)
-    [SFinite μ] [SFinite ν] [SFinite σ] :
-    MeasurePreserving (fun p : (A×B)×C ↦ ((p.1.1,p.2),p.1.2))
-      ((μ.prod ν).prod σ) ((μ.prod σ).prod ν) := by
-  have h1 := measurePreserving_prodAssoc μ ν σ
-  have h2 := (MeasurePreserving.id μ).prod (Measure.measurePreserving_swap (μ := ν) (ν := σ))
-  have h3 := (measurePreserving_prodAssoc μ σ ν).symm
-  exact h3.comp (h2.comp h1)
-
-noncomputable def gaussianAddLaw (c : ℝ) (μ : Measure ℝ) : Measure ℝ :=
-  (standardGaussian.prod μ).map (fun p : ℝ×ℝ ↦ c*p.1+p.2)
-
-instance gaussianAddLaw_probability (c : ℝ) (μ : Measure ℝ) [IsProbabilityMeasure μ] :
-    IsProbabilityMeasure (gaussianAddLaw c μ) := by
-  unfold gaussianAddLaw
-  infer_instance
-
-lemma gaussianAddLaw_integrable (c : ℝ) {μ : Measure ℝ} [IsProbabilityMeasure μ]
-    (hi : Integrable (fun x : ℝ ↦ x) μ) :
-    Integrable (fun x : ℝ ↦ x) (gaussianAddLaw c μ) := by
-  rw [gaussianAddLaw,integrable_map_measure (by fun_prop) (by fun_prop)]
-  exact ((standardGaussian_integrable_id.const_mul c).comp_fst μ).add (hi.comp_snd _)
-
-lemma gaussianAddLaw_exp_integrable (c a : ℝ) {μ : Measure ℝ} [IsProbabilityMeasure μ]
-    (hi : Integrable (fun x : ℝ ↦ Real.exp (a*x)) μ) :
-    Integrable (fun x : ℝ ↦ Real.exp (a*x)) (gaussianAddLaw c μ) := by
-  rw [gaussianAddLaw,integrable_map_measure (by fun_prop) (by fun_prop)]
-  have hg : Integrable (fun z : ℝ ↦ Real.exp ((a*c)*z)) standardGaussian :=
-    integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (a*c)
-  convert hg.mul_prod hi using 1
-  funext p
-  dsimp only [Function.comp_def]
-  rw [←Real.exp_add]
-  congr 1
-  ring
-
-lemma gaussianAddLaw_exp (c a : ℝ) {μ : Measure ℝ} [IsProbabilityMeasure μ] :
-    (∫ x,Real.exp (a*x) ∂gaussianAddLaw c μ)=
-      Real.exp ((a*c)^2/2)*(∫ x,Real.exp (a*x) ∂μ) := by
-  rw [gaussianAddLaw,integral_map (by fun_prop) (by fun_prop)]
-  have he (p : ℝ×ℝ) : Real.exp (a*(c*p.1+p.2))=
-      Real.exp ((a*c)*p.1)*Real.exp (a*p.2) := by
-    rw [←Real.exp_add]; congr 1; ring
-  simp_rw [he]
-  rw [integral_prod_mul (fun z : ℝ ↦ Real.exp ((a*c)*z)) (fun x : ℝ ↦ Real.exp (a*x))]
-  have heq := congrFun (mgf_fun_id_gaussianReal (μ := 0) (v := 1)) (a*c)
-  simpa [mgf,standardGaussian] using congrArg (fun x ↦ x*(∫ x,Real.exp (a*x) ∂μ)) heq
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-noncomputable def edgeNoiseLaw : Measure (ℝ×ℝ) := standardGaussian.prod gumbelLaw
-instance edgeNoiseLaw_probability : IsProbabilityMeasure edgeNoiseLaw := by
-  unfold edgeNoiseLaw; infer_instance
-
-noncomputable def scalarTreeLaw (c : ℝ → TreeLevel → ℝ) (q : ℝ) : GaussianTree → Measure ℝ
-  | [] => Measure.dirac 0
-  | l::T => branchLaw (gaussianAddLaw (c q l) (scalarTreeLaw c l.endpoint T)) l.height l.branchExcess
-
-instance scalarTreeLaw_probability (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
-    IsProbabilityMeasure (scalarTreeLaw c q T) := by
-  induction T generalizing q with
-  | nil => exact inferInstanceAs (IsProbabilityMeasure (Measure.dirac (0 : ℝ)))
-  | cons l T ih =>
-    have := ih l.endpoint
-    exact inferInstanceAs (IsProbabilityMeasure
-      (branchLaw (gaussianAddLaw (c q l) (scalarTreeLaw c l.endpoint T)) l.height l.branchExcess))
-
-noncomputable def scalarTreeEval (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree)
-    (w : TreeEdges T → ℝ×ℝ) : ℝ :=
-  scalarTreeValue c q T (fun e ↦ (w e).1) (fun e ↦ (w e).2)
-
-lemma finiteMaximum_const {I : Type*} [Fintype I] [Nonempty I] (a : ℝ) :
-    finiteMaximum (fun _ : I ↦ a)=a := by
-  obtain ⟨i,hi⟩ := exists_finiteMaximum (fun _ : I ↦ a)
-  exact hi.symm
-
-lemma scalarTreeEval_nil (c : ℝ → TreeLevel → ℝ) (q : ℝ) (w : TreeEdges [] → ℝ×ℝ) :
-    scalarTreeEval c q [] w=0 := by
-  simp only [scalarTreeEval,scalarTreeValue,treeOffset,treeCenter,linearProcess]
-  simp only [show ∀ f : PEmpty → ℝ, (∑ e,f e)=0 from fun f ↦ Fintype.sum_empty f,
-    sub_self,zero_add,finiteMaximum_const]
-
-lemma continuous_scalarTreeEval (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
-    Continuous (scalarTreeEval c q T) := by
-  unfold scalarTreeEval scalarTreeValue treeOffset
-  apply continuous_finiteMaximum.comp
-  apply continuous_pi
-  intro α
-  unfold linearProcess
-  fun_prop
-
-lemma marked_gaussianAddLaw {S : Type*} [MeasurableSpace S] (σ : Measure S) [IsProbabilityMeasure σ]
-    {μ : Measure ℝ} [IsProbabilityMeasure μ] {f : S → ℝ}
-    (hf : MeasurePreserving f σ μ) (c : ℝ) :
-    MeasurePreserving (fun p : (ℝ×ℝ)×S ↦ (c*p.1.1+f p.2,p.1.2))
-      (edgeNoiseLaw.prod σ) ((gaussianAddLaw c μ).prod gumbelLaw) := by
-  have hadd : MeasurePreserving (fun p : ℝ×ℝ ↦ c*p.1+p.2)
-      (standardGaussian.prod μ) (gaussianAddLaw c μ) := ⟨by fun_prop,rfl⟩
-  exact ((hadd.prod (MeasurePreserving.id gumbelLaw)).comp
-    (((MeasurePreserving.id standardGaussian).prod hf).prod (MeasurePreserving.id gumbelLaw))).comp
-    (measurePreserving_midSwap standardGaussian gumbelLaw σ)
-
-lemma scalarTreeEval_step (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
-    (w : TreeEdges (l::T) → ℝ×ℝ) :
-    scalarTreeEval c q (l::T) w=branchMax l.height l.branchExcess
-      ((fun i ↦ c q l*(w (Sum.inl i)).1+
-        scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))),
-       fun i ↦ (w (Sum.inl i)).2) := by
-  rw [scalarTreeEval,scalarTreeValue_step]
-  unfold branchMax
-  have he (i : Fin (l.branchExcess+1)) :
-      c q l*(w (Sum.inl i)).1+scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))+
-        ((w (Sum.inl i)).2-Real.log (l.branchExcess+1 : ℝ))/l.height=
-      (c q l*(w (Sum.inl i)).1+scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))+
-        (w (Sum.inl i)).2/l.height)+(-Real.log (l.branchExcess+1 : ℝ)/l.height) := by ring
-  change finiteMaximum (fun i ↦ c q l*(w (Sum.inl i)).1+
-    scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))+
-      ((w (Sum.inl i)).2-Real.log (l.branchExcess+1 : ℝ))/l.height)=_
-  conv_lhs => arg 1; ext i; rw [he i]
-  rw [finiteMaximum_add_const]
-  ring
-
-theorem scalarTreeEval_law (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
-    MeasurePreserving (scalarTreeEval c q T) (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw))
-      (scalarTreeLaw c q T) := by
-  induction T generalizing q with
-  | nil =>
-    have he : scalarTreeEval c q []=fun _ ↦ (0 : ℝ) := funext (scalarTreeEval_nil c q)
-    rw [he,scalarTreeLaw]
-    exact ⟨measurable_const,by simp⟩
-  | cons l T ih =>
-    let ν := gaussianAddLaw (c q l) (scalarTreeLaw c l.endpoint T)
-    have hblock := measurePreserving_nodeBlocks (I := Fin (l.branchExcess+1)) (E := TreeEdges T) edgeNoiseLaw
-    have hchild := marked_gaussianAddLaw (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)) (ih l.endpoint) (c q l)
-    have hchildren := measurePreserving_pi (fun _ : Fin (l.branchExcess+1) ↦
-      edgeNoiseLaw.prod (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)))
-      (fun _ ↦ ν.prod gumbelLaw) (fun _ ↦ hchild)
-    have hpair := measurePreserving_arrowProdEquivProdArrow ℝ ℝ (Fin (l.branchExcess+1))
-      (fun _ ↦ ν) (fun _ ↦ gumbelLaw)
-    have hmax : MeasurePreserving (branchMax l.height l.branchExcess)
-        ((Measure.pi (fun _ : Fin (l.branchExcess+1) ↦ ν)).prod
-          (Measure.pi (fun _ : Fin (l.branchExcess+1) ↦ gumbelLaw)))
-        (scalarTreeLaw c q (l::T)) := ⟨(continuous_branchMax _ _).measurable,rfl⟩
-    have hall := hmax.comp (hpair.comp (hchildren.comp hblock))
-    convert hall using 1
-    funext w
-    exact scalarTreeEval_step c q l T w
-
-lemma scalarTreeLaw_integrable (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
-    Integrable (fun x : ℝ ↦ x) (scalarTreeLaw c q T) := by
-  induction T generalizing q with
-  | nil =>
-    change Integrable (fun x : ℝ ↦ x) (Measure.dirac (0 : ℝ))
-    exact integrable_dirac (by simp)
-  | cons l T ih => exact branchLaw_integrable (gaussianAddLaw_integrable _ (ih l.endpoint)) _ _
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.TreeGuerra
+import OAI.Probability.SKValue.GroundState.BranchLaw
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma measurePreserving_finite_curry {I E A : Type*} [Fintype I] [Fintype E]
+    [MeasurableSpace A] (μ : Measure A) [IsProbabilityMeasure μ] :
+    MeasurePreserving (fun z : I×E → A ↦ fun i e ↦ z (i,e))
+      (Measure.pi (fun _ ↦ μ)) (Measure.pi (fun _ : I ↦ Measure.pi (fun _ : E ↦ μ))) := by
+  refine ⟨by fun_prop,?_⟩
+  have h := Measure.infinitePi_map_curry (fun (_ : I) (_ : E) ↦ μ)
+  simp only [Measure.infinitePi_eq_pi,MeasurableEquiv.coe_curry] at h
+  convert h using 1
+  rfl
+
+lemma measurePreserving_nodeBlocks {I E A : Type*} [Fintype I] [Fintype E]
+    [MeasurableSpace A] (μ : Measure A) [IsProbabilityMeasure μ] :
+    MeasurePreserving (fun z : I⊕(I×E) → A ↦ fun i ↦ (z (Sum.inl i),fun e ↦ z (Sum.inr (i,e))))
+      (Measure.pi (fun _ ↦ μ))
+      (Measure.pi (fun _ : I ↦ μ.prod (Measure.pi (fun _ : E ↦ μ)))) := by
+  have hsum := measurePreserving_sumPiEquivProdPi (fun _ : I⊕(I×E) ↦ μ)
+  have hc := (MeasurePreserving.id (Measure.pi (fun _ : I ↦ μ))).prod
+    (measurePreserving_finite_curry (I := I) (E := E) μ)
+  have hp := (measurePreserving_arrowProdEquivProdArrow A (E → A) I
+    (fun _ ↦ μ) (fun _ ↦ Measure.pi (fun _ : E ↦ μ))).symm
+  exact hp.comp (hc.comp hsum)
+
+lemma measurePreserving_midSwap {A B C : Type*} [MeasurableSpace A]
+    [MeasurableSpace B] [MeasurableSpace C] (μ : Measure A) (ν : Measure B) (σ : Measure C)
+    [SFinite μ] [SFinite ν] [SFinite σ] :
+    MeasurePreserving (fun p : (A×B)×C ↦ ((p.1.1,p.2),p.1.2))
+      ((μ.prod ν).prod σ) ((μ.prod σ).prod ν) := by
+  have h1 := measurePreserving_prodAssoc μ ν σ
+  have h2 := (MeasurePreserving.id μ).prod (Measure.measurePreserving_swap (μ := ν) (ν := σ))
+  have h3 := (measurePreserving_prodAssoc μ σ ν).symm
+  exact h3.comp (h2.comp h1)
+
+noncomputable def gaussianAddLaw (c : ℝ) (μ : Measure ℝ) : Measure ℝ :=
+  (standardGaussian.prod μ).map (fun p : ℝ×ℝ ↦ c*p.1+p.2)
+
+instance gaussianAddLaw_probability (c : ℝ) (μ : Measure ℝ) [IsProbabilityMeasure μ] :
+    IsProbabilityMeasure (gaussianAddLaw c μ) := by
+  unfold gaussianAddLaw
+  infer_instance
+
+lemma gaussianAddLaw_integrable (c : ℝ) {μ : Measure ℝ} [IsProbabilityMeasure μ]
+    (hi : Integrable (fun x : ℝ ↦ x) μ) :
+    Integrable (fun x : ℝ ↦ x) (gaussianAddLaw c μ) := by
+  rw [gaussianAddLaw,integrable_map_measure (by fun_prop) (by fun_prop)]
+  exact ((standardGaussian_integrable_id.const_mul c).comp_fst μ).add (hi.comp_snd _)
+
+lemma gaussianAddLaw_exp_integrable (c a : ℝ) {μ : Measure ℝ} [IsProbabilityMeasure μ]
+    (hi : Integrable (fun x : ℝ ↦ Real.exp (a*x)) μ) :
+    Integrable (fun x : ℝ ↦ Real.exp (a*x)) (gaussianAddLaw c μ) := by
+  rw [gaussianAddLaw,integrable_map_measure (by fun_prop) (by fun_prop)]
+  have hg : Integrable (fun z : ℝ ↦ Real.exp ((a*c)*z)) standardGaussian :=
+    integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (a*c)
+  convert hg.mul_prod hi using 1
+  funext p
+  dsimp only [Function.comp_def]
+  rw [←Real.exp_add]
+  congr 1
+  ring
+
+lemma gaussianAddLaw_exp (c a : ℝ) {μ : Measure ℝ} [IsProbabilityMeasure μ] :
+    (∫ x,Real.exp (a*x) ∂gaussianAddLaw c μ)=
+      Real.exp ((a*c)^2/2)*(∫ x,Real.exp (a*x) ∂μ) := by
+  rw [gaussianAddLaw,integral_map (by fun_prop) (by fun_prop)]
+  have he (p : ℝ×ℝ) : Real.exp (a*(c*p.1+p.2))=
+      Real.exp ((a*c)*p.1)*Real.exp (a*p.2) := by
+    rw [←Real.exp_add]; congr 1; ring
+  simp_rw [he]
+  rw [integral_prod_mul (fun z : ℝ ↦ Real.exp ((a*c)*z)) (fun x : ℝ ↦ Real.exp (a*x))]
+  have heq := congrFun (mgf_fun_id_gaussianReal (μ := 0) (v := 1)) (a*c)
+  simpa [mgf,standardGaussian] using congrArg (fun x ↦ x*(∫ x,Real.exp (a*x) ∂μ)) heq
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+noncomputable def edgeNoiseLaw : Measure (ℝ×ℝ) := standardGaussian.prod gumbelLaw
+instance edgeNoiseLaw_probability : IsProbabilityMeasure edgeNoiseLaw := by
+  unfold edgeNoiseLaw; infer_instance
+
+noncomputable def scalarTreeLaw (c : ℝ → TreeLevel → ℝ) (q : ℝ) : GaussianTree → Measure ℝ
+  | [] => Measure.dirac 0
+  | l::T => branchLaw (gaussianAddLaw (c q l) (scalarTreeLaw c l.endpoint T)) l.height l.branchExcess
+
+instance scalarTreeLaw_probability (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
+    IsProbabilityMeasure (scalarTreeLaw c q T) := by
+  induction T generalizing q with
+  | nil => exact inferInstanceAs (IsProbabilityMeasure (Measure.dirac (0 : ℝ)))
+  | cons l T ih =>
+    have := ih l.endpoint
+    exact inferInstanceAs (IsProbabilityMeasure
+      (branchLaw (gaussianAddLaw (c q l) (scalarTreeLaw c l.endpoint T)) l.height l.branchExcess))
+
+noncomputable def scalarTreeEval (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree)
+    (w : TreeEdges T → ℝ×ℝ) : ℝ :=
+  scalarTreeValue c q T (fun e ↦ (w e).1) (fun e ↦ (w e).2)
+
+lemma finiteMaximum_const {I : Type*} [Fintype I] [Nonempty I] (a : ℝ) :
+    finiteMaximum (fun _ : I ↦ a)=a := by
+  obtain ⟨i,hi⟩ := exists_finiteMaximum (fun _ : I ↦ a)
+  exact hi.symm
+
+lemma scalarTreeEval_nil (c : ℝ → TreeLevel → ℝ) (q : ℝ) (w : TreeEdges [] → ℝ×ℝ) :
+    scalarTreeEval c q [] w=0 := by
+  simp only [scalarTreeEval,scalarTreeValue,treeOffset,treeCenter,linearProcess]
+  simp only [show ∀ f : PEmpty → ℝ, (∑ e,f e)=0 from fun f ↦ Fintype.sum_empty f,
+    sub_self,zero_add,finiteMaximum_const]
+
+lemma continuous_scalarTreeEval (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
+    Continuous (scalarTreeEval c q T) := by
+  unfold scalarTreeEval scalarTreeValue treeOffset
+  apply continuous_finiteMaximum.comp
+  apply continuous_pi
+  intro α
+  unfold linearProcess
+  fun_prop
+
+lemma marked_gaussianAddLaw {S : Type*} [MeasurableSpace S] (σ : Measure S) [IsProbabilityMeasure σ]
+    {μ : Measure ℝ} [IsProbabilityMeasure μ] {f : S → ℝ}
+    (hf : MeasurePreserving f σ μ) (c : ℝ) :
+    MeasurePreserving (fun p : (ℝ×ℝ)×S ↦ (c*p.1.1+f p.2,p.1.2))
+      (edgeNoiseLaw.prod σ) ((gaussianAddLaw c μ).prod gumbelLaw) := by
+  have hadd : MeasurePreserving (fun p : ℝ×ℝ ↦ c*p.1+p.2)
+      (standardGaussian.prod μ) (gaussianAddLaw c μ) := ⟨by fun_prop,rfl⟩
+  exact ((hadd.prod (MeasurePreserving.id gumbelLaw)).comp
+    (((MeasurePreserving.id standardGaussian).prod hf).prod (MeasurePreserving.id gumbelLaw))).comp
+    (measurePreserving_midSwap standardGaussian gumbelLaw σ)
+
+lemma scalarTreeEval_step (c : ℝ → TreeLevel → ℝ) (q : ℝ) (l : TreeLevel) (T : GaussianTree)
+    (w : TreeEdges (l::T) → ℝ×ℝ) :
+    scalarTreeEval c q (l::T) w=branchMax l.height l.branchExcess
+      ((fun i ↦ c q l*(w (Sum.inl i)).1+
+        scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))),
+       fun i ↦ (w (Sum.inl i)).2) := by
+  rw [scalarTreeEval,scalarTreeValue_step]
+  unfold branchMax
+  have he (i : Fin (l.branchExcess+1)) :
+      c q l*(w (Sum.inl i)).1+scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))+
+        ((w (Sum.inl i)).2-Real.log (l.branchExcess+1 : ℝ))/l.height=
+      (c q l*(w (Sum.inl i)).1+scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))+
+        (w (Sum.inl i)).2/l.height)+(-Real.log (l.branchExcess+1 : ℝ)/l.height) := by ring
+  change finiteMaximum (fun i ↦ c q l*(w (Sum.inl i)).1+
+    scalarTreeEval c l.endpoint T (fun e ↦ w (Sum.inr (i,e)))+
+      ((w (Sum.inl i)).2-Real.log (l.branchExcess+1 : ℝ))/l.height)=_
+  conv_lhs => arg 1; ext i; rw [he i]
+  rw [finiteMaximum_add_const]
+  ring
+
+theorem scalarTreeEval_law (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
+    MeasurePreserving (scalarTreeEval c q T) (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw))
+      (scalarTreeLaw c q T) := by
+  induction T generalizing q with
+  | nil =>
+    have he : scalarTreeEval c q []=fun _ ↦ (0 : ℝ) := funext (scalarTreeEval_nil c q)
+    rw [he,scalarTreeLaw]
+    exact ⟨measurable_const,by simp⟩
+  | cons l T ih =>
+    let ν := gaussianAddLaw (c q l) (scalarTreeLaw c l.endpoint T)
+    have hblock := measurePreserving_nodeBlocks (I := Fin (l.branchExcess+1)) (E := TreeEdges T) edgeNoiseLaw
+    have hchild := marked_gaussianAddLaw (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)) (ih l.endpoint) (c q l)
+    have hchildren := measurePreserving_pi (fun _ : Fin (l.branchExcess+1) ↦
+      edgeNoiseLaw.prod (Measure.pi (fun _ : TreeEdges T ↦ edgeNoiseLaw)))
+      (fun _ ↦ ν.prod gumbelLaw) (fun _ ↦ hchild)
+    have hpair := measurePreserving_arrowProdEquivProdArrow ℝ ℝ (Fin (l.branchExcess+1))
+      (fun _ ↦ ν) (fun _ ↦ gumbelLaw)
+    have hmax : MeasurePreserving (branchMax l.height l.branchExcess)
+        ((Measure.pi (fun _ : Fin (l.branchExcess+1) ↦ ν)).prod
+          (Measure.pi (fun _ : Fin (l.branchExcess+1) ↦ gumbelLaw)))
+        (scalarTreeLaw c q (l::T)) := ⟨(continuous_branchMax _ _).measurable,rfl⟩
+    have hall := hmax.comp (hpair.comp (hchildren.comp hblock))
+    convert hall using 1
+    funext w
+    exact scalarTreeEval_step c q l T w
+
+lemma scalarTreeLaw_integrable (c : ℝ → TreeLevel → ℝ) (q : ℝ) (T : GaussianTree) :
+    Integrable (fun x : ℝ ↦ x) (scalarTreeLaw c q T) := by
+  induction T generalizing q with
+  | nil =>
+    change Integrable (fun x : ℝ ↦ x) (Measure.dirac (0 : ℝ))
+    exact integrable_dirac (by simp)
+  | cons l T ih => exact branchLaw_integrable (gaussianAddLaw_integrable _ (ih l.endpoint)) _ _
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/TreeWidths.lean b/lean/OAI/Probability/SKValue/GroundState/TreeWidths.lean
index bb177de84c807c7ae3013c905b43f984ebf00a3e..fe9b0904153b4693de520e944dc973eea861570c 100644
--- a/lean/OAI/Probability/SKValue/GroundState/TreeWidths.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/TreeWidths.lean
@@ -1,119 +1,119 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.BranchMeanUpper
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-noncomputable def auxAmplitude (n : ℕ) (q : ℝ) (l : TreeLevel) : ℝ :=
-  Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)
-
-noncomputable def auxShape (n : ℕ) (q : ℝ) : GaussianTree → CascadeShape
-  | [] => []
-  | l::T => (auxAmplitude n q l,l.height)::auxShape n l.endpoint T
-
-noncomputable def resizeTree (n : ℕ) (q : ℝ) : (T : GaussianTree) → CascadeWidths (auxShape n q T) → GaussianTree
-  | [],_ => []
-  | l::T,w => {l with branchExcess := w.1}::resizeTree n l.endpoint T w.2
-
-lemma resizeTree_valid (n : ℕ) {q : ℝ} {T : GaussianTree} (h : ValidTreeGrid q T)
-    (w : CascadeWidths (auxShape n q T)) : ValidTreeGrid q (resizeTree n q T w) := by
-  induction T generalizing q with
-  | nil => exact h
-  | cons l T ih => exact ⟨h.1,ih h.2 w.2⟩
-
-lemma resizeTree_scale (n : ℕ) (q : ℝ) {a : ℝ} {T : GaussianTree} (h : TreeScale a T)
-    (w : CascadeWidths (auxShape n q T)) : TreeScale a (resizeTree n q T w) := by
-  induction T generalizing a q with
-  | nil => exact h
-  | cons l T ih => exact ⟨h.1,h.2.1,ih _ h.2.2 w.2⟩
-
-lemma auxShape_scale (n : ℕ) (q : ℝ) {a : ℝ} {T : GaussianTree} (h : TreeScale a T) :
-    CascadeScale a (auxShape n q T) := by
-  induction T generalizing a q with
-  | nil => exact h
-  | cons l T ih => exact ⟨h.1,h.2.1,ih _ h.2.2⟩
-
-lemma resizeTree_profile (n : ℕ) (q : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
-    fieldProfile q (resizeTree n q T w)=fieldProfile q T := by
-  induction T generalizing q with
-  | nil => rfl
-  | cons l T ih => simp only [resizeTree,fieldProfile,ih]
-
-lemma resizeTree_gumbelMoment (n : ℕ) (q a : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
-    gumbelTreeMoment a (resizeTree n q T w)=gumbelTreeMoment a T := by
-  induction T generalizing q a with
-  | nil => rfl
-  | cons l T ih => simp only [resizeTree,gumbelTreeMoment,ih]
-
-lemma resizeTree_gumbelMean (n : ℕ) (q : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
-    gumbelTreeMean (resizeTree n q T w)=gumbelTreeMean T := by
-  cases T with
-  | nil => rfl
-  | cons l T => simp only [resizeTree,gumbelTreeMean,resizeTree_gumbelMoment]
-
-lemma resizeTree_scalarLaw (n : ℕ) (q : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
-    scalarTreeLaw (auxAmplitude n) q (resizeTree n q T w)=cascadeLaw (auxShape n q T) w := by
-  induction T generalizing q with
-  | nil => rfl
-  | cons l T ih => simp only [resizeTree,scalarTreeLaw,auxShape,cascadeLaw,ih,auxAmplitude]
-
-noncomputable def treeCorrection (q : ℝ) : GaussianTree → ℝ
-  | [] => 0
-  | l::T => l.height*(l.endpoint^2-q^2)/4+treeCorrection l.endpoint T
-
-lemma auxAmplitude_sq (n : ℕ) {q : ℝ} {l : TreeLevel} (hq : 0 ≤ q) (hql : q ≤ l.endpoint) :
-    (auxAmplitude n q l)^2=(n : ℝ)/2*(l.endpoint^2-q^2) := by
-  have hl := hq.trans hql
-  have hs : 0 ≤ l.endpoint^2-q^2 := by nlinarith
-  rw [auxAmplitude,mul_pow,Real.sq_sqrt (by positivity),Real.sq_sqrt hs]
-
-lemma auxShape_moment (n : ℕ) {q a : ℝ} {T : GaussianTree} (hq : 0 ≤ q)
-    (hgrid : ValidTreeGrid q T) (h : TreeScale a T) :
-    cascadeMoment a (auxShape n q T)=gumbelTreeMoment a T*Real.exp (a*(n : ℝ)*treeCorrection q T) := by
-  induction T generalizing q a with
-  | nil => simp [auxShape,cascadeMoment,gumbelTreeMoment,treeCorrection]
-  | cons l T ih =>
-    obtain ⟨ha,ham,hT⟩ := h
-    have hm : 0 < l.height := ha.trans_lt ham
-    have hs := auxAmplitude_sq n hq hgrid.1
-    rw [auxShape,cascadeMoment,ih (hq.trans hgrid.1) hgrid.2 hT]
-    rw [mul_pow,hs]
-    have he : Real.exp (l.height^2*((n : ℝ)/2*(l.endpoint^2-q^2))/2)*
-        (gumbelTreeMoment l.height T*Real.exp (l.height*(n : ℝ)*treeCorrection l.endpoint T))=
-        gumbelTreeMoment l.height T*Real.exp (l.height*(n : ℝ)*treeCorrection q (l::T)) := by
-      rw [mul_left_comm,←Real.exp_add]
-      congr 2
-      simp only [treeCorrection]
-      ring
-    rw [he,Real.mul_rpow (gumbelTreeMoment_pos hT).le (Real.exp_pos _).le,←Real.exp_mul]
-    have he2 : l.height*(n : ℝ)*treeCorrection q (l::T)*(a/l.height)=a*(n : ℝ)*treeCorrection q (l::T) := by
-      field_simp
-    rw [he2]
-    simp only [gumbelTreeMoment,mul_assoc]
-
-lemma auxShape_mean (n : ℕ) {q : ℝ} {T : GaussianTree} (hq : 0 ≤ q)
-    (hgrid : ValidTreeGrid q T) (h : TreeScale 0 T) :
-    cascadeMean (auxShape n q T)=gumbelTreeMean T+(n : ℝ)*treeCorrection q T := by
-  cases T with
-  | nil => simp [auxShape,cascadeMean,gumbelTreeMean,treeCorrection]
-  | cons l T =>
-    obtain ⟨_,hm,hT⟩ := h
-    have hs := auxAmplitude_sq n hq hgrid.1
-    rw [auxShape,cascadeMean,auxShape_moment n (hq.trans hgrid.1) hgrid.2 hT]
-    rw [Real.log_mul (Real.exp_pos _).ne' (mul_pos (gumbelTreeMoment_pos hT) (Real.exp_pos _)).ne',
-      Real.log_exp,Real.log_mul (gumbelTreeMoment_pos hT).ne' (Real.exp_pos _).ne',Real.log_exp]
-    rw [mul_pow,hs]
-    simp only [gumbelTreeMean,treeCorrection]
-    field_simp
-    ring
-
-end SKValueG
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.BranchMeanUpper
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+noncomputable def auxAmplitude (n : ℕ) (q : ℝ) (l : TreeLevel) : ℝ :=
+  Real.sqrt ((n : ℝ)/2)*Real.sqrt (l.endpoint^2-q^2)
+
+noncomputable def auxShape (n : ℕ) (q : ℝ) : GaussianTree → CascadeShape
+  | [] => []
+  | l::T => (auxAmplitude n q l,l.height)::auxShape n l.endpoint T
+
+noncomputable def resizeTree (n : ℕ) (q : ℝ) : (T : GaussianTree) → CascadeWidths (auxShape n q T) → GaussianTree
+  | [],_ => []
+  | l::T,w => {l with branchExcess := w.1}::resizeTree n l.endpoint T w.2
+
+lemma resizeTree_valid (n : ℕ) {q : ℝ} {T : GaussianTree} (h : ValidTreeGrid q T)
+    (w : CascadeWidths (auxShape n q T)) : ValidTreeGrid q (resizeTree n q T w) := by
+  induction T generalizing q with
+  | nil => exact h
+  | cons l T ih => exact ⟨h.1,ih h.2 w.2⟩
+
+lemma resizeTree_scale (n : ℕ) (q : ℝ) {a : ℝ} {T : GaussianTree} (h : TreeScale a T)
+    (w : CascadeWidths (auxShape n q T)) : TreeScale a (resizeTree n q T w) := by
+  induction T generalizing a q with
+  | nil => exact h
+  | cons l T ih => exact ⟨h.1,h.2.1,ih _ h.2.2 w.2⟩
+
+lemma auxShape_scale (n : ℕ) (q : ℝ) {a : ℝ} {T : GaussianTree} (h : TreeScale a T) :
+    CascadeScale a (auxShape n q T) := by
+  induction T generalizing a q with
+  | nil => exact h
+  | cons l T ih => exact ⟨h.1,h.2.1,ih _ h.2.2⟩
+
+lemma resizeTree_profile (n : ℕ) (q : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
+    fieldProfile q (resizeTree n q T w)=fieldProfile q T := by
+  induction T generalizing q with
+  | nil => rfl
+  | cons l T ih => simp only [resizeTree,fieldProfile,ih]
+
+lemma resizeTree_gumbelMoment (n : ℕ) (q a : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
+    gumbelTreeMoment a (resizeTree n q T w)=gumbelTreeMoment a T := by
+  induction T generalizing q a with
+  | nil => rfl
+  | cons l T ih => simp only [resizeTree,gumbelTreeMoment,ih]
+
+lemma resizeTree_gumbelMean (n : ℕ) (q : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
+    gumbelTreeMean (resizeTree n q T w)=gumbelTreeMean T := by
+  cases T with
+  | nil => rfl
+  | cons l T => simp only [resizeTree,gumbelTreeMean,resizeTree_gumbelMoment]
+
+lemma resizeTree_scalarLaw (n : ℕ) (q : ℝ) (T : GaussianTree) (w : CascadeWidths (auxShape n q T)) :
+    scalarTreeLaw (auxAmplitude n) q (resizeTree n q T w)=cascadeLaw (auxShape n q T) w := by
+  induction T generalizing q with
+  | nil => rfl
+  | cons l T ih => simp only [resizeTree,scalarTreeLaw,auxShape,cascadeLaw,ih,auxAmplitude]
+
+noncomputable def treeCorrection (q : ℝ) : GaussianTree → ℝ
+  | [] => 0
+  | l::T => l.height*(l.endpoint^2-q^2)/4+treeCorrection l.endpoint T
+
+lemma auxAmplitude_sq (n : ℕ) {q : ℝ} {l : TreeLevel} (hq : 0 ≤ q) (hql : q ≤ l.endpoint) :
+    (auxAmplitude n q l)^2=(n : ℝ)/2*(l.endpoint^2-q^2) := by
+  have hl := hq.trans hql
+  have hs : 0 ≤ l.endpoint^2-q^2 := by nlinarith
+  rw [auxAmplitude,mul_pow,Real.sq_sqrt (by positivity),Real.sq_sqrt hs]
+
+lemma auxShape_moment (n : ℕ) {q a : ℝ} {T : GaussianTree} (hq : 0 ≤ q)
+    (hgrid : ValidTreeGrid q T) (h : TreeScale a T) :
+    cascadeMoment a (auxShape n q T)=gumbelTreeMoment a T*Real.exp (a*(n : ℝ)*treeCorrection q T) := by
+  induction T generalizing q a with
+  | nil => simp [auxShape,cascadeMoment,gumbelTreeMoment,treeCorrection]
+  | cons l T ih =>
+    obtain ⟨ha,ham,hT⟩ := h
+    have hm : 0 < l.height := ha.trans_lt ham
+    have hs := auxAmplitude_sq n hq hgrid.1
+    rw [auxShape,cascadeMoment,ih (hq.trans hgrid.1) hgrid.2 hT]
+    rw [mul_pow,hs]
+    have he : Real.exp (l.height^2*((n : ℝ)/2*(l.endpoint^2-q^2))/2)*
+        (gumbelTreeMoment l.height T*Real.exp (l.height*(n : ℝ)*treeCorrection l.endpoint T))=
+        gumbelTreeMoment l.height T*Real.exp (l.height*(n : ℝ)*treeCorrection q (l::T)) := by
+      rw [mul_left_comm,←Real.exp_add]
+      congr 2
+      simp only [treeCorrection]
+      ring
+    rw [he,Real.mul_rpow (gumbelTreeMoment_pos hT).le (Real.exp_pos _).le,←Real.exp_mul]
+    have he2 : l.height*(n : ℝ)*treeCorrection q (l::T)*(a/l.height)=a*(n : ℝ)*treeCorrection q (l::T) := by
+      field_simp
+    rw [he2]
+    simp only [gumbelTreeMoment,mul_assoc]
+
+lemma auxShape_mean (n : ℕ) {q : ℝ} {T : GaussianTree} (hq : 0 ≤ q)
+    (hgrid : ValidTreeGrid q T) (h : TreeScale 0 T) :
+    cascadeMean (auxShape n q T)=gumbelTreeMean T+(n : ℝ)*treeCorrection q T := by
+  cases T with
+  | nil => simp [auxShape,cascadeMean,gumbelTreeMean,treeCorrection]
+  | cons l T =>
+    obtain ⟨_,hm,hT⟩ := h
+    have hs := auxAmplitude_sq n hq hgrid.1
+    rw [auxShape,cascadeMean,auxShape_moment n (hq.trans hgrid.1) hgrid.2 hT]
+    rw [Real.log_mul (Real.exp_pos _).ne' (mul_pos (gumbelTreeMoment_pos hT) (Real.exp_pos _)).ne',
+      Real.log_exp,Real.log_mul (gumbelTreeMoment_pos hT).ne' (Real.exp_pos _).ne',Real.log_exp]
+    rw [mul_pow,hs]
+    simp only [gumbelTreeMean,treeCorrection]
+    field_simp
+    ring
+
+end SKValueG
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/GroundState/WeakGuerra.lean b/lean/OAI/Probability/SKValue/GroundState/WeakGuerra.lean
index 36232fe0a067af9954662141e9c4f24090ca77c3..7913b2c8e3540b27e8a1822df8cf03b779eb676e 100644
--- a/lean/OAI/Probability/SKValue/GroundState/WeakGuerra.lean
+++ b/lean/OAI/Probability/SKValue/GroundState/WeakGuerra.lean
@@ -1,147 +1,147 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.CascadePassage
-import OAI.Probability.SKValue.Equations.StrictProfile
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValueG
-
-lemma profile_penalty_mono (ε : ℝ≥0) (k : ℕ) {l : SKValue.HeatProfile}
-    (hT : SKValue.profileTime l=1) :
-    (∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t) ≤
-      ∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff (SKValue.liftProfile ε k l) t := by
-  have h1 := profile_weighted_integrable 0 l
-  have h2 := profile_weighted_integrable 0 (SKValue.liftProfile ε k l)
-  simp only [zero_add,SKValue.liftProfile_time,hT] at h1 h2
-  apply intervalIntegral.integral_mono_on (by norm_num) h1 h2
-  intro t ht
-  exact mul_le_mul_of_nonneg_left (SKValue.liftProfile_coeff_lower ε k l t) ht.1
-
-lemma weak_profile_guerra_upper (W : SKValue.BrownianSpace) {n : ℕ} (hn : 0<n)
-    {ψ : ℝ → ℝ} (hψ : SKValue.SmoothTerminal ψ) (hb : ∀ x,|x|≤ψ x)
-    {l : SKValue.HeatProfile} (hl : l.Pairwise (fun p r ↦ p.2≤r.2)) (hT : SKValue.profileTime l=1) :
-    expectedMaximum n ≤ (n : ℝ)*(SKValue.profileValue ψ l 0 0-
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)) := by
-  have he : ∀ ε : ℝ,0<ε → expectedMaximum n ≤ (n : ℝ)*(SKValue.profileValue ψ l 0 0-
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)+(3/2 : ℝ)*ε*(1+l.length)) := by
-    intro ε hε
-    let e : ℝ≥0 := ⟨ε,hε.le⟩
-    have hepos : 0<e := hε
-    have hs := profileTree_scale (q := 0) (by norm_num : (0 : ℝ)≤0)
-      (SKValue.liftProfile_strict hepos 1 hl) (SKValue.liftProfile_positive hepos (by norm_num : 0<1) l)
-    have hg := profileTree_valid (q := 0) (l := SKValue.liftProfile e 1 l)
-      (by rw [zero_add,SKValue.liftProfile_time,hT])
-    have hu := finite_step_guerra_upper hn _ hg hs
-    have hv := profileTree_le_smooth hψ hb hs 0
-    have hvb := abs_le.mp (hψ.liftProfile_bound W hl hT hepos 1 0)
-    have hc := profile_penalty_mono e 1 hT
-    rw [profileTree_correction] at hu
-    simp only [SKValue.liftProfile_time,hT,zero_add] at hu
-    have hnn : (0 : ℝ)≤n := by positivity
-    have hvb' : SKValue.profileValue ψ (SKValue.liftProfile e 1 l) 0 0-SKValue.profileValue ψ l 0 0 ≤
-        (3/2 : ℝ)*ε*(1+l.length) := by
-      have h := hvb.2
-      have hcoe : (e : ℝ)=ε := rfl
-      rw [Nat.cast_one,hcoe] at h
-      exact h
-    apply hu.trans
-    apply mul_le_mul_of_nonneg_left _ hnn
-    linarith
-  have hh : Tendsto (fun j : ℕ ↦ (n : ℝ)*(SKValue.profileValue ψ l 0 0-
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)+
-        (3/2 : ℝ)*(1/(j+1 : ℝ))*(1+l.length))) atTop
-      (𝓝 ((n : ℝ)*(SKValue.profileValue ψ l 0 0-
-        (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)))) := by
-    have hr : Tendsto (fun j : ℕ ↦ (3/2 : ℝ)*(1/(j+1 : ℝ))*(1+l.length)) atTop (𝓝 0) := by
-      simpa only [mul_zero,zero_mul] using
-        (tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (3/2 : ℝ)).mul_const (1+(l.length : ℝ))
-    simpa only [add_zero] using (hr.const_add (SKValue.profileValue ψ l 0 0-
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t))).const_mul (n : ℝ)
-  exact ge_of_tendsto hh (Eventually.of_forall (fun (j : ℕ) ↦ he (1/(j+1 : ℝ)) (one_div_pos.mpr (by exact_mod_cast Nat.succ_pos j))))
-
-end SKValueG
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma logCoshTerminal_ge_abs {M : ℝ} (hM : 0<M) (x : ℝ) :
-    |x|≤logCoshTerminal M x := by
-  have hh := (log_two_cosh_bounds (M*x)).1
-  rw [abs_mul,abs_of_pos hM] at hh
-  exact (le_div_iff₀ hM).mpr (by simpa only [mul_comm] using hh)
-
-lemma OrderParameter.weighted_cutoff_integral (γ : OrderParameter) :
-    (∫ t in (0 : ℝ)..1,t*γ.cutoff t)=∫ t in (0 : ℝ)..1,t*γ.coeff t := by
-  apply intervalIntegral.integral_congr_ae
-  filter_upwards [Measure.ae_ne volume (1 : ℝ)] with t hne ht
-  rw [uIoc_of_le (by norm_num : (0 : ℝ)≤1)] at ht
-  have ht' : t∈Ico (0 : ℝ) 1 := ⟨ht.1.le,lt_of_le_of_ne ht.2 hne⟩
-  simp only [OrderParameter.cutoff,indicator_of_mem ht']
-
-lemma OrderParameter.weighted_grid_error (γ : OrderParameter) (j : ℕ) :
-    |(∫ t in (0 : ℝ)..1,t*profileCoeff (γ.grid j) t)-(∫ t in (0 : ℝ)..1,t*γ.cutoff t)|≤γ.gridError j := by
-  have hi : IntervalIntegrable (profileCoeff (γ.grid j)) volume 0 1 :=
-    γ.grid_time j ▸ profileCoeff_integrable (γ.grid j)
-  have hj : IntervalIntegrable γ.cutoff volume 0 1 := γ.cutoff_integrable.intervalIntegrable
-  have hiw := hi.continuousOn_mul (continuous_id : Continuous (fun t : ℝ ↦ t)).continuousOn
-  have hjw := hj.continuousOn_mul (continuous_id : Continuous (fun t : ℝ ↦ t)).continuousOn
-  simp only [id_eq] at hiw hjw
-  rw [←intervalIntegral.integral_sub hiw hjw]
-  calc
-    _ ≤ ∫ t in (0 : ℝ)..1,|t*profileCoeff (γ.grid j) t-t*γ.cutoff t| :=
-      intervalIntegral.abs_integral_le_integral_abs (by norm_num)
-    _ ≤ ∫ t in (0 : ℝ)..1,|profileCoeff (γ.grid j) t-γ.cutoff t| := by
-      apply intervalIntegral.integral_mono_on (by norm_num) (hiw.sub hjw).abs (hi.sub hj).abs
-      intro t ht
-      rw [←mul_sub,abs_mul,abs_of_nonneg ht.1]
-      exact mul_le_of_le_one_left (abs_nonneg _) ht.2
-    _ = γ.gridError j := rfl
-
-lemma OrderParameter.weighted_grid_tendsto (γ : OrderParameter) :
-    Tendsto (fun j ↦ ∫ t in (0 : ℝ)..1,t*profileCoeff (γ.grid j) t) atTop
-      (𝓝 (∫ t in (0 : ℝ)..1,t*γ.coeff t)) := by
-  rw [←γ.weighted_cutoff_integral]
-  apply (tendsto_iff_dist_tendsto_zero).mpr
-  simp only [Real.dist_eq]
-  exact squeeze_zero (fun _ ↦ abs_nonneg _) γ.weighted_grid_error γ.gridError_tendsto
-
-theorem expectedMaximum_le_parisi (W : BrownianSpace) (γ : OrderParameter) {n : ℕ} (hn : 0<n) :
-    SKValueG.expectedMaximum n≤(n : ℝ)*parisi W γ := by
-  have ht := (smoothApprox_tendsto W γ).tendsto_at ((⟨0,by norm_num⟩ : Ico (0 : ℝ) 1),0)
-  have hp := γ.weighted_grid_tendsto
-  have hf : Tendsto (fun j ↦ (n : ℝ)*(smoothApprox γ j 0 0-
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*profileCoeff (γ.grid j) t))) atTop (𝓝 ((n : ℝ)*parisi W γ)) := by
-    simpa only [parisi] using tendsto_const_nhds.mul (ht.sub (tendsto_const_nhds.mul hp))
-  apply ge_of_tendsto hf
-  exact Eventually.of_forall (fun j ↦ SKValueG.weak_profile_guerra_upper W hn
-    (logCoshTerminal_smoothTerminal (by positivity : (0 : ℝ)<j+1))
-    (logCoshTerminal_ge_abs (by positivity)) (γ.grid_mono j) (γ.grid_time j))
-
-theorem groundStateValue_le_parisi (W : BrownianSpace) (γ : OrderParameter) :
-    groundStateValue≤parisi W γ := by
-  have he : SKValueG.groundStateSequence=groundStateSequence := rfl
-  have hev : SKValueG.groundStateValue=groundStateValue := rfl
-  have ht := SKValueG.groundStateSequence_tendsto
-  rw [he,hev] at ht
-  apply le_of_tendsto ht
-  filter_upwards [eventually_gt_atTop (0 : ℕ)] with n hn
-  change SKValueG.groundStateSequence n≤_
-  rw [SKValueG.groundStateSequence_eq]
-  have hn' : (0 : ℝ)<n := by exact_mod_cast hn
-  have hh := mul_le_mul_of_nonneg_left (expectedMaximum_le_parisi W γ hn) (inv_nonneg.mpr hn'.le)
-  simpa only [←mul_assoc,inv_mul_cancel₀ hn'.ne',one_mul] using hh
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.CascadePassage
+import OAI.Probability.SKValue.Equations.StrictProfile
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValueG
+
+lemma profile_penalty_mono (ε : ℝ≥0) (k : ℕ) {l : SKValue.HeatProfile}
+    (hT : SKValue.profileTime l=1) :
+    (∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t) ≤
+      ∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff (SKValue.liftProfile ε k l) t := by
+  have h1 := profile_weighted_integrable 0 l
+  have h2 := profile_weighted_integrable 0 (SKValue.liftProfile ε k l)
+  simp only [zero_add,SKValue.liftProfile_time,hT] at h1 h2
+  apply intervalIntegral.integral_mono_on (by norm_num) h1 h2
+  intro t ht
+  exact mul_le_mul_of_nonneg_left (SKValue.liftProfile_coeff_lower ε k l t) ht.1
+
+lemma weak_profile_guerra_upper (W : SKValue.BrownianSpace) {n : ℕ} (hn : 0<n)
+    {ψ : ℝ → ℝ} (hψ : SKValue.SmoothTerminal ψ) (hb : ∀ x,|x|≤ψ x)
+    {l : SKValue.HeatProfile} (hl : l.Pairwise (fun p r ↦ p.2≤r.2)) (hT : SKValue.profileTime l=1) :
+    expectedMaximum n ≤ (n : ℝ)*(SKValue.profileValue ψ l 0 0-
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)) := by
+  have he : ∀ ε : ℝ,0<ε → expectedMaximum n ≤ (n : ℝ)*(SKValue.profileValue ψ l 0 0-
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)+(3/2 : ℝ)*ε*(1+l.length)) := by
+    intro ε hε
+    let e : ℝ≥0 := ⟨ε,hε.le⟩
+    have hepos : 0<e := hε
+    have hs := profileTree_scale (q := 0) (by norm_num : (0 : ℝ)≤0)
+      (SKValue.liftProfile_strict hepos 1 hl) (SKValue.liftProfile_positive hepos (by norm_num : 0<1) l)
+    have hg := profileTree_valid (q := 0) (l := SKValue.liftProfile e 1 l)
+      (by rw [zero_add,SKValue.liftProfile_time,hT])
+    have hu := finite_step_guerra_upper hn _ hg hs
+    have hv := profileTree_le_smooth hψ hb hs 0
+    have hvb := abs_le.mp (hψ.liftProfile_bound W hl hT hepos 1 0)
+    have hc := profile_penalty_mono e 1 hT
+    rw [profileTree_correction] at hu
+    simp only [SKValue.liftProfile_time,hT,zero_add] at hu
+    have hnn : (0 : ℝ)≤n := by positivity
+    have hvb' : SKValue.profileValue ψ (SKValue.liftProfile e 1 l) 0 0-SKValue.profileValue ψ l 0 0 ≤
+        (3/2 : ℝ)*ε*(1+l.length) := by
+      have h := hvb.2
+      have hcoe : (e : ℝ)=ε := rfl
+      rw [Nat.cast_one,hcoe] at h
+      exact h
+    apply hu.trans
+    apply mul_le_mul_of_nonneg_left _ hnn
+    linarith
+  have hh : Tendsto (fun j : ℕ ↦ (n : ℝ)*(SKValue.profileValue ψ l 0 0-
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)+
+        (3/2 : ℝ)*(1/(j+1 : ℝ))*(1+l.length))) atTop
+      (𝓝 ((n : ℝ)*(SKValue.profileValue ψ l 0 0-
+        (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t)))) := by
+    have hr : Tendsto (fun j : ℕ ↦ (3/2 : ℝ)*(1/(j+1 : ℝ))*(1+l.length)) atTop (𝓝 0) := by
+      simpa only [mul_zero,zero_mul] using
+        (tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (3/2 : ℝ)).mul_const (1+(l.length : ℝ))
+    simpa only [add_zero] using (hr.const_add (SKValue.profileValue ψ l 0 0-
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*SKValue.profileCoeff l t))).const_mul (n : ℝ)
+  exact ge_of_tendsto hh (Eventually.of_forall (fun (j : ℕ) ↦ he (1/(j+1 : ℝ)) (one_div_pos.mpr (by exact_mod_cast Nat.succ_pos j))))
+
+end SKValueG
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma logCoshTerminal_ge_abs {M : ℝ} (hM : 0<M) (x : ℝ) :
+    |x|≤logCoshTerminal M x := by
+  have hh := (log_two_cosh_bounds (M*x)).1
+  rw [abs_mul,abs_of_pos hM] at hh
+  exact (le_div_iff₀ hM).mpr (by simpa only [mul_comm] using hh)
+
+lemma OrderParameter.weighted_cutoff_integral (γ : OrderParameter) :
+    (∫ t in (0 : ℝ)..1,t*γ.cutoff t)=∫ t in (0 : ℝ)..1,t*γ.coeff t := by
+  apply intervalIntegral.integral_congr_ae
+  filter_upwards [Measure.ae_ne volume (1 : ℝ)] with t hne ht
+  rw [uIoc_of_le (by norm_num : (0 : ℝ)≤1)] at ht
+  have ht' : t∈Ico (0 : ℝ) 1 := ⟨ht.1.le,lt_of_le_of_ne ht.2 hne⟩
+  simp only [OrderParameter.cutoff,indicator_of_mem ht']
+
+lemma OrderParameter.weighted_grid_error (γ : OrderParameter) (j : ℕ) :
+    |(∫ t in (0 : ℝ)..1,t*profileCoeff (γ.grid j) t)-(∫ t in (0 : ℝ)..1,t*γ.cutoff t)|≤γ.gridError j := by
+  have hi : IntervalIntegrable (profileCoeff (γ.grid j)) volume 0 1 :=
+    γ.grid_time j ▸ profileCoeff_integrable (γ.grid j)
+  have hj : IntervalIntegrable γ.cutoff volume 0 1 := γ.cutoff_integrable.intervalIntegrable
+  have hiw := hi.continuousOn_mul (continuous_id : Continuous (fun t : ℝ ↦ t)).continuousOn
+  have hjw := hj.continuousOn_mul (continuous_id : Continuous (fun t : ℝ ↦ t)).continuousOn
+  simp only [id_eq] at hiw hjw
+  rw [←intervalIntegral.integral_sub hiw hjw]
+  calc
+    _ ≤ ∫ t in (0 : ℝ)..1,|t*profileCoeff (γ.grid j) t-t*γ.cutoff t| :=
+      intervalIntegral.abs_integral_le_integral_abs (by norm_num)
+    _ ≤ ∫ t in (0 : ℝ)..1,|profileCoeff (γ.grid j) t-γ.cutoff t| := by
+      apply intervalIntegral.integral_mono_on (by norm_num) (hiw.sub hjw).abs (hi.sub hj).abs
+      intro t ht
+      rw [←mul_sub,abs_mul,abs_of_nonneg ht.1]
+      exact mul_le_of_le_one_left (abs_nonneg _) ht.2
+    _ = γ.gridError j := rfl
+
+lemma OrderParameter.weighted_grid_tendsto (γ : OrderParameter) :
+    Tendsto (fun j ↦ ∫ t in (0 : ℝ)..1,t*profileCoeff (γ.grid j) t) atTop
+      (𝓝 (∫ t in (0 : ℝ)..1,t*γ.coeff t)) := by
+  rw [←γ.weighted_cutoff_integral]
+  apply (tendsto_iff_dist_tendsto_zero).mpr
+  simp only [Real.dist_eq]
+  exact squeeze_zero (fun _ ↦ abs_nonneg _) γ.weighted_grid_error γ.gridError_tendsto
+
+theorem expectedMaximum_le_parisi (W : BrownianSpace) (γ : OrderParameter) {n : ℕ} (hn : 0<n) :
+    SKValueG.expectedMaximum n≤(n : ℝ)*parisi W γ := by
+  have ht := (smoothApprox_tendsto W γ).tendsto_at ((⟨0,by norm_num⟩ : Ico (0 : ℝ) 1),0)
+  have hp := γ.weighted_grid_tendsto
+  have hf : Tendsto (fun j ↦ (n : ℝ)*(smoothApprox γ j 0 0-
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*profileCoeff (γ.grid j) t))) atTop (𝓝 ((n : ℝ)*parisi W γ)) := by
+    simpa only [parisi] using tendsto_const_nhds.mul (ht.sub (tendsto_const_nhds.mul hp))
+  apply ge_of_tendsto hf
+  exact Eventually.of_forall (fun j ↦ SKValueG.weak_profile_guerra_upper W hn
+    (logCoshTerminal_smoothTerminal (by positivity : (0 : ℝ)<j+1))
+    (logCoshTerminal_ge_abs (by positivity)) (γ.grid_mono j) (γ.grid_time j))
+
+theorem groundStateValue_le_parisi (W : BrownianSpace) (γ : OrderParameter) :
+    groundStateValue≤parisi W γ := by
+  have he : SKValueG.groundStateSequence=groundStateSequence := rfl
+  have hev : SKValueG.groundStateValue=groundStateValue := rfl
+  have ht := SKValueG.groundStateSequence_tendsto
+  rw [he,hev] at ht
+  apply le_of_tendsto ht
+  filter_upwards [eventually_gt_atTop (0 : ℕ)] with n hn
+  change SKValueG.groundStateSequence n≤_
+  rw [SKValueG.groundStateSequence_eq]
+  have hn' : (0 : ℝ)<n := by exact_mod_cast hn
+  have hh := mul_le_mul_of_nonneg_left (expectedMaximum_le_parisi W γ hn) (inv_nonneg.mpr hn'.le)
+  simpa only [←mul_assoc,inv_mul_cancel₀ hn'.ne',one_mul] using hh
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Main.lean b/lean/OAI/Probability/SKValue/Main.lean
index 7af6c791d4a703c79c49c0c60533dc474260bcc5..0fcd7ca01b0f99b90f579f69a7bed99074bcdd99 100644
--- a/lean/OAI/Probability/SKValue/Main.lean
+++ b/lean/OAI/Probability/SKValue/Main.lean
@@ -1,28 +1,28 @@
-import Mathlib
-import OAI.Probability.SKValue.Gaussian.IntegralPositiveMulGaussian
-import OAI.Probability.SKValue.Processes.TestExpectation
-import OAI.Probability.SKValue.Processes.ForcedExpectation
-import OAI.Probability.SKValue.Processes.ForcedDiffusion
-import OAI.Probability.SKValue.Equations.PolynomialForced
-import OAI.Probability.SKValue.Evolution.HeatConvexity
-import OAI.Probability.SKValue.Evolution.TerminalAlgebra
-import OAI.Probability.SKValue.Evolution.TestStrip
-import OAI.Probability.SKValue.GroundState.MainConclusions
-
-namespace OAI
-
-namespace SKValue
-
-theorem value_consequences :
-    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ),
-      IsMinimizer W γ → IsDiffusion W γ X → ScalarValueConclusion W γ X) ∧
-    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) (T : ℝ),
-      0 < T → T < 1 → IsMinimizer W γ → IsDiffusion W γ X →
-      FiniteGaussianConclusion W γ X T) ∧
-    (∀ (W : BrownianSpace) (γ : OrderParameter),
-      IsMinimizer W γ → groundStateValue = parisi W γ) := by
-  exact ⟨scalar_value_target, finite_gaussian_target, ground_state_parisi_target⟩
-
-end SKValue
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Gaussian.IntegralPositiveMulGaussian
+import OAI.Probability.SKValue.Processes.TestExpectation
+import OAI.Probability.SKValue.Processes.ForcedExpectation
+import OAI.Probability.SKValue.Processes.ForcedDiffusion
+import OAI.Probability.SKValue.Equations.PolynomialForced
+import OAI.Probability.SKValue.Evolution.HeatConvexity
+import OAI.Probability.SKValue.Evolution.TerminalAlgebra
+import OAI.Probability.SKValue.Evolution.TestStrip
+import OAI.Probability.SKValue.GroundState.MainConclusions
+
+namespace OAI
+
+namespace SKValue
+
+theorem value_consequences :
+    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ),
+      IsMinimizer W γ → IsDiffusion W γ X → ScalarValueConclusion W γ X) ∧
+    (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) (T : ℝ),
+      0 < T → T < 1 → IsMinimizer W γ → IsDiffusion W γ X →
+      FiniteGaussianConclusion W γ X T) ∧
+    (∀ (W : BrownianSpace) (γ : OrderParameter),
+      IsMinimizer W γ → groundStateValue = parisi W γ) := by
+  exact ⟨scalar_value_target, finite_gaussian_target, ground_state_parisi_target⟩
+
+end SKValue
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Model.lean b/lean/OAI/Probability/SKValue/Model.lean
index 19d54e0e2e8b348aa467da522265c7e53ebfa620..f0cc58f200a7e23fa7f43110dcd6e0794591dfce 100644
--- a/lean/OAI/Probability/SKValue/Model.lean
+++ b/lean/OAI/Probability/SKValue/Model.lean
@@ -1,165 +1,165 @@
-import Mathlib
-
-namespace OAI
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology ENNReal NNReal BigOperators
-
-namespace SKValue
-
-noncomputable def standardGaussian : Measure ℝ := gaussianReal 0 1
-
-instance : IsProbabilityMeasure standardGaussian := by unfold standardGaussian; infer_instance
-
-noncomputable def gaussianProduct (ι : Type) [Fintype ι] : Measure (ι → ℝ) :=
-  Measure.pi (fun _ : ι ↦ standardGaussian)
-instance gaussianProduct_probability (ι : Type) [Fintype ι] :
-    IsProbabilityMeasure (gaussianProduct ι) := by
-  unfold gaussianProduct
-  infer_instance
-
-def coordinate (N : ℕ) (j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
-  z ⟨j % (N+1), Nat.mod_lt _ (Nat.succ_pos _)⟩
-
-noncomputable def stepSize (T : ℝ) (N : ℕ) : ℝ := T / N
-noncomputable def meshTime (T : ℝ) (N j : ℕ) : ℝ := j * stepSize T N
-
-noncomputable def euler (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (z : Fin (N+1) → ℝ) : ℕ → ℝ
-  | 0 => 0
-  | j+1 => euler T N γ u z j + Real.sqrt (stepSize T N) * coordinate N j z +
-      stepSize T N * γ (meshTime T N j) * u (meshTime T N j) (euler T N γ u z j)
-
-structure OrderParameter where
-  coeff : ℝ → ℝ
-  nonneg : ∀ t ∈ Ico (0 : ℝ) 1, 0 ≤ coeff t
-  monotone : MonotoneOn coeff (Ico (0 : ℝ) 1)
-  rightContinuous : ∀ t ∈ Ico (0 : ℝ) 1, ContinuousWithinAt coeff (Ici t) t
-  integrable : IntegrableOn coeff (Ioc (0 : ℝ) 1)
-
-structure BrownianSpace where
-  Ω : Type
-  measurableSpace : MeasurableSpace Ω
-  μ : @Measure Ω measurableSpace
-  probability : @IsProbabilityMeasure Ω measurableSpace μ
-  B : ℝ≥0 → Ω → ℝ
-  measurable : ∀ t, @StronglyMeasurable Ω ℝ _ measurableSpace (B t)
-  brownian : @IsBrownianReal Ω measurableSpace B μ
-
-attribute [instance] BrownianSpace.measurableSpace BrownianSpace.probability
-
-noncomputable def BrownianSpace.filtration (W : BrownianSpace) :
-    Filtration ℝ≥0 W.measurableSpace := Filtration.natural W.B W.measurable
-
-noncomputable def shiftedFiltration (W : BrownianSpace) (t : ℝ) :
-    Filtration ℝ≥0 W.measurableSpace :=
-  Filtration.natural (fun s ω ↦ W.B (Real.toNNReal t + s) ω - W.B (Real.toNNReal t) ω)
-    (fun _ ↦ (W.measurable _).sub (W.measurable _))
-
-def Admissible (W : BrownianSpace) (t : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : Prop :=
-  IsProgressive (shiftedFiltration W t) α ∧ ∀ s ω, |α s ω| ≤ 1
-
-noncomputable def controlPayoff (W : BrownianSpace) (γ : OrderParameter)
-    (t x : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : ℝ :=
-  ∫ ω, (|x + W.B 1 ω - W.B (Real.toNNReal t) ω +
-    ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω| -
-    (1/2 : ℝ) * ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω ^ 2) ∂W.μ
-
-noncomputable def phi (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=
-  sSup {r | ∃ α, Admissible W t α ∧ r = controlPayoff W γ t x α}
-
-noncomputable def gradient (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=
-  deriv (phi W γ t) x
-
-noncomputable def curvature (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=
-  deriv (gradient W γ t) x
-
-noncomputable def parisi (W : BrownianSpace) (γ : OrderParameter) : ℝ :=
-  phi W γ 0 0 - (1/2 : ℝ) * ∫ t in (0 : ℝ)..1, t * γ.coeff t
-
-def IsMinimizer (W : BrownianSpace) (γ : OrderParameter) : Prop :=
-  ∀ η : OrderParameter, parisi W γ ≤ parisi W η
-
-def IsDiffusion (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) : Prop :=
-  Adapted W.filtration (fun t ω ↦ X (min (t : ℝ) 1) ω) ∧
-  (∀ᵐ ω ∂W.μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) 1) ∧
-    ∀ t ∈ Icc (0 : ℝ) 1,
-      X t ω = W.B (Real.toNNReal t) ω +
-        ∫ s in (0 : ℝ)..t, γ.coeff s * gradient W γ s (X s ω))
-
-noncomputable def spin (b : Bool) : ℝ := if b then 1 else -1
-
-noncomputable def hamiltonian (n : ℕ) (J : (Fin n × Fin n) → ℝ)
-    (σ : Fin n → Bool) : ℝ :=
-  (Real.sqrt (n : ℝ))⁻¹ *
-    ∑ i : Fin n, ∑ j : Fin n, if i < j then J (i,j) * spin (σ i) * spin (σ j) else 0
-
-noncomputable def groundStateSequence (n : ℕ) : ℝ :=
-  (n : ℝ)⁻¹ * ∫ J, sSup (range (hamiltonian n J)) ∂gaussianProduct (Fin n × Fin n)
-
-noncomputable def groundStateValue : ℝ := limUnder atTop groundStateSequence
-
-noncomputable def gradientStopped (W : BrownianSpace) (γ : OrderParameter)
-    (X : ℝ → W.Ω → ℝ) (U : W.Ω → ℝ) (t : ℝ≥0) (ω : W.Ω) : ℝ :=
-  if (t : ℝ) < 1 then gradient W γ t (X t ω) else U ω
-
-def ScalarValueConclusion (W : BrownianSpace) (γ : OrderParameter)
-    (X : ℝ → W.Ω → ℝ) : Prop :=
-  Tendsto groundStateSequence atTop (𝓝 groundStateValue) ∧
-  ∃ U : W.Ω → ℝ,
-    Martingale (gradientStopped W γ X U) W.filtration W.μ ∧
-    (∀ᵐ ω ∂W.μ, |U ω| ≤ 1 ∧ (U ω)^2 = 1 ∧
-      Tendsto (fun t ↦ gradient W γ t (X t ω)) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) ∧
-    Tendsto (fun t ↦ ∫ ω, (gradient W γ t (X t ω) - U ω)^2 ∂W.μ)
-      (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) ∧
-    groundStateValue = (∫ ω, |X 1 ω| ∂W.μ) - ∫ t in (0 : ℝ)..1, t * γ.coeff t ∧
-    groundStateValue = ∫ ω, U ω * W.B 1 ω ∂W.μ ∧
-    groundStateValue = ∫ t in (0 : ℝ)..1, ∫ ω, curvature W γ t (X t ω) ∂W.μ ∧
-    IntervalIntegrable (fun t ↦ ∫ ω, curvature W γ t (X t ω) ∂W.μ) volume 0 1
-
-noncomputable def eulerPath (W : BrownianSpace) (γ : OrderParameter) (T : ℝ) (N : ℕ)
-    (z : Fin (N+1) → ℝ) (j : ℕ) : ℝ :=
-  euler T N γ.coeff (gradient W γ) z j
-
-noncomputable def rawCoefficient (W : BrownianSpace) (γ : OrderParameter)
-    (T : ℝ) (N j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
-  curvature W γ (meshTime T N j) (eulerPath W γ T N z j)
-
-noncomputable def normalizer (W : BrownianSpace) (γ : OrderParameter)
-    (T : ℝ) (N j : ℕ) : ℝ :=
-  (Real.sqrt (∫ z, (rawCoefficient W γ T N j z)^2 ∂gaussianProduct (Fin (N+1))))⁻¹
-
-noncomputable def gaussianIncrement (W : BrownianSpace) (γ : OrderParameter)
-    (T : ℝ) (N j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
-  normalizer W γ T N j * rawCoefficient W γ T N j z * coordinate N j z
-
-noncomputable def roundedSign (W : BrownianSpace) (γ : OrderParameter)
-    (T : ℝ) (N : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
-  Real.sign (gradient W γ T (eulerPath W γ T N z N) +
-    2 * cdf standardGaussian (coordinate N N z) - 1)
-
-noncomputable def coefficientA (W : BrownianSpace) (γ : OrderParameter)
-    (T : ℝ) (N j : ℕ) : ℝ :=
-  if j = 0 then 0 else
-    ∫ z, roundedSign W γ T N z * gaussianIncrement W γ T N (j-1) z
-      ∂gaussianProduct (Fin (N+1))
-
-noncomputable def coefficientB (W : BrownianSpace) (γ : OrderParameter)
-    (T : ℝ) (N i : ℕ) : ℝ :=
-  ∫ z, roundedSign W γ T N z * coordinate N (i-1) z ∂gaussianProduct (Fin (N+1))
-
-noncomputable def shiftedCoefficientSum (W : BrownianSpace) (γ : OrderParameter)
-    (T : ℝ) (N : ℕ) : ℝ :=
-  ∑ j ∈ Finset.range (N+1), coefficientA W γ T N j * coefficientB W γ T N (j+1)
-
-def FiniteGaussianConclusion (W : BrownianSpace) (γ : OrderParameter)
-    (X : ℝ → W.Ω → ℝ) (T : ℝ) : Prop :=
-  (∀ᶠ N in atTop, ∀ j < N, 0 < normalizer W γ T N j) ∧
-  Tendsto (shiftedCoefficientSum W γ T) atTop
-    (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, curvature W γ t (X t ω) ∂W.μ)) ∧
-  Tendsto (fun S ↦ ∫ t in (0 : ℝ)..S, ∫ ω, curvature W γ t (X t ω) ∂W.μ)
-    (𝓝[<] (1 : ℝ)) (𝓝 groundStateValue)
-
-end SKValue
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology ENNReal NNReal BigOperators
+
+namespace SKValue
+
+noncomputable def standardGaussian : Measure ℝ := gaussianReal 0 1
+
+instance : IsProbabilityMeasure standardGaussian := by unfold standardGaussian; infer_instance
+
+noncomputable def gaussianProduct (ι : Type) [Fintype ι] : Measure (ι → ℝ) :=
+  Measure.pi (fun _ : ι ↦ standardGaussian)
+instance gaussianProduct_probability (ι : Type) [Fintype ι] :
+    IsProbabilityMeasure (gaussianProduct ι) := by
+  unfold gaussianProduct
+  infer_instance
+
+def coordinate (N : ℕ) (j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
+  z ⟨j % (N+1), Nat.mod_lt _ (Nat.succ_pos _)⟩
+
+noncomputable def stepSize (T : ℝ) (N : ℕ) : ℝ := T / N
+noncomputable def meshTime (T : ℝ) (N j : ℕ) : ℝ := j * stepSize T N
+
+noncomputable def euler (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (z : Fin (N+1) → ℝ) : ℕ → ℝ
+  | 0 => 0
+  | j+1 => euler T N γ u z j + Real.sqrt (stepSize T N) * coordinate N j z +
+      stepSize T N * γ (meshTime T N j) * u (meshTime T N j) (euler T N γ u z j)
+
+structure OrderParameter where
+  coeff : ℝ → ℝ
+  nonneg : ∀ t ∈ Ico (0 : ℝ) 1, 0 ≤ coeff t
+  monotone : MonotoneOn coeff (Ico (0 : ℝ) 1)
+  rightContinuous : ∀ t ∈ Ico (0 : ℝ) 1, ContinuousWithinAt coeff (Ici t) t
+  integrable : IntegrableOn coeff (Ioc (0 : ℝ) 1)
+
+structure BrownianSpace where
+  Ω : Type
+  measurableSpace : MeasurableSpace Ω
+  μ : @Measure Ω measurableSpace
+  probability : @IsProbabilityMeasure Ω measurableSpace μ
+  B : ℝ≥0 → Ω → ℝ
+  measurable : ∀ t, @StronglyMeasurable Ω ℝ _ measurableSpace (B t)
+  brownian : @IsBrownianReal Ω measurableSpace B μ
+
+attribute [instance] BrownianSpace.measurableSpace BrownianSpace.probability
+
+noncomputable def BrownianSpace.filtration (W : BrownianSpace) :
+    Filtration ℝ≥0 W.measurableSpace := Filtration.natural W.B W.measurable
+
+noncomputable def shiftedFiltration (W : BrownianSpace) (t : ℝ) :
+    Filtration ℝ≥0 W.measurableSpace :=
+  Filtration.natural (fun s ω ↦ W.B (Real.toNNReal t + s) ω - W.B (Real.toNNReal t) ω)
+    (fun _ ↦ (W.measurable _).sub (W.measurable _))
+
+def Admissible (W : BrownianSpace) (t : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : Prop :=
+  IsProgressive (shiftedFiltration W t) α ∧ ∀ s ω, |α s ω| ≤ 1
+
+noncomputable def controlPayoff (W : BrownianSpace) (γ : OrderParameter)
+    (t x : ℝ) (α : ℝ≥0 → W.Ω → ℝ) : ℝ :=
+  ∫ ω, (|x + W.B 1 ω - W.B (Real.toNNReal t) ω +
+    ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω| -
+    (1/2 : ℝ) * ∫ s in (0 : ℝ)..(1-t), γ.coeff (t+s) * α (Real.toNNReal s) ω ^ 2) ∂W.μ
+
+noncomputable def phi (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=
+  sSup {r | ∃ α, Admissible W t α ∧ r = controlPayoff W γ t x α}
+
+noncomputable def gradient (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=
+  deriv (phi W γ t) x
+
+noncomputable def curvature (W : BrownianSpace) (γ : OrderParameter) (t x : ℝ) : ℝ :=
+  deriv (gradient W γ t) x
+
+noncomputable def parisi (W : BrownianSpace) (γ : OrderParameter) : ℝ :=
+  phi W γ 0 0 - (1/2 : ℝ) * ∫ t in (0 : ℝ)..1, t * γ.coeff t
+
+def IsMinimizer (W : BrownianSpace) (γ : OrderParameter) : Prop :=
+  ∀ η : OrderParameter, parisi W γ ≤ parisi W η
+
+def IsDiffusion (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) : Prop :=
+  Adapted W.filtration (fun t ω ↦ X (min (t : ℝ) 1) ω) ∧
+  (∀ᵐ ω ∂W.μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) 1) ∧
+    ∀ t ∈ Icc (0 : ℝ) 1,
+      X t ω = W.B (Real.toNNReal t) ω +
+        ∫ s in (0 : ℝ)..t, γ.coeff s * gradient W γ s (X s ω))
+
+noncomputable def spin (b : Bool) : ℝ := if b then 1 else -1
+
+noncomputable def hamiltonian (n : ℕ) (J : (Fin n × Fin n) → ℝ)
+    (σ : Fin n → Bool) : ℝ :=
+  (Real.sqrt (n : ℝ))⁻¹ *
+    ∑ i : Fin n, ∑ j : Fin n, if i < j then J (i,j) * spin (σ i) * spin (σ j) else 0
+
+noncomputable def groundStateSequence (n : ℕ) : ℝ :=
+  (n : ℝ)⁻¹ * ∫ J, sSup (range (hamiltonian n J)) ∂gaussianProduct (Fin n × Fin n)
+
+noncomputable def groundStateValue : ℝ := limUnder atTop groundStateSequence
+
+noncomputable def gradientStopped (W : BrownianSpace) (γ : OrderParameter)
+    (X : ℝ → W.Ω → ℝ) (U : W.Ω → ℝ) (t : ℝ≥0) (ω : W.Ω) : ℝ :=
+  if (t : ℝ) < 1 then gradient W γ t (X t ω) else U ω
+
+def ScalarValueConclusion (W : BrownianSpace) (γ : OrderParameter)
+    (X : ℝ → W.Ω → ℝ) : Prop :=
+  Tendsto groundStateSequence atTop (𝓝 groundStateValue) ∧
+  ∃ U : W.Ω → ℝ,
+    Martingale (gradientStopped W γ X U) W.filtration W.μ ∧
+    (∀ᵐ ω ∂W.μ, |U ω| ≤ 1 ∧ (U ω)^2 = 1 ∧
+      Tendsto (fun t ↦ gradient W γ t (X t ω)) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) ∧
+    Tendsto (fun t ↦ ∫ ω, (gradient W γ t (X t ω) - U ω)^2 ∂W.μ)
+      (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) ∧
+    groundStateValue = (∫ ω, |X 1 ω| ∂W.μ) - ∫ t in (0 : ℝ)..1, t * γ.coeff t ∧
+    groundStateValue = ∫ ω, U ω * W.B 1 ω ∂W.μ ∧
+    groundStateValue = ∫ t in (0 : ℝ)..1, ∫ ω, curvature W γ t (X t ω) ∂W.μ ∧
+    IntervalIntegrable (fun t ↦ ∫ ω, curvature W γ t (X t ω) ∂W.μ) volume 0 1
+
+noncomputable def eulerPath (W : BrownianSpace) (γ : OrderParameter) (T : ℝ) (N : ℕ)
+    (z : Fin (N+1) → ℝ) (j : ℕ) : ℝ :=
+  euler T N γ.coeff (gradient W γ) z j
+
+noncomputable def rawCoefficient (W : BrownianSpace) (γ : OrderParameter)
+    (T : ℝ) (N j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
+  curvature W γ (meshTime T N j) (eulerPath W γ T N z j)
+
+noncomputable def normalizer (W : BrownianSpace) (γ : OrderParameter)
+    (T : ℝ) (N j : ℕ) : ℝ :=
+  (Real.sqrt (∫ z, (rawCoefficient W γ T N j z)^2 ∂gaussianProduct (Fin (N+1))))⁻¹
+
+noncomputable def gaussianIncrement (W : BrownianSpace) (γ : OrderParameter)
+    (T : ℝ) (N j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
+  normalizer W γ T N j * rawCoefficient W γ T N j z * coordinate N j z
+
+noncomputable def roundedSign (W : BrownianSpace) (γ : OrderParameter)
+    (T : ℝ) (N : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
+  Real.sign (gradient W γ T (eulerPath W γ T N z N) +
+    2 * cdf standardGaussian (coordinate N N z) - 1)
+
+noncomputable def coefficientA (W : BrownianSpace) (γ : OrderParameter)
+    (T : ℝ) (N j : ℕ) : ℝ :=
+  if j = 0 then 0 else
+    ∫ z, roundedSign W γ T N z * gaussianIncrement W γ T N (j-1) z
+      ∂gaussianProduct (Fin (N+1))
+
+noncomputable def coefficientB (W : BrownianSpace) (γ : OrderParameter)
+    (T : ℝ) (N i : ℕ) : ℝ :=
+  ∫ z, roundedSign W γ T N z * coordinate N (i-1) z ∂gaussianProduct (Fin (N+1))
+
+noncomputable def shiftedCoefficientSum (W : BrownianSpace) (γ : OrderParameter)
+    (T : ℝ) (N : ℕ) : ℝ :=
+  ∑ j ∈ Finset.range (N+1), coefficientA W γ T N j * coefficientB W γ T N (j+1)
+
+def FiniteGaussianConclusion (W : BrownianSpace) (γ : OrderParameter)
+    (X : ℝ → W.Ω → ℝ) (T : ℝ) : Prop :=
+  (∀ᶠ N in atTop, ∀ j < N, 0 < normalizer W γ T N j) ∧
+  Tendsto (shiftedCoefficientSum W γ T) atTop
+    (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, curvature W γ t (X t ω) ∂W.μ)) ∧
+  Tendsto (fun S ↦ ∫ t in (0 : ℝ)..S, ∫ ω, curvature W γ t (X t ω) ∂W.μ)
+    (𝓝[<] (1 : ℝ)) (𝓝 groundStateValue)
+
+end SKValue
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/CrossMesh.lean b/lean/OAI/Probability/SKValue/Processes/CrossMesh.lean
index 74570a4712c2a992c0a69525add7bfb63a65c0da..7657c91e35ec9514bb6e753ee12e90d12d9c8d1c 100644
--- a/lean/OAI/Probability/SKValue/Processes/CrossMesh.lean
+++ b/lean/OAI/Probability/SKValue/Processes/CrossMesh.lean
@@ -1,153 +1,153 @@
-import Mathlib
-import OAI.Probability.SKValue.GroundState.CrossExpectation
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma abs_telescope_source {N : ℕ} {A b Γ Δ : ℕ → ℝ} {c d e : ℝ}
-    (h : ∀ j<N,|A (j+1)-A j+b j|≤c+d*(Γ (j+1)-Γ j)+e*(Δ (j+1)-Δ j)) :
-    |A N-A 0+∑ j∈Finset.range N,b j|≤
-      (N : ℝ)*c+d*(Γ N-Γ 0)+e*(Δ N-Δ 0) := by
-  have hs := (Finset.abs_sum_le_sum_abs (fun j ↦ A (j+1)-A j+b j) (Finset.range N)).trans
-    (Finset.sum_le_sum (fun j hj ↦ h j (Finset.mem_range.mp hj)))
-  simpa only [Finset.sum_add_distrib,Finset.sum_range_sub,←Finset.mul_sum,
-    Finset.sum_const,Finset.card_range,nsmul_eq_mul] using hs
-
-noncomputable def crossMeshSource (T : ℝ) (N : ℕ) (γ β : ℝ → ℝ) (V U : ℝ → ℝ → ℝ) : ℝ :=
-  ∑ j∈Finset.range N,(stepSize T N/2)*(β (meshTime T N j)-γ (meshTime T N j))*
-    (∫ z,(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2
-      ∂gaussianProduct (Fin (N+1)))
-
-lemma cross_value_mesh_expectation {T K L K' L' G : ℝ}
-    {γ β : ℝ → ℝ} {V U : ℝ → ℝ → ℝ}
-    (hV : ValueStrip T γ V K L) (hU : ValueStrip T β U K' L')
-    (hT : 0<T) (hT1 : T≤1) (hG : γ T≤G) (hG' : β T≤G) {N : ℕ} (hN : 0<N) :
-    |(∫ z,U T (euler T N β (crossFeedback V U) z N)-
-      V T (euler T N β (crossFeedback V U) z N) ∂gaussianProduct (Fin (N+1)))-
-      (U 0 0-V 0 0)+crossMeshSource T N γ β V U|≤
-    T*Real.sqrt (stepSize T N)*(cubicEnvelopeMean G K ((1/2+G)*L)+
-      cubicEnvelopeMean G K' ((1/2+G)*L'))+
-      stepSize T N*K*(γ T-γ 0)+stepSize T N*K'*(β T-β 0) := by
-  let δ := stepSize T N
-  let Y := euler T N β (crossFeedback V U)
-  let A := fun j ↦ ∫ z,U (meshTime T N j) (Y z j)-V (meshTime T N j) (Y z j)
-    ∂gaussianProduct (Fin (N+1))
-  let b := fun j ↦ (δ/2)*(β (meshTime T N j)-γ (meshTime T N j))*
-    (∫ z,(deriv (V (meshTime T N j)) (Y z j))^2 ∂gaussianProduct (Fin (N+1)))
-  have hNr : 0<(N : ℝ) := by exact_mod_cast hN
-  have hδ : 0≤δ := div_nonneg hT.le hNr.le
-  have hδ1 : δ≤1 := by
-    apply (div_le_iff₀ hNr).mpr
-    have hN1 : (1 : ℝ)≤N := by exact_mod_cast hN
-    simpa only [one_mul] using hT1.trans hN1
-  have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT.le hN hj
-  have hs (j : ℕ) : meshTime T N j+δ=meshTime T N (j+1) := by
-    dsimp [meshTime]; push_cast; ring
-  have hlast : meshTime T N N=T := by dsimp [meshTime,stepSize]; field_simp
-  have hzero : meshTime T N 0=0 := by simp [meshTime]
-  have hfm (j : ℕ) (hj : j<N) : Measurable (crossFeedback V U (meshTime T N j)) :=
-    (((hU.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable.add
-      ((hV.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable).div_const 2
-  have hfb (j : ℕ) (hj : j<N) (x : ℝ) : |crossFeedback V U (meshTime T N j) x|≤1 :=
-    crossFeedback_bound (hV.bounded _ (htime j hj.le) x) (hU.bounded _ (htime j hj.le) x)
-  have hstep (j : ℕ) (hj : j<N) : |A (j+1)-A j+b j|≤
-      δ*Real.sqrt δ*(cubicEnvelopeMean G K ((1/2+G)*L)+cubicEnvelopeMean G K' ((1/2+G)*L'))+
-      δ*K*(γ (meshTime T N (j+1))-γ (meshTime T N j))+
-      δ*K'*(β (meshTime T N (j+1))-β (meshTime T N j)) := by
-    have hYp : DependsBefore j (fun z ↦ Y z j) :=
-      eulerCoefficient_dependsBefore T hj.le β (crossFeedback V U) (fun _ y ↦ y)
-    have he := cross_value_expectation_step hV hU (htime j hj.le).1 hδ hδ1
-      (by rw [hs]; exact (htime (j+1) (by omega)).2) hG hG' (by omega : j<N+1)
-      (measurable_euler T N β (crossFeedback V U) hfm j hj.le) hYp
-      (euler_memLp T N β (crossFeedback V U) hfm hfb j hj.le)
-    dsimp only at he
-    rw [hs] at he
-    simpa only [A,b,Y,euler,δ,mul_assoc] using he
-  have hh := abs_telescope_source (A := A) (b := b)
-    (Γ := fun j ↦ γ (meshTime T N j)) (Δ := fun j ↦ β (meshTime T N j)) hstep
-  have hA0 : A 0=U 0 0-V 0 0 := by simp [A,Y,meshTime,euler]
-  rw [hA0,show A N=(∫ z,U T (Y z N)-V T (Y z N) ∂gaussianProduct (Fin (N+1))) by
-    dsimp only [A]; rw [hlast],hlast,hzero] at hh
-  have hNd : (N : ℝ)*δ=T := by dsimp [δ,stepSize]; field_simp
-  convert hh using 1
-  · rfl
-  · dsimp only [δ] at hNd ⊢
-    simp only [←mul_assoc,hNd]
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma euler_same_noise_bound {T G L d e : ℝ} {N : ℕ} {γ β : ℝ → ℝ}
-    {u v : ℝ → ℝ → ℝ} (hT : 0≤T) (hN : 0<N)
-    (hG : 0≤G) (hL : 0≤L) (hd : 0≤d) (he : 0≤e)
-    (hg : ∀ j<N,|γ (meshTime T N j)|≤G)
-    (hc : ∀ j<N,|β (meshTime T N j)-γ (meshTime T N j)|≤d)
-    (hv : ∀ j<N,∀ x,|v (meshTime T N j) x|≤1)
-    (hs : ∀ j<N,∀ x,|v (meshTime T N j) x-u (meshTime T N j) x|≤e)
-    (hu : ∀ j<N,∀ x y,|u (meshTime T N j) x-u (meshTime T N j) y|≤L*|x-y|)
-    (z : Fin (N+1) → ℝ) {j : ℕ} (hj : j≤N) :
-    |euler T N β v z j-euler T N γ u z j|≤T*(d+G*e)*Real.exp (T*G*L) := by
-  let δ := stepSize T N
-  have hδ : 0≤δ := div_nonneg hT (Nat.cast_nonneg _)
-  have hNd : (N : ℝ)*δ=T := by dsimp [δ,stepSize]; field_simp
-  have hrec (k : ℕ) (hk : k<N) :
-      |euler T N β v z (k+1)-euler T N γ u z (k+1)|≤
-        (1+δ*G*L)*|euler T N β v z k-euler T N γ u z k|+δ*(d+G*e) := by
-    let y := euler T N β v z k
-    let x := euler T N γ u z k
-    let t := meshTime T N k
-    have hb : |β t*v t y-γ t*u t x|≤d+G*e+G*L*|y-x| := by
-      calc
-        |β t*v t y-γ t*u t x| = |(β t-γ t)*v t y+γ t*(v t y-u t y)+γ t*(u t y-u t x)| := by congr 1; ring
-        _ ≤ |(β t-γ t)*v t y|+|γ t*(v t y-u t y)|+|γ t*(u t y-u t x)| := (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
-        _ ≤ d*1+G*e+G*(L*|y-x|) := by
-          simp only [abs_mul]
-          gcongr
-          · exact hc k hk
-          · exact hv k hk y
-          · exact hg k hk
-          · exact hs k hk y
-          · exact hg k hk
-          · exact hu k hk y x
-        _ = _ := by ring
-    change |y+Real.sqrt δ*coordinate N k z+δ*β t*v t y-
-      (x+Real.sqrt δ*coordinate N k z+δ*γ t*u t x)|≤_
-    calc
-      _ = |(y-x)+δ*(β t*v t y-γ t*u t x)| := by congr 1; ring
-      _ ≤ |y-x|+δ*|β t*v t y-γ t*u t x| := by simpa only [abs_mul,abs_of_nonneg hδ] using abs_add_le (y-x) (δ*(β t*v t y-γ t*u t x))
-      _ ≤ |y-x|+δ*(d+G*e+G*L*|y-x|) := add_le_add_right (mul_le_mul_of_nonneg_left hb hδ) _
-      _ = _ := by ring
-  have h := finite_discrete_gronwall (e := fun k ↦ |euler T N β v z k-euler T N γ u z k|) (mul_nonneg (mul_nonneg hδ hG) hL)
-    (show |euler T N β v z 0-euler T N γ u z 0|=0 by simp [euler])
-    (fun k hk ↦ mul_nonneg hδ (add_nonneg hd (mul_nonneg hG he))) hrec j hj
-  have hsum : (∑ k∈Finset.range j,δ*(d+G*e))≤T*(d+G*e) := by
-    simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
-    rw [←hNd]
-    have hjs : (j : ℝ)≤N := by exact_mod_cast hj
-    nlinarith [mul_nonneg hG he,mul_nonneg hd hδ,mul_nonneg hδ (mul_nonneg hG he)]
-  apply h.trans
-  apply mul_le_mul hsum
-  · apply Real.exp_le_exp.mpr
-    calc
-      (j : ℝ)*(δ*G*L)≤(N : ℝ)*(δ*G*L) := mul_le_mul_of_nonneg_right (by exact_mod_cast hj) (mul_nonneg (mul_nonneg hδ hG) hL)
-      _=T*G*L := by rw [←hNd]; ring
-  · positivity
-  · positivity
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.GroundState.CrossExpectation
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma abs_telescope_source {N : ℕ} {A b Γ Δ : ℕ → ℝ} {c d e : ℝ}
+    (h : ∀ j<N,|A (j+1)-A j+b j|≤c+d*(Γ (j+1)-Γ j)+e*(Δ (j+1)-Δ j)) :
+    |A N-A 0+∑ j∈Finset.range N,b j|≤
+      (N : ℝ)*c+d*(Γ N-Γ 0)+e*(Δ N-Δ 0) := by
+  have hs := (Finset.abs_sum_le_sum_abs (fun j ↦ A (j+1)-A j+b j) (Finset.range N)).trans
+    (Finset.sum_le_sum (fun j hj ↦ h j (Finset.mem_range.mp hj)))
+  simpa only [Finset.sum_add_distrib,Finset.sum_range_sub,←Finset.mul_sum,
+    Finset.sum_const,Finset.card_range,nsmul_eq_mul] using hs
+
+noncomputable def crossMeshSource (T : ℝ) (N : ℕ) (γ β : ℝ → ℝ) (V U : ℝ → ℝ → ℝ) : ℝ :=
+  ∑ j∈Finset.range N,(stepSize T N/2)*(β (meshTime T N j)-γ (meshTime T N j))*
+    (∫ z,(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2
+      ∂gaussianProduct (Fin (N+1)))
+
+lemma cross_value_mesh_expectation {T K L K' L' G : ℝ}
+    {γ β : ℝ → ℝ} {V U : ℝ → ℝ → ℝ}
+    (hV : ValueStrip T γ V K L) (hU : ValueStrip T β U K' L')
+    (hT : 0<T) (hT1 : T≤1) (hG : γ T≤G) (hG' : β T≤G) {N : ℕ} (hN : 0<N) :
+    |(∫ z,U T (euler T N β (crossFeedback V U) z N)-
+      V T (euler T N β (crossFeedback V U) z N) ∂gaussianProduct (Fin (N+1)))-
+      (U 0 0-V 0 0)+crossMeshSource T N γ β V U|≤
+    T*Real.sqrt (stepSize T N)*(cubicEnvelopeMean G K ((1/2+G)*L)+
+      cubicEnvelopeMean G K' ((1/2+G)*L'))+
+      stepSize T N*K*(γ T-γ 0)+stepSize T N*K'*(β T-β 0) := by
+  let δ := stepSize T N
+  let Y := euler T N β (crossFeedback V U)
+  let A := fun j ↦ ∫ z,U (meshTime T N j) (Y z j)-V (meshTime T N j) (Y z j)
+    ∂gaussianProduct (Fin (N+1))
+  let b := fun j ↦ (δ/2)*(β (meshTime T N j)-γ (meshTime T N j))*
+    (∫ z,(deriv (V (meshTime T N j)) (Y z j))^2 ∂gaussianProduct (Fin (N+1)))
+  have hNr : 0<(N : ℝ) := by exact_mod_cast hN
+  have hδ : 0≤δ := div_nonneg hT.le hNr.le
+  have hδ1 : δ≤1 := by
+    apply (div_le_iff₀ hNr).mpr
+    have hN1 : (1 : ℝ)≤N := by exact_mod_cast hN
+    simpa only [one_mul] using hT1.trans hN1
+  have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT.le hN hj
+  have hs (j : ℕ) : meshTime T N j+δ=meshTime T N (j+1) := by
+    dsimp [meshTime]; push_cast; ring
+  have hlast : meshTime T N N=T := by dsimp [meshTime,stepSize]; field_simp
+  have hzero : meshTime T N 0=0 := by simp [meshTime]
+  have hfm (j : ℕ) (hj : j<N) : Measurable (crossFeedback V U (meshTime T N j)) :=
+    (((hU.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable.add
+      ((hV.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable).div_const 2
+  have hfb (j : ℕ) (hj : j<N) (x : ℝ) : |crossFeedback V U (meshTime T N j) x|≤1 :=
+    crossFeedback_bound (hV.bounded _ (htime j hj.le) x) (hU.bounded _ (htime j hj.le) x)
+  have hstep (j : ℕ) (hj : j<N) : |A (j+1)-A j+b j|≤
+      δ*Real.sqrt δ*(cubicEnvelopeMean G K ((1/2+G)*L)+cubicEnvelopeMean G K' ((1/2+G)*L'))+
+      δ*K*(γ (meshTime T N (j+1))-γ (meshTime T N j))+
+      δ*K'*(β (meshTime T N (j+1))-β (meshTime T N j)) := by
+    have hYp : DependsBefore j (fun z ↦ Y z j) :=
+      eulerCoefficient_dependsBefore T hj.le β (crossFeedback V U) (fun _ y ↦ y)
+    have he := cross_value_expectation_step hV hU (htime j hj.le).1 hδ hδ1
+      (by rw [hs]; exact (htime (j+1) (by omega)).2) hG hG' (by omega : j<N+1)
+      (measurable_euler T N β (crossFeedback V U) hfm j hj.le) hYp
+      (euler_memLp T N β (crossFeedback V U) hfm hfb j hj.le)
+    dsimp only at he
+    rw [hs] at he
+    simpa only [A,b,Y,euler,δ,mul_assoc] using he
+  have hh := abs_telescope_source (A := A) (b := b)
+    (Γ := fun j ↦ γ (meshTime T N j)) (Δ := fun j ↦ β (meshTime T N j)) hstep
+  have hA0 : A 0=U 0 0-V 0 0 := by simp [A,Y,meshTime,euler]
+  rw [hA0,show A N=(∫ z,U T (Y z N)-V T (Y z N) ∂gaussianProduct (Fin (N+1))) by
+    dsimp only [A]; rw [hlast],hlast,hzero] at hh
+  have hNd : (N : ℝ)*δ=T := by dsimp [δ,stepSize]; field_simp
+  convert hh using 1
+  · rfl
+  · dsimp only [δ] at hNd ⊢
+    simp only [←mul_assoc,hNd]
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma euler_same_noise_bound {T G L d e : ℝ} {N : ℕ} {γ β : ℝ → ℝ}
+    {u v : ℝ → ℝ → ℝ} (hT : 0≤T) (hN : 0<N)
+    (hG : 0≤G) (hL : 0≤L) (hd : 0≤d) (he : 0≤e)
+    (hg : ∀ j<N,|γ (meshTime T N j)|≤G)
+    (hc : ∀ j<N,|β (meshTime T N j)-γ (meshTime T N j)|≤d)
+    (hv : ∀ j<N,∀ x,|v (meshTime T N j) x|≤1)
+    (hs : ∀ j<N,∀ x,|v (meshTime T N j) x-u (meshTime T N j) x|≤e)
+    (hu : ∀ j<N,∀ x y,|u (meshTime T N j) x-u (meshTime T N j) y|≤L*|x-y|)
+    (z : Fin (N+1) → ℝ) {j : ℕ} (hj : j≤N) :
+    |euler T N β v z j-euler T N γ u z j|≤T*(d+G*e)*Real.exp (T*G*L) := by
+  let δ := stepSize T N
+  have hδ : 0≤δ := div_nonneg hT (Nat.cast_nonneg _)
+  have hNd : (N : ℝ)*δ=T := by dsimp [δ,stepSize]; field_simp
+  have hrec (k : ℕ) (hk : k<N) :
+      |euler T N β v z (k+1)-euler T N γ u z (k+1)|≤
+        (1+δ*G*L)*|euler T N β v z k-euler T N γ u z k|+δ*(d+G*e) := by
+    let y := euler T N β v z k
+    let x := euler T N γ u z k
+    let t := meshTime T N k
+    have hb : |β t*v t y-γ t*u t x|≤d+G*e+G*L*|y-x| := by
+      calc
+        |β t*v t y-γ t*u t x| = |(β t-γ t)*v t y+γ t*(v t y-u t y)+γ t*(u t y-u t x)| := by congr 1; ring
+        _ ≤ |(β t-γ t)*v t y|+|γ t*(v t y-u t y)|+|γ t*(u t y-u t x)| := (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
+        _ ≤ d*1+G*e+G*(L*|y-x|) := by
+          simp only [abs_mul]
+          gcongr
+          · exact hc k hk
+          · exact hv k hk y
+          · exact hg k hk
+          · exact hs k hk y
+          · exact hg k hk
+          · exact hu k hk y x
+        _ = _ := by ring
+    change |y+Real.sqrt δ*coordinate N k z+δ*β t*v t y-
+      (x+Real.sqrt δ*coordinate N k z+δ*γ t*u t x)|≤_
+    calc
+      _ = |(y-x)+δ*(β t*v t y-γ t*u t x)| := by congr 1; ring
+      _ ≤ |y-x|+δ*|β t*v t y-γ t*u t x| := by simpa only [abs_mul,abs_of_nonneg hδ] using abs_add_le (y-x) (δ*(β t*v t y-γ t*u t x))
+      _ ≤ |y-x|+δ*(d+G*e+G*L*|y-x|) := add_le_add_right (mul_le_mul_of_nonneg_left hb hδ) _
+      _ = _ := by ring
+  have h := finite_discrete_gronwall (e := fun k ↦ |euler T N β v z k-euler T N γ u z k|) (mul_nonneg (mul_nonneg hδ hG) hL)
+    (show |euler T N β v z 0-euler T N γ u z 0|=0 by simp [euler])
+    (fun k hk ↦ mul_nonneg hδ (add_nonneg hd (mul_nonneg hG he))) hrec j hj
+  have hsum : (∑ k∈Finset.range j,δ*(d+G*e))≤T*(d+G*e) := by
+    simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
+    rw [←hNd]
+    have hjs : (j : ℝ)≤N := by exact_mod_cast hj
+    nlinarith [mul_nonneg hG he,mul_nonneg hd hδ,mul_nonneg hδ (mul_nonneg hG he)]
+  apply h.trans
+  apply mul_le_mul hsum
+  · apply Real.exp_le_exp.mpr
+    calc
+      (j : ℝ)*(δ*G*L)≤(N : ℝ)*(δ*G*L) := mul_le_mul_of_nonneg_right (by exact_mod_cast hj) (mul_nonneg (mul_nonneg hδ hG) hL)
+      _=T*G*L := by rw [←hNd]; ring
+  · positivity
+  · positivity
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/CrossMeshComparison.lean b/lean/OAI/Probability/SKValue/Processes/CrossMeshComparison.lean
index db4b7749d9049f1bce376c919701ba32fc54f509..8cdc14f756c4ff22756bfe2368ba70f3a1cef9ac 100644
--- a/lean/OAI/Probability/SKValue/Processes/CrossMeshComparison.lean
+++ b/lean/OAI/Probability/SKValue/Processes/CrossMeshComparison.lean
@@ -1,90 +1,90 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.CrossMesh
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def ordinaryMeshSource (T : ℝ) (N : ℕ) (γ β : ℝ → ℝ) (V : ℝ → ℝ → ℝ) : ℝ :=
-  stepSize T N*∑ j : Fin N,(β (meshTime T N j)-γ (meshTime T N j))*
-    (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j))
-      (euler T N γ (fun t ↦ deriv (V t)) z j))^2 ∂gaussianProduct (Fin (N+1)))
-
-lemma crossMeshSource_half_form (T : ℝ) (N : ℕ) (γ β : ℝ → ℝ) (V U : ℝ → ℝ → ℝ) :
-    crossMeshSource T N γ β V U=stepSize T N*∑ j : Fin N,(β (meshTime T N j)-γ (meshTime T N j))*
-      (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j))
-        (euler T N β (crossFeedback V U) z j))^2 ∂gaussianProduct (Fin (N+1))) := by
-  unfold crossMeshSource
-  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ ↦ (β (meshTime T N j)-γ (meshTime T N j))*
-    (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j))
-      (euler T N β (crossFeedback V U) z j))^2 ∂gaussianProduct (Fin (N+1))))]
-  simp only [integral_const_mul,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro j hj
-  ring
-
-lemma crossMeshSource_difference_bound {T K L K' L' d R : ℝ}
-    {γ β : ℝ → ℝ} {V U : ℝ → ℝ → ℝ}
-    (hV : ValueStrip T γ V K L) (hU : ValueStrip T β U K' L')
-    (hT : 0≤T) {N : ℕ} (hN : 0<N) (hd : 0≤d) (hR : 0≤R)
-    (hc : ∀ t∈Icc (0 : ℝ) T,|β t-γ t|≤d)
-    (hY : ∀ z : Fin (N+1) → ℝ,∀ j≤N,
-      |euler T N β (crossFeedback V U) z j-euler T N γ (fun t ↦ deriv (V t)) z j|≤R) :
-    |crossMeshSource T N γ β V U-ordinaryMeshSource T N γ β V|≤T*d*L*R := by
-  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg _)
-  have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT hN hj
-  have hfm (j : ℕ) (hj : j<N) : Measurable (crossFeedback V U (meshTime T N j)) :=
-    (((hU.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable.add
-      ((hV.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable).div_const 2
-  have hfb (j : ℕ) (hj : j<N) (x : ℝ) : |crossFeedback V U (meshTime T N j) x|≤1 :=
-    crossFeedback_bound (hV.bounded _ (htime j hj.le) x) (hU.bounded _ (htime j hj.le) x)
-  have hm (j : Fin N) :
-      |(∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2
-        ∂gaussianProduct (Fin (N+1)))-
-       (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N γ (fun t ↦ deriv (V t)) z j))^2
-        ∂gaussianProduct (Fin (N+1)))|≤L*R := by
-    have hLip := hV.product_lipschitzWith (htime j j.isLt.le)
-    have hM1 := lipschitz_comp_memLp_finite hLip (euler_memLp T N β (crossFeedback V U) hfm hfb j j.isLt.le)
-    have hM2 := lipschitz_comp_memLp_finite hLip (euler_memLp T N γ (fun t ↦ deriv (V t))
-      (fun k hk ↦ ((hV.smooth _ (htime k hk.le)).continuous_deriv (by norm_num)).measurable)
-      (fun k hk x ↦ hV.bounded _ (htime k hk.le) x) j j.isLt.le)
-    rw [←integral_sub (f := fun z ↦ (1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2)
-      (g := fun z ↦ (1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N γ (fun t ↦ deriv (V t)) z j))^2)
-      (hM1.integrable (by norm_num)) (hM2.integrable (by norm_num))]
-    apply (abs_integral_le_integral_abs).trans
-    calc
-      _ ≤ ∫ _ : Fin (N+1) → ℝ,|L*R| ∂gaussianProduct (Fin (N+1)) := by
-        apply integral_mono ((hM1.integrable (by norm_num)).sub (hM2.integrable (by norm_num))).norm (integrable_const _)
-        intro z
-        have hh := hLip.dist_le_mul (euler T N β (crossFeedback V U) z j) (euler T N γ (fun t ↦ deriv (V t)) z j)
-        rw [Real.dist_eq,Real.dist_eq,Real.coe_toNNReal _ hV.L_nonneg] at hh
-        exact (hh.trans (mul_le_mul_of_nonneg_left (hY z j j.isLt.le) hV.L_nonneg)).trans
-          (le_abs_self (L*R))
-      _=L*R := by simp [abs_of_nonneg (mul_nonneg hV.L_nonneg hR)]
-  rw [crossMeshSource_half_form,ordinaryMeshSource,←mul_sub,←Finset.sum_sub_distrib,
-    abs_mul,abs_of_nonneg hδ]
-  calc
-    _ ≤ stepSize T N*∑ j : Fin N,|(β (meshTime T N j)-γ (meshTime T N j))*
-      ((∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2 ∂gaussianProduct (Fin (N+1)))-
-       (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N γ (fun t ↦ deriv (V t)) z j))^2 ∂gaussianProduct (Fin (N+1))))| := by
-      simp_rw [←mul_sub]
-      exact mul_le_mul_of_nonneg_left (Finset.abs_sum_le_sum_abs _ _) hδ
-    _ ≤ stepSize T N*∑ _ : Fin N,d*(L*R) := by
-      apply mul_le_mul_of_nonneg_left _ hδ
-      apply Finset.sum_le_sum
-      intro j hj
-      rw [abs_mul]
-      exact mul_le_mul (hc _ (htime j j.isLt.le)) (hm j) (abs_nonneg _) hd
-    _=T*d*L*R := by
-      simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,stepSize]
-      field_simp
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.CrossMesh
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def ordinaryMeshSource (T : ℝ) (N : ℕ) (γ β : ℝ → ℝ) (V : ℝ → ℝ → ℝ) : ℝ :=
+  stepSize T N*∑ j : Fin N,(β (meshTime T N j)-γ (meshTime T N j))*
+    (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j))
+      (euler T N γ (fun t ↦ deriv (V t)) z j))^2 ∂gaussianProduct (Fin (N+1)))
+
+lemma crossMeshSource_half_form (T : ℝ) (N : ℕ) (γ β : ℝ → ℝ) (V U : ℝ → ℝ → ℝ) :
+    crossMeshSource T N γ β V U=stepSize T N*∑ j : Fin N,(β (meshTime T N j)-γ (meshTime T N j))*
+      (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j))
+        (euler T N β (crossFeedback V U) z j))^2 ∂gaussianProduct (Fin (N+1))) := by
+  unfold crossMeshSource
+  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ ↦ (β (meshTime T N j)-γ (meshTime T N j))*
+    (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j))
+      (euler T N β (crossFeedback V U) z j))^2 ∂gaussianProduct (Fin (N+1))))]
+  simp only [integral_const_mul,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro j hj
+  ring
+
+lemma crossMeshSource_difference_bound {T K L K' L' d R : ℝ}
+    {γ β : ℝ → ℝ} {V U : ℝ → ℝ → ℝ}
+    (hV : ValueStrip T γ V K L) (hU : ValueStrip T β U K' L')
+    (hT : 0≤T) {N : ℕ} (hN : 0<N) (hd : 0≤d) (hR : 0≤R)
+    (hc : ∀ t∈Icc (0 : ℝ) T,|β t-γ t|≤d)
+    (hY : ∀ z : Fin (N+1) → ℝ,∀ j≤N,
+      |euler T N β (crossFeedback V U) z j-euler T N γ (fun t ↦ deriv (V t)) z j|≤R) :
+    |crossMeshSource T N γ β V U-ordinaryMeshSource T N γ β V|≤T*d*L*R := by
+  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg _)
+  have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT hN hj
+  have hfm (j : ℕ) (hj : j<N) : Measurable (crossFeedback V U (meshTime T N j)) :=
+    (((hU.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable.add
+      ((hV.smooth _ (htime j hj.le)).continuous_deriv (by norm_num)).measurable).div_const 2
+  have hfb (j : ℕ) (hj : j<N) (x : ℝ) : |crossFeedback V U (meshTime T N j) x|≤1 :=
+    crossFeedback_bound (hV.bounded _ (htime j hj.le) x) (hU.bounded _ (htime j hj.le) x)
+  have hm (j : Fin N) :
+      |(∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2
+        ∂gaussianProduct (Fin (N+1)))-
+       (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N γ (fun t ↦ deriv (V t)) z j))^2
+        ∂gaussianProduct (Fin (N+1)))|≤L*R := by
+    have hLip := hV.product_lipschitzWith (htime j j.isLt.le)
+    have hM1 := lipschitz_comp_memLp_finite hLip (euler_memLp T N β (crossFeedback V U) hfm hfb j j.isLt.le)
+    have hM2 := lipschitz_comp_memLp_finite hLip (euler_memLp T N γ (fun t ↦ deriv (V t))
+      (fun k hk ↦ ((hV.smooth _ (htime k hk.le)).continuous_deriv (by norm_num)).measurable)
+      (fun k hk x ↦ hV.bounded _ (htime k hk.le) x) j j.isLt.le)
+    rw [←integral_sub (f := fun z ↦ (1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2)
+      (g := fun z ↦ (1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N γ (fun t ↦ deriv (V t)) z j))^2)
+      (hM1.integrable (by norm_num)) (hM2.integrable (by norm_num))]
+    apply (abs_integral_le_integral_abs).trans
+    calc
+      _ ≤ ∫ _ : Fin (N+1) → ℝ,|L*R| ∂gaussianProduct (Fin (N+1)) := by
+        apply integral_mono ((hM1.integrable (by norm_num)).sub (hM2.integrable (by norm_num))).norm (integrable_const _)
+        intro z
+        have hh := hLip.dist_le_mul (euler T N β (crossFeedback V U) z j) (euler T N γ (fun t ↦ deriv (V t)) z j)
+        rw [Real.dist_eq,Real.dist_eq,Real.coe_toNNReal _ hV.L_nonneg] at hh
+        exact (hh.trans (mul_le_mul_of_nonneg_left (hY z j j.isLt.le) hV.L_nonneg)).trans
+          (le_abs_self (L*R))
+      _=L*R := by simp [abs_of_nonneg (mul_nonneg hV.L_nonneg hR)]
+  rw [crossMeshSource_half_form,ordinaryMeshSource,←mul_sub,←Finset.sum_sub_distrib,
+    abs_mul,abs_of_nonneg hδ]
+  calc
+    _ ≤ stepSize T N*∑ j : Fin N,|(β (meshTime T N j)-γ (meshTime T N j))*
+      ((∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N β (crossFeedback V U) z j))^2 ∂gaussianProduct (Fin (N+1)))-
+       (∫ z,(1/2 : ℝ)*(deriv (V (meshTime T N j)) (euler T N γ (fun t ↦ deriv (V t)) z j))^2 ∂gaussianProduct (Fin (N+1))))| := by
+      simp_rw [←mul_sub]
+      exact mul_le_mul_of_nonneg_left (Finset.abs_sum_le_sum_abs _ _) hδ
+    _ ≤ stepSize T N*∑ _ : Fin N,d*(L*R) := by
+      apply mul_le_mul_of_nonneg_left _ hδ
+      apply Finset.sum_le_sum
+      intro j hj
+      rw [abs_mul]
+      exact mul_le_mul (hc _ (htime j j.isLt.le)) (hm j) (abs_nonneg _) hd
+    _=T*d*L*R := by
+      simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,stepSize]
+      field_simp
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/CrossValueBound.lean b/lean/OAI/Probability/SKValue/Processes/CrossValueBound.lean
index dbd4b4d9299ea05a824e8e604813a570f680407d..d4c349feb528775f84d68788d12b2939331d935c 100644
--- a/lean/OAI/Probability/SKValue/Processes/CrossValueBound.lean
+++ b/lean/OAI/Probability/SKValue/Processes/CrossValueBound.lean
@@ -1,108 +1,108 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.CrossMeshComparison
-import OAI.Probability.SKValue.Variation.WeightedMoments
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma ordinaryMeshSource_eq (W : BrownianSpace) (γ β : OrderParameter) (T : ℝ) (N : ℕ) :
-    ordinaryMeshSource T N γ.coeff β.coeff (phi W γ)=
-      weightedEulerMoment W γ β T N-weightedEulerMoment W γ γ T N := by
-  have he : gradient W γ=(fun t ↦ deriv (phi W γ t)) := rfl
-  simp only [ordinaryMeshSource,weightedEulerMoment,he,←mul_sub,←Finset.sum_sub_distrib]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro j hj
-  ring
-
-lemma IsDiffusion.cross_value_bound {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (β : OrderParameter)
-    {T Kv Lv Lu d e : ℝ} (hT : 0<T) (hT1 : T<1)
-    (hV : ValueStrip T γ.coeff (phi W γ) Kv Lv)
-    (hLu : 0≤Lu) (hd : 0≤d) (he : 0≤e)
-    (hc : ∀ t∈Icc (0 : ℝ) T,|(β.coeff t)-(γ.coeff t)|≤d)
-    (hu : ∀ t∈Icc (0 : ℝ) T,∀ x y,|gradient W γ t x-gradient W γ t y|≤Lu*|x-y|)
-    (hs : ∀ t∈Icc (0 : ℝ) T,∀ x,|gradient W β t x-gradient W γ t x|≤e) :
-    phi W β 0 0-phi W γ 0 0≤
-      (3/2 : ℝ)*(∫ t in T..1,|β.cutoff t-γ.cutoff t|)+
-      ((∫ t in (0 : ℝ)..T,β.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))-
-       (∫ t in (0 : ℝ)..T,γ.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ)))+
-      T*d*Lv*(T*(d+γ.coeff T*e)*Real.exp (T*γ.coeff T*Lu)) := by
-  obtain ⟨Kv',Lv',K',L',D',Lu',La',hU,rest⟩ := sourceStripRegularity W β T ⟨hT.le,hT1⟩
-  let G := max (γ.coeff T) (β.coeff T)
-  let R := T*(d+γ.coeff T*e)*Real.exp (T*γ.coeff T*Lu)
-  let B := (3/2 : ℝ)*(∫ t in T..1,|β.cutoff t-γ.cutoff t|)
-  let err (N : ℕ) :=
-    T*Real.sqrt (stepSize T N)*(cubicEnvelopeMean G Kv ((1/2+G)*Lv)+
-      cubicEnvelopeMean G Kv' ((1/2+G)*Lv'))+
-      stepSize T N*Kv*(γ.coeff T-γ.coeff 0)+stepSize T N*Kv'*(β.coeff T-β.coeff 0)
-  have hγT : 0≤γ.coeff T := γ.nonneg T ⟨hT.le,hT1⟩
-  have hR : 0≤R := by dsimp [R]; positivity
-  have heN (N : ℕ) (hN : 0<N) : phi W β 0 0-phi W γ 0 0≤
-      B+(weightedEulerMoment W γ β T N-weightedEulerMoment W γ γ T N)+T*d*Lv*R+err N := by
-    have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT.le hN hj
-    have hfb (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
-        |crossFeedback (phi W γ) (phi W β) t x|≤1 :=
-      crossFeedback_bound (hV.bounded t ht x) (hU.bounded t ht x)
-    have hfs (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
-        |crossFeedback (phi W γ) (phi W β) t x-gradient W γ t x|≤e := by
-      have hh := hs t ht x
-      have hh' : |crossFeedback (phi W γ) (phi W β) t x-gradient W γ t x|=
-          |gradient W β t x-gradient W γ t x|/2 := by
-        dsimp only [crossFeedback,gradient]
-        rw [show (deriv (phi W β t) x+deriv (phi W γ t) x)/2-deriv (phi W γ t) x=
-          (deriv (phi W β t) x-deriv (phi W γ t) x)/2 by ring,abs_div,abs_of_pos (by norm_num : (0 : ℝ)<2)]
-      rw [hh']
-      linarith
-    have hY (z : Fin (N+1) → ℝ) (j : ℕ) (hj : j≤N) :
-        |euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z j-
-          euler T N γ.coeff (gradient W γ) z j|≤R := by
-      apply euler_same_noise_bound hT.le hN hγT hLu hd he
-      · intro k hk
-        rw [abs_of_nonneg (γ.nonneg _ ⟨(htime k hk.le).1,(htime k hk.le).2.trans_lt hT1⟩)]
-        exact γ.monotone ⟨(htime k hk.le).1,(htime k hk.le).2.trans_lt hT1⟩ ⟨hT.le,hT1⟩ (htime k hk.le).2
-      · intro k hk; exact hc _ (htime k hk.le)
-      · intro k hk; exact hfb _ (htime k hk.le)
-      · intro k hk; exact hfs _ (htime k hk.le)
-      · intro k hk; exact hu _ (htime k hk.le)
-      · exact hj
-    have htelescope := cross_value_mesh_expectation hV hU hT hT1.le
-      (le_max_left _ _ : γ.coeff T≤G) (le_max_right _ _ : β.coeff T≤G) hN
-    have hcompare := crossMeshSource_difference_bound hV hU hT.le hN hd hR hc hY
-    rw [ordinaryMeshSource_eq] at hcompare
-    have hterminal : (∫ z,phi W β T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N)-
-        phi W γ T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N)
-        ∂gaussianProduct (Fin (N+1)))≤B := by
-      apply (le_abs_self _).trans
-      simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using
-        (norm_integral_le_of_norm_le_const (μ := gaussianProduct (Fin (N+1))) (C := B)
-          (f := fun z ↦ phi W β T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N)-
-            phi W γ T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N))
-          (Eventually.of_forall (fun z ↦ phi_difference_bound W γ β ⟨hT.le,hT1.le⟩ _)))
-    have htlow := (abs_le.mp htelescope).1
-    have hcup := (abs_le.mp hcompare).2
-    dsimp only [err]
-    linarith
-  have hδ : Tendsto (fun N : ℕ ↦ stepSize T N) atTop (𝓝 (0 : ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
-  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
-  have herr : Tendsto err atTop (𝓝 0) := by
-    simpa only [err,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,zero_add] using
-      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean G Kv ((1/2+G)*Lv)+
-        cubicEnvelopeMean G Kv' ((1/2+G)*Lv'))).add
-        ((hδ.mul_const Kv).mul_const (γ.coeff T-γ.coeff 0))).add
-        ((hδ.mul_const Kv').mul_const (β.coeff T-β.coeff 0))
-  have hlim := (((((hX.weightedEulerMoment_tendsto β hT hT1).sub
-    (hX.weightedEulerMoment_tendsto γ hT hT1)).const_add B).add_const (T*d*Lv*R)).add herr)
-  have hh := ge_of_tendsto hlim (by filter_upwards [eventually_gt_atTop 0] with N hN using heN N hN)
-  simpa only [add_zero,B,R] using hh
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.CrossMeshComparison
+import OAI.Probability.SKValue.Variation.WeightedMoments
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma ordinaryMeshSource_eq (W : BrownianSpace) (γ β : OrderParameter) (T : ℝ) (N : ℕ) :
+    ordinaryMeshSource T N γ.coeff β.coeff (phi W γ)=
+      weightedEulerMoment W γ β T N-weightedEulerMoment W γ γ T N := by
+  have he : gradient W γ=(fun t ↦ deriv (phi W γ t)) := rfl
+  simp only [ordinaryMeshSource,weightedEulerMoment,he,←mul_sub,←Finset.sum_sub_distrib]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro j hj
+  ring
+
+lemma IsDiffusion.cross_value_bound {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (β : OrderParameter)
+    {T Kv Lv Lu d e : ℝ} (hT : 0<T) (hT1 : T<1)
+    (hV : ValueStrip T γ.coeff (phi W γ) Kv Lv)
+    (hLu : 0≤Lu) (hd : 0≤d) (he : 0≤e)
+    (hc : ∀ t∈Icc (0 : ℝ) T,|(β.coeff t)-(γ.coeff t)|≤d)
+    (hu : ∀ t∈Icc (0 : ℝ) T,∀ x y,|gradient W γ t x-gradient W γ t y|≤Lu*|x-y|)
+    (hs : ∀ t∈Icc (0 : ℝ) T,∀ x,|gradient W β t x-gradient W γ t x|≤e) :
+    phi W β 0 0-phi W γ 0 0≤
+      (3/2 : ℝ)*(∫ t in T..1,|β.cutoff t-γ.cutoff t|)+
+      ((∫ t in (0 : ℝ)..T,β.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))-
+       (∫ t in (0 : ℝ)..T,γ.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ)))+
+      T*d*Lv*(T*(d+γ.coeff T*e)*Real.exp (T*γ.coeff T*Lu)) := by
+  obtain ⟨Kv',Lv',K',L',D',Lu',La',hU,rest⟩ := sourceStripRegularity W β T ⟨hT.le,hT1⟩
+  let G := max (γ.coeff T) (β.coeff T)
+  let R := T*(d+γ.coeff T*e)*Real.exp (T*γ.coeff T*Lu)
+  let B := (3/2 : ℝ)*(∫ t in T..1,|β.cutoff t-γ.cutoff t|)
+  let err (N : ℕ) :=
+    T*Real.sqrt (stepSize T N)*(cubicEnvelopeMean G Kv ((1/2+G)*Lv)+
+      cubicEnvelopeMean G Kv' ((1/2+G)*Lv'))+
+      stepSize T N*Kv*(γ.coeff T-γ.coeff 0)+stepSize T N*Kv'*(β.coeff T-β.coeff 0)
+  have hγT : 0≤γ.coeff T := γ.nonneg T ⟨hT.le,hT1⟩
+  have hR : 0≤R := by dsimp [R]; positivity
+  have heN (N : ℕ) (hN : 0<N) : phi W β 0 0-phi W γ 0 0≤
+      B+(weightedEulerMoment W γ β T N-weightedEulerMoment W γ γ T N)+T*d*Lv*R+err N := by
+    have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT.le hN hj
+    have hfb (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
+        |crossFeedback (phi W γ) (phi W β) t x|≤1 :=
+      crossFeedback_bound (hV.bounded t ht x) (hU.bounded t ht x)
+    have hfs (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
+        |crossFeedback (phi W γ) (phi W β) t x-gradient W γ t x|≤e := by
+      have hh := hs t ht x
+      have hh' : |crossFeedback (phi W γ) (phi W β) t x-gradient W γ t x|=
+          |gradient W β t x-gradient W γ t x|/2 := by
+        dsimp only [crossFeedback,gradient]
+        rw [show (deriv (phi W β t) x+deriv (phi W γ t) x)/2-deriv (phi W γ t) x=
+          (deriv (phi W β t) x-deriv (phi W γ t) x)/2 by ring,abs_div,abs_of_pos (by norm_num : (0 : ℝ)<2)]
+      rw [hh']
+      linarith
+    have hY (z : Fin (N+1) → ℝ) (j : ℕ) (hj : j≤N) :
+        |euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z j-
+          euler T N γ.coeff (gradient W γ) z j|≤R := by
+      apply euler_same_noise_bound hT.le hN hγT hLu hd he
+      · intro k hk
+        rw [abs_of_nonneg (γ.nonneg _ ⟨(htime k hk.le).1,(htime k hk.le).2.trans_lt hT1⟩)]
+        exact γ.monotone ⟨(htime k hk.le).1,(htime k hk.le).2.trans_lt hT1⟩ ⟨hT.le,hT1⟩ (htime k hk.le).2
+      · intro k hk; exact hc _ (htime k hk.le)
+      · intro k hk; exact hfb _ (htime k hk.le)
+      · intro k hk; exact hfs _ (htime k hk.le)
+      · intro k hk; exact hu _ (htime k hk.le)
+      · exact hj
+    have htelescope := cross_value_mesh_expectation hV hU hT hT1.le
+      (le_max_left _ _ : γ.coeff T≤G) (le_max_right _ _ : β.coeff T≤G) hN
+    have hcompare := crossMeshSource_difference_bound hV hU hT.le hN hd hR hc hY
+    rw [ordinaryMeshSource_eq] at hcompare
+    have hterminal : (∫ z,phi W β T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N)-
+        phi W γ T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N)
+        ∂gaussianProduct (Fin (N+1)))≤B := by
+      apply (le_abs_self _).trans
+      simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using
+        (norm_integral_le_of_norm_le_const (μ := gaussianProduct (Fin (N+1))) (C := B)
+          (f := fun z ↦ phi W β T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N)-
+            phi W γ T (euler T N β.coeff (crossFeedback (phi W γ) (phi W β)) z N))
+          (Eventually.of_forall (fun z ↦ phi_difference_bound W γ β ⟨hT.le,hT1.le⟩ _)))
+    have htlow := (abs_le.mp htelescope).1
+    have hcup := (abs_le.mp hcompare).2
+    dsimp only [err]
+    linarith
+  have hδ : Tendsto (fun N : ℕ ↦ stepSize T N) atTop (𝓝 (0 : ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
+  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
+  have herr : Tendsto err atTop (𝓝 0) := by
+    simpa only [err,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,zero_add] using
+      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean G Kv ((1/2+G)*Lv)+
+        cubicEnvelopeMean G Kv' ((1/2+G)*Lv'))).add
+        ((hδ.mul_const Kv).mul_const (γ.coeff T-γ.coeff 0))).add
+        ((hδ.mul_const Kv').mul_const (β.coeff T-β.coeff 0))
+  have hlim := (((((hX.weightedEulerMoment_tendsto β hT hT1).sub
+    (hX.weightedEulerMoment_tendsto γ hT hT1)).const_add B).add_const (T*d*Lv*R)).add herr)
+  have hh := ge_of_tendsto hlim (by filter_upwards [eventually_gt_atTop 0] with N hN using heN N hN)
+  simpa only [add_zero,B,R] using hh
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/CurvatureNonneg.lean b/lean/OAI/Probability/SKValue/Processes/CurvatureNonneg.lean
index 861a52a5402a348c867e85d7ab44d8784123699c..0f28a304dbca1dc8ad6442a22b9a731199634589 100644
--- a/lean/OAI/Probability/SKValue/Processes/CurvatureNonneg.lean
+++ b/lean/OAI/Probability/SKValue/Processes/CurvatureNonneg.lean
@@ -1,43 +1,43 @@
-import Mathlib
-import OAI.Probability.SKValue.Coercivity.LiteralCoercivity
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal
-namespace SKValue
-lemma IsDiffusion.curvature_generator_nonneg {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
-    (hX:IsDiffusion W γ X) {t:ℝ} (ht:t∈Ioo (0:ℝ) 1):
-    0≤∫ z,(iteratedDeriv 3 (deriv (phi W γ t)) (X t z))^2-
-      12*γ.coeff t*iteratedDeriv 1 (deriv (phi W γ t)) (X t z)*(iteratedDeriv 2 (deriv (phi W γ t)) (X t z))^2+
-      6*(γ.coeff t)^2*(iteratedDeriv 1 (deriv (phi W γ t)) (X t z))^4 ∂W.μ := by
-  have hc:=γ.nonneg t ⟨ht.1.le,ht.2⟩
-  rcases hc.eq_or_lt with hc|hc
-  · rw [←hc]
-    simp only [mul_zero,zero_mul,sub_zero,zero_pow (by norm_num: (2:ℕ)≠0),add_zero]
-    exact integral_nonneg (fun z ↦ sq_nonneg _)
-  · have h:=hX.literal_coercivity ⟨t,ht⟩
-    have hn:0≤(1/1000:ℝ)*(∫ z,(spatialJet (γ.coeff t) (phi W γ t) 1 (X t z))^4 ∂W.μ) := by
-      apply mul_nonneg (by norm_num)
-      exact integral_nonneg (fun z ↦ by positivity)
-    have hn:=hn.trans h
-    have he:(fun z ↦ (spatialJet (γ.coeff t) (phi W γ t) 3 (X t z))^2-
-        12*spatialJet (γ.coeff t) (phi W γ t) 1 (X t z)*(spatialJet (γ.coeff t) (phi W γ t) 2 (X t z))^2+
-        6*(spatialJet (γ.coeff t) (phi W γ t) 1 (X t z))^4)=
-      (fun z ↦ (γ.coeff t)^2*((iteratedDeriv 3 (deriv (phi W γ t)) (X t z))^2-
-        12*γ.coeff t*iteratedDeriv 1 (deriv (phi W γ t)) (X t z)*(iteratedDeriv 2 (deriv (phi W γ t)) (X t z))^2+
-        6*(γ.coeff t)^2*(iteratedDeriv 1 (deriv (phi W γ t)) (X t z))^4)) := by
-      funext z
-      simp only [spatialJet]
-      ring
-    change 0≤∫ z,(spatialJet (γ.coeff t) (phi W γ t) 3 (X t z))^2-
-        12*spatialJet (γ.coeff t) (phi W γ t) 1 (X t z)*(spatialJet (γ.coeff t) (phi W γ t) 2 (X t z))^2+
-        6*(spatialJet (γ.coeff t) (phi W γ t) 1 (X t z))^4 ∂W.μ at hn
-    rw [he,integral_const_mul] at hn
-    exact nonneg_of_mul_nonneg_right hn (sq_pos_of_pos hc)
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Coercivity.LiteralCoercivity
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal
+namespace SKValue
+lemma IsDiffusion.curvature_generator_nonneg {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
+    (hX:IsDiffusion W γ X) {t:ℝ} (ht:t∈Ioo (0:ℝ) 1):
+    0≤∫ z,(iteratedDeriv 3 (deriv (phi W γ t)) (X t z))^2-
+      12*γ.coeff t*iteratedDeriv 1 (deriv (phi W γ t)) (X t z)*(iteratedDeriv 2 (deriv (phi W γ t)) (X t z))^2+
+      6*(γ.coeff t)^2*(iteratedDeriv 1 (deriv (phi W γ t)) (X t z))^4 ∂W.μ := by
+  have hc:=γ.nonneg t ⟨ht.1.le,ht.2⟩
+  rcases hc.eq_or_lt with hc|hc
+  · rw [←hc]
+    simp only [mul_zero,zero_mul,sub_zero,zero_pow (by norm_num: (2:ℕ)≠0),add_zero]
+    exact integral_nonneg (fun z ↦ sq_nonneg _)
+  · have h:=hX.literal_coercivity ⟨t,ht⟩
+    have hn:0≤(1/1000:ℝ)*(∫ z,(spatialJet (γ.coeff t) (phi W γ t) 1 (X t z))^4 ∂W.μ) := by
+      apply mul_nonneg (by norm_num)
+      exact integral_nonneg (fun z ↦ by positivity)
+    have hn:=hn.trans h
+    have he:(fun z ↦ (spatialJet (γ.coeff t) (phi W γ t) 3 (X t z))^2-
+        12*spatialJet (γ.coeff t) (phi W γ t) 1 (X t z)*(spatialJet (γ.coeff t) (phi W γ t) 2 (X t z))^2+
+        6*(spatialJet (γ.coeff t) (phi W γ t) 1 (X t z))^4)=
+      (fun z ↦ (γ.coeff t)^2*((iteratedDeriv 3 (deriv (phi W γ t)) (X t z))^2-
+        12*γ.coeff t*iteratedDeriv 1 (deriv (phi W γ t)) (X t z)*(iteratedDeriv 2 (deriv (phi W γ t)) (X t z))^2+
+        6*(γ.coeff t)^2*(iteratedDeriv 1 (deriv (phi W γ t)) (X t z))^4)) := by
+      funext z
+      simp only [spatialJet]
+      ring
+    change 0≤∫ z,(spatialJet (γ.coeff t) (phi W γ t) 3 (X t z))^2-
+        12*spatialJet (γ.coeff t) (phi W γ t) 1 (X t z)*(spatialJet (γ.coeff t) (phi W γ t) 2 (X t z))^2+
+        6*(spatialJet (γ.coeff t) (phi W γ t) 1 (X t z))^4 ∂W.μ at hn
+    rw [he,integral_const_mul] at hn
+    exact nonneg_of_mul_nonneg_right hn (sq_pos_of_pos hc)
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/DiffusionExistence.lean b/lean/OAI/Probability/SKValue/Processes/DiffusionExistence.lean
index 36b1a488236e340ea665f46715970e25494fa17f..62dcae10a72346cf666ef59aee8804f0b9e968dc 100644
--- a/lean/OAI/Probability/SKValue/Processes/DiffusionExistence.lean
+++ b/lean/OAI/Probability/SKValue/Processes/DiffusionExistence.lean
@@ -1,84 +1,84 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.EulerLimitProcess
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter Function
-open scoped Topology NNReal
-namespace SKValue
-noncomputable def terminalEulerProcess (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω):ℝ :=
-  limsup (fun n:ℕ ↦ eulerLimitProcess W γ (terminalTime n) z) atTop
-noncomputable def diffusionProcess (W:BrownianSpace) (γ:OrderParameter) (t:ℝ) (z:W.Ω):ℝ :=
-  if t<1 then eulerLimitProcess W γ t z else terminalEulerProcess W γ z
-lemma terminalEulerProcess_measurable (W:BrownianSpace) (γ:OrderParameter):
-    StronglyMeasurable[W.filtration 1] (terminalEulerProcess W γ) :=by
-  apply Measurable.stronglyMeasurable
-  apply Measurable.limsup
-  intro n
-  apply ((eulerLimitProcess_measurable W γ (terminalTime n)).mono ?_).measurable
-  apply W.filtration.mono
-  exact Real.toNNReal_le_iff_le_coe.mpr (terminalTime_mem n).2.le
-lemma diffusionProcess_measurable (W:BrownianSpace) (γ:OrderParameter) {t:ℝ}:
-    StronglyMeasurable[W.filtration t.toNNReal] (diffusionProcess W γ t) :=by
-  change StronglyMeasurable[W.filtration t.toNNReal] (fun z ↦ if t<1 then eulerLimitProcess W γ t z else terminalEulerProcess W γ z)
-  by_cases ht:t<1
-  · simp only [ite_eq_left ht]
-    exact eulerLimitProcess_measurable W γ t
-  · simp only [ite_eq_right ht]
-    apply (terminalEulerProcess_measurable W γ).mono (W.filtration.mono ?_)
-    exact (Real.le_toNNReal_iff_coe_le (zero_le_one.trans (not_lt.mp ht))).mpr (not_lt.mp ht)
-lemma diffusionProcess_adapted (W:BrownianSpace) (γ:OrderParameter):
-    Adapted W.filtration (fun t z ↦ diffusionProcess W γ (min (t:ℝ) 1) z) :=by
-  intro t
-  apply ((diffusionProcess_measurable W γ (t:=min (t:ℝ) 1)).mono ?_).measurable
-  apply W.filtration.mono
-  exact Real.toNNReal_le_iff_le_coe.mpr (min_le_left _ _)
-lemma diffusionProcess_path (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
-    (hb:Continuous (fun t:ℝ ↦ W.B t.toNNReal z)) (h0:W.B 0 z=0):
-    ContinuousOn (fun t ↦ diffusionProcess W γ t z) (Icc (0:ℝ) 1) ∧
-    ∀t∈Icc (0:ℝ) 1,diffusionProcess W γ t z=W.B t.toNNReal z+
-      ∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (diffusionProcess W γ s z) :=by
-  let f:=fun s ↦ γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z)
-  let F:=fun t ↦ W.B t.toNNReal z+∫s in (0:ℝ)..t,f s
-  have hi:IntervalIntegrable f volume 0 1:=eulerLimitProcess_drift_integrable W γ z hb h0
-  have hc:ContinuousOn F (Icc (0:ℝ) 1):=by
-    apply hb.continuousOn.add
-    simpa only [uIcc_of_le (by norm_num:(0:ℝ)≤1)] using
-      intervalIntegral.continuousOn_primitive_interval' hi left_mem_uIcc
-  have he:∀t∈Ico (0:ℝ) 1,eulerLimitProcess W γ t z=F t:=by
-    intro t ht
-    exact (eulerLimitProcess_local_path W γ z hb h0 ht).2.2 t ⟨ht.1,le_rfl⟩
-  have hl:Tendsto (fun n ↦ eulerLimitProcess W γ (terminalTime n) z) atTop (𝓝 (F 1)):=by
-    have htime:Tendsto terminalTime atTop (𝓝[Icc (0:ℝ) 1] (1:ℝ)):=
-      tendsto_nhdsWithin_iff.mpr ⟨(tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1,
-        Eventually.of_forall (fun n ↦ ⟨(terminalTime_mem n).1,(terminalTime_mem n).2.le⟩)⟩
-    apply ((hc 1 ⟨by norm_num,le_rfl⟩).tendsto.comp htime).congr
-    intro n
-    exact (he _ (terminalTime_mem n)).symm
-  have hU:terminalEulerProcess W γ z=F 1:=hl.limsup_eq
-  have hEq (t:ℝ) (ht:t∈Icc (0:ℝ) 1):diffusionProcess W γ t z=F t:=by
-    by_cases ht1:t<1
-    · simpa only [diffusionProcess,ite_eq_left ht1] using he t ⟨ht.1,ht1⟩
-    · have ht':t=1:=le_antisymm ht.2 (not_lt.mp ht1)
-      subst t
-      simpa only [diffusionProcess,lt_self_iff_false,ite_false] using hU
-  refine ⟨hc.congr hEq,?_⟩
-  intro t ht
-  rw [hEq t ht]
-  change W.B t.toNNReal z+(∫s in (0:ℝ)..t,f s)=_
-  congr 1
-  apply intervalIntegral.integral_congr_Ioo_of_le ht.1
-  intro s hs
-  simp only [f,diffusionProcess,ite_eq_left (hs.2.trans_le ht.2)]
-theorem diffusionProcess_isDiffusion (W:BrownianSpace) (γ:OrderParameter):IsDiffusion W γ (diffusionProcess W γ) :=by
-  refine ⟨diffusionProcess_adapted W γ,?_⟩
-  filter_upwards [W.brownian.cont,W.brownian.toIsPreBrownianReal.eval_zero_ae_eq_zero] with z hz h0
-  exact diffusionProcess_path W γ z (hz.comp (by fun_prop:Continuous (fun t:ℝ ↦ t.toNNReal))) h0
-lemma exists_diffusion (W:BrownianSpace) (γ:OrderParameter):∃X:ℝ → W.Ω → ℝ,IsDiffusion W γ X:=
-  ⟨diffusionProcess W γ,diffusionProcess_isDiffusion W γ⟩
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.EulerLimitProcess
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter Function
+open scoped Topology NNReal
+namespace SKValue
+noncomputable def terminalEulerProcess (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω):ℝ :=
+  limsup (fun n:ℕ ↦ eulerLimitProcess W γ (terminalTime n) z) atTop
+noncomputable def diffusionProcess (W:BrownianSpace) (γ:OrderParameter) (t:ℝ) (z:W.Ω):ℝ :=
+  if t<1 then eulerLimitProcess W γ t z else terminalEulerProcess W γ z
+lemma terminalEulerProcess_measurable (W:BrownianSpace) (γ:OrderParameter):
+    StronglyMeasurable[W.filtration 1] (terminalEulerProcess W γ) :=by
+  apply Measurable.stronglyMeasurable
+  apply Measurable.limsup
+  intro n
+  apply ((eulerLimitProcess_measurable W γ (terminalTime n)).mono ?_).measurable
+  apply W.filtration.mono
+  exact Real.toNNReal_le_iff_le_coe.mpr (terminalTime_mem n).2.le
+lemma diffusionProcess_measurable (W:BrownianSpace) (γ:OrderParameter) {t:ℝ}:
+    StronglyMeasurable[W.filtration t.toNNReal] (diffusionProcess W γ t) :=by
+  change StronglyMeasurable[W.filtration t.toNNReal] (fun z ↦ if t<1 then eulerLimitProcess W γ t z else terminalEulerProcess W γ z)
+  by_cases ht:t<1
+  · simp only [ite_eq_left ht]
+    exact eulerLimitProcess_measurable W γ t
+  · simp only [ite_eq_right ht]
+    apply (terminalEulerProcess_measurable W γ).mono (W.filtration.mono ?_)
+    exact (Real.le_toNNReal_iff_coe_le (zero_le_one.trans (not_lt.mp ht))).mpr (not_lt.mp ht)
+lemma diffusionProcess_adapted (W:BrownianSpace) (γ:OrderParameter):
+    Adapted W.filtration (fun t z ↦ diffusionProcess W γ (min (t:ℝ) 1) z) :=by
+  intro t
+  apply ((diffusionProcess_measurable W γ (t:=min (t:ℝ) 1)).mono ?_).measurable
+  apply W.filtration.mono
+  exact Real.toNNReal_le_iff_le_coe.mpr (min_le_left _ _)
+lemma diffusionProcess_path (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
+    (hb:Continuous (fun t:ℝ ↦ W.B t.toNNReal z)) (h0:W.B 0 z=0):
+    ContinuousOn (fun t ↦ diffusionProcess W γ t z) (Icc (0:ℝ) 1) ∧
+    ∀t∈Icc (0:ℝ) 1,diffusionProcess W γ t z=W.B t.toNNReal z+
+      ∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (diffusionProcess W γ s z) :=by
+  let f:=fun s ↦ γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z)
+  let F:=fun t ↦ W.B t.toNNReal z+∫s in (0:ℝ)..t,f s
+  have hi:IntervalIntegrable f volume 0 1:=eulerLimitProcess_drift_integrable W γ z hb h0
+  have hc:ContinuousOn F (Icc (0:ℝ) 1):=by
+    apply hb.continuousOn.add
+    simpa only [uIcc_of_le (by norm_num:(0:ℝ)≤1)] using
+      intervalIntegral.continuousOn_primitive_interval' hi left_mem_uIcc
+  have he:∀t∈Ico (0:ℝ) 1,eulerLimitProcess W γ t z=F t:=by
+    intro t ht
+    exact (eulerLimitProcess_local_path W γ z hb h0 ht).2.2 t ⟨ht.1,le_rfl⟩
+  have hl:Tendsto (fun n ↦ eulerLimitProcess W γ (terminalTime n) z) atTop (𝓝 (F 1)):=by
+    have htime:Tendsto terminalTime atTop (𝓝[Icc (0:ℝ) 1] (1:ℝ)):=
+      tendsto_nhdsWithin_iff.mpr ⟨(tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1,
+        Eventually.of_forall (fun n ↦ ⟨(terminalTime_mem n).1,(terminalTime_mem n).2.le⟩)⟩
+    apply ((hc 1 ⟨by norm_num,le_rfl⟩).tendsto.comp htime).congr
+    intro n
+    exact (he _ (terminalTime_mem n)).symm
+  have hU:terminalEulerProcess W γ z=F 1:=hl.limsup_eq
+  have hEq (t:ℝ) (ht:t∈Icc (0:ℝ) 1):diffusionProcess W γ t z=F t:=by
+    by_cases ht1:t<1
+    · simpa only [diffusionProcess,ite_eq_left ht1] using he t ⟨ht.1,ht1⟩
+    · have ht':t=1:=le_antisymm ht.2 (not_lt.mp ht1)
+      subst t
+      simpa only [diffusionProcess,lt_self_iff_false,ite_false] using hU
+  refine ⟨hc.congr hEq,?_⟩
+  intro t ht
+  rw [hEq t ht]
+  change W.B t.toNNReal z+(∫s in (0:ℝ)..t,f s)=_
+  congr 1
+  apply intervalIntegral.integral_congr_Ioo_of_le ht.1
+  intro s hs
+  simp only [f,diffusionProcess,ite_eq_left (hs.2.trans_le ht.2)]
+theorem diffusionProcess_isDiffusion (W:BrownianSpace) (γ:OrderParameter):IsDiffusion W γ (diffusionProcess W γ) :=by
+  refine ⟨diffusionProcess_adapted W γ,?_⟩
+  filter_upwards [W.brownian.cont,W.brownian.toIsPreBrownianReal.eval_zero_ae_eq_zero] with z hz h0
+  exact diffusionProcess_path W γ z (hz.comp (by fun_prop:Continuous (fun t:ℝ ↦ t.toNNReal))) h0
+lemma exists_diffusion (W:BrownianSpace) (γ:OrderParameter):∃X:ℝ → W.Ω → ℝ,IsDiffusion W γ X:=
+  ⟨diffusionProcess W γ,diffusionProcess_isDiffusion W γ⟩
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/DiffusionGradientCondExp.lean b/lean/OAI/Probability/SKValue/Processes/DiffusionGradientCondExp.lean
index f067cd869d7167578958a195e0e093fb42de3b97..de924b0dfd36873e7481325d18ebc420ec857cec 100644
--- a/lean/OAI/Probability/SKValue/Processes/DiffusionGradientCondExp.lean
+++ b/lean/OAI/Probability/SKValue/Processes/DiffusionGradientCondExp.lean
@@ -1,143 +1,143 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.Shift
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal BigOperators
-
-lemma diffusion_gradient_condExp
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    (hBm : ∀ t, StronglyMeasurable (B t))
-    {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {X : ℝ → Ω → ℝ}
-    (hT1 : T≤1) (hLu : 0≤Lu) (h : GradientStripCore T γ u K L)
-    (hLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (hXM : ∀ t∈Icc (0 : ℝ) T,
-      StronglyMeasurable[Filtration.natural B hBm t.toNNReal] (X t))
-    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      ∀ t∈Icc (0 : ℝ) T, X t ω=B t.toNNReal ω+∫ s in (0 : ℝ)..t, γ s*u s (X s ω))
-    {a b : ℝ} (ha : a∈Icc (0 : ℝ) T) (hb : b∈Icc (0 : ℝ) T) (hab : a≤b) :
-    μ[(fun ω ↦ u b (X b ω)) | Filtration.natural B hBm a.toNNReal]=ᵐ[μ](fun ω ↦ u a (X a ω)) := by
-  let ℱ := Filtration.natural B hBm
-  by_cases heq : a=b
-  · subst b
-    have hh := (h.smooth a ha).continuous.comp_stronglyMeasurable (hXM a ha)
-    apply condExp_of_aestronglyMeasurable' (ℱ.le _) hh.aestronglyMeasurable
-    exact Integrable.of_bound (hh.mono (ℱ.le _)).aestronglyMeasurable 1
-      (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le (h.bounded a ha _)))
-  have hlt : a<b := lt_of_le_of_ne hab heq
-  have hba : 0<b-a := sub_pos.mpr hlt
-  have hba1 : b-a≤1 := by linarith [hb.2,ha.1]
-  let Y := fun t : ℝ ↦ X (a+t)
-  let W := fun t : ℝ ↦ fun ω ↦ X a ω+B (a.toNNReal+t.toNNReal) ω-B a.toNNReal ω
-  have hmem (t : ℝ) (ht : t∈Icc (0 : ℝ) (b-a)) : a+t∈Icc (0 : ℝ) T := by
-    constructor <;> linarith [ha.1,hb.2,ht.1,ht.2]
-  have hLip' : ∀ s∈Icc (0 : ℝ) (b-a), ∀ t∈Icc (0 : ℝ) (b-a), ∀ x y,
-      |u (a+s) x-u (a+t) y|≤Lu*(|s-t|+|x-y|) := by
-    intro s hs t ht x y
-    simpa only [add_sub_add_left_eq_sub] using hLip _ (hmem s hs) _ (hmem t ht) x y
-  have hYM (t : ℝ) (ht : t∈Icc (0 : ℝ) (b-a)) :
-      StronglyMeasurable[ℱ (a.toNNReal+t.toNNReal)] (Y t) := by
-    have hh := hXM _ (hmem t ht)
-    rw [Real.toNNReal_add ha.1 ht.1] at hh
-    exact hh
-  have hWM (t : ℝ) (_ht : t∈Icc (0 : ℝ) (b-a)) : AEStronglyMeasurable (W t) μ :=
-    ((((hXM a ha).mono (ℱ.le _)).add (hBm _)).sub (hBm _)).aestronglyMeasurable
-  have hW : ∀ s∈Icc (0 : ℝ) (b-a), ∀ t∈Icc (0 : ℝ) (b-a), ∀ ω,
-      W t ω-W s ω=B (a.toNNReal+t.toNNReal) ω-B (a.toNNReal+s.toNNReal) ω := by
-    intro s _ t _ ω
-    dsimp [W]
-    ring
-  have hpaths' : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ Y t ω) (Icc (0 : ℝ) (b-a)) ∧
-      IntervalIntegrable (fun s ↦ γ (a+s)*u (a+s) (Y s ω)) volume 0 (b-a) ∧
-      ∀ t∈Icc (0 : ℝ) (b-a), Y t ω=W t ω+∫ s in (0 : ℝ)..t, γ (a+s)*u (a+s) (Y s ω) := by
-    filter_upwards [hpaths] with ω hω
-    have hT : 0≤T := ha.1.trans ha.2
-    have hi := intervalIntegrable_substrip hT hω.2.1 ha hb
-    refine ⟨hω.1.comp (continuous_const.add continuous_id).continuousOn hmem,?_,?_⟩
-    · simpa only [sub_self,Y] using hi.comp_add_left a
-    · intro t ht
-      have hhalf := intervalIntegrable_substrip hT hω.2.1 ha (hmem t ht)
-      have hleft := intervalIntegrable_substrip hT hω.2.1 ⟨le_rfl,hT⟩ ha
-      have hsum := intervalIntegral.integral_add_adjacent_intervals hleft hhalf
-      dsimp only [Y,W]
-      rw [intervalIntegral.integral_comp_add_left (fun s ↦ γ s*u s (X s ω)) a,add_zero]
-      rw [hω.2.2 _ (hmem t ht),hω.2.2 _ ha,Real.toNNReal_add ha.1 ht.1]
-      linarith
-  have hh := diffusion_gradient_condExp_local hB hBm a.toNNReal hba hba1 hLu
-    (h.shift ha.1 hab hb.2) hLip' hYM hWM hW hpaths'
-  simpa only [Y,add_sub_cancel,add_zero] using hh
-
-end SKValue
-
-namespace SKValue
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal BigOperators
-
-lemma diffusion_gradient_stopped_martingale
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    (hBm : ∀ t, StronglyMeasurable (B t))
-    {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {X : ℝ → Ω → ℝ}
-    (hT : 0≤T) (hT1 : T≤1) (hLu : 0≤Lu) (h : GradientStripCore T γ u K L)
-    (hLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (hXM : ∀ t∈Icc (0 : ℝ) T,
-      StronglyMeasurable[Filtration.natural B hBm t.toNNReal] (X t))
-    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      ∀ t∈Icc (0 : ℝ) T, X t ω=B t.toNNReal ω+∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) :
-    Martingale (fun t : ℝ≥0 ↦ fun ω ↦ u (min (t : ℝ) T) (X (min (t : ℝ) T) ω))
-      (Filtration.natural B hBm) μ := by
-  let ℱ := Filtration.natural B hBm
-  have hmem (t : ℝ≥0) : min (t : ℝ) T∈Icc (0 : ℝ) T :=
-    ⟨le_min t.coe_nonneg hT,min_le_right _ _⟩
-  have htime (t : ℝ≥0) : (min (t : ℝ) T).toNNReal≤t := by
-    have hh := Real.toNNReal_mono (min_le_left (t : ℝ) T)
-    rw [Real.toNNReal_coe] at hh
-    exact hh
-  have hm (t : ℝ≥0) : StronglyMeasurable[ℱ t]
-      (fun ω ↦ u (min (t : ℝ) T) (X (min (t : ℝ) T) ω)) :=
-    ((h.smooth _ (hmem t)).continuous.comp_stronglyMeasurable (hXM _ (hmem t))).mono (ℱ.mono (htime t))
-  have hi (t : ℝ≥0) : Integrable (fun ω ↦ u (min (t : ℝ) T) (X (min (t : ℝ) T) ω)) μ :=
-    Integrable.of_bound ((hm t).mono (ℱ.le _)).aestronglyMeasurable 1
-      (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le (h.bounded _ (hmem t) _)))
-  refine ⟨hm,?_⟩
-  intro s t hst
-  by_cases hs : (s : ℝ)≤T
-  · have ha : (s : ℝ)∈Icc (0 : ℝ) T := ⟨s.coe_nonneg,hs⟩
-    have hh := diffusion_gradient_condExp hB hBm hT1 hLu h hLip hXM hpaths ha (hmem t)
-      (le_min (by exact_mod_cast hst) hs)
-    rw [Real.toNNReal_coe] at hh
-    simpa only [min_eq_left hs] using hh
-  · have hs' : T≤(s : ℝ) := (lt_of_not_ge hs).le
-    have ht : T≤(t : ℝ) := hs'.trans (by exact_mod_cast hst)
-    have hh := condExp_of_aestronglyMeasurable' (ℱ.le s) (hm s).aestronglyMeasurable (hi s)
-    simpa only [min_eq_right hs',min_eq_right ht] using hh
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.Shift
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal BigOperators
+
+lemma diffusion_gradient_condExp
+    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    (hBm : ∀ t, StronglyMeasurable (B t))
+    {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {X : ℝ → Ω → ℝ}
+    (hT1 : T≤1) (hLu : 0≤Lu) (h : GradientStripCore T γ u K L)
+    (hLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (hXM : ∀ t∈Icc (0 : ℝ) T,
+      StronglyMeasurable[Filtration.natural B hBm t.toNNReal] (X t))
+    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      ∀ t∈Icc (0 : ℝ) T, X t ω=B t.toNNReal ω+∫ s in (0 : ℝ)..t, γ s*u s (X s ω))
+    {a b : ℝ} (ha : a∈Icc (0 : ℝ) T) (hb : b∈Icc (0 : ℝ) T) (hab : a≤b) :
+    μ[(fun ω ↦ u b (X b ω)) | Filtration.natural B hBm a.toNNReal]=ᵐ[μ](fun ω ↦ u a (X a ω)) := by
+  let ℱ := Filtration.natural B hBm
+  by_cases heq : a=b
+  · subst b
+    have hh := (h.smooth a ha).continuous.comp_stronglyMeasurable (hXM a ha)
+    apply condExp_of_aestronglyMeasurable' (ℱ.le _) hh.aestronglyMeasurable
+    exact Integrable.of_bound (hh.mono (ℱ.le _)).aestronglyMeasurable 1
+      (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le (h.bounded a ha _)))
+  have hlt : a<b := lt_of_le_of_ne hab heq
+  have hba : 0<b-a := sub_pos.mpr hlt
+  have hba1 : b-a≤1 := by linarith [hb.2,ha.1]
+  let Y := fun t : ℝ ↦ X (a+t)
+  let W := fun t : ℝ ↦ fun ω ↦ X a ω+B (a.toNNReal+t.toNNReal) ω-B a.toNNReal ω
+  have hmem (t : ℝ) (ht : t∈Icc (0 : ℝ) (b-a)) : a+t∈Icc (0 : ℝ) T := by
+    constructor <;> linarith [ha.1,hb.2,ht.1,ht.2]
+  have hLip' : ∀ s∈Icc (0 : ℝ) (b-a), ∀ t∈Icc (0 : ℝ) (b-a), ∀ x y,
+      |u (a+s) x-u (a+t) y|≤Lu*(|s-t|+|x-y|) := by
+    intro s hs t ht x y
+    simpa only [add_sub_add_left_eq_sub] using hLip _ (hmem s hs) _ (hmem t ht) x y
+  have hYM (t : ℝ) (ht : t∈Icc (0 : ℝ) (b-a)) :
+      StronglyMeasurable[ℱ (a.toNNReal+t.toNNReal)] (Y t) := by
+    have hh := hXM _ (hmem t ht)
+    rw [Real.toNNReal_add ha.1 ht.1] at hh
+    exact hh
+  have hWM (t : ℝ) (_ht : t∈Icc (0 : ℝ) (b-a)) : AEStronglyMeasurable (W t) μ :=
+    ((((hXM a ha).mono (ℱ.le _)).add (hBm _)).sub (hBm _)).aestronglyMeasurable
+  have hW : ∀ s∈Icc (0 : ℝ) (b-a), ∀ t∈Icc (0 : ℝ) (b-a), ∀ ω,
+      W t ω-W s ω=B (a.toNNReal+t.toNNReal) ω-B (a.toNNReal+s.toNNReal) ω := by
+    intro s _ t _ ω
+    dsimp [W]
+    ring
+  have hpaths' : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ Y t ω) (Icc (0 : ℝ) (b-a)) ∧
+      IntervalIntegrable (fun s ↦ γ (a+s)*u (a+s) (Y s ω)) volume 0 (b-a) ∧
+      ∀ t∈Icc (0 : ℝ) (b-a), Y t ω=W t ω+∫ s in (0 : ℝ)..t, γ (a+s)*u (a+s) (Y s ω) := by
+    filter_upwards [hpaths] with ω hω
+    have hT : 0≤T := ha.1.trans ha.2
+    have hi := intervalIntegrable_substrip hT hω.2.1 ha hb
+    refine ⟨hω.1.comp (continuous_const.add continuous_id).continuousOn hmem,?_,?_⟩
+    · simpa only [sub_self,Y] using hi.comp_add_left a
+    · intro t ht
+      have hhalf := intervalIntegrable_substrip hT hω.2.1 ha (hmem t ht)
+      have hleft := intervalIntegrable_substrip hT hω.2.1 ⟨le_rfl,hT⟩ ha
+      have hsum := intervalIntegral.integral_add_adjacent_intervals hleft hhalf
+      dsimp only [Y,W]
+      rw [intervalIntegral.integral_comp_add_left (fun s ↦ γ s*u s (X s ω)) a,add_zero]
+      rw [hω.2.2 _ (hmem t ht),hω.2.2 _ ha,Real.toNNReal_add ha.1 ht.1]
+      linarith
+  have hh := diffusion_gradient_condExp_local hB hBm a.toNNReal hba hba1 hLu
+    (h.shift ha.1 hab hb.2) hLip' hYM hWM hW hpaths'
+  simpa only [Y,add_sub_cancel,add_zero] using hh
+
+end SKValue
+
+namespace SKValue
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal BigOperators
+
+lemma diffusion_gradient_stopped_martingale
+    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    (hBm : ∀ t, StronglyMeasurable (B t))
+    {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {X : ℝ → Ω → ℝ}
+    (hT : 0≤T) (hT1 : T≤1) (hLu : 0≤Lu) (h : GradientStripCore T γ u K L)
+    (hLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (hXM : ∀ t∈Icc (0 : ℝ) T,
+      StronglyMeasurable[Filtration.natural B hBm t.toNNReal] (X t))
+    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      ∀ t∈Icc (0 : ℝ) T, X t ω=B t.toNNReal ω+∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) :
+    Martingale (fun t : ℝ≥0 ↦ fun ω ↦ u (min (t : ℝ) T) (X (min (t : ℝ) T) ω))
+      (Filtration.natural B hBm) μ := by
+  let ℱ := Filtration.natural B hBm
+  have hmem (t : ℝ≥0) : min (t : ℝ) T∈Icc (0 : ℝ) T :=
+    ⟨le_min t.coe_nonneg hT,min_le_right _ _⟩
+  have htime (t : ℝ≥0) : (min (t : ℝ) T).toNNReal≤t := by
+    have hh := Real.toNNReal_mono (min_le_left (t : ℝ) T)
+    rw [Real.toNNReal_coe] at hh
+    exact hh
+  have hm (t : ℝ≥0) : StronglyMeasurable[ℱ t]
+      (fun ω ↦ u (min (t : ℝ) T) (X (min (t : ℝ) T) ω)) :=
+    ((h.smooth _ (hmem t)).continuous.comp_stronglyMeasurable (hXM _ (hmem t))).mono (ℱ.mono (htime t))
+  have hi (t : ℝ≥0) : Integrable (fun ω ↦ u (min (t : ℝ) T) (X (min (t : ℝ) T) ω)) μ :=
+    Integrable.of_bound ((hm t).mono (ℱ.le _)).aestronglyMeasurable 1
+      (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le (h.bounded _ (hmem t) _)))
+  refine ⟨hm,?_⟩
+  intro s t hst
+  by_cases hs : (s : ℝ)≤T
+  · have ha : (s : ℝ)∈Icc (0 : ℝ) T := ⟨s.coe_nonneg,hs⟩
+    have hh := diffusion_gradient_condExp hB hBm hT1 hLu h hLip hXM hpaths ha (hmem t)
+      (le_min (by exact_mod_cast hst) hs)
+    rw [Real.toNNReal_coe] at hh
+    simpa only [min_eq_left hs] using hh
+  · have hs' : T≤(s : ℝ) := (lt_of_not_ge hs).le
+    have ht : T≤(t : ℝ) := hs'.trans (by exact_mod_cast hst)
+    have hh := condExp_of_aestronglyMeasurable' (ℱ.le s) (hm s).aestronglyMeasurable (hi s)
+    simpa only [min_eq_right hs',min_eq_right ht] using hh
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/DiffusionGradientWeakMartingaleLocal.lean b/lean/OAI/Probability/SKValue/Processes/DiffusionGradientWeakMartingaleLocal.lean
index 959dbbfbe8ce698598e9a4c912c0d97cdcc99507..48c7bc876e5e9f61cda2ece1b133c01dd0637f7b 100644
--- a/lean/OAI/Probability/SKValue/Processes/DiffusionGradientWeakMartingaleLocal.lean
+++ b/lean/OAI/Probability/SKValue/Processes/DiffusionGradientWeakMartingaleLocal.lean
@@ -1,131 +1,131 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.CubicEnvelopeMean
-import OAI.Probability.SKValue.Gaussian.ShiftedBrownianCoordinates
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal BigOperators
-
-lemma diffusion_gradient_weak_martingale_local
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    (hBm : ∀ t, StronglyMeasurable (B t)) (a : ℝ≥0)
-    {T K L Lu H : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {X W : ℝ → Ω → ℝ}
-    {F : Ω → ℝ} (hT : 0<T) (hT1 : T≤1) (hLu : 0≤Lu) (hH : 0≤H)
-    (h : GradientStripCore T γ u K L)
-    (hLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (hXM : ∀ t∈Icc (0 : ℝ) T,
-      StronglyMeasurable[Filtration.natural B hBm (a+t.toNNReal)] (X t))
-    (hWM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (W t) μ)
-    (hW : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ ω,
-      W t ω-W s ω=B (a+t.toNNReal) ω-B (a+s.toNNReal) ω)
-    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      ∀ t∈Icc (0 : ℝ) T, X t ω=W t ω+∫ s in (0 : ℝ)..t, γ s*u s (X s ω))
-    (hFm : StronglyMeasurable[Filtration.natural B hBm a] F) (hFb : ∀ ω, |F ω|≤H) :
-    (∫ ω, F ω*u T (X T ω) ∂μ)=(∫ ω, F ω*u 0 (X 0 ω) ∂μ) := by
-  let ℱ := Filtration.natural B hBm
-  have hFM : AEStronglyMeasurable F μ := (hFm.mono (ℱ.le a)).aestronglyMeasurable
-  have hXM' (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : AEStronglyMeasurable (X t) μ :=
-    ((hXM t ht).mono (ℱ.le _)).aestronglyMeasurable
-  have hsp (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x y : ℝ) :
-      |u t x-u t y|≤Lu*|x-y| := by
-    simpa only [sub_self,abs_zero,zero_add] using hLip t ht t ht x y
-  have hVi (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : Integrable (fun ω ↦ F ω*u t (X t ω)) μ := by
-    apply Integrable.of_bound (hFM.mul ((h.smooth t ht).continuous.comp_aestronglyMeasurable (hXM' t ht))) H
-    filter_upwards [] with ω
-    simp only [Pi.mul_apply, Real.norm_eq_abs, abs_mul]
-    exact (mul_le_mul (hFb ω) (h.bounded t ht _) (abs_nonneg _) hH).trans_eq (mul_one H)
-  have hlim := weighted_pathGradientResidual_tendsto (Z:=shiftedBrownianCoordinates B a T)
-    hT hT1 hLu hH h hLip hXM' hWM
-    (fun N hN i ↦ shiftedBrownianCoordinates_hasLaw hB a hT hN i)
-    (fun N hN i ω ↦ shiftedBrownianCoordinates_exact hT hN hW i ω) hpaths hFM hFb
-  have hmean (N : ℕ) : (∫ ω, F ω*pathGradientResidual T N u (fun t ↦ X t ω)
-      (fun t ↦ W t ω) ∂μ)=(∫ ω, F ω*u T (X T ω) ∂μ)-(∫ ω, F ω*u 0 (X 0 ω) ∂μ) := by
-    have hδ : 0≤ stepSize T N := div_nonneg hT.le (Nat.cast_nonneg N)
-    have htime (j : ℕ) (hj : j≤N) := mesh_time_mem_total hT.le hj
-    let τ := fun j : ℕ ↦ a+(meshTime T N j).toNNReal
-    have hτ (i : Fin N) : τ i≤τ (i+1) := by
-      dsimp [τ]
-      apply add_le_add le_rfl
-      apply Real.toNNReal_mono
-      dsimp [meshTime]
-      push_cast
-      nlinarith
-    have hτdiff (i : Fin N) : (τ (i+1) : ℝ)-(τ i : ℝ)=stepSize T N := by
-      dsimp only [τ]
-      rw [NNReal.coe_add,NNReal.coe_add,Real.coe_toNNReal _ (htime (i+1) i.isLt).1,
-        Real.coe_toNNReal _ (htime i i.isLt.le).1]
-      dsimp [meshTime]
-      push_cast
-      ring
-    have hWeq (i : Fin N) (ω : Ω) :
-        W (meshTime T N (i+1)) ω-W (meshTime T N i) ω=B (τ (i+1)) ω-B (τ i) ω :=
-      hW _ (htime i i.isLt.le) _ (htime (i+1) i.isLt) ω
-    have hFm' (i : Fin N) : StronglyMeasurable[ℱ (τ i)] F :=
-      hFm.mono (ℱ.mono (le_add_of_nonneg_right (by positivity)))
-    have hDm (i : Fin N) : StronglyMeasurable[ℱ (τ i)]
-        (fun ω ↦ F ω*deriv (u (meshTime T N i)) (X (meshTime T N i) ω)) :=
-      (hFm' i).mul ((h.continuous_deriv (htime i i.isLt.le)).comp_stronglyMeasurable
-        (hXM _ (htime i i.isLt.le)))
-    have hD2m (i : Fin N) : StronglyMeasurable[ℱ (τ i)]
-        (fun ω ↦ F ω*deriv (deriv (u (meshTime T N i))) (X (meshTime T N i) ω)) :=
-      (hFm' i).mul ((h.continuous_second (htime i i.isLt.le)).comp_stronglyMeasurable
-        (hXM _ (htime i i.isLt.le)))
-    have hDi (i : Fin N) : Integrable
-        (fun ω ↦ F ω*deriv (u (meshTime T N i)) (X (meshTime T N i) ω)) μ := by
-      apply Integrable.of_bound ((hDm i).mono (ℱ.le _)).aestronglyMeasurable (H*Lu)
-      filter_upwards [] with ω
-      rw [Real.norm_eq_abs,abs_mul]
-      exact mul_le_mul (hFb ω) (GradientStripCore.deriv_bound hLu hsp (htime i i.isLt.le) _)
-        (abs_nonneg _) hH
-    have hD2i (i : Fin N) : Integrable
-        (fun ω ↦ F ω*deriv (deriv (u (meshTime T N i))) (X (meshTime T N i) ω)) μ := by
-      apply Integrable.of_bound ((hD2m i).mono (ℱ.le _)).aestronglyMeasurable (H*K)
-      filter_upwards [] with ω
-      rw [Real.norm_eq_abs,abs_mul]
-      exact mul_le_mul (hFb ω) (h.second_bound _ (htime i i.isLt.le) _) (abs_nonneg _) hH
-    apply weighted_pathGradientResidual_integral (hVi T ⟨hT.le,le_rfl⟩) (hVi 0 ⟨le_rfl,hT.le⟩)
-    · intro i
-      simpa only [hWeq,mul_assoc] using
-        brownian_predictable_increment_integrable hB hBm (hτ i) (hDm i) (hDi i)
-    · intro i
-      simpa only [hWeq,hτdiff,mul_assoc] using
-        brownian_predictable_quadratic_integrable hB hBm (hτ i) (hD2m i) (hD2i i)
-    · intro i
-      simpa only [hWeq,mul_assoc] using brownian_predictable_increment_mean hB hBm (hτ i) (hDm i)
-    · intro i
-      simpa only [hWeq,hτdiff,mul_assoc] using
-        brownian_predictable_quadratic_mean hB hBm (hτ i) (hD2m i)
-  have hh : (∫ ω, F ω*u T (X T ω) ∂μ)-(∫ ω, F ω*u 0 (X 0 ω) ∂μ)=0 := by
-    have hc : Tendsto (fun _ : ℕ ↦ (∫ ω, F ω*u T (X T ω) ∂μ)-
-        (∫ ω, F ω*u 0 (X 0 ω) ∂μ)) atTop (𝓝 (0 : ℝ)) := by
-      simpa only [hmean] using hlim
-    exact tendsto_nhds_unique tendsto_const_nhds hc
-  exact sub_eq_zero.mp hh
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.CubicEnvelopeMean
+import OAI.Probability.SKValue.Gaussian.ShiftedBrownianCoordinates
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal BigOperators
+
+lemma diffusion_gradient_weak_martingale_local
+    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    (hBm : ∀ t, StronglyMeasurable (B t)) (a : ℝ≥0)
+    {T K L Lu H : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {X W : ℝ → Ω → ℝ}
+    {F : Ω → ℝ} (hT : 0<T) (hT1 : T≤1) (hLu : 0≤Lu) (hH : 0≤H)
+    (h : GradientStripCore T γ u K L)
+    (hLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (hXM : ∀ t∈Icc (0 : ℝ) T,
+      StronglyMeasurable[Filtration.natural B hBm (a+t.toNNReal)] (X t))
+    (hWM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (W t) μ)
+    (hW : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ ω,
+      W t ω-W s ω=B (a+t.toNNReal) ω-B (a+s.toNNReal) ω)
+    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      ∀ t∈Icc (0 : ℝ) T, X t ω=W t ω+∫ s in (0 : ℝ)..t, γ s*u s (X s ω))
+    (hFm : StronglyMeasurable[Filtration.natural B hBm a] F) (hFb : ∀ ω, |F ω|≤H) :
+    (∫ ω, F ω*u T (X T ω) ∂μ)=(∫ ω, F ω*u 0 (X 0 ω) ∂μ) := by
+  let ℱ := Filtration.natural B hBm
+  have hFM : AEStronglyMeasurable F μ := (hFm.mono (ℱ.le a)).aestronglyMeasurable
+  have hXM' (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : AEStronglyMeasurable (X t) μ :=
+    ((hXM t ht).mono (ℱ.le _)).aestronglyMeasurable
+  have hsp (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x y : ℝ) :
+      |u t x-u t y|≤Lu*|x-y| := by
+    simpa only [sub_self,abs_zero,zero_add] using hLip t ht t ht x y
+  have hVi (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : Integrable (fun ω ↦ F ω*u t (X t ω)) μ := by
+    apply Integrable.of_bound (hFM.mul ((h.smooth t ht).continuous.comp_aestronglyMeasurable (hXM' t ht))) H
+    filter_upwards [] with ω
+    simp only [Pi.mul_apply, Real.norm_eq_abs, abs_mul]
+    exact (mul_le_mul (hFb ω) (h.bounded t ht _) (abs_nonneg _) hH).trans_eq (mul_one H)
+  have hlim := weighted_pathGradientResidual_tendsto (Z:=shiftedBrownianCoordinates B a T)
+    hT hT1 hLu hH h hLip hXM' hWM
+    (fun N hN i ↦ shiftedBrownianCoordinates_hasLaw hB a hT hN i)
+    (fun N hN i ω ↦ shiftedBrownianCoordinates_exact hT hN hW i ω) hpaths hFM hFb
+  have hmean (N : ℕ) : (∫ ω, F ω*pathGradientResidual T N u (fun t ↦ X t ω)
+      (fun t ↦ W t ω) ∂μ)=(∫ ω, F ω*u T (X T ω) ∂μ)-(∫ ω, F ω*u 0 (X 0 ω) ∂μ) := by
+    have hδ : 0≤ stepSize T N := div_nonneg hT.le (Nat.cast_nonneg N)
+    have htime (j : ℕ) (hj : j≤N) := mesh_time_mem_total hT.le hj
+    let τ := fun j : ℕ ↦ a+(meshTime T N j).toNNReal
+    have hτ (i : Fin N) : τ i≤τ (i+1) := by
+      dsimp [τ]
+      apply add_le_add le_rfl
+      apply Real.toNNReal_mono
+      dsimp [meshTime]
+      push_cast
+      nlinarith
+    have hτdiff (i : Fin N) : (τ (i+1) : ℝ)-(τ i : ℝ)=stepSize T N := by
+      dsimp only [τ]
+      rw [NNReal.coe_add,NNReal.coe_add,Real.coe_toNNReal _ (htime (i+1) i.isLt).1,
+        Real.coe_toNNReal _ (htime i i.isLt.le).1]
+      dsimp [meshTime]
+      push_cast
+      ring
+    have hWeq (i : Fin N) (ω : Ω) :
+        W (meshTime T N (i+1)) ω-W (meshTime T N i) ω=B (τ (i+1)) ω-B (τ i) ω :=
+      hW _ (htime i i.isLt.le) _ (htime (i+1) i.isLt) ω
+    have hFm' (i : Fin N) : StronglyMeasurable[ℱ (τ i)] F :=
+      hFm.mono (ℱ.mono (le_add_of_nonneg_right (by positivity)))
+    have hDm (i : Fin N) : StronglyMeasurable[ℱ (τ i)]
+        (fun ω ↦ F ω*deriv (u (meshTime T N i)) (X (meshTime T N i) ω)) :=
+      (hFm' i).mul ((h.continuous_deriv (htime i i.isLt.le)).comp_stronglyMeasurable
+        (hXM _ (htime i i.isLt.le)))
+    have hD2m (i : Fin N) : StronglyMeasurable[ℱ (τ i)]
+        (fun ω ↦ F ω*deriv (deriv (u (meshTime T N i))) (X (meshTime T N i) ω)) :=
+      (hFm' i).mul ((h.continuous_second (htime i i.isLt.le)).comp_stronglyMeasurable
+        (hXM _ (htime i i.isLt.le)))
+    have hDi (i : Fin N) : Integrable
+        (fun ω ↦ F ω*deriv (u (meshTime T N i)) (X (meshTime T N i) ω)) μ := by
+      apply Integrable.of_bound ((hDm i).mono (ℱ.le _)).aestronglyMeasurable (H*Lu)
+      filter_upwards [] with ω
+      rw [Real.norm_eq_abs,abs_mul]
+      exact mul_le_mul (hFb ω) (GradientStripCore.deriv_bound hLu hsp (htime i i.isLt.le) _)
+        (abs_nonneg _) hH
+    have hD2i (i : Fin N) : Integrable
+        (fun ω ↦ F ω*deriv (deriv (u (meshTime T N i))) (X (meshTime T N i) ω)) μ := by
+      apply Integrable.of_bound ((hD2m i).mono (ℱ.le _)).aestronglyMeasurable (H*K)
+      filter_upwards [] with ω
+      rw [Real.norm_eq_abs,abs_mul]
+      exact mul_le_mul (hFb ω) (h.second_bound _ (htime i i.isLt.le) _) (abs_nonneg _) hH
+    apply weighted_pathGradientResidual_integral (hVi T ⟨hT.le,le_rfl⟩) (hVi 0 ⟨le_rfl,hT.le⟩)
+    · intro i
+      simpa only [hWeq,mul_assoc] using
+        brownian_predictable_increment_integrable hB hBm (hτ i) (hDm i) (hDi i)
+    · intro i
+      simpa only [hWeq,hτdiff,mul_assoc] using
+        brownian_predictable_quadratic_integrable hB hBm (hτ i) (hD2m i) (hD2i i)
+    · intro i
+      simpa only [hWeq,mul_assoc] using brownian_predictable_increment_mean hB hBm (hτ i) (hDm i)
+    · intro i
+      simpa only [hWeq,hτdiff,mul_assoc] using
+        brownian_predictable_quadratic_mean hB hBm (hτ i) (hD2m i)
+  have hh : (∫ ω, F ω*u T (X T ω) ∂μ)-(∫ ω, F ω*u 0 (X 0 ω) ∂μ)=0 := by
+    have hc : Tendsto (fun _ : ℕ ↦ (∫ ω, F ω*u T (X T ω) ∂μ)-
+        (∫ ω, F ω*u 0 (X 0 ω) ∂μ)) atTop (𝓝 (0 : ℝ)) := by
+      simpa only [hmean] using hlim
+    exact tendsto_nhds_unique tendsto_const_nhds hc
+  exact sub_eq_zero.mp hh
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/EulerL1.lean b/lean/OAI/Probability/SKValue/Processes/EulerL1.lean
index 8ad33e2fb6c6060380c09ea959ccee765c458201..ca04e7e15c117cd93c6c1a02fd2834c3731538de 100644
--- a/lean/OAI/Probability/SKValue/Processes/EulerL1.lean
+++ b/lean/OAI/Probability/SKValue/Processes/EulerL1.lean
@@ -1,202 +1,202 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.LipschitzCompMemLpFinite
-import OAI.Probability.SKValue.Evolution.BoundedSmooth
-import OAI.Probability.SKValue.Equations.TestGenerator
-import OAI.Probability.SKValue.Processes.TestExpectation
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal BigOperators
-namespace SKValue
-lemma euler_same_noise_sum_bound {T G L e : ℝ} {d : ℕ → ℝ} {N : ℕ} {γ β : ℝ → ℝ}
-    {u v : ℝ → ℝ → ℝ} (hT : 0≤T) (hN : 0<N)
-    (hG : 0≤G) (hL : 0≤L) (hd : ∀ k<N,0≤d k) (he : 0≤e)
-    (hg : ∀ j<N,|γ (meshTime T N j)|≤G)
-    (hc : ∀ j<N,|β (meshTime T N j)-γ (meshTime T N j)|≤d j)
-    (hv : ∀ j<N,∀ x,|v (meshTime T N j) x|≤1)
-    (hs : ∀ j<N,∀ x,|v (meshTime T N j) x-u (meshTime T N j) x|≤e)
-    (hu : ∀ j<N,∀ x y,|u (meshTime T N j) x-u (meshTime T N j) y|≤L*|x-y|)
-    (z : Fin (N+1) → ℝ) {j : ℕ} (hj : j≤N) :
-    |euler T N β v z j-euler T N γ u z j|≤(∑ k∈Finset.range j,stepSize T N*(d k+G*e))*Real.exp (T*G*L) := by
-  let δ := stepSize T N
-  have hδ : 0≤δ := div_nonneg hT (Nat.cast_nonneg _)
-  have hNd : (N : ℝ)*δ=T := by dsimp [δ,stepSize]; field_simp
-  have hrec (k : ℕ) (hk : k<N) :
-      |euler T N β v z (k+1)-euler T N γ u z (k+1)|≤
-        (1+δ*G*L)*|euler T N β v z k-euler T N γ u z k|+δ*(d k+G*e) := by
-    let y := euler T N β v z k
-    let x := euler T N γ u z k
-    let t := meshTime T N k
-    have hb : |β t*v t y-γ t*u t x|≤d k+G*e+G*L*|y-x| := by
-      calc
-        |β t*v t y-γ t*u t x| = |(β t-γ t)*v t y+γ t*(v t y-u t y)+γ t*(u t y-u t x)| := by congr 1; ring
-        _ ≤ |(β t-γ t)*v t y|+|γ t*(v t y-u t y)|+|γ t*(u t y-u t x)| := (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
-        _ ≤ d k*1+G*e+G*(L*|y-x|) := by
-          simp only [abs_mul]
-          gcongr
-          · exact hd k hk
-          · exact hc k hk
-          · exact hv k hk y
-          · exact hg k hk
-          · exact hs k hk y
-          · exact hg k hk
-          · exact hu k hk y x
-        _ = _ := by ring
-    change |y+Real.sqrt δ*coordinate N k z+δ*β t*v t y-
-      (x+Real.sqrt δ*coordinate N k z+δ*γ t*u t x)|≤_
-    calc
-      _ = |(y-x)+δ*(β t*v t y-γ t*u t x)| := by congr 1; ring
-      _ ≤ |y-x|+δ*|β t*v t y-γ t*u t x| := by simpa only [abs_mul,abs_of_nonneg hδ] using abs_add_le (y-x) (δ*(β t*v t y-γ t*u t x))
-      _ ≤ |y-x|+δ*(d k+G*e+G*L*|y-x|) := add_le_add_right (mul_le_mul_of_nonneg_left hb hδ) _
-      _ = _ := by ring
-  have h := finite_discrete_gronwall (e := fun k ↦ |euler T N β v z k-euler T N γ u z k|) (mul_nonneg (mul_nonneg hδ hG) hL)
-    (show |euler T N β v z 0-euler T N γ u z 0|=0 by simp [euler])
-    (fun k hk ↦ mul_nonneg hδ (add_nonneg (hd k hk) (mul_nonneg hG he))) hrec j hj
-  apply h.trans
-  apply mul_le_mul le_rfl
-  · apply Real.exp_le_exp.mpr
-    calc
-      (j : ℝ)*(δ*G*L)≤(N : ℝ)*(δ*G*L) := mul_le_mul_of_nonneg_right (by exact_mod_cast hj) (mul_nonneg (mul_nonneg hδ hG) hL)
-      _=T*G*L := by rw [←hNd]; ring
-  · positivity
-  · exact Finset.sum_nonneg (fun k hk ↦ mul_nonneg hδ (add_nonneg (hd k ((Finset.mem_range.mp hk).trans_le hj)) (mul_nonneg hG he)))
-
-end SKValue
-
-end
-
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal BigOperators
-namespace SKValue
-lemma monotone_grid_sum_tendsto {γ:ℝ → ℝ} {T:ℝ} (hT:0<T)
-    (hm:MonotoneOn γ (Icc (0:ℝ) T)) (h0:0≤γ 0) :
-    Tendsto (fun N:ℕ ↦ stepSize T N*∑ j∈Finset.range N,γ (meshTime T N j)) atTop
-      (𝓝 (∫ t in (0:ℝ)..T,γ t)) := by
-  simpa only [mul_one,stepSize,meshTime] using
-    monotone_weighted_grid_tendsto (γ:=γ) (f:=fun _ ↦ (1:ℝ)) (K:=1) hT (by norm_num) hm h0
-      continuousOn_const (fun _ _ ↦ by norm_num)
-lemma euler_ordered_coeff_bound {T G L a b:ℝ} {β γ:ℝ → ℝ} {u v:ℝ → ℝ → ℝ}
-    {N:ℕ} (hT:0≤T) (hN:0<N) (hG:0≤G) (hL:0≤L) (ha:0≤a) (hb:0≤b)
-    (hg:∀ j<N,|γ (meshTime T N j)|≤G)
-    (hc:∀ j<N,β (meshTime T N j)≤γ (meshTime T N j)+a)
-    (hv:∀ j<N,∀ x,|v (meshTime T N j) x|≤1)
-    (hs:∀ j<N,∀ x,|v (meshTime T N j) x-u (meshTime T N j) x|≤b)
-    (hu:∀ j<N,∀ x y,|u (meshTime T N j) x-u (meshTime T N j) y|≤L*|x-y|)
-    (z:Fin (N+1) → ℝ) :
-    |euler T N β v z N-euler T N γ u z N|≤
-      (stepSize T N*(∑ j∈Finset.range N,γ (meshTime T N j))-
-        stepSize T N*(∑ j∈Finset.range N,β (meshTime T N j))+T*(2*a+G*b))*Real.exp (T*G*L) := by
-  have hd (j:ℕ) (hj:j<N) : 0≤γ (meshTime T N j)-β (meshTime T N j)+2*a := by linarith [hc j hj]
-  have h:=euler_same_noise_sum_bound (β:=β) (γ:=γ) (u:=u) (v:=v) hT hN hG hL hd hb hg
-    (fun j hj ↦ by apply abs_le.mpr;constructor <;> linarith [hc j hj]) hv hs hu z le_rfl
-  have he:(∑ j∈Finset.range N,stepSize T N*(γ (meshTime T N j)-β (meshTime T N j)+2*a+G*b))=
-      stepSize T N*(∑ j∈Finset.range N,γ (meshTime T N j))-
-        stepSize T N*(∑ j∈Finset.range N,β (meshTime T N j))+T*(2*a+G*b) := by
-    simp only [mul_add,mul_sub,Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,
-      Finset.sum_const,Finset.card_range,nsmul_eq_mul]
-    have hNd:(N:ℝ)*stepSize T N=T := by dsimp [stepSize];field_simp
-    nlinarith only [congrArg (fun x:ℝ ↦ x*(2*a+G*b)) hNd]
-  rwa [he] at h
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal BigOperators
-namespace SKValue
-lemma BackwardTest.diffusion_comparison {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
-    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
-    {X:ℝ → Ω → ℝ} {T K L K' L' Lu a b H:ℝ} {γ β:ℝ → ℝ} {u v V:ℝ → ℝ → ℝ}
-    (hT:0<T) (hT1:T≤1) (hu:GradientStrip T γ u K L) (h:BackwardTest T β v V K' L')
-    (hLu:0≤Lu) (hLip:∀ s∈Icc (0:ℝ) T,∀ t∈Icc (0:ℝ) T,∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (ha:0≤a) (hb:0≤b) (hH:0≤H)
-    (hc:∀ t∈Icc (0:ℝ) T,β t≤γ t+a)
-    (hs:∀ t∈Icc (0:ℝ) T,∀ x,|v t x-u t x|≤b)
-    (hf:∀ x y,|V T x-V T y|≤H*|x-y|)
-    (hXM:∀ t∈Icc (0:ℝ) T,MemLp (X t) 2 μ)
-    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      (∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) :
-    |(∫ ω,V T (X T ω) ∂μ)-V 0 0|≤
-      H*((∫ t in (0:ℝ)..T,γ t)-(∫ t in (0:ℝ)..T,β t)+T*(2*a+γ T*b))*Real.exp (T*γ T*Lu) := by
-  let D:=fun N:ℕ ↦ stepSize T N*(∑ j∈Finset.range N,γ (meshTime T N j))-
-      stepSize T N*(∑ j∈Finset.range N,β (meshTime T N j))+T*(2*a+γ T*b)
-  let E:=fun N:ℕ ↦ T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (β T) K' ((1/2+β T)*L')+
-    stepSize T N*K'*(β T-β 0)
-  let F:=fun N:ℕ ↦ H*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)
-  have hG:0≤γ T := hu.gamma_nonneg T ⟨hT.le,le_rfl⟩
-  have hD:Tendsto D atTop (𝓝 ((∫ t in (0:ℝ)..T,γ t)-(∫ t in (0:ℝ)..T,β t)+T*(2*a+γ T*b))) :=
-    ((monotone_grid_sum_tendsto hT hu.gamma_mono (hu.gamma_nonneg 0 ⟨le_rfl,hT.le⟩)).sub
-      (monotone_grid_sum_tendsto hT h.gamma_mono (h.gamma_nonneg 0 ⟨le_rfl,hT.le⟩))).add_const _
-  have hδ:Tendsto (fun N:ℕ ↦ stepSize T N) atTop (𝓝 (0:ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
-  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
-  have hE:Tendsto E atTop (𝓝 0) := by
-    simpa only [E,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,add_zero] using
-      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean (β T) K' ((1/2+β T)*L'))).add
-        ((hδ.mul_const K').mul_const (β T-β 0)))
-  have hF:Tendsto F atTop (𝓝 0) := by
-    simpa only [F,Function.comp_def,Real.sqrt_zero,mul_zero] using
-      (((Real.continuous_sqrt.tendsto 0).comp
-        (coupled_euler_L2_convergence hB hT hu hLu hLip (fun t ht ↦ (hXM t ht).aestronglyMeasurable) hpaths)).const_mul H)
-  have hlip:LipschitzWith ⟨H,hH⟩ (V T) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro x y
-    exact hf x y
-  have hbound:∀ᶠ N in atTop,|(∫ ω,V T (X T ω) ∂μ)-V 0 0|≤E N+H*D N*Real.exp (T*γ T*Lu)+F N := by
-    filter_upwards [eventually_gt_atTop 0] with N hN
-    let P:=fun z:Fin (N+1) → ℝ ↦ V T (euler T N β v z N)
-    let Q:=fun z:Fin (N+1) → ℝ ↦ V T (euler T N γ u z N)
-    have htime (j:ℕ) (hj:j≤N):meshTime T N j∈Icc (0:ℝ) T := mesh_time_mem hT.le hN hj
-    have hp:Integrable P (gaussianProduct (Fin (N+1))) := by
-      apply Integrable.of_bound ((hlip.continuous.measurable.comp
-        (measurable_euler T N β v (fun j hj ↦ h.drift_measurable _ (htime j hj.le)) N le_rfl)).aestronglyMeasurable) K'
-      exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_apply] using h.bound T ⟨hT.le,le_rfl⟩ _)
-    have hq:Integrable Q (gaussianProduct (Fin (N+1))) := by
-      apply Integrable.of_bound ((hlip.continuous.measurable.comp
-        (measurable_euler T N γ u (fun j hj ↦ (hu.smooth _ (htime j hj.le)).continuous.measurable) N le_rfl)).aestronglyMeasurable) K'
-      exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_apply] using h.bound T ⟨hT.le,le_rfl⟩ _)
-    have hpq:|(∫ z,P z ∂gaussianProduct (Fin (N+1)))-(∫ z,Q z ∂gaussianProduct (Fin (N+1)))|≤H*D N*Real.exp (T*γ T*Lu) := by
-      rw [←integral_sub hp hq]
-      apply (show |∫ z,P z-Q z ∂gaussianProduct (Fin (N+1))|≤H*D N*Real.exp (T*γ T*Lu) from ?_)
-      have hnorm:=norm_integral_le_of_norm_le_const (μ:=gaussianProduct (Fin (N+1)))
-        (C:=H*D N*Real.exp (T*γ T*Lu)) (f:=fun z ↦ P z-Q z) (Eventually.of_forall (fun z ↦ by
-          rw [Real.norm_eq_abs]
-          apply (hf _ _).trans
-          have he:=euler_ordered_coeff_bound (β:=β) (γ:=γ) (u:=u) (v:=v) hT.le hN hG hLu ha hb
-            (fun j hj ↦ by rw [abs_of_nonneg (hu.gamma_nonneg _ (htime j hj.le))]; exact hu.gamma_mono (htime j hj.le) ⟨hT.le,le_rfl⟩ (htime j hj.le).2)
-            (fun j hj ↦ hc _ (htime j hj.le)) (fun j hj ↦ h.drift_bound _ (htime j hj.le))
-            (fun j hj ↦ hs _ (htime j hj.le))
-            (fun j hj x y ↦ by simpa only [sub_self,abs_zero,zero_add] using hLip _ (htime j hj.le) _ (htime j hj.le) x y) z
-          simpa only [D,mul_assoc] using mul_le_mul_of_nonneg_left he hH))
-      simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using hnorm
-    have he:=h.mesh_expectation hT hT1 hN
-    have hqX:=coupled_lipschitz_expectation_bound hB hT hu hXM
-      (hpaths.mono (fun _ hω ↦ hω.2.2)) hN le_rfl hlip
-    have hend:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
-    rw [hend] at hqX
-    change |(∫ z,Q z ∂gaussianProduct (Fin (N+1)))-(∫ ω,V T (X T ω) ∂μ)|≤F N at hqX
-    change |(∫ z,P z ∂gaussianProduct (Fin (N+1)))-V 0 0|≤E N at he
-    calc
-      |(∫ ω,V T (X T ω) ∂μ)-V 0 0| ≤
-          |(∫ ω,V T (X T ω) ∂μ)-(∫ z,Q z ∂gaussianProduct (Fin (N+1)))|+
-          |(∫ z,Q z ∂gaussianProduct (Fin (N+1)))-V 0 0| := abs_sub_le _ _ _
-      _ ≤ F N+(H*D N*Real.exp (T*γ T*Lu)+E N) := by
-        apply add_le_add
-        · rw [abs_sub_comm]; exact hqX
-        · exact (abs_sub_le _ (∫ z,P z ∂gaussianProduct (Fin (N+1))) _).trans
-            (add_le_add (by rw [abs_sub_comm]; exact hpq) he)
-      _ = E N+H*D N*Real.exp (T*γ T*Lu)+F N := by ring
-  have he:=le_of_tendsto_of_tendsto tendsto_const_nhds
-    ((hE.add ((hD.const_mul H).mul_const (Real.exp (T*γ T*Lu)))).add hF) hbound
-  simpa only [zero_add,add_zero] using he
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.LipschitzCompMemLpFinite
+import OAI.Probability.SKValue.Evolution.BoundedSmooth
+import OAI.Probability.SKValue.Equations.TestGenerator
+import OAI.Probability.SKValue.Processes.TestExpectation
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal BigOperators
+namespace SKValue
+lemma euler_same_noise_sum_bound {T G L e : ℝ} {d : ℕ → ℝ} {N : ℕ} {γ β : ℝ → ℝ}
+    {u v : ℝ → ℝ → ℝ} (hT : 0≤T) (hN : 0<N)
+    (hG : 0≤G) (hL : 0≤L) (hd : ∀ k<N,0≤d k) (he : 0≤e)
+    (hg : ∀ j<N,|γ (meshTime T N j)|≤G)
+    (hc : ∀ j<N,|β (meshTime T N j)-γ (meshTime T N j)|≤d j)
+    (hv : ∀ j<N,∀ x,|v (meshTime T N j) x|≤1)
+    (hs : ∀ j<N,∀ x,|v (meshTime T N j) x-u (meshTime T N j) x|≤e)
+    (hu : ∀ j<N,∀ x y,|u (meshTime T N j) x-u (meshTime T N j) y|≤L*|x-y|)
+    (z : Fin (N+1) → ℝ) {j : ℕ} (hj : j≤N) :
+    |euler T N β v z j-euler T N γ u z j|≤(∑ k∈Finset.range j,stepSize T N*(d k+G*e))*Real.exp (T*G*L) := by
+  let δ := stepSize T N
+  have hδ : 0≤δ := div_nonneg hT (Nat.cast_nonneg _)
+  have hNd : (N : ℝ)*δ=T := by dsimp [δ,stepSize]; field_simp
+  have hrec (k : ℕ) (hk : k<N) :
+      |euler T N β v z (k+1)-euler T N γ u z (k+1)|≤
+        (1+δ*G*L)*|euler T N β v z k-euler T N γ u z k|+δ*(d k+G*e) := by
+    let y := euler T N β v z k
+    let x := euler T N γ u z k
+    let t := meshTime T N k
+    have hb : |β t*v t y-γ t*u t x|≤d k+G*e+G*L*|y-x| := by
+      calc
+        |β t*v t y-γ t*u t x| = |(β t-γ t)*v t y+γ t*(v t y-u t y)+γ t*(u t y-u t x)| := by congr 1; ring
+        _ ≤ |(β t-γ t)*v t y|+|γ t*(v t y-u t y)|+|γ t*(u t y-u t x)| := (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
+        _ ≤ d k*1+G*e+G*(L*|y-x|) := by
+          simp only [abs_mul]
+          gcongr
+          · exact hd k hk
+          · exact hc k hk
+          · exact hv k hk y
+          · exact hg k hk
+          · exact hs k hk y
+          · exact hg k hk
+          · exact hu k hk y x
+        _ = _ := by ring
+    change |y+Real.sqrt δ*coordinate N k z+δ*β t*v t y-
+      (x+Real.sqrt δ*coordinate N k z+δ*γ t*u t x)|≤_
+    calc
+      _ = |(y-x)+δ*(β t*v t y-γ t*u t x)| := by congr 1; ring
+      _ ≤ |y-x|+δ*|β t*v t y-γ t*u t x| := by simpa only [abs_mul,abs_of_nonneg hδ] using abs_add_le (y-x) (δ*(β t*v t y-γ t*u t x))
+      _ ≤ |y-x|+δ*(d k+G*e+G*L*|y-x|) := add_le_add_right (mul_le_mul_of_nonneg_left hb hδ) _
+      _ = _ := by ring
+  have h := finite_discrete_gronwall (e := fun k ↦ |euler T N β v z k-euler T N γ u z k|) (mul_nonneg (mul_nonneg hδ hG) hL)
+    (show |euler T N β v z 0-euler T N γ u z 0|=0 by simp [euler])
+    (fun k hk ↦ mul_nonneg hδ (add_nonneg (hd k hk) (mul_nonneg hG he))) hrec j hj
+  apply h.trans
+  apply mul_le_mul le_rfl
+  · apply Real.exp_le_exp.mpr
+    calc
+      (j : ℝ)*(δ*G*L)≤(N : ℝ)*(δ*G*L) := mul_le_mul_of_nonneg_right (by exact_mod_cast hj) (mul_nonneg (mul_nonneg hδ hG) hL)
+      _=T*G*L := by rw [←hNd]; ring
+  · positivity
+  · exact Finset.sum_nonneg (fun k hk ↦ mul_nonneg hδ (add_nonneg (hd k ((Finset.mem_range.mp hk).trans_le hj)) (mul_nonneg hG he)))
+
+end SKValue
+
+end
+
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal BigOperators
+namespace SKValue
+lemma monotone_grid_sum_tendsto {γ:ℝ → ℝ} {T:ℝ} (hT:0<T)
+    (hm:MonotoneOn γ (Icc (0:ℝ) T)) (h0:0≤γ 0) :
+    Tendsto (fun N:ℕ ↦ stepSize T N*∑ j∈Finset.range N,γ (meshTime T N j)) atTop
+      (𝓝 (∫ t in (0:ℝ)..T,γ t)) := by
+  simpa only [mul_one,stepSize,meshTime] using
+    monotone_weighted_grid_tendsto (γ:=γ) (f:=fun _ ↦ (1:ℝ)) (K:=1) hT (by norm_num) hm h0
+      continuousOn_const (fun _ _ ↦ by norm_num)
+lemma euler_ordered_coeff_bound {T G L a b:ℝ} {β γ:ℝ → ℝ} {u v:ℝ → ℝ → ℝ}
+    {N:ℕ} (hT:0≤T) (hN:0<N) (hG:0≤G) (hL:0≤L) (ha:0≤a) (hb:0≤b)
+    (hg:∀ j<N,|γ (meshTime T N j)|≤G)
+    (hc:∀ j<N,β (meshTime T N j)≤γ (meshTime T N j)+a)
+    (hv:∀ j<N,∀ x,|v (meshTime T N j) x|≤1)
+    (hs:∀ j<N,∀ x,|v (meshTime T N j) x-u (meshTime T N j) x|≤b)
+    (hu:∀ j<N,∀ x y,|u (meshTime T N j) x-u (meshTime T N j) y|≤L*|x-y|)
+    (z:Fin (N+1) → ℝ) :
+    |euler T N β v z N-euler T N γ u z N|≤
+      (stepSize T N*(∑ j∈Finset.range N,γ (meshTime T N j))-
+        stepSize T N*(∑ j∈Finset.range N,β (meshTime T N j))+T*(2*a+G*b))*Real.exp (T*G*L) := by
+  have hd (j:ℕ) (hj:j<N) : 0≤γ (meshTime T N j)-β (meshTime T N j)+2*a := by linarith [hc j hj]
+  have h:=euler_same_noise_sum_bound (β:=β) (γ:=γ) (u:=u) (v:=v) hT hN hG hL hd hb hg
+    (fun j hj ↦ by apply abs_le.mpr;constructor <;> linarith [hc j hj]) hv hs hu z le_rfl
+  have he:(∑ j∈Finset.range N,stepSize T N*(γ (meshTime T N j)-β (meshTime T N j)+2*a+G*b))=
+      stepSize T N*(∑ j∈Finset.range N,γ (meshTime T N j))-
+        stepSize T N*(∑ j∈Finset.range N,β (meshTime T N j))+T*(2*a+G*b) := by
+    simp only [mul_add,mul_sub,Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,
+      Finset.sum_const,Finset.card_range,nsmul_eq_mul]
+    have hNd:(N:ℝ)*stepSize T N=T := by dsimp [stepSize];field_simp
+    nlinarith only [congrArg (fun x:ℝ ↦ x*(2*a+G*b)) hNd]
+  rwa [he] at h
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal BigOperators
+namespace SKValue
+lemma BackwardTest.diffusion_comparison {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
+    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
+    {X:ℝ → Ω → ℝ} {T K L K' L' Lu a b H:ℝ} {γ β:ℝ → ℝ} {u v V:ℝ → ℝ → ℝ}
+    (hT:0<T) (hT1:T≤1) (hu:GradientStrip T γ u K L) (h:BackwardTest T β v V K' L')
+    (hLu:0≤Lu) (hLip:∀ s∈Icc (0:ℝ) T,∀ t∈Icc (0:ℝ) T,∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (ha:0≤a) (hb:0≤b) (hH:0≤H)
+    (hc:∀ t∈Icc (0:ℝ) T,β t≤γ t+a)
+    (hs:∀ t∈Icc (0:ℝ) T,∀ x,|v t x-u t x|≤b)
+    (hf:∀ x y,|V T x-V T y|≤H*|x-y|)
+    (hXM:∀ t∈Icc (0:ℝ) T,MemLp (X t) 2 μ)
+    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      (∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) :
+    |(∫ ω,V T (X T ω) ∂μ)-V 0 0|≤
+      H*((∫ t in (0:ℝ)..T,γ t)-(∫ t in (0:ℝ)..T,β t)+T*(2*a+γ T*b))*Real.exp (T*γ T*Lu) := by
+  let D:=fun N:ℕ ↦ stepSize T N*(∑ j∈Finset.range N,γ (meshTime T N j))-
+      stepSize T N*(∑ j∈Finset.range N,β (meshTime T N j))+T*(2*a+γ T*b)
+  let E:=fun N:ℕ ↦ T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (β T) K' ((1/2+β T)*L')+
+    stepSize T N*K'*(β T-β 0)
+  let F:=fun N:ℕ ↦ H*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)
+  have hG:0≤γ T := hu.gamma_nonneg T ⟨hT.le,le_rfl⟩
+  have hD:Tendsto D atTop (𝓝 ((∫ t in (0:ℝ)..T,γ t)-(∫ t in (0:ℝ)..T,β t)+T*(2*a+γ T*b))) :=
+    ((monotone_grid_sum_tendsto hT hu.gamma_mono (hu.gamma_nonneg 0 ⟨le_rfl,hT.le⟩)).sub
+      (monotone_grid_sum_tendsto hT h.gamma_mono (h.gamma_nonneg 0 ⟨le_rfl,hT.le⟩))).add_const _
+  have hδ:Tendsto (fun N:ℕ ↦ stepSize T N) atTop (𝓝 (0:ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
+  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
+  have hE:Tendsto E atTop (𝓝 0) := by
+    simpa only [E,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,add_zero] using
+      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean (β T) K' ((1/2+β T)*L'))).add
+        ((hδ.mul_const K').mul_const (β T-β 0)))
+  have hF:Tendsto F atTop (𝓝 0) := by
+    simpa only [F,Function.comp_def,Real.sqrt_zero,mul_zero] using
+      (((Real.continuous_sqrt.tendsto 0).comp
+        (coupled_euler_L2_convergence hB hT hu hLu hLip (fun t ht ↦ (hXM t ht).aestronglyMeasurable) hpaths)).const_mul H)
+  have hlip:LipschitzWith ⟨H,hH⟩ (V T) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro x y
+    exact hf x y
+  have hbound:∀ᶠ N in atTop,|(∫ ω,V T (X T ω) ∂μ)-V 0 0|≤E N+H*D N*Real.exp (T*γ T*Lu)+F N := by
+    filter_upwards [eventually_gt_atTop 0] with N hN
+    let P:=fun z:Fin (N+1) → ℝ ↦ V T (euler T N β v z N)
+    let Q:=fun z:Fin (N+1) → ℝ ↦ V T (euler T N γ u z N)
+    have htime (j:ℕ) (hj:j≤N):meshTime T N j∈Icc (0:ℝ) T := mesh_time_mem hT.le hN hj
+    have hp:Integrable P (gaussianProduct (Fin (N+1))) := by
+      apply Integrable.of_bound ((hlip.continuous.measurable.comp
+        (measurable_euler T N β v (fun j hj ↦ h.drift_measurable _ (htime j hj.le)) N le_rfl)).aestronglyMeasurable) K'
+      exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_apply] using h.bound T ⟨hT.le,le_rfl⟩ _)
+    have hq:Integrable Q (gaussianProduct (Fin (N+1))) := by
+      apply Integrable.of_bound ((hlip.continuous.measurable.comp
+        (measurable_euler T N γ u (fun j hj ↦ (hu.smooth _ (htime j hj.le)).continuous.measurable) N le_rfl)).aestronglyMeasurable) K'
+      exact Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs,Function.comp_apply] using h.bound T ⟨hT.le,le_rfl⟩ _)
+    have hpq:|(∫ z,P z ∂gaussianProduct (Fin (N+1)))-(∫ z,Q z ∂gaussianProduct (Fin (N+1)))|≤H*D N*Real.exp (T*γ T*Lu) := by
+      rw [←integral_sub hp hq]
+      apply (show |∫ z,P z-Q z ∂gaussianProduct (Fin (N+1))|≤H*D N*Real.exp (T*γ T*Lu) from ?_)
+      have hnorm:=norm_integral_le_of_norm_le_const (μ:=gaussianProduct (Fin (N+1)))
+        (C:=H*D N*Real.exp (T*γ T*Lu)) (f:=fun z ↦ P z-Q z) (Eventually.of_forall (fun z ↦ by
+          rw [Real.norm_eq_abs]
+          apply (hf _ _).trans
+          have he:=euler_ordered_coeff_bound (β:=β) (γ:=γ) (u:=u) (v:=v) hT.le hN hG hLu ha hb
+            (fun j hj ↦ by rw [abs_of_nonneg (hu.gamma_nonneg _ (htime j hj.le))]; exact hu.gamma_mono (htime j hj.le) ⟨hT.le,le_rfl⟩ (htime j hj.le).2)
+            (fun j hj ↦ hc _ (htime j hj.le)) (fun j hj ↦ h.drift_bound _ (htime j hj.le))
+            (fun j hj ↦ hs _ (htime j hj.le))
+            (fun j hj x y ↦ by simpa only [sub_self,abs_zero,zero_add] using hLip _ (htime j hj.le) _ (htime j hj.le) x y) z
+          simpa only [D,mul_assoc] using mul_le_mul_of_nonneg_left he hH))
+      simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using hnorm
+    have he:=h.mesh_expectation hT hT1 hN
+    have hqX:=coupled_lipschitz_expectation_bound hB hT hu hXM
+      (hpaths.mono (fun _ hω ↦ hω.2.2)) hN le_rfl hlip
+    have hend:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
+    rw [hend] at hqX
+    change |(∫ z,Q z ∂gaussianProduct (Fin (N+1)))-(∫ ω,V T (X T ω) ∂μ)|≤F N at hqX
+    change |(∫ z,P z ∂gaussianProduct (Fin (N+1)))-V 0 0|≤E N at he
+    calc
+      |(∫ ω,V T (X T ω) ∂μ)-V 0 0| ≤
+          |(∫ ω,V T (X T ω) ∂μ)-(∫ z,Q z ∂gaussianProduct (Fin (N+1)))|+
+          |(∫ z,Q z ∂gaussianProduct (Fin (N+1)))-V 0 0| := abs_sub_le _ _ _
+      _ ≤ F N+(H*D N*Real.exp (T*γ T*Lu)+E N) := by
+        apply add_le_add
+        · rw [abs_sub_comm]; exact hqX
+        · exact (abs_sub_le _ (∫ z,P z ∂gaussianProduct (Fin (N+1))) _).trans
+            (add_le_add (by rw [abs_sub_comm]; exact hpq) he)
+      _ = E N+H*D N*Real.exp (T*γ T*Lu)+F N := by ring
+  have he:=le_of_tendsto_of_tendsto tendsto_const_nhds
+    ((hE.add ((hD.const_mul H).mul_const (Real.exp (T*γ T*Lu)))).add hF) hbound
+  simpa only [zero_add,add_zero] using he
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/EulerLimitProcess.lean b/lean/OAI/Probability/SKValue/Processes/EulerLimitProcess.lean
index dd35c813b13dfc4d93b1dda4dd7405a798a3f495..e25455b6cc4c6dc755029caab74b0b6d7b94ae53 100644
--- a/lean/OAI/Probability/SKValue/Processes/EulerLimitProcess.lean
+++ b/lean/OAI/Probability/SKValue/Processes/EulerLimitProcess.lean
@@ -1,128 +1,128 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.Volterra
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter Function
-open scoped Topology NNReal
-namespace SKValue
-noncomputable def eulerLimitProcess (W:BrownianSpace) (γ:OrderParameter) (t:ℝ) (z:W.Ω):ℝ :=
-  if 0<t ∧ t<1 then limsup (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop else 0
-lemma eulerLimitProcess_zero (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω):eulerLimitProcess W γ 0 z=0 :=by
-  simp [eulerLimitProcess]
-lemma eulerLimitProcess_measurable (W:BrownianSpace) (γ:OrderParameter) (t:ℝ):
-    StronglyMeasurable[W.filtration t.toNNReal] (eulerLimitProcess W γ t) := by
-  by_cases ht:0<t ∧ t<1
-  · change StronglyMeasurable[W.filtration t.toNNReal] (fun z ↦ if 0<t ∧ t<1 then limsup (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop else 0)
-    simp only [ite_eq_left ht]
-    apply Measurable.stronglyMeasurable
-    apply Measurable.limsup
-    intro N
-    by_cases hN:N=0
-    · subst N
-      exact measurable_const
-    have hN':0<N:=Nat.pos_of_ne_zero hN
-    have hh:=coupled_euler_past_measurable W.measurable ht.1 hN' γ.coeff (gradient W γ)
-      (fun j hj ↦ ((phi_evolution W γ ⟨ht.1.le,ht.2⟩).slices _ (mesh_time_mem ht.1.le hN' hj.le)).jets.smooth.continuous.measurable) N le_rfl
-    have he:meshTime t N N=t:=by dsimp [meshTime,stepSize];field_simp
-    rw [he] at hh
-    exact hh.measurable
-  · change StronglyMeasurable[W.filtration t.toNNReal] (fun z ↦ if 0<t ∧ t<1 then limsup (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop else 0)
-    simp only [ite_eq_right ht]
-    exact stronglyMeasurable_const
-lemma eulerLimitProcess_eq_path (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
-    {T:ℝ} (hT:T∈Ico (0:ℝ) 1) {X:ℝ → ℝ}
-    (hc:ContinuousOn X (Icc (0:ℝ) T))
-    (hi:IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s)) volume 0 T)
-    (he:∀t∈Icc (0:ℝ) T,X t=W.B t.toNNReal z+∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (X s))
-    (h0:X 0=0) {t:ℝ} (ht:t∈Icc (0:ℝ) T):eulerLimitProcess W γ t z=X t := by
-  rcases ht.1.eq_or_lt with htz|htz
-  · subst t
-    rw [eulerLimitProcess_zero,h0]
-  have ht1:t<1:=ht.2.trans_lt hT.2
-  obtain ⟨K,L,Lu,hLu,hG,hu⟩:=(sourceStripRegularity W γ).core t ⟨htz.le,ht1⟩
-  have hsub:Icc (0:ℝ) t⊆Icc (0:ℝ) T:=Icc_subset_Icc le_rfl ht.2
-  have hi':IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s)) volume 0 t:=hi.mono_set (by
-    rw [uIcc_of_le htz.le,uIcc_of_le hT.1];exact hsub)
-  have hv:=driven_euler_uniform_convergence
-    (Y:=fun N j ↦ coupledEuler t γ.coeff (gradient W γ) W.B N j z) htz.le hLu
-    (γ.monotone.mono (fun s hs ↦ ⟨hs.1,hs.2.trans_lt ht1⟩))
-    (γ.nonneg 0 (by norm_num)) (fun s hs x ↦ gradient_bound W γ ⟨hs.1,hs.2.trans ht1.le⟩ x)
-    hu (hc.mono hsub) hi' (fun s hs ↦ he s (hsub hs)) h0
-    (fun N ↦ coupled_euler_zero t γ.coeff (gradient W γ) W.B N z)
-    (fun N hN j hj ↦ coupled_euler_step W.B htz γ.coeff (gradient W γ) hN hj z)
-  have hl:Tendsto (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop (𝓝 (X t)) :=by
-    apply Metric.tendsto_atTop.mpr
-    intro ε hε
-    obtain ⟨N,hN⟩:=eventually_atTop.mp (hv ε hε)
-    refine ⟨max N 1,?_⟩
-    intro n hn
-    have hnpos:0<n:= lt_of_lt_of_le (by norm_num:(0:ℕ)<1) ((le_max_right N 1).trans hn)
-    have hh:=hN n ((le_max_left N 1).trans hn) n le_rfl
-    have hnt:(n:ℝ)*(t/n)=t:=by field_simp
-    simpa only [hnt,Real.dist_eq] using hh
-  simpa only [eulerLimitProcess,ite_eq_left (show 0<t ∧ t<1 from ⟨htz,ht1⟩)] using hl.limsup_eq
-end SKValue
-
-end
-
-section
-open MeasureTheory ProbabilityTheory Set Filter Function
-open scoped Topology NNReal
-namespace SKValue
-lemma eulerLimitProcess_local_path (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
-    (hb:Continuous (fun t:ℝ ↦ W.B t.toNNReal z)) (h0:W.B 0 z=0)
-    {T:ℝ} (hT:T∈Ico (0:ℝ) 1):
-    ContinuousOn (fun t ↦ eulerLimitProcess W γ t z) (Icc (0:ℝ) T) ∧
-    IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z)) volume 0 T ∧
-    ∀t∈Icc (0:ℝ) T,eulerLimitProcess W γ t z=W.B t.toNNReal z+
-      ∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z) :=by
-  obtain ⟨X,hc,hi,he⟩:=exists_local_diffusion_path W γ hT _ hb.continuousOn
-  have hz:X 0=0:=by simpa only [Real.toNNReal_zero,h0,intervalIntegral.integral_same,add_zero] using he 0 ⟨le_rfl,hT.1⟩
-  have heq (t:ℝ) (ht:t∈Icc (0:ℝ) T):eulerLimitProcess W γ t z=X t:=
-    eulerLimitProcess_eq_path W γ z hT hc hi he hz ht
-  refine ⟨hc.congr heq,hi.congr (fun s hs ↦ ?_),?_⟩
-  · rw [uIoc_of_le hT.1] at hs
-    rw [heq s ⟨hs.1.le,hs.2⟩]
-  intro t ht
-  rw [heq t ht,he t ht]
-  congr 1
-  apply intervalIntegral.integral_congr
-  intro s hs
-  rw [uIcc_of_le ht.1] at hs
-  dsimp only
-  rw [heq s ⟨hs.1,hs.2.trans ht.2⟩]
-lemma continuousOn_preterminal_of_strips {f:ℝ → ℝ}
-    (h:∀T∈Ico (0:ℝ) 1,ContinuousOn f (Icc (0:ℝ) T)):
-    ContinuousOn f (Ico (0:ℝ) 1) :=by
-  intro t ht
-  obtain ⟨T,htT,hT⟩:=exists_between ht.2
-  apply (h T ⟨ht.1.trans htT.le,hT⟩ t ⟨ht.1,htT.le⟩).mono_of_mem_nhdsWithin
-  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds htT)] with s hs hsT
-  exact ⟨hs.1,hsT.le⟩
-lemma eulerLimitProcess_drift_integrable (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
-    (hb:Continuous (fun t:ℝ ↦ W.B t.toNNReal z)) (h0:W.B 0 z=0):
-    IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z)) volume 0 1 :=by
-  have hu:ContinuousOn (fun s ↦ gradient W γ s (eulerLimitProcess W γ s z)) (Ico (0:ℝ) 1):=by
-    apply continuousOn_preterminal_of_strips
-    intro T hT
-    obtain ⟨K,L,Lu,hLu,hG,hLip⟩:=(sourceStripRegularity W γ).core T hT
-    have hj:ContinuousOn (fun p:ℝ×ℝ ↦ gradient W γ p.1 p.2) {p | p.1∈Icc (0:ℝ) T}:=
-      derivative_joint_continuous hLu hLip
-    have hp:ContinuousOn (fun s ↦ (s,eulerLimitProcess W γ s z)) (Icc (0:ℝ) T):=
-      continuousOn_id.prodMk (eulerLimitProcess_local_path W γ z hb h0 hT).1
-    exact ContinuousOn.comp (g:=fun p:ℝ×ℝ ↦ gradient W γ p.1 p.2)
-      (f:=fun s ↦ (s,eulerLimitProcess W γ s z)) hj hp (fun s hs ↦ hs)
-  apply (intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num:(0:ℝ)≤1)).mpr
-  have hm:AEStronglyMeasurable (fun s ↦ gradient W γ s (eulerLimitProcess W γ s z)) (volume.restrict (Ioc (0:ℝ) 1)) :=by
-    rw [←Measure.restrict_congr_set Ioo_ae_eq_Ioc]
-    exact (hu.mono Ioo_subset_Ico_self).aestronglyMeasurable measurableSet_Ioo
-  apply γ.integrable.mul_bdd hm (c:=(1:ℝ))
-  filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
-  simpa only [Real.norm_eq_abs] using gradient_bound W γ ⟨hs.1.le,hs.2⟩ (eulerLimitProcess W γ s z)
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.Volterra
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter Function
+open scoped Topology NNReal
+namespace SKValue
+noncomputable def eulerLimitProcess (W:BrownianSpace) (γ:OrderParameter) (t:ℝ) (z:W.Ω):ℝ :=
+  if 0<t ∧ t<1 then limsup (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop else 0
+lemma eulerLimitProcess_zero (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω):eulerLimitProcess W γ 0 z=0 :=by
+  simp [eulerLimitProcess]
+lemma eulerLimitProcess_measurable (W:BrownianSpace) (γ:OrderParameter) (t:ℝ):
+    StronglyMeasurable[W.filtration t.toNNReal] (eulerLimitProcess W γ t) := by
+  by_cases ht:0<t ∧ t<1
+  · change StronglyMeasurable[W.filtration t.toNNReal] (fun z ↦ if 0<t ∧ t<1 then limsup (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop else 0)
+    simp only [ite_eq_left ht]
+    apply Measurable.stronglyMeasurable
+    apply Measurable.limsup
+    intro N
+    by_cases hN:N=0
+    · subst N
+      exact measurable_const
+    have hN':0<N:=Nat.pos_of_ne_zero hN
+    have hh:=coupled_euler_past_measurable W.measurable ht.1 hN' γ.coeff (gradient W γ)
+      (fun j hj ↦ ((phi_evolution W γ ⟨ht.1.le,ht.2⟩).slices _ (mesh_time_mem ht.1.le hN' hj.le)).jets.smooth.continuous.measurable) N le_rfl
+    have he:meshTime t N N=t:=by dsimp [meshTime,stepSize];field_simp
+    rw [he] at hh
+    exact hh.measurable
+  · change StronglyMeasurable[W.filtration t.toNNReal] (fun z ↦ if 0<t ∧ t<1 then limsup (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop else 0)
+    simp only [ite_eq_right ht]
+    exact stronglyMeasurable_const
+lemma eulerLimitProcess_eq_path (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
+    {T:ℝ} (hT:T∈Ico (0:ℝ) 1) {X:ℝ → ℝ}
+    (hc:ContinuousOn X (Icc (0:ℝ) T))
+    (hi:IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s)) volume 0 T)
+    (he:∀t∈Icc (0:ℝ) T,X t=W.B t.toNNReal z+∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (X s))
+    (h0:X 0=0) {t:ℝ} (ht:t∈Icc (0:ℝ) T):eulerLimitProcess W γ t z=X t := by
+  rcases ht.1.eq_or_lt with htz|htz
+  · subst t
+    rw [eulerLimitProcess_zero,h0]
+  have ht1:t<1:=ht.2.trans_lt hT.2
+  obtain ⟨K,L,Lu,hLu,hG,hu⟩:=(sourceStripRegularity W γ).core t ⟨htz.le,ht1⟩
+  have hsub:Icc (0:ℝ) t⊆Icc (0:ℝ) T:=Icc_subset_Icc le_rfl ht.2
+  have hi':IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s)) volume 0 t:=hi.mono_set (by
+    rw [uIcc_of_le htz.le,uIcc_of_le hT.1];exact hsub)
+  have hv:=driven_euler_uniform_convergence
+    (Y:=fun N j ↦ coupledEuler t γ.coeff (gradient W γ) W.B N j z) htz.le hLu
+    (γ.monotone.mono (fun s hs ↦ ⟨hs.1,hs.2.trans_lt ht1⟩))
+    (γ.nonneg 0 (by norm_num)) (fun s hs x ↦ gradient_bound W γ ⟨hs.1,hs.2.trans ht1.le⟩ x)
+    hu (hc.mono hsub) hi' (fun s hs ↦ he s (hsub hs)) h0
+    (fun N ↦ coupled_euler_zero t γ.coeff (gradient W γ) W.B N z)
+    (fun N hN j hj ↦ coupled_euler_step W.B htz γ.coeff (gradient W γ) hN hj z)
+  have hl:Tendsto (fun N:ℕ ↦ coupledEuler t γ.coeff (gradient W γ) W.B N N z) atTop (𝓝 (X t)) :=by
+    apply Metric.tendsto_atTop.mpr
+    intro ε hε
+    obtain ⟨N,hN⟩:=eventually_atTop.mp (hv ε hε)
+    refine ⟨max N 1,?_⟩
+    intro n hn
+    have hnpos:0<n:= lt_of_lt_of_le (by norm_num:(0:ℕ)<1) ((le_max_right N 1).trans hn)
+    have hh:=hN n ((le_max_left N 1).trans hn) n le_rfl
+    have hnt:(n:ℝ)*(t/n)=t:=by field_simp
+    simpa only [hnt,Real.dist_eq] using hh
+  simpa only [eulerLimitProcess,ite_eq_left (show 0<t ∧ t<1 from ⟨htz,ht1⟩)] using hl.limsup_eq
+end SKValue
+
+end
+
+section
+open MeasureTheory ProbabilityTheory Set Filter Function
+open scoped Topology NNReal
+namespace SKValue
+lemma eulerLimitProcess_local_path (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
+    (hb:Continuous (fun t:ℝ ↦ W.B t.toNNReal z)) (h0:W.B 0 z=0)
+    {T:ℝ} (hT:T∈Ico (0:ℝ) 1):
+    ContinuousOn (fun t ↦ eulerLimitProcess W γ t z) (Icc (0:ℝ) T) ∧
+    IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z)) volume 0 T ∧
+    ∀t∈Icc (0:ℝ) T,eulerLimitProcess W γ t z=W.B t.toNNReal z+
+      ∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z) :=by
+  obtain ⟨X,hc,hi,he⟩:=exists_local_diffusion_path W γ hT _ hb.continuousOn
+  have hz:X 0=0:=by simpa only [Real.toNNReal_zero,h0,intervalIntegral.integral_same,add_zero] using he 0 ⟨le_rfl,hT.1⟩
+  have heq (t:ℝ) (ht:t∈Icc (0:ℝ) T):eulerLimitProcess W γ t z=X t:=
+    eulerLimitProcess_eq_path W γ z hT hc hi he hz ht
+  refine ⟨hc.congr heq,hi.congr (fun s hs ↦ ?_),?_⟩
+  · rw [uIoc_of_le hT.1] at hs
+    rw [heq s ⟨hs.1.le,hs.2⟩]
+  intro t ht
+  rw [heq t ht,he t ht]
+  congr 1
+  apply intervalIntegral.integral_congr
+  intro s hs
+  rw [uIcc_of_le ht.1] at hs
+  dsimp only
+  rw [heq s ⟨hs.1,hs.2.trans ht.2⟩]
+lemma continuousOn_preterminal_of_strips {f:ℝ → ℝ}
+    (h:∀T∈Ico (0:ℝ) 1,ContinuousOn f (Icc (0:ℝ) T)):
+    ContinuousOn f (Ico (0:ℝ) 1) :=by
+  intro t ht
+  obtain ⟨T,htT,hT⟩:=exists_between ht.2
+  apply (h T ⟨ht.1.trans htT.le,hT⟩ t ⟨ht.1,htT.le⟩).mono_of_mem_nhdsWithin
+  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds htT)] with s hs hsT
+  exact ⟨hs.1,hsT.le⟩
+lemma eulerLimitProcess_drift_integrable (W:BrownianSpace) (γ:OrderParameter) (z:W.Ω)
+    (hb:Continuous (fun t:ℝ ↦ W.B t.toNNReal z)) (h0:W.B 0 z=0):
+    IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (eulerLimitProcess W γ s z)) volume 0 1 :=by
+  have hu:ContinuousOn (fun s ↦ gradient W γ s (eulerLimitProcess W γ s z)) (Ico (0:ℝ) 1):=by
+    apply continuousOn_preterminal_of_strips
+    intro T hT
+    obtain ⟨K,L,Lu,hLu,hG,hLip⟩:=(sourceStripRegularity W γ).core T hT
+    have hj:ContinuousOn (fun p:ℝ×ℝ ↦ gradient W γ p.1 p.2) {p | p.1∈Icc (0:ℝ) T}:=
+      derivative_joint_continuous hLu hLip
+    have hp:ContinuousOn (fun s ↦ (s,eulerLimitProcess W γ s z)) (Icc (0:ℝ) T):=
+      continuousOn_id.prodMk (eulerLimitProcess_local_path W γ z hb h0 hT).1
+    exact ContinuousOn.comp (g:=fun p:ℝ×ℝ ↦ gradient W γ p.1 p.2)
+      (f:=fun s ↦ (s,eulerLimitProcess W γ s z)) hj hp (fun s hs ↦ hs)
+  apply (intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num:(0:ℝ)≤1)).mpr
+  have hm:AEStronglyMeasurable (fun s ↦ gradient W γ s (eulerLimitProcess W γ s z)) (volume.restrict (Ioc (0:ℝ) 1)) :=by
+    rw [←Measure.restrict_congr_set Ioo_ae_eq_Ioc]
+    exact (hu.mono Ioo_subset_Ico_self).aestronglyMeasurable measurableSet_Ioo
+  apply γ.integrable.mul_bdd hm (c:=(1:ℝ))
+  filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
+  simpa only [Real.norm_eq_abs] using gradient_bound W γ ⟨hs.1.le,hs.2⟩ (eulerLimitProcess W γ s z)
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/EulerMemLp.lean b/lean/OAI/Probability/SKValue/Processes/EulerMemLp.lean
index ca755bd1e3eb8ada0052ca25856e6276e14d365e..695bbd993f2313d8633576c0f8f9364401aee43c 100644
--- a/lean/OAI/Probability/SKValue/Processes/EulerMemLp.lean
+++ b/lean/OAI/Probability/SKValue/Processes/EulerMemLp.lean
@@ -1,151 +1,151 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.RawIncrement
-import OAI.Probability.SKValue.Equations.ValueStrip
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-
-lemma euler_memLp (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (huM : ∀ j<N, Measurable (u (meshTime T N j)))
-    (huB : ∀ j<N, ∀ x, |u (meshTime T N j) x|≤1) :
-    ∀ j≤N, MemLp (fun z ↦ euler T N γ u z j) 2 (gaussianProduct (Fin (N+1))) := by
-  intro j
-  induction j with
-  | zero => intro _; exact memLp_const (0 : ℝ)
-  | succ j ih =>
-    intro hj
-    have hlt : j<N := by omega
-    have hU : MemLp (fun z ↦ u (meshTime T N j) (euler T N γ u z j)) 2
-        (gaussianProduct (Fin (N+1))) :=
-      MemLp.of_bound ((huM j hlt).comp (measurable_euler T N γ u huM j hlt.le)).aestronglyMeasurable 1
-        (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using huB j hlt _))
-    exact ((ih hlt.le).add ((gaussian_memLp (coordinate_hasLaw N j) 2 (by norm_num)).const_mul _)).add
-      (hU.const_mul (stepSize T N*γ (meshTime T N j)))
-
-lemma ValueStrip.value_lipschitz {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
-    (h : ValueStrip T γ V K L) {t : ℝ} (ht : t∈Icc (0 : ℝ) T) :
-    LipschitzWith 1 (V t) := by
-  apply lipschitzWith_of_nnnorm_deriv_le ((h.smooth t ht).differentiable (by norm_num))
-  intro x
-  exact_mod_cast (show ‖deriv (V t) x‖≤(1 : ℝ) by simpa only [Real.norm_eq_abs] using h.bounded t ht x)
-
-lemma ValueStrip.value_memLp {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ} (h : ValueStrip T γ V K L)
-    {t : ℝ} (ht : t∈Icc (0 : ℝ) T) {Y : Ω → ℝ} (hY : MemLp Y 2 μ) :
-    MemLp (fun ω ↦ V t (Y ω)) 2 μ := by
-  have hl : LipschitzWith 1 (fun x ↦ V t x - V t 0) := by
-    rw [lipschitzWith_iff_dist_le_mul]
-    intro x y
-    simpa only [dist_sub_right] using (h.value_lipschitz ht).dist_le_mul x y
-  have hh := hl.comp_memLp (by simp) hY
-  convert hh.add (memLp_const (V t 0)) using 1
-  funext ω
-  simp
-
-lemma value_mesh_expectation_error {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
-    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N) :
-    |(∫ z, V T (euler T N γ (fun t ↦ deriv (V t)) z N) ∂gaussianProduct (Fin (N+1)))-V 0 0-
-      (∫ z, valueMeshEnergy T N γ V N z ∂gaussianProduct (Fin (N+1)))| ≤
-      Real.sqrt (valueMeshError T N γ V) := by
-  let u := fun t ↦ deriv (V t)
-  let Y := euler T N γ u
-  let μ := gaussianProduct (Fin (N+1))
-  let A (i : Fin N) (z : Fin (N+1) → ℝ) := u (meshTime T N i) (Y z i)
-  let M (z : Fin (N+1) → ℝ) := ∑ i : Fin N, Real.sqrt (stepSize T N)*A i z*coordinate N i z
-  let R (z : Fin (N+1) → ℝ) := valueMeshCompensated T N γ V N z-V 0 0-M z
-  have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT.le hN hj
-  have huc (j : ℕ) (hj : j≤N) : Continuous (u (meshTime T N j)) :=
-    (h.smooth _ (htime j hj)).continuous_deriv (by norm_num)
-  have hY (j : ℕ) (hj : j≤N) : Measurable (fun z ↦ Y z j) :=
-    measurable_euler T N γ u (fun i hi ↦ (huc i hi.le).measurable) j hj
-  have hAm (i : Fin N) : Measurable (A i) := (huc i i.isLt.le).measurable.comp (hY i i.isLt.le)
-  have hAb (i : Fin N) z : |A i z|≤1 := h.bounded _ (htime i i.isLt.le) _
-  have hAp (i : Fin N) : DependsBefore i (A i) := eulerCoefficient_dependsBefore T i.isLt.le γ u u
-  have hAM (i : Fin N) : MemLp (A i) 2 μ :=
-    MemLp.of_bound (hAm i).aestronglyMeasurable 1
-      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using hAb i z))
-  have hMm : MemLp M 2 μ := finiteRaw_memLp hAm hAp hAM
-  have hMv : (∫ z, M z ∂μ)=0 := by
-    have hMi (i : Fin N) : Integrable (fun z ↦ Real.sqrt (stepSize T N)*A i z*coordinate N i z) μ :=
-      ((rawIncrement_memLp (by omega : (i : ℕ)<N+1) (hAm i) (hAp i) (hAM i)).const_mul
-        (Real.sqrt (stepSize T N))).integrable (by norm_num) |>.congr (by
-          filter_upwards [] with z; simp only [rawIncrement]; ring)
-    rw [show (∫ z, M z ∂μ)=∑ i : Fin N, ∫ z, Real.sqrt (stepSize T N)*A i z*coordinate N i z ∂μ from
-      integral_finsetSum _ (fun i _ ↦ hMi i)]
-    have he (i : Fin N) : (∫ z, Real.sqrt (stepSize T N)*A i z*coordinate N i z ∂μ)=0 := by
-      simp only [mul_assoc]
-      rw [integral_const_mul]
-      change Real.sqrt (stepSize T N)*(∫ z, rawIncrement (A i) i z ∂μ)=0
-      rw [rawIncrement_mean (by omega) (hAm i) (hAp i),mul_zero]
-    simp only [he, Finset.sum_const_zero]
-  have hEm : MemLp (valueMeshEnergy T N γ V N) 2 μ := by
-    apply memLp_finsetSum
-    intro j hj
-    have hjN : j<N := Finset.mem_range.mp hj
-    apply MemLp.const_mul
-    apply MemLp.of_bound (((huc j hjN.le).measurable.comp (hY j hjN.le)).pow_const 2).aestronglyMeasurable 1
-    filter_upwards [] with z
-    rw [Real.norm_eq_abs, abs_pow, sq_le_one_iff_abs_le_one]
-    simpa only [abs_abs, u, Function.comp_apply] using h.bounded _ (htime j hjN.le) (Y z j)
-  have hYM : MemLp (fun z ↦ Y z N) 2 μ :=
-    euler_memLp T N γ u (fun i hi ↦ (huc i hi.le).measurable)
-      (fun i hi x ↦ h.bounded _ (htime i hi.le) x) N le_rfl
-  have hVM := h.value_memLp (t := T) ⟨hT.le,le_rfl⟩ hYM
-  have hlast : meshTime T N N=T := by
-    dsimp only [meshTime, stepSize]
-    have hn : (N : ℝ)≠0 := by exact_mod_cast Nat.ne_of_gt hN
-    field_simp
-  have hRM : MemLp R 2 μ := by
-    dsimp only [R,valueMeshCompensated]
-    rw [hlast]
-    exact ((hVM.sub hEm).sub (memLp_const (V 0 0))).sub hMm
-  have hRint : (∫ z, R z ∂μ)=
-      (∫ z, V T (Y z N) ∂μ)-V 0 0-(∫ z, valueMeshEnergy T N γ V N z ∂μ) := by
-    dsimp only [R,valueMeshCompensated]
-    rw [hlast]
-    have hi₁ := integral_sub (((hVM.sub hEm).sub (memLp_const (V 0 0))).integrable (by norm_num)) (hMm.integrable (by norm_num))
-    have hi₂ := integral_sub ((hVM.sub hEm).integrable (by norm_num)) (integrable_const (V 0 0))
-    simp only [Pi.sub_apply] at hi₁ hi₂
-    change (∫ z, V T (Y z N)-valueMeshEnergy T N γ V N z-V 0 0-M z ∂μ)=_
-    rw [hi₁, hi₂]
-    rw [integral_sub (hVM.integrable (by norm_num)) (hEm.integrable (by norm_num)),hMv,integral_const]
-    simp only [probReal_univ, smul_eq_mul, sub_zero]
-    ring
-  rw [←hRint]
-  exact abs_integral_le_integral_abs.trans (integral_abs_le_sqrt_integral_sq hRM)
-
-lemma value_mesh_expectation_error_tendsto {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
-    (hT : 0<T) (hT1 : T≤1) (h : ValueStrip T γ V K L) :
-    Tendsto (fun N ↦ (∫ z, V T (euler T N γ (fun t ↦ deriv (V t)) z N)
-      ∂gaussianProduct (Fin (N+1)))-V 0 0-
-      (∫ z, valueMeshEnergy T N γ V N z ∂gaussianProduct (Fin (N+1)))) atTop (𝓝 0) := by
-  apply squeeze_zero_norm' (a := fun N ↦ Real.sqrt (valueMeshError T N γ V))
-  · filter_upwards [eventually_gt_atTop 0] with N hN
-    simpa only [Real.norm_eq_abs] using value_mesh_expectation_error hT h hN
-  · simpa only [Real.sqrt_zero, Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp (valueMeshError_tendsto_zero hT hT1 h)
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.RawIncrement
+import OAI.Probability.SKValue.Equations.ValueStrip
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+
+lemma euler_memLp (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (huM : ∀ j<N, Measurable (u (meshTime T N j)))
+    (huB : ∀ j<N, ∀ x, |u (meshTime T N j) x|≤1) :
+    ∀ j≤N, MemLp (fun z ↦ euler T N γ u z j) 2 (gaussianProduct (Fin (N+1))) := by
+  intro j
+  induction j with
+  | zero => intro _; exact memLp_const (0 : ℝ)
+  | succ j ih =>
+    intro hj
+    have hlt : j<N := by omega
+    have hU : MemLp (fun z ↦ u (meshTime T N j) (euler T N γ u z j)) 2
+        (gaussianProduct (Fin (N+1))) :=
+      MemLp.of_bound ((huM j hlt).comp (measurable_euler T N γ u huM j hlt.le)).aestronglyMeasurable 1
+        (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using huB j hlt _))
+    exact ((ih hlt.le).add ((gaussian_memLp (coordinate_hasLaw N j) 2 (by norm_num)).const_mul _)).add
+      (hU.const_mul (stepSize T N*γ (meshTime T N j)))
+
+lemma ValueStrip.value_lipschitz {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
+    (h : ValueStrip T γ V K L) {t : ℝ} (ht : t∈Icc (0 : ℝ) T) :
+    LipschitzWith 1 (V t) := by
+  apply lipschitzWith_of_nnnorm_deriv_le ((h.smooth t ht).differentiable (by norm_num))
+  intro x
+  exact_mod_cast (show ‖deriv (V t) x‖≤(1 : ℝ) by simpa only [Real.norm_eq_abs] using h.bounded t ht x)
+
+lemma ValueStrip.value_memLp {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ} (h : ValueStrip T γ V K L)
+    {t : ℝ} (ht : t∈Icc (0 : ℝ) T) {Y : Ω → ℝ} (hY : MemLp Y 2 μ) :
+    MemLp (fun ω ↦ V t (Y ω)) 2 μ := by
+  have hl : LipschitzWith 1 (fun x ↦ V t x - V t 0) := by
+    rw [lipschitzWith_iff_dist_le_mul]
+    intro x y
+    simpa only [dist_sub_right] using (h.value_lipschitz ht).dist_le_mul x y
+  have hh := hl.comp_memLp (by simp) hY
+  convert hh.add (memLp_const (V t 0)) using 1
+  funext ω
+  simp
+
+lemma value_mesh_expectation_error {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
+    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N) :
+    |(∫ z, V T (euler T N γ (fun t ↦ deriv (V t)) z N) ∂gaussianProduct (Fin (N+1)))-V 0 0-
+      (∫ z, valueMeshEnergy T N γ V N z ∂gaussianProduct (Fin (N+1)))| ≤
+      Real.sqrt (valueMeshError T N γ V) := by
+  let u := fun t ↦ deriv (V t)
+  let Y := euler T N γ u
+  let μ := gaussianProduct (Fin (N+1))
+  let A (i : Fin N) (z : Fin (N+1) → ℝ) := u (meshTime T N i) (Y z i)
+  let M (z : Fin (N+1) → ℝ) := ∑ i : Fin N, Real.sqrt (stepSize T N)*A i z*coordinate N i z
+  let R (z : Fin (N+1) → ℝ) := valueMeshCompensated T N γ V N z-V 0 0-M z
+  have htime (j : ℕ) (hj : j≤N) := mesh_time_mem hT.le hN hj
+  have huc (j : ℕ) (hj : j≤N) : Continuous (u (meshTime T N j)) :=
+    (h.smooth _ (htime j hj)).continuous_deriv (by norm_num)
+  have hY (j : ℕ) (hj : j≤N) : Measurable (fun z ↦ Y z j) :=
+    measurable_euler T N γ u (fun i hi ↦ (huc i hi.le).measurable) j hj
+  have hAm (i : Fin N) : Measurable (A i) := (huc i i.isLt.le).measurable.comp (hY i i.isLt.le)
+  have hAb (i : Fin N) z : |A i z|≤1 := h.bounded _ (htime i i.isLt.le) _
+  have hAp (i : Fin N) : DependsBefore i (A i) := eulerCoefficient_dependsBefore T i.isLt.le γ u u
+  have hAM (i : Fin N) : MemLp (A i) 2 μ :=
+    MemLp.of_bound (hAm i).aestronglyMeasurable 1
+      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using hAb i z))
+  have hMm : MemLp M 2 μ := finiteRaw_memLp hAm hAp hAM
+  have hMv : (∫ z, M z ∂μ)=0 := by
+    have hMi (i : Fin N) : Integrable (fun z ↦ Real.sqrt (stepSize T N)*A i z*coordinate N i z) μ :=
+      ((rawIncrement_memLp (by omega : (i : ℕ)<N+1) (hAm i) (hAp i) (hAM i)).const_mul
+        (Real.sqrt (stepSize T N))).integrable (by norm_num) |>.congr (by
+          filter_upwards [] with z; simp only [rawIncrement]; ring)
+    rw [show (∫ z, M z ∂μ)=∑ i : Fin N, ∫ z, Real.sqrt (stepSize T N)*A i z*coordinate N i z ∂μ from
+      integral_finsetSum _ (fun i _ ↦ hMi i)]
+    have he (i : Fin N) : (∫ z, Real.sqrt (stepSize T N)*A i z*coordinate N i z ∂μ)=0 := by
+      simp only [mul_assoc]
+      rw [integral_const_mul]
+      change Real.sqrt (stepSize T N)*(∫ z, rawIncrement (A i) i z ∂μ)=0
+      rw [rawIncrement_mean (by omega) (hAm i) (hAp i),mul_zero]
+    simp only [he, Finset.sum_const_zero]
+  have hEm : MemLp (valueMeshEnergy T N γ V N) 2 μ := by
+    apply memLp_finsetSum
+    intro j hj
+    have hjN : j<N := Finset.mem_range.mp hj
+    apply MemLp.const_mul
+    apply MemLp.of_bound (((huc j hjN.le).measurable.comp (hY j hjN.le)).pow_const 2).aestronglyMeasurable 1
+    filter_upwards [] with z
+    rw [Real.norm_eq_abs, abs_pow, sq_le_one_iff_abs_le_one]
+    simpa only [abs_abs, u, Function.comp_apply] using h.bounded _ (htime j hjN.le) (Y z j)
+  have hYM : MemLp (fun z ↦ Y z N) 2 μ :=
+    euler_memLp T N γ u (fun i hi ↦ (huc i hi.le).measurable)
+      (fun i hi x ↦ h.bounded _ (htime i hi.le) x) N le_rfl
+  have hVM := h.value_memLp (t := T) ⟨hT.le,le_rfl⟩ hYM
+  have hlast : meshTime T N N=T := by
+    dsimp only [meshTime, stepSize]
+    have hn : (N : ℝ)≠0 := by exact_mod_cast Nat.ne_of_gt hN
+    field_simp
+  have hRM : MemLp R 2 μ := by
+    dsimp only [R,valueMeshCompensated]
+    rw [hlast]
+    exact ((hVM.sub hEm).sub (memLp_const (V 0 0))).sub hMm
+  have hRint : (∫ z, R z ∂μ)=
+      (∫ z, V T (Y z N) ∂μ)-V 0 0-(∫ z, valueMeshEnergy T N γ V N z ∂μ) := by
+    dsimp only [R,valueMeshCompensated]
+    rw [hlast]
+    have hi₁ := integral_sub (((hVM.sub hEm).sub (memLp_const (V 0 0))).integrable (by norm_num)) (hMm.integrable (by norm_num))
+    have hi₂ := integral_sub ((hVM.sub hEm).integrable (by norm_num)) (integrable_const (V 0 0))
+    simp only [Pi.sub_apply] at hi₁ hi₂
+    change (∫ z, V T (Y z N)-valueMeshEnergy T N γ V N z-V 0 0-M z ∂μ)=_
+    rw [hi₁, hi₂]
+    rw [integral_sub (hVM.integrable (by norm_num)) (hEm.integrable (by norm_num)),hMv,integral_const]
+    simp only [probReal_univ, smul_eq_mul, sub_zero]
+    ring
+  rw [←hRint]
+  exact abs_integral_le_integral_abs.trans (integral_abs_le_sqrt_integral_sq hRM)
+
+lemma value_mesh_expectation_error_tendsto {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
+    (hT : 0<T) (hT1 : T≤1) (h : ValueStrip T γ V K L) :
+    Tendsto (fun N ↦ (∫ z, V T (euler T N γ (fun t ↦ deriv (V t)) z N)
+      ∂gaussianProduct (Fin (N+1)))-V 0 0-
+      (∫ z, valueMeshEnergy T N γ V N z ∂gaussianProduct (Fin (N+1)))) atTop (𝓝 0) := by
+  apply squeeze_zero_norm' (a := fun N ↦ Real.sqrt (valueMeshError T N γ V))
+  · filter_upwards [eventually_gt_atTop 0] with N hN
+    simpa only [Real.norm_eq_abs] using value_mesh_expectation_error hT h hN
+  · simpa only [Real.sqrt_zero, Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp (valueMeshError_tendsto_zero hT hT1 h)
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/Feedback.lean b/lean/OAI/Probability/SKValue/Processes/Feedback.lean
index b19fe036cf7f87ac4d82afe220db3c244d785d91..67a5e5a612883c19f1938e8c862570b458646cb6 100644
--- a/lean/OAI/Probability/SKValue/Processes/Feedback.lean
+++ b/lean/OAI/Probability/SKValue/Processes/Feedback.lean
@@ -1,299 +1,299 @@
-import Mathlib
-import OAI.Probability.SKValue.Gaussian.CoupledCoordinates
-import OAI.Probability.SKValue.Equations.ValueStrip
-import OAI.Probability.SKValue.Equations.CubicEnvelopeMean
-import OAI.Probability.SKValue.Control.Controls
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def meshControl {Ω : Type*} (T : ℝ) (N : ℕ) (H : Fin N → Ω → ℝ)
-    (t : ℝ≥0) (ω : Ω) : ℝ :=
-  ∑ j : Fin N, if meshTime T N j < t ∧ (t : ℝ)  ≤  meshTime T N (j+1) then H j ω else 0
-
-lemma progressive_interval {Ω : Type*} [m : MeasurableSpace Ω]
-    {f : Filtration ℝ≥0 m} {a b : ℝ} {H : Ω → ℝ} (_ : 0 ≤ a)
-    (hH : Measurable[f (Real.toNNReal a)] H) :
-    IsProgressive f (fun t ω ↦ if a < (t : ℝ) ∧ (t : ℝ)  ≤  b then H ω else 0) := by
-  intro t
-  by_cases hat : Real.toNNReal a  ≤  t
-  · let : MeasurableSpace Ω := f t
-    have hHm : Measurable H := hH.mono (f.mono hat) le_rfl
-    exact (hHm.comp measurable_snd).ite
-      ((measurableSet_lt measurable_const (measurable_coe_nnreal_real.comp
-        (measurable_subtype_coe.comp measurable_fst))).inter
-        (measurableSet_le (measurable_coe_nnreal_real.comp
-          (measurable_subtype_coe.comp measurable_fst)) measurable_const)) measurable_const
-  · have he : (fun p : Iic t × Ω ↦ if a < (p.1 : ℝ≥0) ∧ ((p.1 : ℝ≥0) : ℝ)  ≤  b then H p.2 else 0) =
-        fun _ ↦ (0 : ℝ) := by
-      funext p
-      apply ite_eq_right
-      intro hp
-      exact hat ((Real.toNNReal_le_iff_le_coe).2 (hp.1.le.trans (by exact_mod_cast p.1.2)))
-    rw [he]
-    exact measurable_const
-
-lemma meshControl_progressive {Ω : Type*} [m : MeasurableSpace Ω]
-    {f : Filtration ℝ≥0 m} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} {H : Fin N → Ω → ℝ}
-    (hH : ∀ j : Fin N, Measurable[f (Real.toNNReal (meshTime T N j))] (H j)) :
-    IsProgressive f (meshControl T N H) := by
-  apply IsProgressive.finsetSum
-  intro j _
-  exact progressive_interval (mesh_time_mem_total hT j.isLt.le).1 (hH j)
-
-lemma mesh_intervals_disjoint {T : ℝ} (hT : 0 ≤ T) {N : ℕ} {i j : Fin N} {t : ℝ}
-    (hi : meshTime T N i < t ∧ t  ≤  meshTime T N (i+1))
-    (hj : meshTime T N j < t ∧ t  ≤  meshTime T N (j+1)) : i=j := by
-  have hδ : 0  ≤  stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
-  have hlt : ∀ a b : ℕ, a+1 ≤ b → meshTime T N (a+1) ≤ meshTime T N b := by
-    intro a b hab
-    exact mul_le_mul_of_nonneg_right (by exact_mod_cast hab) hδ
-  apply Fin.ext
-  by_contra he
-  rcases lt_or_gt_of_ne he with hij | hji
-  · have hh := hlt i j hij
-    linarith [hi.2,hj.1]
-  · have hh := hlt j i hji
-    linarith [hi.1,hj.2]
-
-lemma meshControl_eq {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (H : Fin N → Ω → ℝ)
-    {j : Fin N} {t : ℝ≥0} (ht : meshTime T N j < t ∧ (t : ℝ)  ≤  meshTime T N (j+1)) (ω : Ω) :
-    meshControl T N H t ω=H j ω := by
-  unfold meshControl
-  rw [Finset.sum_eq_single j]
-  · simp only [ite_eq_left ht]
-  · intro i _ hij
-    exact ite_eq_right (fun hi ↦ hij (mesh_intervals_disjoint hT hi ht))
-  · simp
-
-lemma meshControl_bound {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (H : Fin N → Ω → ℝ)
-    (hH : ∀ j ω, |H j ω| ≤ 1) (t : ℝ≥0) (ω : Ω) : |meshControl T N H t ω| ≤ 1 := by
-  by_cases he : ∃ j : Fin N, meshTime T N j < t ∧ (t : ℝ)  ≤  meshTime T N (j+1)
-  · obtain ⟨j,hj⟩ := he
-    rw [meshControl_eq hT H hj]
-    exact hH j ω
-  · have hh : meshControl T N H t ω=0 := by
-      unfold meshControl
-      apply Finset.sum_eq_zero
-      intro j _
-      exact ite_eq_right (fun hj ↦ he ⟨j,hj⟩)
-    rw [hh]
-    norm_num
-
-lemma meshControl_momentIntegral {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (hN : 0<N)
-    (H : Fin N → Ω → ℝ) {γ : ℝ → ℝ} (hγ : IntervalIntegrable γ volume 0 T)
-    (k : ℕ) (ω : Ω) :
-    controlAccum γ (meshControl T N H) k T ω =
-      ∑ j : Fin N, (∫ s in meshTime T N j..meshTime T N (j+1), γ s)*(H j ω)^k := by
-  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
-  have htime (j : ℕ) (hj : j ≤ N) := mesh_time_mem_total hT hj
-  have horder (j : ℕ) : meshTime T N j ≤ meshTime T N (j+1) := by
-    dsimp [meshTime]
-    exact mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.le_succ j) hδ
-  have hlast : meshTime T N N=T := by
-    dsimp [meshTime,stepSize]
-    field_simp
-  have he (j : Fin N) :
-      (fun s ↦ γ s*(meshControl T N H (Real.toNNReal s) ω)^k) =ᵐ[
-        volume.restrict (uIoc (meshTime T N j) (meshTime T N (j+1)))]
-      (fun s ↦ γ s*(H j ω)^k) := by
-    filter_upwards [ae_restrict_mem measurableSet_uIoc] with s hs
-    rw [uIoc_of_le (horder j)] at hs
-    have hs0 : 0 ≤ s := (htime j j.isLt.le).1.trans hs.1.le
-    rw [meshControl_eq hT H (by simpa only [Real.coe_toNNReal _ hs0,Set.mem_Ioc] using hs)]
-  have hi (j : ℕ) (hj : j<N) : IntervalIntegrable
-      (fun s ↦ γ s*(meshControl T N H (Real.toNNReal s) ω)^k) volume
-      (meshTime T N j) (meshTime T N (j+1)) := by
-    apply ((intervalIntegrable_substrip hT hγ (htime j hj.le)
-      (htime (j+1) hj)).mul_const ((H ⟨j,hj⟩ ω)^k)).congr_ae
-    exact (he ⟨j,hj⟩).symm
-  have hs := intervalIntegral.sum_integral_adjacent_intervals hi
-  rw [hlast,show meshTime T N 0=0 by simp [meshTime]] at hs
-  rw [controlAccum,←hs,←Fin.sum_univ_eq_sum_range]
-  apply Finset.sum_congr rfl
-  intro j _
-  rw [intervalIntegral.integral_congr_ae_restrict (he j),intervalIntegral.integral_mul_const]
-
-lemma monotone_interval_quadrature {γ : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b)
-    (hγ : MonotoneOn γ (Icc a b)) :
-    |(∫ s in a..b, γ s)-(b-a)*γ a| ≤ (b-a)*(γ b-γ a) := by
-  have hi : IntervalIntegrable γ volume a b := by
-    apply MonotoneOn.intervalIntegrable
-    simpa only [uIcc_of_le hab] using hγ
-  have hdiff : (∫ s in a..b, γ s)-(b-a)*γ a =∫ s in a..b, γ s-γ a := by
-    rw [intervalIntegral.integral_sub hi intervalIntegrable_const]
-    simp only [intervalIntegral.integral_const,smul_eq_mul]
-  rw [hdiff]
-  have hh := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := a) (b := b) (f := fun s ↦ γ s-γ a) (C := γ b-γ a) (fun s hs ↦ by
-      rw [uIoc_of_le hab] at hs
-      have hs' : s∈Icc a b := ⟨hs.1.le,hs.2⟩
-      rw [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr (hγ ⟨le_rfl,hab⟩ hs' hs.1.le))]
-      exact sub_le_sub_right (hγ hs' ⟨hab,le_rfl⟩ hs.2) _)
-  simpa only [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hab),mul_comm] using hh
-
-lemma meshControl_quadrature {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (hN : 0<N)
-    (H : Fin N → Ω → ℝ) (hH : ∀ j ω, |H j ω| ≤ 1)
-    {γ : ℝ → ℝ} (hγ : MonotoneOn γ (Icc (0 : ℝ) T)) (k : ℕ) (ω : Ω) :
-    |controlAccum γ (meshControl T N H) k T ω-
-      ∑ j : Fin N, stepSize T N*γ (meshTime T N j)*(H j ω)^k| ≤
-      stepSize T N*(γ T-γ 0) := by
-  have hi : IntervalIntegrable γ volume 0 T := by
-    apply MonotoneOn.intervalIntegrable
-    simpa only [uIcc_of_le hT] using hγ
-  rw [meshControl_momentIntegral hT hN H hi, ←Finset.sum_sub_distrib]
-  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
-  have htime (j : ℕ) (hj : j ≤ N) := mesh_time_mem_total hT hj
-  have hgap (j : ℕ) : meshTime T N (j+1)-meshTime T N j=stepSize T N := by
-    dsimp [meshTime]
-    push_cast
-    ring
-  have horder (j : ℕ) : meshTime T N j ≤ meshTime T N (j+1) := by
-    linarith [hgap j]
-  have hloc (j : Fin N) :
-      |(∫ s in meshTime T N j..meshTime T N (j+1), γ s)*(H j ω)^k-
-        stepSize T N*γ (meshTime T N j)*(H j ω)^k| ≤
-      stepSize T N*(γ (meshTime T N (j+1))-γ (meshTime T N j)) := by
-    rw [←sub_mul,abs_mul]
-    have hh := monotone_interval_quadrature (horder j)
-      (hγ.mono (Icc_subset_Icc (htime j j.isLt.le).1 (htime (j+1) j.isLt).2))
-    rw [hgap] at hh
-    have hp : |(H j ω)^k| ≤ 1 := by
-      rw [abs_pow]
-      exact pow_le_one₀ (abs_nonneg _) (hH j ω)
-    exact (mul_le_mul_of_nonneg_left hp (abs_nonneg _)).trans (by simpa only [mul_one] using hh)
-  calc
-    _ ≤ ∑ j : Fin N, |(∫ s in meshTime T N j..meshTime T N (j+1), γ s)*(H j ω)^k-
-        stepSize T N*γ (meshTime T N j)*(H j ω)^k| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ j : Fin N, stepSize T N*(γ (meshTime T N (j+1))-γ (meshTime T N j)) :=
-      Finset.sum_le_sum (fun j _ ↦ hloc j)
-    _ = _ := by
-      rw [←Finset.mul_sum, sum_steps_sub (fun j ↦ γ (meshTime T N j))]
-      have hlast : meshTime T N N=T := by
-        dsimp [meshTime,stepSize]
-        field_simp
-      change stepSize T N*(γ (meshTime T N N)-γ (meshTime T N 0)) = _
-      rw [hlast]
-      simp only [meshTime,Nat.cast_zero,zero_mul]
-
-lemma coupled_euler_past_measurable {Ω : Type*} [m : MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} (hBm : ∀ t, StronglyMeasurable (B t))
-    {T : ℝ} (hT : 0<T) {N : ℕ} (hN : 0<N) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (hu : ∀ j<N, Measurable (u (meshTime T N j))) :
-    ∀ j ≤ N, StronglyMeasurable[Filtration.natural B hBm (Real.toNNReal (meshTime T N j))]
-      (coupledEuler T γ u B N j) := by
-  let f := Filtration.natural B hBm
-  intro j
-  induction j with
-  | zero =>
-    intro _
-    convert (stronglyMeasurable_const : StronglyMeasurable[f (Real.toNNReal (meshTime T N 0))] (fun _ : Ω ↦ (0 : ℝ))) using 1
-    funext ω
-    exact coupled_euler_zero T γ u B N ω
-  | succ j ih =>
-    intro hjN
-    have hj : j<N := hjN
-    have hδ : 0 ≤ stepSize T N := div_nonneg hT.le (Nat.cast_nonneg N)
-    have ht : Real.toNNReal (meshTime T N j) ≤ Real.toNNReal (meshTime T N (j+1)) := by
-      apply Real.toNNReal_mono
-      apply mul_le_mul_of_nonneg_right _ hδ
-      exact_mod_cast Nat.le_succ j
-    have hY := (ih hj.le).mono (f.mono ht)
-    have hB0 := (Filtration.stronglyAdapted_natural hBm (Real.toNNReal (meshTime T N j))).mono (f.mono ht)
-    have hB1 := Filtration.stronglyAdapted_natural hBm (Real.toNNReal (meshTime T N (j+1)))
-    have hU := ((hu j hj).comp hY.measurable).stronglyMeasurable.const_mul
-      (stepSize T N*γ (meshTime T N j))
-    convert ((hY.add hB1).sub hB0).add hU using 1
-    funext ω
-    rw [coupled_euler_step B hT γ u hN hj ω]
-    rfl
-
-lemma coupled_euler_drift_sum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    {T : ℝ} (hT : 0<T) {N : ℕ} (hN : 0<N) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) (ω : Ω) :
-    coupledEuler T γ u B N N ω = B (Real.toNNReal T) ω-B 0 ω+
-      ∑ j : Fin N, stepSize T N*γ (meshTime T N j)*u (meshTime T N j) (coupledEuler T γ u B N j ω) := by
-  have he (j : Fin N) :
-      coupledEuler T γ u B N (j+1) ω-coupledEuler T γ u B N j ω =
-        (B (Real.toNNReal (meshTime T N (j+1))) ω-B (Real.toNNReal (meshTime T N j)) ω)+
-          stepSize T N*γ (meshTime T N j)*u (meshTime T N j) (coupledEuler T γ u B N j ω) := by
-    rw [coupled_euler_step B hT γ u hN j.isLt ω]
-    dsimp [meshTime,stepSize]
-    ring
-  have hh := Finset.sum_congr (rfl : (Finset.univ : Finset (Fin N))=Finset.univ) (fun j _ ↦ he j)
-  rw [sum_steps_sub (fun j ↦ coupledEuler T γ u B N j ω),Finset.sum_add_distrib,
-    sum_steps_sub (fun j ↦ B (Real.toNNReal (meshTime T N j)) ω),coupled_euler_zero] at hh
-  have hlast : meshTime T N N=T := by dsimp [meshTime,stepSize]; field_simp
-  simpa only [sub_zero,hlast,show meshTime T N 0=0 by simp [meshTime],Real.toNNReal_zero] using hh
-
-noncomputable def feedbackControl {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (γ : ℝ → ℝ)
-    (V : ℝ → ℝ → ℝ) (T : ℝ) (N : ℕ) : ℝ≥0 → Ω → ℝ :=
-  meshControl T N (fun j ω ↦ deriv (V (meshTime T N j))
-    (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))
-
-lemma ValueStrip.feedback_progressive {Ω : Type*} [m : MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} (hBm : ∀ t, StronglyMeasurable (B t))
-    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
-    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N) :
-    IsProgressive (Filtration.natural B hBm) (feedbackControl B γ V T N) := by
-  apply meshControl_progressive hT.le
-  intro j
-  have hu (i : ℕ) (hi : i ≤ N) : Continuous (deriv (V (meshTime T N i))) :=
-    (h.smooth _ (mesh_time_mem_total hT.le hi)).continuous_deriv (by norm_num)
-  exact (hu j j.isLt.le).measurable.comp
-    (coupled_euler_past_measurable hBm hT hN γ (fun t ↦ deriv (V t))
-      (fun i hi ↦ (hu i hi.le).measurable) j j.isLt.le).measurable
-
-lemma ValueStrip.feedback_bound {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
-    (hT : 0 ≤ T) (h : ValueStrip T γ V K L) (N : ℕ) (s : ℝ≥0) (ω : Ω) :
-    |feedbackControl B γ V T N s ω| ≤ 1 :=
-  meshControl_bound hT _ (fun j _ ↦ h.bounded _ (mesh_time_mem_total hT j.isLt.le) _) s ω
-
-lemma ValueStrip.feedback_state_error {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
-    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N)
-    (ω : Ω) (hB0 : B 0 ω=0) :
-    |controlState B γ (feedbackControl B γ V T N) 0 T ω-
-      coupledEuler T γ (fun t ↦ deriv (V t)) B N N ω| ≤ stepSize T N*(γ T-γ 0) := by
-  have hh := meshControl_quadrature hT.le hN
-    (fun (j : Fin N) ω ↦ deriv (V (meshTime T N j))
-      (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))
-    (fun j ω ↦ h.bounded _ (mesh_time_mem_total hT.le j.isLt.le) _)
-    h.gamma_mono 1 ω
-  rw [controlState,coupled_euler_drift_sum B hT hN,hB0]
-  simpa only [pow_one,zero_add,sub_zero,add_sub_add_left_eq_sub,feedbackControl] using hh
-
-lemma ValueStrip.feedback_energy_error {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
-    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N) (ω : Ω) :
-    |(1/2 : ℝ)*controlAccum γ (feedbackControl B γ V T N) 2 T ω-
-      valueMeshEnergy T N γ V N (coupledCoordinates T N B ω)| ≤
-        (1/2 : ℝ)*stepSize T N*(γ T-γ 0) := by
-  have hh := meshControl_quadrature hT.le hN
-    (fun (j : Fin N) ω ↦ deriv (V (meshTime T N j))
-      (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))
-    (fun j ω ↦ h.bounded _ (mesh_time_mem_total hT.le j.isLt.le) _)
-    h.gamma_mono 2 ω
-  have he : valueMeshEnergy T N γ V N (coupledCoordinates T N B ω) = (1/2 : ℝ)*
-      ∑ j : Fin N, stepSize T N*γ (meshTime T N j)*
-        (deriv (V (meshTime T N j)) (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))^2 := by
-    rw [Fin.sum_univ_eq_sum_range (fun j ↦ stepSize T N*γ (meshTime T N j)*
-      (deriv (V (meshTime T N j)) (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))^2)]
-    dsimp only [valueMeshEnergy,coupledEuler]
-    simp only [ite_eq_right hN.ne',Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro j _
-    ring
-  rw [he,←mul_sub,abs_mul,abs_of_nonneg (by norm_num : 0 ≤ (1/2 : ℝ))]
-  exact (mul_le_mul_of_nonneg_left hh (by norm_num)).trans_eq (by ring)
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Gaussian.CoupledCoordinates
+import OAI.Probability.SKValue.Equations.ValueStrip
+import OAI.Probability.SKValue.Equations.CubicEnvelopeMean
+import OAI.Probability.SKValue.Control.Controls
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def meshControl {Ω : Type*} (T : ℝ) (N : ℕ) (H : Fin N → Ω → ℝ)
+    (t : ℝ≥0) (ω : Ω) : ℝ :=
+  ∑ j : Fin N, if meshTime T N j < t ∧ (t : ℝ)  ≤  meshTime T N (j+1) then H j ω else 0
+
+lemma progressive_interval {Ω : Type*} [m : MeasurableSpace Ω]
+    {f : Filtration ℝ≥0 m} {a b : ℝ} {H : Ω → ℝ} (_ : 0 ≤ a)
+    (hH : Measurable[f (Real.toNNReal a)] H) :
+    IsProgressive f (fun t ω ↦ if a < (t : ℝ) ∧ (t : ℝ)  ≤  b then H ω else 0) := by
+  intro t
+  by_cases hat : Real.toNNReal a  ≤  t
+  · let : MeasurableSpace Ω := f t
+    have hHm : Measurable H := hH.mono (f.mono hat) le_rfl
+    exact (hHm.comp measurable_snd).ite
+      ((measurableSet_lt measurable_const (measurable_coe_nnreal_real.comp
+        (measurable_subtype_coe.comp measurable_fst))).inter
+        (measurableSet_le (measurable_coe_nnreal_real.comp
+          (measurable_subtype_coe.comp measurable_fst)) measurable_const)) measurable_const
+  · have he : (fun p : Iic t × Ω ↦ if a < (p.1 : ℝ≥0) ∧ ((p.1 : ℝ≥0) : ℝ)  ≤  b then H p.2 else 0) =
+        fun _ ↦ (0 : ℝ) := by
+      funext p
+      apply ite_eq_right
+      intro hp
+      exact hat ((Real.toNNReal_le_iff_le_coe).2 (hp.1.le.trans (by exact_mod_cast p.1.2)))
+    rw [he]
+    exact measurable_const
+
+lemma meshControl_progressive {Ω : Type*} [m : MeasurableSpace Ω]
+    {f : Filtration ℝ≥0 m} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} {H : Fin N → Ω → ℝ}
+    (hH : ∀ j : Fin N, Measurable[f (Real.toNNReal (meshTime T N j))] (H j)) :
+    IsProgressive f (meshControl T N H) := by
+  apply IsProgressive.finsetSum
+  intro j _
+  exact progressive_interval (mesh_time_mem_total hT j.isLt.le).1 (hH j)
+
+lemma mesh_intervals_disjoint {T : ℝ} (hT : 0 ≤ T) {N : ℕ} {i j : Fin N} {t : ℝ}
+    (hi : meshTime T N i < t ∧ t  ≤  meshTime T N (i+1))
+    (hj : meshTime T N j < t ∧ t  ≤  meshTime T N (j+1)) : i=j := by
+  have hδ : 0  ≤  stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
+  have hlt : ∀ a b : ℕ, a+1 ≤ b → meshTime T N (a+1) ≤ meshTime T N b := by
+    intro a b hab
+    exact mul_le_mul_of_nonneg_right (by exact_mod_cast hab) hδ
+  apply Fin.ext
+  by_contra he
+  rcases lt_or_gt_of_ne he with hij | hji
+  · have hh := hlt i j hij
+    linarith [hi.2,hj.1]
+  · have hh := hlt j i hji
+    linarith [hi.1,hj.2]
+
+lemma meshControl_eq {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (H : Fin N → Ω → ℝ)
+    {j : Fin N} {t : ℝ≥0} (ht : meshTime T N j < t ∧ (t : ℝ)  ≤  meshTime T N (j+1)) (ω : Ω) :
+    meshControl T N H t ω=H j ω := by
+  unfold meshControl
+  rw [Finset.sum_eq_single j]
+  · simp only [ite_eq_left ht]
+  · intro i _ hij
+    exact ite_eq_right (fun hi ↦ hij (mesh_intervals_disjoint hT hi ht))
+  · simp
+
+lemma meshControl_bound {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (H : Fin N → Ω → ℝ)
+    (hH : ∀ j ω, |H j ω| ≤ 1) (t : ℝ≥0) (ω : Ω) : |meshControl T N H t ω| ≤ 1 := by
+  by_cases he : ∃ j : Fin N, meshTime T N j < t ∧ (t : ℝ)  ≤  meshTime T N (j+1)
+  · obtain ⟨j,hj⟩ := he
+    rw [meshControl_eq hT H hj]
+    exact hH j ω
+  · have hh : meshControl T N H t ω=0 := by
+      unfold meshControl
+      apply Finset.sum_eq_zero
+      intro j _
+      exact ite_eq_right (fun hj ↦ he ⟨j,hj⟩)
+    rw [hh]
+    norm_num
+
+lemma meshControl_momentIntegral {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (hN : 0<N)
+    (H : Fin N → Ω → ℝ) {γ : ℝ → ℝ} (hγ : IntervalIntegrable γ volume 0 T)
+    (k : ℕ) (ω : Ω) :
+    controlAccum γ (meshControl T N H) k T ω =
+      ∑ j : Fin N, (∫ s in meshTime T N j..meshTime T N (j+1), γ s)*(H j ω)^k := by
+  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
+  have htime (j : ℕ) (hj : j ≤ N) := mesh_time_mem_total hT hj
+  have horder (j : ℕ) : meshTime T N j ≤ meshTime T N (j+1) := by
+    dsimp [meshTime]
+    exact mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.le_succ j) hδ
+  have hlast : meshTime T N N=T := by
+    dsimp [meshTime,stepSize]
+    field_simp
+  have he (j : Fin N) :
+      (fun s ↦ γ s*(meshControl T N H (Real.toNNReal s) ω)^k) =ᵐ[
+        volume.restrict (uIoc (meshTime T N j) (meshTime T N (j+1)))]
+      (fun s ↦ γ s*(H j ω)^k) := by
+    filter_upwards [ae_restrict_mem measurableSet_uIoc] with s hs
+    rw [uIoc_of_le (horder j)] at hs
+    have hs0 : 0 ≤ s := (htime j j.isLt.le).1.trans hs.1.le
+    rw [meshControl_eq hT H (by simpa only [Real.coe_toNNReal _ hs0,Set.mem_Ioc] using hs)]
+  have hi (j : ℕ) (hj : j<N) : IntervalIntegrable
+      (fun s ↦ γ s*(meshControl T N H (Real.toNNReal s) ω)^k) volume
+      (meshTime T N j) (meshTime T N (j+1)) := by
+    apply ((intervalIntegrable_substrip hT hγ (htime j hj.le)
+      (htime (j+1) hj)).mul_const ((H ⟨j,hj⟩ ω)^k)).congr_ae
+    exact (he ⟨j,hj⟩).symm
+  have hs := intervalIntegral.sum_integral_adjacent_intervals hi
+  rw [hlast,show meshTime T N 0=0 by simp [meshTime]] at hs
+  rw [controlAccum,←hs,←Fin.sum_univ_eq_sum_range]
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [intervalIntegral.integral_congr_ae_restrict (he j),intervalIntegral.integral_mul_const]
+
+lemma monotone_interval_quadrature {γ : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b)
+    (hγ : MonotoneOn γ (Icc a b)) :
+    |(∫ s in a..b, γ s)-(b-a)*γ a| ≤ (b-a)*(γ b-γ a) := by
+  have hi : IntervalIntegrable γ volume a b := by
+    apply MonotoneOn.intervalIntegrable
+    simpa only [uIcc_of_le hab] using hγ
+  have hdiff : (∫ s in a..b, γ s)-(b-a)*γ a =∫ s in a..b, γ s-γ a := by
+    rw [intervalIntegral.integral_sub hi intervalIntegrable_const]
+    simp only [intervalIntegral.integral_const,smul_eq_mul]
+  rw [hdiff]
+  have hh := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := a) (b := b) (f := fun s ↦ γ s-γ a) (C := γ b-γ a) (fun s hs ↦ by
+      rw [uIoc_of_le hab] at hs
+      have hs' : s∈Icc a b := ⟨hs.1.le,hs.2⟩
+      rw [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr (hγ ⟨le_rfl,hab⟩ hs' hs.1.le))]
+      exact sub_le_sub_right (hγ hs' ⟨hab,le_rfl⟩ hs.2) _)
+  simpa only [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hab),mul_comm] using hh
+
+lemma meshControl_quadrature {Ω : Type*} {T : ℝ} (hT : 0 ≤ T) {N : ℕ} (hN : 0<N)
+    (H : Fin N → Ω → ℝ) (hH : ∀ j ω, |H j ω| ≤ 1)
+    {γ : ℝ → ℝ} (hγ : MonotoneOn γ (Icc (0 : ℝ) T)) (k : ℕ) (ω : Ω) :
+    |controlAccum γ (meshControl T N H) k T ω-
+      ∑ j : Fin N, stepSize T N*γ (meshTime T N j)*(H j ω)^k| ≤
+      stepSize T N*(γ T-γ 0) := by
+  have hi : IntervalIntegrable γ volume 0 T := by
+    apply MonotoneOn.intervalIntegrable
+    simpa only [uIcc_of_le hT] using hγ
+  rw [meshControl_momentIntegral hT hN H hi, ←Finset.sum_sub_distrib]
+  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
+  have htime (j : ℕ) (hj : j ≤ N) := mesh_time_mem_total hT hj
+  have hgap (j : ℕ) : meshTime T N (j+1)-meshTime T N j=stepSize T N := by
+    dsimp [meshTime]
+    push_cast
+    ring
+  have horder (j : ℕ) : meshTime T N j ≤ meshTime T N (j+1) := by
+    linarith [hgap j]
+  have hloc (j : Fin N) :
+      |(∫ s in meshTime T N j..meshTime T N (j+1), γ s)*(H j ω)^k-
+        stepSize T N*γ (meshTime T N j)*(H j ω)^k| ≤
+      stepSize T N*(γ (meshTime T N (j+1))-γ (meshTime T N j)) := by
+    rw [←sub_mul,abs_mul]
+    have hh := monotone_interval_quadrature (horder j)
+      (hγ.mono (Icc_subset_Icc (htime j j.isLt.le).1 (htime (j+1) j.isLt).2))
+    rw [hgap] at hh
+    have hp : |(H j ω)^k| ≤ 1 := by
+      rw [abs_pow]
+      exact pow_le_one₀ (abs_nonneg _) (hH j ω)
+    exact (mul_le_mul_of_nonneg_left hp (abs_nonneg _)).trans (by simpa only [mul_one] using hh)
+  calc
+    _ ≤ ∑ j : Fin N, |(∫ s in meshTime T N j..meshTime T N (j+1), γ s)*(H j ω)^k-
+        stepSize T N*γ (meshTime T N j)*(H j ω)^k| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ j : Fin N, stepSize T N*(γ (meshTime T N (j+1))-γ (meshTime T N j)) :=
+      Finset.sum_le_sum (fun j _ ↦ hloc j)
+    _ = _ := by
+      rw [←Finset.mul_sum, sum_steps_sub (fun j ↦ γ (meshTime T N j))]
+      have hlast : meshTime T N N=T := by
+        dsimp [meshTime,stepSize]
+        field_simp
+      change stepSize T N*(γ (meshTime T N N)-γ (meshTime T N 0)) = _
+      rw [hlast]
+      simp only [meshTime,Nat.cast_zero,zero_mul]
+
+lemma coupled_euler_past_measurable {Ω : Type*} [m : MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} (hBm : ∀ t, StronglyMeasurable (B t))
+    {T : ℝ} (hT : 0<T) {N : ℕ} (hN : 0<N) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (hu : ∀ j<N, Measurable (u (meshTime T N j))) :
+    ∀ j ≤ N, StronglyMeasurable[Filtration.natural B hBm (Real.toNNReal (meshTime T N j))]
+      (coupledEuler T γ u B N j) := by
+  let f := Filtration.natural B hBm
+  intro j
+  induction j with
+  | zero =>
+    intro _
+    convert (stronglyMeasurable_const : StronglyMeasurable[f (Real.toNNReal (meshTime T N 0))] (fun _ : Ω ↦ (0 : ℝ))) using 1
+    funext ω
+    exact coupled_euler_zero T γ u B N ω
+  | succ j ih =>
+    intro hjN
+    have hj : j<N := hjN
+    have hδ : 0 ≤ stepSize T N := div_nonneg hT.le (Nat.cast_nonneg N)
+    have ht : Real.toNNReal (meshTime T N j) ≤ Real.toNNReal (meshTime T N (j+1)) := by
+      apply Real.toNNReal_mono
+      apply mul_le_mul_of_nonneg_right _ hδ
+      exact_mod_cast Nat.le_succ j
+    have hY := (ih hj.le).mono (f.mono ht)
+    have hB0 := (Filtration.stronglyAdapted_natural hBm (Real.toNNReal (meshTime T N j))).mono (f.mono ht)
+    have hB1 := Filtration.stronglyAdapted_natural hBm (Real.toNNReal (meshTime T N (j+1)))
+    have hU := ((hu j hj).comp hY.measurable).stronglyMeasurable.const_mul
+      (stepSize T N*γ (meshTime T N j))
+    convert ((hY.add hB1).sub hB0).add hU using 1
+    funext ω
+    rw [coupled_euler_step B hT γ u hN hj ω]
+    rfl
+
+lemma coupled_euler_drift_sum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    {T : ℝ} (hT : 0<T) {N : ℕ} (hN : 0<N) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) (ω : Ω) :
+    coupledEuler T γ u B N N ω = B (Real.toNNReal T) ω-B 0 ω+
+      ∑ j : Fin N, stepSize T N*γ (meshTime T N j)*u (meshTime T N j) (coupledEuler T γ u B N j ω) := by
+  have he (j : Fin N) :
+      coupledEuler T γ u B N (j+1) ω-coupledEuler T γ u B N j ω =
+        (B (Real.toNNReal (meshTime T N (j+1))) ω-B (Real.toNNReal (meshTime T N j)) ω)+
+          stepSize T N*γ (meshTime T N j)*u (meshTime T N j) (coupledEuler T γ u B N j ω) := by
+    rw [coupled_euler_step B hT γ u hN j.isLt ω]
+    dsimp [meshTime,stepSize]
+    ring
+  have hh := Finset.sum_congr (rfl : (Finset.univ : Finset (Fin N))=Finset.univ) (fun j _ ↦ he j)
+  rw [sum_steps_sub (fun j ↦ coupledEuler T γ u B N j ω),Finset.sum_add_distrib,
+    sum_steps_sub (fun j ↦ B (Real.toNNReal (meshTime T N j)) ω),coupled_euler_zero] at hh
+  have hlast : meshTime T N N=T := by dsimp [meshTime,stepSize]; field_simp
+  simpa only [sub_zero,hlast,show meshTime T N 0=0 by simp [meshTime],Real.toNNReal_zero] using hh
+
+noncomputable def feedbackControl {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (γ : ℝ → ℝ)
+    (V : ℝ → ℝ → ℝ) (T : ℝ) (N : ℕ) : ℝ≥0 → Ω → ℝ :=
+  meshControl T N (fun j ω ↦ deriv (V (meshTime T N j))
+    (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))
+
+lemma ValueStrip.feedback_progressive {Ω : Type*} [m : MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} (hBm : ∀ t, StronglyMeasurable (B t))
+    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
+    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N) :
+    IsProgressive (Filtration.natural B hBm) (feedbackControl B γ V T N) := by
+  apply meshControl_progressive hT.le
+  intro j
+  have hu (i : ℕ) (hi : i ≤ N) : Continuous (deriv (V (meshTime T N i))) :=
+    (h.smooth _ (mesh_time_mem_total hT.le hi)).continuous_deriv (by norm_num)
+  exact (hu j j.isLt.le).measurable.comp
+    (coupled_euler_past_measurable hBm hT hN γ (fun t ↦ deriv (V t))
+      (fun i hi ↦ (hu i hi.le).measurable) j j.isLt.le).measurable
+
+lemma ValueStrip.feedback_bound {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
+    (hT : 0 ≤ T) (h : ValueStrip T γ V K L) (N : ℕ) (s : ℝ≥0) (ω : Ω) :
+    |feedbackControl B γ V T N s ω| ≤ 1 :=
+  meshControl_bound hT _ (fun j _ ↦ h.bounded _ (mesh_time_mem_total hT j.isLt.le) _) s ω
+
+lemma ValueStrip.feedback_state_error {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
+    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N)
+    (ω : Ω) (hB0 : B 0 ω=0) :
+    |controlState B γ (feedbackControl B γ V T N) 0 T ω-
+      coupledEuler T γ (fun t ↦ deriv (V t)) B N N ω| ≤ stepSize T N*(γ T-γ 0) := by
+  have hh := meshControl_quadrature hT.le hN
+    (fun (j : Fin N) ω ↦ deriv (V (meshTime T N j))
+      (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))
+    (fun j ω ↦ h.bounded _ (mesh_time_mem_total hT.le j.isLt.le) _)
+    h.gamma_mono 1 ω
+  rw [controlState,coupled_euler_drift_sum B hT hN,hB0]
+  simpa only [pow_one,zero_add,sub_zero,add_sub_add_left_eq_sub,feedbackControl] using hh
+
+lemma ValueStrip.feedback_energy_error {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    {T K L : ℝ} {γ : ℝ → ℝ} {V : ℝ → ℝ → ℝ}
+    (hT : 0<T) (h : ValueStrip T γ V K L) {N : ℕ} (hN : 0<N) (ω : Ω) :
+    |(1/2 : ℝ)*controlAccum γ (feedbackControl B γ V T N) 2 T ω-
+      valueMeshEnergy T N γ V N (coupledCoordinates T N B ω)| ≤
+        (1/2 : ℝ)*stepSize T N*(γ T-γ 0) := by
+  have hh := meshControl_quadrature hT.le hN
+    (fun (j : Fin N) ω ↦ deriv (V (meshTime T N j))
+      (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))
+    (fun j ω ↦ h.bounded _ (mesh_time_mem_total hT.le j.isLt.le) _)
+    h.gamma_mono 2 ω
+  have he : valueMeshEnergy T N γ V N (coupledCoordinates T N B ω) = (1/2 : ℝ)*
+      ∑ j : Fin N, stepSize T N*γ (meshTime T N j)*
+        (deriv (V (meshTime T N j)) (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))^2 := by
+    rw [Fin.sum_univ_eq_sum_range (fun j ↦ stepSize T N*γ (meshTime T N j)*
+      (deriv (V (meshTime T N j)) (coupledEuler T γ (fun t ↦ deriv (V t)) B N j ω))^2)]
+    dsimp only [valueMeshEnergy,coupledEuler]
+    simp only [ite_eq_right hN.ne',Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    ring
+  rw [he,←mul_sub,abs_mul,abs_of_nonneg (by norm_num : 0 ≤ (1/2 : ℝ))]
+  exact (mul_le_mul_of_nonneg_left hh (by norm_num)).trans_eq (by ring)
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/FiniteGaussianFixedStrip.lean b/lean/OAI/Probability/SKValue/Processes/FiniteGaussianFixedStrip.lean
index 171635bd22860734fa4def246529d1087ac66bee..b4f450ec567dd0d4a8de45ed6af69cb467acaa04 100644
--- a/lean/OAI/Probability/SKValue/Processes/FiniteGaussianFixedStrip.lean
+++ b/lean/OAI/Probability/SKValue/Processes/FiniteGaussianFixedStrip.lean
@@ -1,55 +1,55 @@
-import Mathlib
-import OAI.Probability.SKValue.Evolution.MeanDerivativeContinuous
-import OAI.Probability.SKValue.Processes.StripPaths
-import OAI.Probability.SKValue.Equations.LiteralNormalizer
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-
-lemma IsDiffusion.finite_gaussian_fixed_strip {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T K L D Lu La : ℝ}
-    (hT : 0<T) (hT1 : T<1) (hg : GradientStrip T γ.coeff (gradient W γ) K L)
-    (hD : 0≤D) (hLu : 0≤Lu) (hLa : 0≤La)
-    (ha : ∀ t∈Icc (0 : ℝ) T, ∀ x, |curvature W γ t x|≤D)
-    (huLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |gradient W γ s x-gradient W γ t y|≤Lu*(|s-t|+|x-y|))
-    (haLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |curvature W γ s x-curvature W γ t y|≤La*(|s-t|+|x-y|))
-    (hmom : ∀ t∈Icc (0 : ℝ) T, (∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ)=1) :
-    (∀ᶠ N in atTop, ∀ j<N, 0<normalizer W γ T N j) ∧
-    Tendsto (shiftedCoefficientSum W γ T) atTop
-      (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, curvature W γ t (X t ω) ∂W.μ)) := by
-  have hh := finiteGaussian_from_diffusion W.brownian.toIsPreBrownianReal hT hT1.le hg
-    hD hLu hLa ha huLip haLip
-    (fun t ht ↦ (hX.measurable ⟨ht.1,ht.2.trans hT1.le⟩).aestronglyMeasurable)
-    (hX.strip_paths hT.le hT1 hLu (fun t ht s hs x y ↦ huLip s hs t ht y x)) hmom
-  constructor
-  · simpa only [literal_normalizer] using hh.1
-  · apply hh.2.congr'
-    exact Eventually.of_forall (fun N ↦ (literal_shiftedCoefficientSum W γ T N).symm)
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Evolution.MeanDerivativeContinuous
+import OAI.Probability.SKValue.Processes.StripPaths
+import OAI.Probability.SKValue.Equations.LiteralNormalizer
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+
+lemma IsDiffusion.finite_gaussian_fixed_strip {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T K L D Lu La : ℝ}
+    (hT : 0<T) (hT1 : T<1) (hg : GradientStrip T γ.coeff (gradient W γ) K L)
+    (hD : 0≤D) (hLu : 0≤Lu) (hLa : 0≤La)
+    (ha : ∀ t∈Icc (0 : ℝ) T, ∀ x, |curvature W γ t x|≤D)
+    (huLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |gradient W γ s x-gradient W γ t y|≤Lu*(|s-t|+|x-y|))
+    (haLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |curvature W γ s x-curvature W γ t y|≤La*(|s-t|+|x-y|))
+    (hmom : ∀ t∈Icc (0 : ℝ) T, (∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ)=1) :
+    (∀ᶠ N in atTop, ∀ j<N, 0<normalizer W γ T N j) ∧
+    Tendsto (shiftedCoefficientSum W γ T) atTop
+      (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, curvature W γ t (X t ω) ∂W.μ)) := by
+  have hh := finiteGaussian_from_diffusion W.brownian.toIsPreBrownianReal hT hT1.le hg
+    hD hLu hLa ha huLip haLip
+    (fun t ht ↦ (hX.measurable ⟨ht.1,ht.2.trans hT1.le⟩).aestronglyMeasurable)
+    (hX.strip_paths hT.le hT1 hLu (fun t ht s hs x y ↦ huLip s hs t ht y x)) hmom
+  constructor
+  · simpa only [literal_normalizer] using hh.1
+  · apply hh.2.congr'
+    exact Eventually.of_forall (fun N ↦ (literal_shiftedCoefficientSum W γ T N).symm)
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/FiniteProfileLaw.lean b/lean/OAI/Probability/SKValue/Processes/FiniteProfileLaw.lean
index f0aebb27b91379331fa21855a3dc13bce26a0622..50b23cb6d9304a1155e19663df9d0efd0c17d07e 100644
--- a/lean/OAI/Probability/SKValue/Processes/FiniteProfileLaw.lean
+++ b/lean/OAI/Probability/SKValue/Processes/FiniteProfileLaw.lean
@@ -1,40 +1,40 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.TestDiffusion
-import OAI.Probability.SKValue.Equations.TestPatch
-import OAI.Probability.SKValue.Evolution.TestStrip
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal
-namespace SKValue
-lemma SmoothTerminal.profile_diffusion_test {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
-    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
-    {X:ℝ → Ω → ℝ} {ψ f:ℝ → ℝ} (hψ:SmoothTerminal ψ) (heven:∀ x,ψ (-x)=ψ x)
-    (hf:BoundedSmooth f) {l:HeatProfile} (hl:l.Pairwise (fun p q ↦ p.2≤q.2))
-    (hT:0<profileTime l) (hT1:profileTime l≤1)
-    (hXM:∀ t∈Icc (0:ℝ) (profileTime l),AEStronglyMeasurable (X t) μ)
-    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) (profileTime l)) ∧
-      IntervalIntegrable (fun s ↦ profileCoeff l s*deriv (profileValue ψ l s) (X s ω)) volume 0 (profileTime l) ∧
-      (∀ t∈Icc (0:ℝ) (profileTime l),X t ω=B t.toNNReal ω+
-        ∫ s in (0:ℝ)..t,profileCoeff l s*deriv (profileValue ψ l s) (X s ω)) ∧ X 0 ω=0) :
-    (∫ ω,f (X (profileTime l) ω) ∂μ)=profileResponse ψ f l 0 0 := by
-  have hA := hψ.profile_evolution l
-  have hV := hψ.linear_profile hf l
-  obtain ⟨K,L,hu⟩ := hψ.profile_gradient_strip heven hl
-  obtain ⟨K',L',hv⟩ := hV.toBackwardTest (profileTime_nonneg l) hA
-    (fun t _ ↦ profileCoeff_nonneg l t) (profileCoeff_mono hl)
-  obtain ⟨Lu,hLu,hLip⟩ := hA.jet_joint 0
-  have hLip' : ∀ s∈Icc (0:ℝ) (profileTime l),∀ t∈Icc (0:ℝ) (profileTime l),∀ x y,
-      |deriv (profileValue ψ l s) x-deriv (profileValue ψ l t) y|≤Lu*(|s-t|+|x-y|) := by
-    intro s hs t ht x y
-    simpa only [iteratedDeriv_zero] using hLip s hs t ht y x
-  have he := hv.diffusion_expectation hB hT hT1 hu hLu hLip' hXM hpaths
-  rwa [profileResponse_terminal] at he
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.TestDiffusion
+import OAI.Probability.SKValue.Equations.TestPatch
+import OAI.Probability.SKValue.Evolution.TestStrip
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal
+namespace SKValue
+lemma SmoothTerminal.profile_diffusion_test {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
+    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
+    {X:ℝ → Ω → ℝ} {ψ f:ℝ → ℝ} (hψ:SmoothTerminal ψ) (heven:∀ x,ψ (-x)=ψ x)
+    (hf:BoundedSmooth f) {l:HeatProfile} (hl:l.Pairwise (fun p q ↦ p.2≤q.2))
+    (hT:0<profileTime l) (hT1:profileTime l≤1)
+    (hXM:∀ t∈Icc (0:ℝ) (profileTime l),AEStronglyMeasurable (X t) μ)
+    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) (profileTime l)) ∧
+      IntervalIntegrable (fun s ↦ profileCoeff l s*deriv (profileValue ψ l s) (X s ω)) volume 0 (profileTime l) ∧
+      (∀ t∈Icc (0:ℝ) (profileTime l),X t ω=B t.toNNReal ω+
+        ∫ s in (0:ℝ)..t,profileCoeff l s*deriv (profileValue ψ l s) (X s ω)) ∧ X 0 ω=0) :
+    (∫ ω,f (X (profileTime l) ω) ∂μ)=profileResponse ψ f l 0 0 := by
+  have hA := hψ.profile_evolution l
+  have hV := hψ.linear_profile hf l
+  obtain ⟨K,L,hu⟩ := hψ.profile_gradient_strip heven hl
+  obtain ⟨K',L',hv⟩ := hV.toBackwardTest (profileTime_nonneg l) hA
+    (fun t _ ↦ profileCoeff_nonneg l t) (profileCoeff_mono hl)
+  obtain ⟨Lu,hLu,hLip⟩ := hA.jet_joint 0
+  have hLip' : ∀ s∈Icc (0:ℝ) (profileTime l),∀ t∈Icc (0:ℝ) (profileTime l),∀ x y,
+      |deriv (profileValue ψ l s) x-deriv (profileValue ψ l t) y|≤Lu*(|s-t|+|x-y|) := by
+    intro s hs t ht x y
+    simpa only [iteratedDeriv_zero] using hLip s hs t ht y x
+  have he := hv.diffusion_expectation hB hT hT1 hu hLu hLip' hXM hpaths
+  rwa [profileResponse_terminal] at he
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/ForcedDiffusion.lean b/lean/OAI/Probability/SKValue/Processes/ForcedDiffusion.lean
index ec510356e18bc1b1ca871f0cc9aca0d0e3bfdd54..ac02e2effc801cc66e4cbacd2a267ec6755a95ba 100644
--- a/lean/OAI/Probability/SKValue/Processes/ForcedDiffusion.lean
+++ b/lean/OAI/Probability/SKValue/Processes/ForcedDiffusion.lean
@@ -1,66 +1,66 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.ForcedGenerator
-import OAI.Probability.SKValue.Processes.ForcedExpectation
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal BigOperators
-namespace SKValue
-lemma ForcedTest.diffusion_generator_of_source_limit {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
-    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
-    {X:ℝ → Ω → ℝ} {T K L K' L' Lu:ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
-    (hT:0<T) (hT1:T≤1) (hu:GradientStrip T γ u K L) (h:ForcedTest T γ u V b p K' L')
-    (hLu:0≤Lu) (hLip:∀ s∈Icc (0:ℝ) T,∀ t∈Icc (0:ℝ) T,∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
-    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      (∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0)
-    {I:ℝ} (hsource:Tendsto (fun N ↦ stepSize T N*∑ j∈Finset.range N,
-      (∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))))
-        atTop (𝓝 I)) :
-    (∫ ω,V T (X T ω) ∂μ)=V 0 0+I := by
-  let S := fun N ↦ stepSize T N*∑ j∈Finset.range N,
-      (∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1)))
-  let A := fun N ↦ ∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))
-  let E := fun N ↦ T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L')+
-    stepSize T N*K'*(γ T-γ 0)
-  let F := fun N ↦ K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)
-  have hδ:Tendsto (fun N:ℕ ↦ stepSize T N) atTop (𝓝 (0:ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
-  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
-  have hE:Tendsto E atTop (𝓝 0) := by
-    simpa only [E,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,add_zero] using
-      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L'))).add
-        ((hδ.mul_const K').mul_const (γ T-γ 0)))
-  have hF:Tendsto F atTop (𝓝 0) := by
-    simpa only [F,Function.comp_def,Real.sqrt_zero,mul_zero] using
-      (((Real.continuous_sqrt.tendsto 0).comp
-        (coupled_euler_L2_convergence hB hT hu hLu hLip hXM hpaths)).const_mul K')
-  have hbound:∀ᶠ N in atTop,|(∫ ω,V T (X T ω) ∂μ)-V 0 0-S N|≤E N+F N := by
-    filter_upwards [eventually_gt_atTop 0] with N hN
-    have he := h.mesh_expectation hT hT1 hN
-    have hf := h.coupled_terminal_error hB hT hu hXM
-      (by filter_upwards [hpaths] with ω hω using hω.2.2) hN
-    have ht := abs_sub_le ((∫ ω,V T (X T ω) ∂μ)-V 0 0-S N) (A N-V 0 0-S N) 0
-    change |A N-V 0 0-S N|≤E N at he
-    change |A N-(∫ ω,V T (X T ω) ∂μ)|≤F N at hf
-    have hf' : |(∫ ω,V T (X T ω) ∂μ)-A N|≤F N := by
-      rw [abs_sub_comm];exact hf
-    have hid:((∫ ω,V T (X T ω) ∂μ)-V 0 0-S N)-(A N-V 0 0-S N)=
-      (∫ ω,V T (X T ω) ∂μ)-A N := by ring
-    rw [hid,sub_zero,sub_zero] at ht
-    linarith
-  have hleft:Tendsto (fun N ↦ |(∫ ω,V T (X T ω) ∂μ)-V 0 0-S N|) atTop
-      (𝓝 |(∫ ω,V T (X T ω) ∂μ)-V 0 0-I|) := (tendsto_const_nhds.sub hsource).abs
-  have hz := le_of_tendsto_of_tendsto hleft (hE.add hF) hbound
-  have habs:|(∫ ω,V T (X T ω) ∂μ)-V 0 0-I|=0 := le_antisymm (by simpa using hz) (abs_nonneg _)
-  have he := abs_eq_zero.mp habs
-  linarith
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.ForcedGenerator
+import OAI.Probability.SKValue.Processes.ForcedExpectation
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal BigOperators
+namespace SKValue
+lemma ForcedTest.diffusion_generator_of_source_limit {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
+    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
+    {X:ℝ → Ω → ℝ} {T K L K' L' Lu:ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
+    (hT:0<T) (hT1:T≤1) (hu:GradientStrip T γ u K L) (h:ForcedTest T γ u V b p K' L')
+    (hLu:0≤Lu) (hLip:∀ s∈Icc (0:ℝ) T,∀ t∈Icc (0:ℝ) T,∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
+    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      (∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0)
+    {I:ℝ} (hsource:Tendsto (fun N ↦ stepSize T N*∑ j∈Finset.range N,
+      (∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))))
+        atTop (𝓝 I)) :
+    (∫ ω,V T (X T ω) ∂μ)=V 0 0+I := by
+  let S := fun N ↦ stepSize T N*∑ j∈Finset.range N,
+      (∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1)))
+  let A := fun N ↦ ∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))
+  let E := fun N ↦ T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L')+
+    stepSize T N*K'*(γ T-γ 0)
+  let F := fun N ↦ K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)
+  have hδ:Tendsto (fun N:ℕ ↦ stepSize T N) atTop (𝓝 (0:ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
+  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
+  have hE:Tendsto E atTop (𝓝 0) := by
+    simpa only [E,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,add_zero] using
+      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L'))).add
+        ((hδ.mul_const K').mul_const (γ T-γ 0)))
+  have hF:Tendsto F atTop (𝓝 0) := by
+    simpa only [F,Function.comp_def,Real.sqrt_zero,mul_zero] using
+      (((Real.continuous_sqrt.tendsto 0).comp
+        (coupled_euler_L2_convergence hB hT hu hLu hLip hXM hpaths)).const_mul K')
+  have hbound:∀ᶠ N in atTop,|(∫ ω,V T (X T ω) ∂μ)-V 0 0-S N|≤E N+F N := by
+    filter_upwards [eventually_gt_atTop 0] with N hN
+    have he := h.mesh_expectation hT hT1 hN
+    have hf := h.coupled_terminal_error hB hT hu hXM
+      (by filter_upwards [hpaths] with ω hω using hω.2.2) hN
+    have ht := abs_sub_le ((∫ ω,V T (X T ω) ∂μ)-V 0 0-S N) (A N-V 0 0-S N) 0
+    change |A N-V 0 0-S N|≤E N at he
+    change |A N-(∫ ω,V T (X T ω) ∂μ)|≤F N at hf
+    have hf' : |(∫ ω,V T (X T ω) ∂μ)-A N|≤F N := by
+      rw [abs_sub_comm];exact hf
+    have hid:((∫ ω,V T (X T ω) ∂μ)-V 0 0-S N)-(A N-V 0 0-S N)=
+      (∫ ω,V T (X T ω) ∂μ)-A N := by ring
+    rw [hid,sub_zero,sub_zero] at ht
+    linarith
+  have hleft:Tendsto (fun N ↦ |(∫ ω,V T (X T ω) ∂μ)-V 0 0-S N|) atTop
+      (𝓝 |(∫ ω,V T (X T ω) ∂μ)-V 0 0-I|) := (tendsto_const_nhds.sub hsource).abs
+  have hz := le_of_tendsto_of_tendsto hleft (hE.add hF) hbound
+  have habs:|(∫ ω,V T (X T ω) ∂μ)-V 0 0-I|=0 := le_antisymm (by simpa using hz) (abs_nonneg _)
+  have he := abs_eq_zero.mp habs
+  linarith
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/ForcedExpectation.lean b/lean/OAI/Probability/SKValue/Processes/ForcedExpectation.lean
index 45ea9cb829c372e4e27f4691026e9bf0c24f1f8d..cf78ed9cf79bc2eea21314c51a3bb0ff3f7edc26 100644
--- a/lean/OAI/Probability/SKValue/Processes/ForcedExpectation.lean
+++ b/lean/OAI/Probability/SKValue/Processes/ForcedExpectation.lean
@@ -1,202 +1,202 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.CrossMesh
-import OAI.Probability.SKValue.Control.Forced
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators NNReal
-namespace SKValue
-lemma ForcedTest.expectation_step {T K L t δ:ℝ} {γ:ℝ → ℝ} {u V b₀ p₀:ℝ → ℝ → ℝ}
-    (h:ForcedTest T γ u V b₀ p₀ K L) (ht:0≤t) (hδ:0≤δ) (hδ1:δ≤1) (htδ:t+δ≤T)
-    {N j:ℕ} (hj:j<N+1) {Y:(Fin (N+1) → ℝ) → ℝ}
-    (hYm:Measurable Y) (hYp:DependsBefore j Y) :
-    let μ := gaussianProduct (Fin (N+1))
-    let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
-    |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)-δ*(∫ z,forcedSource γ u V b₀ p₀ t (Y z) ∂μ)|≤
-      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
-  let μ := gaussianProduct (Fin (N+1))
-  let p := fun z ↦ deriv (V t) (Y z)
-  let a := fun z ↦ deriv (deriv (V t)) (Y z)
-  let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
-  let P := fun z ↦ Real.sqrt δ*rawIncrement p j z
-  let Q := fun z ↦ (δ/2)*(a z*((coordinate N j z)^2-1))
-  let S := fun z ↦ forcedSource γ u V b₀ p₀ t (Y z)
-  let D := fun z ↦ V (t+δ) (Y' z)-V t (Y z)-δ*S z
-  let R := fun z ↦ D z-P z-Q z
-  let E := fun z ↦ δ*Real.sqrt δ*cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z)+δ*K*(γ (t+δ)-γ t)
-  have htT:t≤T := (le_add_of_nonneg_right hδ).trans htδ
-  have htm:t∈Icc (0:ℝ) T := ⟨ht,htT⟩
-  have hem:t+δ∈Icc (0:ℝ) T := ⟨by linarith,htδ⟩
-  have hpm:Measurable p := ((h.smooth t htm).continuous_deriv (by norm_num)).measurable.comp hYm
-  have ham:Measurable a := by
-    have hh := (h.smooth t htm).continuous_iteratedDeriv 2 (by norm_num)
-    simp only [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ] at hh
-    exact hh.measurable.comp hYm
-  have hpM:MemLp p 2 μ := MemLp.of_bound hpm.aestronglyMeasurable K
-    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound t htm (Y z)))
-  have haM:MemLp a 2 μ := MemLp.of_bound ham.aestronglyMeasurable K
-    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.second_bound t htm (Y z)))
-  have hpp:DependsBefore j p := fun z z' hz ↦ by dsimp only [p];rw [hYp z z' hz]
-  have hap:DependsBefore j a := fun z z' hz ↦ by dsimp only [a];rw [hYp z z' hz]
-  have hPi:Integrable P μ := (((rawIncrement_memLp hj hpm hpp hpM).const_mul _).integrable (by norm_num))
-  have hQi:Integrable Q μ := (((predictableCenteredSquare_memLp ham (fun z ↦ h.second_bound t htm (Y z))).const_mul _).integrable (by norm_num))
-  have hP0:(∫ z,P z ∂μ)=0 := by
-    dsimp only [P]
-    rw [integral_const_mul,rawIncrement_mean hj hpm hpp,mul_zero]
-  have hQ0:(∫ z,Q z ∂μ)=0 := by
-    dsimp only [Q]
-    rw [integral_const_mul,predictableCenteredSquare_mean hj ham hap,mul_zero]
-  have hY'm:Measurable Y' := (hYm.add ((measurable_coordinate N j).const_mul _)).add
-    (((h.drift_measurable t htm).comp hYm).const_mul _)
-  have hcur:Integrable (fun z ↦ V t (Y z)) μ :=
-    (MemLp.of_bound ((h.smooth t htm).continuous.measurable.comp hYm).aestronglyMeasurable K
-      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound t htm (Y z))) :
-        MemLp (fun z ↦ V t (Y z)) 2 μ).integrable (by norm_num)
-  have hnext:Integrable (fun z ↦ V (t+δ) (Y' z)) μ :=
-    (MemLp.of_bound ((h.smooth (t+δ) hem).continuous.measurable.comp hY'm).aestronglyMeasurable K
-      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound (t+δ) hem (Y' z))) :
-        MemLp (fun z ↦ V (t+δ) (Y' z)) 2 μ).integrable (by norm_num)
-  have hSi:Integrable S μ :=
-    (MemLp.of_bound ((h.source_measurable htm).comp hYm).aestronglyMeasurable ((1+2*γ T)*K)
-      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.source_bound htm (Y z))) :
-        MemLp S 2 μ).integrable (by norm_num)
-  have hDi:Integrable D μ := (hnext.sub hcur).sub (hSi.const_mul δ)
-  have hRi:Integrable R μ := (hDi.sub hPi).sub hQi
-  have hEi0 := (cubicEnvelope_memLp_two (coordinate_hasLaw N j)
-    (h.gamma_nonneg T ⟨ht.trans htT,le_rfl⟩) K ((1/2+γ T)*L)).integrable (by norm_num)
-  have hEi:Integrable E μ := (hEi0.const_mul _).add (integrable_const _)
-  have hRE (z) : |R z|≤E z := by
-    have hh := h.euler_one_step ht hδ hδ1 htδ (x:=Y z) (z:=coordinate N j z)
-    convert! hh using 1
-    dsimp only [R,D,P,Q,p,a,Y',S,E,rawIncrement]
-    congr 1
-    ring_nf
-  have hRmean:(∫ z,R z ∂μ)=(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)-δ*(∫ z,S z ∂μ) := by
-    have hs := integral_sub (hDi.sub hPi) hQi
-    change (∫ z,D z-P z-Q z ∂μ)=(∫ z,D z-P z ∂μ)-(∫ z,Q z ∂μ) at hs
-    have hs1 := integral_sub hDi hPi
-    change (∫ z,D z-P z ∂μ)=(∫ z,D z ∂μ)-(∫ z,P z ∂μ) at hs1
-    change (∫ z,D z-P z-Q z ∂μ)=_
-    rw [hs,hs1,hP0,hQ0,sub_zero,sub_zero]
-    have hs2 := integral_sub (hnext.sub hcur) (hSi.const_mul δ)
-    change (∫ z,V (t+δ) (Y' z)-V t (Y z)-δ*S z ∂μ)=
-      (∫ z,V (t+δ) (Y' z)-V t (Y z) ∂μ)-(∫ z,δ*S z ∂μ) at hs2
-    dsimp only [D]
-    rw [hs2,integral_sub hnext hcur,integral_const_mul]
-  have hEmean:(∫ z,E z ∂μ)=
-      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
-    dsimp only [E]
-    rw [integral_add (hEi0.const_mul _) (integrable_const _),integral_const_mul]
-    simp only [integral_const,probReal_univ,smul_eq_mul,one_mul]
-    have he:(∫ z,cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z) ∂μ)=
-        cubicEnvelopeMean (γ T) K ((1/2+γ T)*L) := by
-      simpa only [Function.comp_def,cubicEnvelopeMean,standardGaussian] using
-        (coordinate_hasLaw N j).integral_comp (measurable_cubicEnvelope (γ T) K ((1/2+γ T)*L)).aestronglyMeasurable
-    rw [he]
-  change |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)-δ*(∫ z,forcedSource γ u V b₀ p₀ t (Y z) ∂μ)|≤_
-  rw [←hRmean,←hEmean]
-  exact abs_integral_le_integral_abs.trans (integral_mono hRi.norm hEi hRE)
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators NNReal
-namespace SKValue
-lemma ForcedTest.mesh_expectation {T K L:ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
-    (h:ForcedTest T γ u V b p K L) (hT:0<T) (hT1:T≤1) {N:ℕ} (hN:0<N) :
-    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-V 0 0-stepSize T N*∑ j∈Finset.range N,
-      (∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1)))|≤
-      T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
-        stepSize T N*K*(γ T-γ 0) := by
-  let δ := stepSize T N
-  let A := fun j ↦ ∫ z,V (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))
-  let S := fun j ↦ ∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))
-  have hNr:0<(N:ℝ) := by exact_mod_cast hN
-  have hδ:0≤δ := div_nonneg hT.le hNr.le
-  have hδ1:δ≤1 := by
-    apply (div_le_iff₀ hNr).mpr
-    have hn:(1:ℝ)≤N := by exact_mod_cast hN
-    simpa only [one_mul] using hT1.trans hn
-  have htime (j:ℕ) (hj:j≤N) := mesh_time_mem hT.le hN hj
-  have hs (j:ℕ) : meshTime T N j+δ=meshTime T N (j+1) := by dsimp [meshTime];push_cast;ring
-  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
-  have hzero:meshTime T N 0=0 := by simp [meshTime]
-  have hum (j:ℕ) (hj:j<N) : Measurable (u (meshTime T N j)) := h.drift_measurable _ (htime j hj.le)
-  have hstep (j:ℕ) (hj:j<N) : |A (j+1)-A j+(-δ*S j)|≤
-      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
-        δ*K*(γ (meshTime T N (j+1))-γ (meshTime T N j))+0*((0:ℝ)-0) := by
-    have hYp:DependsBefore j (fun z ↦ euler T N γ u z j) :=
-      eulerCoefficient_dependsBefore T hj.le γ u (fun _ y ↦ y)
-    have he := h.expectation_step (htime j hj.le).1 hδ hδ1
-      (by rw [hs];exact (htime (j+1) (by omega)).2) (by omega:j<N+1)
-      (measurable_euler T N γ u hum j hj.le) hYp
-    dsimp only at he
-    rw [hs] at he
-    simpa only [A,S,euler,δ,mul_assoc,add_zero,sub_self,mul_zero,sub_eq_add_neg,neg_mul] using he
-  have hh := abs_telescope_source (A:=A) (b:=fun j ↦ -δ*S j) (Γ:=fun j ↦ γ (meshTime T N j))
-    (Δ:=fun _ ↦ 0) (e:=0) hstep
-  have hA0:A 0=V 0 0 := by simp [A,meshTime,euler]
-  rw [hA0,show A N=(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))) by
-    dsimp only [A];rw [hlast],hlast,hzero] at hh
-  have hNd:(N:ℝ)*δ=T := by dsimp [δ,stepSize];field_simp
-  simpa only [neg_mul,Finset.sum_neg_distrib,←Finset.mul_sum,sub_self,mul_zero,add_zero,←mul_assoc,hNd,←sub_eq_add_neg,S] using hh
-
-lemma ForcedTest.terminal_lipschitz {T K L:ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
-    (h:ForcedTest T γ u V b p K L) (hT:0≤T) (x y:ℝ) : |V T x-V T y|≤K*|x-y| := by
-  have hd := (h.smooth T ⟨hT,le_rfl⟩).differentiable (by norm_num)
-  have hb:∀ z,‖deriv (V T) z‖≤K := fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound T ⟨hT,le_rfl⟩ z
-  have hl : LipschitzWith ⟨K,h.K_nonneg⟩ (V T) := by
-    apply lipschitzWith_of_nnnorm_deriv_le hd
-    intro z
-    exact_mod_cast hb z
-  have hh := hl.dist_le_mul x y
-  change |V T x-V T y|≤K*|x-y| at hh
-  exact hh
-
-lemma ForcedTest.coupled_terminal_error {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
-    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
-    {X:ℝ → Ω → ℝ} {T K L K' L':ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
-    (hT:0<T) (hu:GradientStrip T γ u K L) (h:ForcedTest T γ u V b p K' L')
-    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
-    (hpaths:∀ᵐ ω ∂μ,(∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+
-      ∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) {N:ℕ} (hN:0<N) :
-    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-
-      (∫ ω,V T (X T ω) ∂μ)|≤K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ) := by
-  let F := fun ω ↦ V T (coupledEuler T γ u B N N ω)
-  let G := fun ω ↦ V T (X T ω)
-  let E := meshMaxError T X (coupledEuler T γ u B) N
-  have ht:T∈Icc (0:ℝ) T := ⟨hT.le,le_rfl⟩
-  have hm:Continuous (V T) := (h.smooth T ht).continuous
-  have hFM:MemLp F 2 μ := MemLp.of_bound
-    (hm.measurable.comp_aemeasurable (coupled_euler_measurable hB hT hu N N le_rfl).aemeasurable).aestronglyMeasurable K'
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,F] using h.bound T ht _))
-  have hGM:MemLp G 2 μ := MemLp.of_bound
-    (hm.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable K'
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,G] using h.bound T ht _))
-  have hEM:MemLp E 2 μ := coupled_mesh_error_memLp hB hT hu hXM hpaths hN
-  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
-  have hFG (ω:Ω) : |F ω-G ω|≤K'*|E ω| := by
-    have he:|coupledEuler T γ u B N N ω-X T ω|≤E ω := by
-      have hh := le_initialMax (f:=fun k ↦
-        |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
-      change |coupledEuler T γ u B N N ω-X (meshTime T N N) ω|≤E ω at hh
-      rwa [hlast] at hh
-    exact (h.terminal_lipschitz hT.le _ _).trans
-      (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) h.K_nonneg)
-  have heq:(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))=(∫ ω,F ω ∂μ) := by
-    have heum := measurable_euler T N γ u (fun j hj ↦ (hu.smooth _ (mesh_time_mem hT.le hN hj.le)).continuous.measurable) N le_rfl
-    have hh := (coupledCoordinates_hasLaw hB hT hN).integral_comp (hm.measurable.comp heum).aestronglyMeasurable
-    simpa only [Function.comp_apply,F,coupledEuler,ite_eq_right (Nat.ne_of_gt hN)] using hh.symm
-  rw [heq]
-  exact first_moment_perturbation hFM hGM hEM h.K_nonneg (Eventually.of_forall hFG)
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.CrossMesh
+import OAI.Probability.SKValue.Control.Forced
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators NNReal
+namespace SKValue
+lemma ForcedTest.expectation_step {T K L t δ:ℝ} {γ:ℝ → ℝ} {u V b₀ p₀:ℝ → ℝ → ℝ}
+    (h:ForcedTest T γ u V b₀ p₀ K L) (ht:0≤t) (hδ:0≤δ) (hδ1:δ≤1) (htδ:t+δ≤T)
+    {N j:ℕ} (hj:j<N+1) {Y:(Fin (N+1) → ℝ) → ℝ}
+    (hYm:Measurable Y) (hYp:DependsBefore j Y) :
+    let μ := gaussianProduct (Fin (N+1))
+    let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
+    |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)-δ*(∫ z,forcedSource γ u V b₀ p₀ t (Y z) ∂μ)|≤
+      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
+  let μ := gaussianProduct (Fin (N+1))
+  let p := fun z ↦ deriv (V t) (Y z)
+  let a := fun z ↦ deriv (deriv (V t)) (Y z)
+  let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
+  let P := fun z ↦ Real.sqrt δ*rawIncrement p j z
+  let Q := fun z ↦ (δ/2)*(a z*((coordinate N j z)^2-1))
+  let S := fun z ↦ forcedSource γ u V b₀ p₀ t (Y z)
+  let D := fun z ↦ V (t+δ) (Y' z)-V t (Y z)-δ*S z
+  let R := fun z ↦ D z-P z-Q z
+  let E := fun z ↦ δ*Real.sqrt δ*cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z)+δ*K*(γ (t+δ)-γ t)
+  have htT:t≤T := (le_add_of_nonneg_right hδ).trans htδ
+  have htm:t∈Icc (0:ℝ) T := ⟨ht,htT⟩
+  have hem:t+δ∈Icc (0:ℝ) T := ⟨by linarith,htδ⟩
+  have hpm:Measurable p := ((h.smooth t htm).continuous_deriv (by norm_num)).measurable.comp hYm
+  have ham:Measurable a := by
+    have hh := (h.smooth t htm).continuous_iteratedDeriv 2 (by norm_num)
+    simp only [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ] at hh
+    exact hh.measurable.comp hYm
+  have hpM:MemLp p 2 μ := MemLp.of_bound hpm.aestronglyMeasurable K
+    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound t htm (Y z)))
+  have haM:MemLp a 2 μ := MemLp.of_bound ham.aestronglyMeasurable K
+    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.second_bound t htm (Y z)))
+  have hpp:DependsBefore j p := fun z z' hz ↦ by dsimp only [p];rw [hYp z z' hz]
+  have hap:DependsBefore j a := fun z z' hz ↦ by dsimp only [a];rw [hYp z z' hz]
+  have hPi:Integrable P μ := (((rawIncrement_memLp hj hpm hpp hpM).const_mul _).integrable (by norm_num))
+  have hQi:Integrable Q μ := (((predictableCenteredSquare_memLp ham (fun z ↦ h.second_bound t htm (Y z))).const_mul _).integrable (by norm_num))
+  have hP0:(∫ z,P z ∂μ)=0 := by
+    dsimp only [P]
+    rw [integral_const_mul,rawIncrement_mean hj hpm hpp,mul_zero]
+  have hQ0:(∫ z,Q z ∂μ)=0 := by
+    dsimp only [Q]
+    rw [integral_const_mul,predictableCenteredSquare_mean hj ham hap,mul_zero]
+  have hY'm:Measurable Y' := (hYm.add ((measurable_coordinate N j).const_mul _)).add
+    (((h.drift_measurable t htm).comp hYm).const_mul _)
+  have hcur:Integrable (fun z ↦ V t (Y z)) μ :=
+    (MemLp.of_bound ((h.smooth t htm).continuous.measurable.comp hYm).aestronglyMeasurable K
+      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound t htm (Y z))) :
+        MemLp (fun z ↦ V t (Y z)) 2 μ).integrable (by norm_num)
+  have hnext:Integrable (fun z ↦ V (t+δ) (Y' z)) μ :=
+    (MemLp.of_bound ((h.smooth (t+δ) hem).continuous.measurable.comp hY'm).aestronglyMeasurable K
+      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound (t+δ) hem (Y' z))) :
+        MemLp (fun z ↦ V (t+δ) (Y' z)) 2 μ).integrable (by norm_num)
+  have hSi:Integrable S μ :=
+    (MemLp.of_bound ((h.source_measurable htm).comp hYm).aestronglyMeasurable ((1+2*γ T)*K)
+      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.source_bound htm (Y z))) :
+        MemLp S 2 μ).integrable (by norm_num)
+  have hDi:Integrable D μ := (hnext.sub hcur).sub (hSi.const_mul δ)
+  have hRi:Integrable R μ := (hDi.sub hPi).sub hQi
+  have hEi0 := (cubicEnvelope_memLp_two (coordinate_hasLaw N j)
+    (h.gamma_nonneg T ⟨ht.trans htT,le_rfl⟩) K ((1/2+γ T)*L)).integrable (by norm_num)
+  have hEi:Integrable E μ := (hEi0.const_mul _).add (integrable_const _)
+  have hRE (z) : |R z|≤E z := by
+    have hh := h.euler_one_step ht hδ hδ1 htδ (x:=Y z) (z:=coordinate N j z)
+    convert! hh using 1
+    dsimp only [R,D,P,Q,p,a,Y',S,E,rawIncrement]
+    congr 1
+    ring_nf
+  have hRmean:(∫ z,R z ∂μ)=(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)-δ*(∫ z,S z ∂μ) := by
+    have hs := integral_sub (hDi.sub hPi) hQi
+    change (∫ z,D z-P z-Q z ∂μ)=(∫ z,D z-P z ∂μ)-(∫ z,Q z ∂μ) at hs
+    have hs1 := integral_sub hDi hPi
+    change (∫ z,D z-P z ∂μ)=(∫ z,D z ∂μ)-(∫ z,P z ∂μ) at hs1
+    change (∫ z,D z-P z-Q z ∂μ)=_
+    rw [hs,hs1,hP0,hQ0,sub_zero,sub_zero]
+    have hs2 := integral_sub (hnext.sub hcur) (hSi.const_mul δ)
+    change (∫ z,V (t+δ) (Y' z)-V t (Y z)-δ*S z ∂μ)=
+      (∫ z,V (t+δ) (Y' z)-V t (Y z) ∂μ)-(∫ z,δ*S z ∂μ) at hs2
+    dsimp only [D]
+    rw [hs2,integral_sub hnext hcur,integral_const_mul]
+  have hEmean:(∫ z,E z ∂μ)=
+      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
+    dsimp only [E]
+    rw [integral_add (hEi0.const_mul _) (integrable_const _),integral_const_mul]
+    simp only [integral_const,probReal_univ,smul_eq_mul,one_mul]
+    have he:(∫ z,cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z) ∂μ)=
+        cubicEnvelopeMean (γ T) K ((1/2+γ T)*L) := by
+      simpa only [Function.comp_def,cubicEnvelopeMean,standardGaussian] using
+        (coordinate_hasLaw N j).integral_comp (measurable_cubicEnvelope (γ T) K ((1/2+γ T)*L)).aestronglyMeasurable
+    rw [he]
+  change |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)-δ*(∫ z,forcedSource γ u V b₀ p₀ t (Y z) ∂μ)|≤_
+  rw [←hRmean,←hEmean]
+  exact abs_integral_le_integral_abs.trans (integral_mono hRi.norm hEi hRE)
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators NNReal
+namespace SKValue
+lemma ForcedTest.mesh_expectation {T K L:ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
+    (h:ForcedTest T γ u V b p K L) (hT:0<T) (hT1:T≤1) {N:ℕ} (hN:0<N) :
+    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-V 0 0-stepSize T N*∑ j∈Finset.range N,
+      (∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1)))|≤
+      T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
+        stepSize T N*K*(γ T-γ 0) := by
+  let δ := stepSize T N
+  let A := fun j ↦ ∫ z,V (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))
+  let S := fun j ↦ ∫ z,forcedSource γ u V b p (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))
+  have hNr:0<(N:ℝ) := by exact_mod_cast hN
+  have hδ:0≤δ := div_nonneg hT.le hNr.le
+  have hδ1:δ≤1 := by
+    apply (div_le_iff₀ hNr).mpr
+    have hn:(1:ℝ)≤N := by exact_mod_cast hN
+    simpa only [one_mul] using hT1.trans hn
+  have htime (j:ℕ) (hj:j≤N) := mesh_time_mem hT.le hN hj
+  have hs (j:ℕ) : meshTime T N j+δ=meshTime T N (j+1) := by dsimp [meshTime];push_cast;ring
+  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
+  have hzero:meshTime T N 0=0 := by simp [meshTime]
+  have hum (j:ℕ) (hj:j<N) : Measurable (u (meshTime T N j)) := h.drift_measurable _ (htime j hj.le)
+  have hstep (j:ℕ) (hj:j<N) : |A (j+1)-A j+(-δ*S j)|≤
+      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
+        δ*K*(γ (meshTime T N (j+1))-γ (meshTime T N j))+0*((0:ℝ)-0) := by
+    have hYp:DependsBefore j (fun z ↦ euler T N γ u z j) :=
+      eulerCoefficient_dependsBefore T hj.le γ u (fun _ y ↦ y)
+    have he := h.expectation_step (htime j hj.le).1 hδ hδ1
+      (by rw [hs];exact (htime (j+1) (by omega)).2) (by omega:j<N+1)
+      (measurable_euler T N γ u hum j hj.le) hYp
+    dsimp only at he
+    rw [hs] at he
+    simpa only [A,S,euler,δ,mul_assoc,add_zero,sub_self,mul_zero,sub_eq_add_neg,neg_mul] using he
+  have hh := abs_telescope_source (A:=A) (b:=fun j ↦ -δ*S j) (Γ:=fun j ↦ γ (meshTime T N j))
+    (Δ:=fun _ ↦ 0) (e:=0) hstep
+  have hA0:A 0=V 0 0 := by simp [A,meshTime,euler]
+  rw [hA0,show A N=(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))) by
+    dsimp only [A];rw [hlast],hlast,hzero] at hh
+  have hNd:(N:ℝ)*δ=T := by dsimp [δ,stepSize];field_simp
+  simpa only [neg_mul,Finset.sum_neg_distrib,←Finset.mul_sum,sub_self,mul_zero,add_zero,←mul_assoc,hNd,←sub_eq_add_neg,S] using hh
+
+lemma ForcedTest.terminal_lipschitz {T K L:ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
+    (h:ForcedTest T γ u V b p K L) (hT:0≤T) (x y:ℝ) : |V T x-V T y|≤K*|x-y| := by
+  have hd := (h.smooth T ⟨hT,le_rfl⟩).differentiable (by norm_num)
+  have hb:∀ z,‖deriv (V T) z‖≤K := fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound T ⟨hT,le_rfl⟩ z
+  have hl : LipschitzWith ⟨K,h.K_nonneg⟩ (V T) := by
+    apply lipschitzWith_of_nnnorm_deriv_le hd
+    intro z
+    exact_mod_cast hb z
+  have hh := hl.dist_le_mul x y
+  change |V T x-V T y|≤K*|x-y| at hh
+  exact hh
+
+lemma ForcedTest.coupled_terminal_error {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
+    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
+    {X:ℝ → Ω → ℝ} {T K L K' L':ℝ} {γ:ℝ → ℝ} {u V b p:ℝ → ℝ → ℝ}
+    (hT:0<T) (hu:GradientStrip T γ u K L) (h:ForcedTest T γ u V b p K' L')
+    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
+    (hpaths:∀ᵐ ω ∂μ,(∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+
+      ∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) {N:ℕ} (hN:0<N) :
+    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-
+      (∫ ω,V T (X T ω) ∂μ)|≤K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ) := by
+  let F := fun ω ↦ V T (coupledEuler T γ u B N N ω)
+  let G := fun ω ↦ V T (X T ω)
+  let E := meshMaxError T X (coupledEuler T γ u B) N
+  have ht:T∈Icc (0:ℝ) T := ⟨hT.le,le_rfl⟩
+  have hm:Continuous (V T) := (h.smooth T ht).continuous
+  have hFM:MemLp F 2 μ := MemLp.of_bound
+    (hm.measurable.comp_aemeasurable (coupled_euler_measurable hB hT hu N N le_rfl).aemeasurable).aestronglyMeasurable K'
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,F] using h.bound T ht _))
+  have hGM:MemLp G 2 μ := MemLp.of_bound
+    (hm.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable K'
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,G] using h.bound T ht _))
+  have hEM:MemLp E 2 μ := coupled_mesh_error_memLp hB hT hu hXM hpaths hN
+  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
+  have hFG (ω:Ω) : |F ω-G ω|≤K'*|E ω| := by
+    have he:|coupledEuler T γ u B N N ω-X T ω|≤E ω := by
+      have hh := le_initialMax (f:=fun k ↦
+        |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
+      change |coupledEuler T γ u B N N ω-X (meshTime T N N) ω|≤E ω at hh
+      rwa [hlast] at hh
+    exact (h.terminal_lipschitz hT.le _ _).trans
+      (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) h.K_nonneg)
+  have heq:(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))=(∫ ω,F ω ∂μ) := by
+    have heum := measurable_euler T N γ u (fun j hj ↦ (hu.smooth _ (mesh_time_mem hT.le hN hj.le)).continuous.measurable) N le_rfl
+    have hh := (coupledCoordinates_hasLaw hB hT hN).integral_comp (hm.measurable.comp heum).aestronglyMeasurable
+    simpa only [Function.comp_apply,F,coupledEuler,ite_eq_right (Nat.ne_of_gt hN)] using hh.symm
+  rw [heq]
+  exact first_moment_perturbation hFM hGM hEM h.K_nonneg (Eventually.of_forall hFG)
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/GradientSquareIntegral.lean b/lean/OAI/Probability/SKValue/Processes/GradientSquareIntegral.lean
index a63e5a3768596aaaed7c32ea971319add293f3d8..a82b5df53758393cbcd605c6a445773e50290818 100644
--- a/lean/OAI/Probability/SKValue/Processes/GradientSquareIntegral.lean
+++ b/lean/OAI/Probability/SKValue/Processes/GradientSquareIntegral.lean
@@ -1,96 +1,96 @@
-import Mathlib
-import OAI.Probability.SKValue.Control.StripRegularity
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal
-
-lemma IsDiffusion.gradient_square_integral {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (hreg : SourceStripRegularity W γ)
-    {T : ℝ} (hT : 0<T) (hT1 : T<1) :
-    (∫ ω, (gradient W γ T (X T ω))^2 ∂W.μ)=
-      ∫ t in (0 : ℝ)..T, ∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ := by
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨hT.le,hT1⟩
-  exact diffusion_gradient_isometry W.brownian.toIsPreBrownianReal hT hT1.le hG
-    hD hLu hLa hDb hu ha
-    (fun t ht ↦ (hX.measurable ⟨ht.1,ht.2.trans hT1.le⟩).aestronglyMeasurable)
-    (hX.strip_paths hT.le hT1 hLu (fun t ht s hs x y ↦ hu s hs t ht y x))
-
-lemma IsDiffusion.curvature_square_mean_continuous {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (hreg : SourceStripRegularity W γ) :
-    ContinuousOn (fun t ↦ ∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ) (Ico (0 : ℝ) 1) := by
-  intro t ht
-  obtain ⟨T,htT,hT1⟩ := exists_between ht.2
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨ht.1.trans htT.le,hT1⟩
-  have hc := mean_derivative_square_continuous hD hLa
-    (fun s hs ↦ (hG.smooth s hs).continuous_deriv (by norm_num)) hDb ha
-    (fun s hs ↦ (hX.measurable ⟨hs.1,hs.2.trans hT1.le⟩).aestronglyMeasurable)
-    (hX.2.mono (fun _ hω ↦ hω.1.mono (Icc_subset_Icc le_rfl hT1.le)))
-  apply (hc t ⟨ht.1,htT.le⟩).mono_of_mem_nhdsWithin
-  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds htT)] with s hs hsT
-  exact ⟨hs.1,hsT.le⟩
-
-lemma continuous_eq_one_of_primitive {f : ℝ → ℝ}
-    (hc : ContinuousOn f (Ico (0 : ℝ) 1))
-    (hi : ∀ t∈Ioo (0 : ℝ) 1, (∫ s in (0 : ℝ)..t, f s)=t) :
-    ∀ t∈Ico (0 : ℝ) 1, f t=1 := by
-  have hint (t : ℝ) (ht : t∈Ioo (0 : ℝ) 1) : f t=1 := by
-    have hf : IntervalIntegrable f volume 0 t := by
-      apply ContinuousOn.intervalIntegrable
-      rw [uIcc_of_le ht.1.le]
-      exact hc.mono (fun s hs ↦ ⟨hs.1,hs.2.trans_lt ht.2⟩)
-    have hs : Ico (0 : ℝ) 1∈𝓝 t := Ico_mem_nhds ht.1 ht.2
-    have hd := intervalIntegral.integral_hasDerivAt_right hf
-      ⟨Ico (0 : ℝ) 1,hs,hc.aestronglyMeasurable measurableSet_Ico⟩
-      ((hc t ⟨ht.1.le,ht.2⟩).continuousAt hs)
-    have he : (fun t ↦ ∫ s in (0 : ℝ)..t, f s)=ᶠ[𝓝 t] (id : ℝ → ℝ) := by
-      filter_upwards [Ioo_mem_nhds ht.1 ht.2] with s hs
-      exact hi s hs
-    exact hd.unique ((hasDerivAt_id t).congr_of_eventuallyEq he)
-  intro t ht
-  obtain h0|h0 := ht.1.eq_or_lt
-  · subst t
-    have hlim : Tendsto f (𝓝[>] (0 : ℝ)) (𝓝 (f 0)) := by
-      apply (hc 0 (by norm_num)).mono_left
-      apply le_inf nhdsWithin_le_nhds
-      apply le_principal_iff.mpr
-      filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds (by norm_num : (0 : ℝ)<1))] with s hs hs1
-      exact ⟨hs.le,hs1⟩
-    apply tendsto_nhds_unique hlim
-    apply tendsto_const_nhds.congr'
-    filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds (by norm_num : (0 : ℝ)<1))] with s hs hs1
-    exact (hint s ⟨hs,hs1⟩).symm
-  · exact hint t ⟨h0,ht.2⟩
-
-lemma IsDiffusion.curvature_unit_moment_of_contact {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (hreg : SourceStripRegularity W γ)
-    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω, (gradient W γ t (X t ω))^2 ∂W.μ)=t) :
-    ∀ t∈Ico (0 : ℝ) 1, (∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ)=1 := by
-  apply continuous_eq_one_of_primitive (hX.curvature_square_mean_continuous hreg)
-  intro t ht
-  rw [←hX.gradient_square_integral hreg ht.1 ht.2,hmom t ⟨ht.1.le,ht.2⟩]
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Control.StripRegularity
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal
+
+lemma IsDiffusion.gradient_square_integral {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (hreg : SourceStripRegularity W γ)
+    {T : ℝ} (hT : 0<T) (hT1 : T<1) :
+    (∫ ω, (gradient W γ T (X T ω))^2 ∂W.μ)=
+      ∫ t in (0 : ℝ)..T, ∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ := by
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨hT.le,hT1⟩
+  exact diffusion_gradient_isometry W.brownian.toIsPreBrownianReal hT hT1.le hG
+    hD hLu hLa hDb hu ha
+    (fun t ht ↦ (hX.measurable ⟨ht.1,ht.2.trans hT1.le⟩).aestronglyMeasurable)
+    (hX.strip_paths hT.le hT1 hLu (fun t ht s hs x y ↦ hu s hs t ht y x))
+
+lemma IsDiffusion.curvature_square_mean_continuous {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (hreg : SourceStripRegularity W γ) :
+    ContinuousOn (fun t ↦ ∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ) (Ico (0 : ℝ) 1) := by
+  intro t ht
+  obtain ⟨T,htT,hT1⟩ := exists_between ht.2
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨ht.1.trans htT.le,hT1⟩
+  have hc := mean_derivative_square_continuous hD hLa
+    (fun s hs ↦ (hG.smooth s hs).continuous_deriv (by norm_num)) hDb ha
+    (fun s hs ↦ (hX.measurable ⟨hs.1,hs.2.trans hT1.le⟩).aestronglyMeasurable)
+    (hX.2.mono (fun _ hω ↦ hω.1.mono (Icc_subset_Icc le_rfl hT1.le)))
+  apply (hc t ⟨ht.1,htT.le⟩).mono_of_mem_nhdsWithin
+  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds htT)] with s hs hsT
+  exact ⟨hs.1,hsT.le⟩
+
+lemma continuous_eq_one_of_primitive {f : ℝ → ℝ}
+    (hc : ContinuousOn f (Ico (0 : ℝ) 1))
+    (hi : ∀ t∈Ioo (0 : ℝ) 1, (∫ s in (0 : ℝ)..t, f s)=t) :
+    ∀ t∈Ico (0 : ℝ) 1, f t=1 := by
+  have hint (t : ℝ) (ht : t∈Ioo (0 : ℝ) 1) : f t=1 := by
+    have hf : IntervalIntegrable f volume 0 t := by
+      apply ContinuousOn.intervalIntegrable
+      rw [uIcc_of_le ht.1.le]
+      exact hc.mono (fun s hs ↦ ⟨hs.1,hs.2.trans_lt ht.2⟩)
+    have hs : Ico (0 : ℝ) 1∈𝓝 t := Ico_mem_nhds ht.1 ht.2
+    have hd := intervalIntegral.integral_hasDerivAt_right hf
+      ⟨Ico (0 : ℝ) 1,hs,hc.aestronglyMeasurable measurableSet_Ico⟩
+      ((hc t ⟨ht.1.le,ht.2⟩).continuousAt hs)
+    have he : (fun t ↦ ∫ s in (0 : ℝ)..t, f s)=ᶠ[𝓝 t] (id : ℝ → ℝ) := by
+      filter_upwards [Ioo_mem_nhds ht.1 ht.2] with s hs
+      exact hi s hs
+    exact hd.unique ((hasDerivAt_id t).congr_of_eventuallyEq he)
+  intro t ht
+  obtain h0|h0 := ht.1.eq_or_lt
+  · subst t
+    have hlim : Tendsto f (𝓝[>] (0 : ℝ)) (𝓝 (f 0)) := by
+      apply (hc 0 (by norm_num)).mono_left
+      apply le_inf nhdsWithin_le_nhds
+      apply le_principal_iff.mpr
+      filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds (by norm_num : (0 : ℝ)<1))] with s hs hs1
+      exact ⟨hs.le,hs1⟩
+    apply tendsto_nhds_unique hlim
+    apply tendsto_const_nhds.congr'
+    filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds (by norm_num : (0 : ℝ)<1))] with s hs hs1
+    exact (hint s ⟨hs,hs1⟩).symm
+  · exact hint t ⟨h0,ht.2⟩
+
+lemma IsDiffusion.curvature_unit_moment_of_contact {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (hreg : SourceStripRegularity W γ)
+    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω, (gradient W γ t (X t ω))^2 ∂W.μ)=t) :
+    ∀ t∈Ico (0 : ℝ) 1, (∫ ω, (curvature W γ t (X t ω))^2 ∂W.μ)=1 := by
+  apply continuous_eq_one_of_primitive (hX.curvature_square_mean_continuous hreg)
+  intro t ht
+  rw [←hX.gradient_square_integral hreg ht.1 ht.2,hmom t ⟨ht.1.le,ht.2⟩]
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/Measurable.lean b/lean/OAI/Probability/SKValue/Processes/Measurable.lean
index 20805245f1d1fe49fbb6235166d57d1fb4caba76..a0cb912aa59fe9ff66000b05103424ced81128db 100644
--- a/lean/OAI/Probability/SKValue/Processes/Measurable.lean
+++ b/lean/OAI/Probability/SKValue/Processes/Measurable.lean
@@ -1,147 +1,147 @@
-import Mathlib
-import OAI.Probability.SKValue.Evolution.TerminalError
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-
-lemma IsDiffusion.measurable {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {t : ℝ} (ht : t∈Icc (0 : ℝ) 1) :
-    Measurable (X t) := by
-  simpa only [Real.coe_toNNReal t ht.1, min_eq_left ht.2] using
-    (hX.1 t.toNNReal).mono (W.filtration.le t.toNNReal) le_rfl
-
-lemma drift_integral_bound (W : BrownianSpace) (γ : OrderParameter) {t : ℝ}
-    (ht : t∈Icc (0 : ℝ) 1) (x : ℝ → ℝ) :
-    |∫ s in (0 : ℝ)..t, γ.coeff s*gradient W γ s (x s)|≤∫ s in (0 : ℝ)..1, γ.cutoff s := by
-  have he : (∫ s in (0 : ℝ)..t, γ.coeff s*gradient W γ s (x s)) =
-      ∫ s in (0 : ℝ)..t, γ.cutoff s*gradient W γ s (x s) := by
-    apply intervalIntegral.integral_congr_Ioo_of_le ht.1
-    intro s hs
-    change γ.coeff s*gradient W γ s (x s)=γ.cutoff s*gradient W γ s (x s)
-    have hs' : s∈Ico (0 : ℝ) 1 := ⟨hs.1.le,hs.2.trans_le ht.2⟩
-    rw [OrderParameter.cutoff, Set.indicator_of_mem hs']
-  rw [he]
-  have hb := intervalIntegral.norm_integral_le_of_norm_le ht.1
-    (f := fun s ↦ γ.cutoff s*gradient W γ s (x s))
-    (Eventually.of_forall (fun s hs ↦ by
-      rw [Real.norm_eq_abs,abs_mul,abs_of_nonneg (γ.cutoff_nonneg s)]
-      exact (mul_le_mul_of_nonneg_left (gradient_bound W γ ⟨hs.1.le,hs.2.trans ht.2⟩ (x s))
-        (γ.cutoff_nonneg s)).trans_eq (mul_one _)))
-    (γ.cutoff_integrable.intervalIntegrable (a := 0) (b := t))
-  have hm := intervalIntegral.integral_mono_interval (c := (0 : ℝ)) (d := 1) le_rfl ht.1 ht.2
-    (Eventually.of_forall (fun s ↦ γ.cutoff_nonneg s)) γ.cutoff_integrable.intervalIntegrable
-  rw [Real.norm_eq_abs] at hb
-  exact hb.trans hm
-
-lemma IsDiffusion.drift_bound {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
-    ∀ᵐ ω ∂W.μ, ∀ t∈Icc (0 : ℝ) 1,
-      |X t ω-W.B t.toNNReal ω|≤∫ s in (0 : ℝ)..1, γ.cutoff s := by
-  filter_upwards [hX.2] with ω hω t ht
-  rw [hω.2 t ht, add_sub_cancel_left]
-  exact drift_integral_bound W γ ht (fun s ↦ X s ω)
-
-lemma IsDiffusion.memLp {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {t : ℝ} (ht : t∈Icc (0 : ℝ) 1)
-    {p : ℝ≥0∞} (hp : p≠∞) : MemLp (X t) p W.μ := by
-  have hd : MemLp (fun ω ↦ X t ω-W.B t.toNNReal ω) p W.μ :=
-    MemLp.of_bound ((hX.measurable ht).sub (W.measurable _).measurable).aestronglyMeasurable
-      (∫ s in (0 : ℝ)..1, γ.cutoff s) (hX.drift_bound.mono (fun ω hω ↦ by
-        simpa only [Real.norm_eq_abs] using hω t ht))
-  have hb := (W.brownian.hasLaw_eval t.toNNReal).hasGaussianLaw.memLp hp
-  convert hd.add hb using 1
-  funext ω
-  simp only [Pi.add_apply, sub_add_cancel]
-
-lemma eventually_time_mem : ∀ᶠ t : ℝ in 𝓝[<] 1, t∈Icc (0 : ℝ) 1 := by
-  filter_upwards [Ico_mem_nhdsLT (by norm_num : (0 : ℝ)<1)] with t ht
-  exact ⟨ht.1,ht.2.le⟩
-
-lemma IsDiffusion.ae_tendsto {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
-    ∀ᵐ ω ∂W.μ, Tendsto (fun t ↦ X t ω) (𝓝[<] (1 : ℝ)) (𝓝 (X 1 ω)) := by
-  have ht : Tendsto (id : ℝ → ℝ) (𝓝[<] 1) (𝓝[Icc (0 : ℝ) 1] 1) :=
-    tendsto_nhdsWithin_iff.mpr ⟨nhdsWithin_le_nhds,eventually_time_mem⟩
-  filter_upwards [hX.2] with ω hω
-  exact Tendsto.comp (hω.1 1 (by norm_num : (1 : ℝ)∈Icc (0 : ℝ) 1)) ht
-
-lemma IsDiffusion.L2_terminal {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
-    Tendsto (fun t ↦ ∫ ω, (X t ω-X 1 ω)^2 ∂W.μ) (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
-  let D (t : ℝ) (ω : W.Ω) := X t ω-W.B t.toNNReal ω
-  let G := ∫ s in (0 : ℝ)..1, γ.cutoff s
-  have hG : 0≤G := intervalIntegral.integral_nonneg_of_forall (by norm_num)
-    (fun s ↦ γ.cutoff_nonneg s)
-  have hD (t : ℝ) (ht : t∈Icc (0 : ℝ) 1) : MemLp (D t) 2 W.μ :=
-    (hX.memLp ht (by norm_num)).sub ((W.brownian.hasLaw_eval t.toNNReal).hasGaussianLaw.memLp_two)
-  have hD1 := hD 1 (by norm_num)
-  have hDb : ∀ᵐ ω ∂W.μ, ∀ t∈Icc (0 : ℝ) 1, |D t ω|≤G := hX.drift_bound
-  have hDC : ∀ᵐ ω ∂W.μ, Tendsto (fun t ↦ D t ω) (𝓝[<] (1 : ℝ)) (𝓝 (D 1 ω)) := by
-    filter_upwards [hX.ae_tendsto,W.brownian.cont] with ω hω hBω
-    exact hω.sub ((hBω.comp continuous_real_toNNReal).continuousAt.tendsto.mono_left nhdsWithin_le_nhds)
-  have hDL : Tendsto (fun t ↦ ∫ ω, (D t ω-D 1 ω)^2 ∂W.μ)
-      (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
-    have h := tendsto_integral_filter_of_dominated_convergence
-      (μ := W.μ) (l := 𝓝[<] (1 : ℝ)) (F := fun t ω ↦ (D t ω-D 1 ω)^2) (f := fun _ ↦ (0 : ℝ))
-      (fun _ ↦ 4*G^2) ?_ ?_ (integrable_const (4*G^2)) ?_
-    · simpa only [integral_zero] using h
-    · filter_upwards [eventually_time_mem] with t ht
-      exact ((hD t ht).sub hD1).aestronglyMeasurable.aemeasurable.pow_const 2 |>.aestronglyMeasurable
-    · filter_upwards [eventually_time_mem] with t ht
-      filter_upwards [hDb] with ω hω
-      rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
-      have h0 := abs_le.mp (hω t ht)
-      have h1 := abs_le.mp (hω 1 (by norm_num))
-      nlinarith [sq_nonneg (D t ω-D 1 ω), sq_nonneg (G-(D t ω-D 1 ω)/2),
-        sq_nonneg (G+(D t ω-D 1 ω)/2), mul_nonneg (show 0≤2*G-(D t ω-D 1 ω) by linarith)
-          (show 0≤2*G+(D t ω-D 1 ω) by linarith)]
-    · filter_upwards [hDC] with ω hω
-      simpa only [sub_self, zero_pow (by norm_num : (2 : ℕ)≠0)] using (hω.sub (tendsto_const_nhds (x := D 1 ω))).pow 2
-  have hb : ∀ t∈Icc (0 : ℝ) 1,
-      (∫ ω, (X t ω-X 1 ω)^2 ∂W.μ) ≤
-      2*(∫ ω, (D t ω-D 1 ω)^2 ∂W.μ)+2*(1-t) := by
-    intro t ht
-    have hxint := ((hX.memLp ht (by norm_num : (2 : ℝ≥0∞)≠∞)).sub
-      (hX.memLp (by norm_num : (1 : ℝ)∈Icc (0 : ℝ) 1) (by norm_num))).integrable_sq
-    have hdint : Integrable (fun ω ↦ (D t ω-D 1 ω)^2) W.μ := ((hD t ht).sub hD1).integrable_sq
-    have hbint := (brownian_increment_memLp W t).integrable_sq
-    calc
-      _ ≤ ∫ ω, 2*(D t ω-D 1 ω)^2+2*(W.B 1 ω-W.B t.toNNReal ω)^2 ∂W.μ := by
-        apply integral_mono hxint ((hdint.const_mul 2).add (hbint.const_mul 2))
-        intro ω
-        dsimp only [D,Pi.add_apply,Pi.sub_apply]
-        simp only [Real.toNNReal_one]
-        nlinarith [sq_nonneg (X t ω-W.B t.toNNReal ω-(X 1 ω-W.B 1 ω)+(W.B 1 ω-W.B t.toNNReal ω))]
-      _ = _ := by
-        rw [integral_add (hdint.const_mul 2) (hbint.const_mul 2), integral_const_mul,
-          integral_const_mul, brownian_increment_second W ht]
-  apply squeeze_zero' (Eventually.of_forall (fun t ↦ integral_nonneg (fun ω ↦ sq_nonneg _)))
-    (eventually_time_mem.mono (fun t ht ↦ hb t ht))
-  simpa only [sub_self, mul_zero, add_zero,id_eq] using
-    (hDL.const_mul 2).add (((tendsto_const_nhds (x := (1 : ℝ))).sub (tendsto_id.mono_left nhdsWithin_le_nhds : Tendsto (id : ℝ → ℝ) (𝓝[<] 1) (𝓝 1))).const_mul 2)
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Evolution.TerminalError
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+
+lemma IsDiffusion.measurable {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {t : ℝ} (ht : t∈Icc (0 : ℝ) 1) :
+    Measurable (X t) := by
+  simpa only [Real.coe_toNNReal t ht.1, min_eq_left ht.2] using
+    (hX.1 t.toNNReal).mono (W.filtration.le t.toNNReal) le_rfl
+
+lemma drift_integral_bound (W : BrownianSpace) (γ : OrderParameter) {t : ℝ}
+    (ht : t∈Icc (0 : ℝ) 1) (x : ℝ → ℝ) :
+    |∫ s in (0 : ℝ)..t, γ.coeff s*gradient W γ s (x s)|≤∫ s in (0 : ℝ)..1, γ.cutoff s := by
+  have he : (∫ s in (0 : ℝ)..t, γ.coeff s*gradient W γ s (x s)) =
+      ∫ s in (0 : ℝ)..t, γ.cutoff s*gradient W γ s (x s) := by
+    apply intervalIntegral.integral_congr_Ioo_of_le ht.1
+    intro s hs
+    change γ.coeff s*gradient W γ s (x s)=γ.cutoff s*gradient W γ s (x s)
+    have hs' : s∈Ico (0 : ℝ) 1 := ⟨hs.1.le,hs.2.trans_le ht.2⟩
+    rw [OrderParameter.cutoff, Set.indicator_of_mem hs']
+  rw [he]
+  have hb := intervalIntegral.norm_integral_le_of_norm_le ht.1
+    (f := fun s ↦ γ.cutoff s*gradient W γ s (x s))
+    (Eventually.of_forall (fun s hs ↦ by
+      rw [Real.norm_eq_abs,abs_mul,abs_of_nonneg (γ.cutoff_nonneg s)]
+      exact (mul_le_mul_of_nonneg_left (gradient_bound W γ ⟨hs.1.le,hs.2.trans ht.2⟩ (x s))
+        (γ.cutoff_nonneg s)).trans_eq (mul_one _)))
+    (γ.cutoff_integrable.intervalIntegrable (a := 0) (b := t))
+  have hm := intervalIntegral.integral_mono_interval (c := (0 : ℝ)) (d := 1) le_rfl ht.1 ht.2
+    (Eventually.of_forall (fun s ↦ γ.cutoff_nonneg s)) γ.cutoff_integrable.intervalIntegrable
+  rw [Real.norm_eq_abs] at hb
+  exact hb.trans hm
+
+lemma IsDiffusion.drift_bound {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
+    ∀ᵐ ω ∂W.μ, ∀ t∈Icc (0 : ℝ) 1,
+      |X t ω-W.B t.toNNReal ω|≤∫ s in (0 : ℝ)..1, γ.cutoff s := by
+  filter_upwards [hX.2] with ω hω t ht
+  rw [hω.2 t ht, add_sub_cancel_left]
+  exact drift_integral_bound W γ ht (fun s ↦ X s ω)
+
+lemma IsDiffusion.memLp {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {t : ℝ} (ht : t∈Icc (0 : ℝ) 1)
+    {p : ℝ≥0∞} (hp : p≠∞) : MemLp (X t) p W.μ := by
+  have hd : MemLp (fun ω ↦ X t ω-W.B t.toNNReal ω) p W.μ :=
+    MemLp.of_bound ((hX.measurable ht).sub (W.measurable _).measurable).aestronglyMeasurable
+      (∫ s in (0 : ℝ)..1, γ.cutoff s) (hX.drift_bound.mono (fun ω hω ↦ by
+        simpa only [Real.norm_eq_abs] using hω t ht))
+  have hb := (W.brownian.hasLaw_eval t.toNNReal).hasGaussianLaw.memLp hp
+  convert hd.add hb using 1
+  funext ω
+  simp only [Pi.add_apply, sub_add_cancel]
+
+lemma eventually_time_mem : ∀ᶠ t : ℝ in 𝓝[<] 1, t∈Icc (0 : ℝ) 1 := by
+  filter_upwards [Ico_mem_nhdsLT (by norm_num : (0 : ℝ)<1)] with t ht
+  exact ⟨ht.1,ht.2.le⟩
+
+lemma IsDiffusion.ae_tendsto {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
+    ∀ᵐ ω ∂W.μ, Tendsto (fun t ↦ X t ω) (𝓝[<] (1 : ℝ)) (𝓝 (X 1 ω)) := by
+  have ht : Tendsto (id : ℝ → ℝ) (𝓝[<] 1) (𝓝[Icc (0 : ℝ) 1] 1) :=
+    tendsto_nhdsWithin_iff.mpr ⟨nhdsWithin_le_nhds,eventually_time_mem⟩
+  filter_upwards [hX.2] with ω hω
+  exact Tendsto.comp (hω.1 1 (by norm_num : (1 : ℝ)∈Icc (0 : ℝ) 1)) ht
+
+lemma IsDiffusion.L2_terminal {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
+    Tendsto (fun t ↦ ∫ ω, (X t ω-X 1 ω)^2 ∂W.μ) (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
+  let D (t : ℝ) (ω : W.Ω) := X t ω-W.B t.toNNReal ω
+  let G := ∫ s in (0 : ℝ)..1, γ.cutoff s
+  have hG : 0≤G := intervalIntegral.integral_nonneg_of_forall (by norm_num)
+    (fun s ↦ γ.cutoff_nonneg s)
+  have hD (t : ℝ) (ht : t∈Icc (0 : ℝ) 1) : MemLp (D t) 2 W.μ :=
+    (hX.memLp ht (by norm_num)).sub ((W.brownian.hasLaw_eval t.toNNReal).hasGaussianLaw.memLp_two)
+  have hD1 := hD 1 (by norm_num)
+  have hDb : ∀ᵐ ω ∂W.μ, ∀ t∈Icc (0 : ℝ) 1, |D t ω|≤G := hX.drift_bound
+  have hDC : ∀ᵐ ω ∂W.μ, Tendsto (fun t ↦ D t ω) (𝓝[<] (1 : ℝ)) (𝓝 (D 1 ω)) := by
+    filter_upwards [hX.ae_tendsto,W.brownian.cont] with ω hω hBω
+    exact hω.sub ((hBω.comp continuous_real_toNNReal).continuousAt.tendsto.mono_left nhdsWithin_le_nhds)
+  have hDL : Tendsto (fun t ↦ ∫ ω, (D t ω-D 1 ω)^2 ∂W.μ)
+      (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
+    have h := tendsto_integral_filter_of_dominated_convergence
+      (μ := W.μ) (l := 𝓝[<] (1 : ℝ)) (F := fun t ω ↦ (D t ω-D 1 ω)^2) (f := fun _ ↦ (0 : ℝ))
+      (fun _ ↦ 4*G^2) ?_ ?_ (integrable_const (4*G^2)) ?_
+    · simpa only [integral_zero] using h
+    · filter_upwards [eventually_time_mem] with t ht
+      exact ((hD t ht).sub hD1).aestronglyMeasurable.aemeasurable.pow_const 2 |>.aestronglyMeasurable
+    · filter_upwards [eventually_time_mem] with t ht
+      filter_upwards [hDb] with ω hω
+      rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
+      have h0 := abs_le.mp (hω t ht)
+      have h1 := abs_le.mp (hω 1 (by norm_num))
+      nlinarith [sq_nonneg (D t ω-D 1 ω), sq_nonneg (G-(D t ω-D 1 ω)/2),
+        sq_nonneg (G+(D t ω-D 1 ω)/2), mul_nonneg (show 0≤2*G-(D t ω-D 1 ω) by linarith)
+          (show 0≤2*G+(D t ω-D 1 ω) by linarith)]
+    · filter_upwards [hDC] with ω hω
+      simpa only [sub_self, zero_pow (by norm_num : (2 : ℕ)≠0)] using (hω.sub (tendsto_const_nhds (x := D 1 ω))).pow 2
+  have hb : ∀ t∈Icc (0 : ℝ) 1,
+      (∫ ω, (X t ω-X 1 ω)^2 ∂W.μ) ≤
+      2*(∫ ω, (D t ω-D 1 ω)^2 ∂W.μ)+2*(1-t) := by
+    intro t ht
+    have hxint := ((hX.memLp ht (by norm_num : (2 : ℝ≥0∞)≠∞)).sub
+      (hX.memLp (by norm_num : (1 : ℝ)∈Icc (0 : ℝ) 1) (by norm_num))).integrable_sq
+    have hdint : Integrable (fun ω ↦ (D t ω-D 1 ω)^2) W.μ := ((hD t ht).sub hD1).integrable_sq
+    have hbint := (brownian_increment_memLp W t).integrable_sq
+    calc
+      _ ≤ ∫ ω, 2*(D t ω-D 1 ω)^2+2*(W.B 1 ω-W.B t.toNNReal ω)^2 ∂W.μ := by
+        apply integral_mono hxint ((hdint.const_mul 2).add (hbint.const_mul 2))
+        intro ω
+        dsimp only [D,Pi.add_apply,Pi.sub_apply]
+        simp only [Real.toNNReal_one]
+        nlinarith [sq_nonneg (X t ω-W.B t.toNNReal ω-(X 1 ω-W.B 1 ω)+(W.B 1 ω-W.B t.toNNReal ω))]
+      _ = _ := by
+        rw [integral_add (hdint.const_mul 2) (hbint.const_mul 2), integral_const_mul,
+          integral_const_mul, brownian_increment_second W ht]
+  apply squeeze_zero' (Eventually.of_forall (fun t ↦ integral_nonneg (fun ω ↦ sq_nonneg _)))
+    (eventually_time_mem.mono (fun t ht ↦ hb t ht))
+  simpa only [sub_self, mul_zero, add_zero,id_eq] using
+    (hDL.const_mul 2).add (((tendsto_const_nhds (x := (1 : ℝ))).sub (tendsto_id.mono_left nhdsWithin_le_nhds : Tendsto (id : ℝ → ℝ) (𝓝[<] 1) (𝓝 1))).const_mul 2)
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/MeshBrownian.lean b/lean/OAI/Probability/SKValue/Processes/MeshBrownian.lean
index c0bc4d3e1623cb354dc30895284da953bcdf0999..1f61f1dc306c5012f85f716e92fec09afef27ac8 100644
--- a/lean/OAI/Probability/SKValue/Processes/MeshBrownian.lean
+++ b/lean/OAI/Probability/SKValue/Processes/MeshBrownian.lean
@@ -1,343 +1,343 @@
-import Mathlib
-import OAI.Probability.SKValue.Evolution.MeanDerivativeContinuous
-import OAI.Probability.SKValue.Equations.RawIncrement
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology BigOperators
-
-lemma coupled_terminal_second_moment_bound {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    [IsProbabilityMeasure μ] {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    {X : ℝ → Ω → ℝ} {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0<T) (h : GradientStrip T γ u K L) (hLu : 0≤Lu)
-    (huLip : ∀ x y, |u T x-u T y|≤Lu*|x-y|)
-    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
-    (hpaths : ∀ᵐ ω ∂μ,
-      (∀ t∈Icc (0 : ℝ) T, X t ω=B (Real.toNNReal t) ω+
-        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0)
-    {N : ℕ} (hN : 0<N) :
-    |(∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))-
-      (∫ ω, (u T (X T ω))^2 ∂μ)| ≤
-      (2*Lu)*Real.sqrt (∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ) := by
-  let F (ω : Ω) := u T (coupledEuler T γ u B N N ω)
-  let G (ω : Ω) := u T (X T ω)
-  let E := meshMaxError T X (coupledEuler T γ u B) N
-  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
-  have hd := (h.smooth T ht).continuous
-  have hFM : MemLp F 2 μ := MemLp.of_bound
-    (hd.measurable.comp_aemeasurable (coupled_euler_measurable hB hT h N N le_rfl).aemeasurable).aestronglyMeasurable 1
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, F] using h.bounded T ht _))
-  have hGM : MemLp G 2 μ := MemLp.of_bound
-    (hd.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable 1
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, G] using h.bounded T ht _))
-  have hEM : MemLp E 2 μ := coupled_mesh_error_memLp hB hT h hXM hpaths hN
-  have hlast : meshTime T N N = T := by dsimp [meshTime, stepSize]; field_simp
-  have hFG (ω : Ω) : |F ω-G ω|≤Lu*|E ω| := by
-    have he : |coupledEuler T γ u B N N ω-X T ω|≤E ω := by
-      change |coupledEuler T γ u B N N ω-X T ω| ≤ initialMax _ N
-      have hm := le_initialMax (f := fun k ↦ |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
-      change |coupledEuler T γ u B N N ω-X (meshTime T N N) ω| ≤ initialMax _ N at hm
-      rw [hlast] at hm
-      exact hm
-
-    exact (huLip _ _).trans (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) hLu)
-  have hlaw := coupledCoordinates_hasLaw hB hT hN
-  have heq : (∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))=∫ ω, (F ω)^2 ∂μ := by
-    rw [←hlaw.integral_comp ((mesh_terminal_measurable hT.le h hN).pow_const 2).aestronglyMeasurable]
-    simp only [Function.comp_apply, F, meshTerminal, coupledEuler, ite_eq_right (Nat.ne_of_gt hN)]
-  rw [heq]
-  simpa only [mul_one] using second_moment_perturbation hFM hGM hEM (by norm_num : (0 : ℝ)≤1) hLu
-    (Eventually.of_forall (fun ω ↦ ⟨h.bounded T ht _, h.bounded T ht _, hFG ω⟩))
-
-theorem diffusion_gradient_isometry
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    {X : ℝ → Ω → ℝ} {T K L D Lu La : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0<T) (hT1 : T≤1) (h : GradientStrip T γ u K L)
-    (hD0 : 0≤D) (hLu : 0≤Lu) (hLa : 0≤La)
-    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
-    (huLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (haLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |deriv (u s) x-deriv (u t) y|≤La*(|s-t|+|x-y|))
-    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
-    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      (∀ t∈Icc (0 : ℝ) T, X t ω = B (Real.toNNReal t) ω+
-        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0) :
-    (∫ ω, (u T (X T ω))^2 ∂μ) =
-      ∫ t in (0 : ℝ)..T, ∫ ω, (deriv (u t) (X t ω))^2 ∂μ := by
-  let E (N : ℕ) := ∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ
-  have hE : Tendsto E atTop (𝓝 0) := coupled_euler_L2_convergence hB hT h hLu huLip hXM hpaths
-  have hEs : Tendsto (fun N ↦ Real.sqrt (E N)) atTop (𝓝 0) := by
-    simpa only [Real.sqrt_zero, Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp hE
-  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
-  have hmoment : Tendsto (fun N ↦ ∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))
-      atTop (𝓝 (∫ ω, (u T (X T ω))^2 ∂μ)) := by
-    apply tendsto_iff_norm_sub_tendsto_zero.mpr
-    apply squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) ?_
-      (by simpa only [mul_zero] using hEs.const_mul (2*Lu))
-    filter_upwards [eventually_gt_atTop 0] with N hN
-    rw [Real.norm_eq_abs]
-    exact coupled_terminal_second_moment_bound hB hT h hLu
-      (fun x y ↦ by simpa only [sub_self, abs_zero, zero_add] using huLip T ht T ht x y)
-      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN
-  have hmean := mean_derivative_square_continuous hD0 hLa
-    (fun t ht ↦ (h.smooth t ht).continuous_deriv (by norm_num)) hD haLip hXM
-    (hpaths.mono (fun _ hω ↦ hω.1))
-  have hquadrature : Tendsto (fun N ↦ stepSize T N * ∑ i : Fin N,
-      ∫ z, (meshRaw T N γ u i z)^2 ∂gaussianProduct (Fin (N+1))) atTop
-      (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, (deriv (u t) (X t ω))^2 ∂μ)) := by
-    apply perturbed_grid_riemann_tendsto (r := fun N ↦ (2*D*La)*Real.sqrt (E N)) hT hmean
-      (by simpa only [mul_zero] using hEs.const_mul (2*D*La))
-    filter_upwards [eventually_gt_atTop 0] with N hN i
-    exact (coupled_mesh_general_moment_bounds hB hT h hD0 hLa hD
-      (fun t ht x y ↦ by simpa only [sub_self, abs_zero, zero_add] using haLip t ht t ht x y)
-      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN i).1
-  have herror : Tendsto (fun N ↦
-      (∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))-
-      stepSize T N * ∑ i : Fin N, ∫ z, (meshRaw T N γ u i z)^2 ∂gaussianProduct (Fin (N+1)))
-      atTop (𝓝 0) := by
-    apply squeeze_zero_norm' (a := fun N ↦
-      Real.sqrt (gradientMeshError T N γ u) * Real.sqrt (2*(1+T*D^2)))
-    · filter_upwards [eventually_gt_atTop 0] with N hN
-      rw [Real.norm_eq_abs]
-      exact mesh_terminal_stochastic_second_moment_error hT h hD0 hD hN
-    · simpa only [Real.sqrt_zero, zero_mul, Function.comp_def] using
-        ((Real.continuous_sqrt.tendsto 0).comp (gradientMeshError_tendsto_zero hT hT1 h)).mul_const
-          (Real.sqrt (2*(1+T*D^2)))
-  have hlimit := herror.add hquadrature
-  simp only [sub_add_cancel, zero_add] at hlimit
-  exact tendsto_nhds_unique hmoment hlimit
-
-lemma weighted_first_moment_perturbation {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    {f g e b : Ω → ℝ} {L : ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ)
-    (he : MemLp e 2 μ) (hb : MemLp b 2 μ) (hL : 0≤L)
-    (hfg : ∀ᵐ ω ∂μ, |f ω-g ω|≤L*|e ω|) :
-    |(∫ ω, f ω*b ω ∂μ)-(∫ ω, g ω*b ω ∂μ)| ≤
-      L*Real.sqrt (∫ ω, (e ω)^2 ∂μ)*Real.sqrt (∫ ω, (b ω)^2 ∂μ) := by
-  rw [←integral_sub (f := fun ω ↦ f ω*b ω) (g := fun ω ↦ g ω*b ω) (hf.integrable_mul hb) (hg.integrable_mul hb)]
-  have heq : (fun ω ↦ f ω*b ω-g ω*b ω)=(fun ω ↦ (f ω-g ω)*b ω) := by funext ω; ring
-  rw [heq]
-  calc
-    _ ≤ ∫ ω, |(f ω-g ω)*b ω| ∂μ := abs_integral_le_integral_abs
-    _ ≤ ∫ ω, L*(|e ω| * |b ω|) ∂μ := by
-      apply integral_mono_ae ((hf.sub hg).integrable_mul hb).abs ((he.abs.integrable_mul hb.abs).const_mul L)
-      filter_upwards [hfg] with ω hω
-      change |(f ω-g ω)*b ω| ≤ L*(|e ω| * |b ω|)
-      rw [abs_mul]
-      exact (mul_le_mul_of_nonneg_right hω (abs_nonneg _)).trans_eq (mul_assoc _ _ _)
-    _ = L*(∫ ω, |e ω| * |b ω| ∂μ) := integral_const_mul _ _
-    _ ≤ L*(Real.sqrt (∫ ω, (e ω)^2 ∂μ)*Real.sqrt (∫ ω, (b ω)^2 ∂μ)) := by
-      have hcs := integral_product_CS he.abs hb.abs
-      change |∫ ω, |e ω| * |b ω| ∂μ|≤Real.sqrt (∫ ω, |e ω|^2 ∂μ)*Real.sqrt (∫ ω, |b ω|^2 ∂μ) at hcs
-      simp only [sq_abs] at hcs
-      exact mul_le_mul_of_nonneg_left ((le_abs_self _).trans hcs) hL
-    _ = _ := by ring
-
-lemma brownian_eval_second_moment {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ) (t : ℝ≥0) :
-    (∫ ω, (B t ω)^2 ∂μ)=(t : ℝ) := by
-  have hv := (hB.hasLaw_eval t).variance_eq
-  rw [variance_eq_integral (hB.aemeasurable t), hB.integral_eval t] at hv
-  simpa using hv
-
-noncomputable def meshBrownian (T : ℝ) (N : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
-  ∑ j : Fin N, Real.sqrt (stepSize T N)*coordinate N j z
-
-lemma meshBrownian_measurable (T : ℝ) (N : ℕ) : Measurable (meshBrownian T N) := by
-  apply Finset.measurable_sum
-  intro i _
-  exact measurable_const.mul (measurable_coordinate N i)
-
-lemma meshBrownian_memLp (T : ℝ) (N : ℕ) :
-    MemLp (meshBrownian T N) 2 (gaussianProduct (Fin (N+1))) := by
-  convert memLp_finsetSum' Finset.univ (fun i : Fin N ↦ fun _ ↦
-    (gaussian_memLp (coordinate_hasLaw N i) 2 (by norm_num)).const_mul (Real.sqrt (stepSize T N))) using 1
-  funext z
-  simp only [meshBrownian, Finset.sum_apply]
-
-lemma coupled_meshBrownian {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) {T : ℝ} (hT : 0<T)
-    {N : ℕ} (hN : 0<N) (ω : Ω) :
-    meshBrownian T N (coupledCoordinates T N B ω)=B (Real.toNNReal T) ω-B 0 ω := by
-  have hlast : (N : ℝ)*(T/N)=T := by field_simp
-  unfold meshBrownian
-  have heq (j : Fin N) := coupled_coordinate_exact B hT hN (show (j : ℕ)<N+1 by omega) ω
-  simp_rw [stepSize, heq]
-  rw [Fin.sum_univ_eq_sum_range (fun j ↦ B (Real.toNNReal (((j+1 : ℕ) : ℝ)*(T/N))) ω-
-    B (Real.toNNReal ((j : ℝ)*(T/N))) ω), Finset.sum_range_sub (fun j : ℕ ↦ B (Real.toNNReal ((j : ℝ)*(T/N))) ω) N]
-  simp only [Nat.cast_zero, zero_mul, Real.toNNReal_zero, hlast]
-
-lemma meshBrownian_second_moment {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ) {T : ℝ} (hT : 0<T)
-    {N : ℕ} (hN : 0<N) :
-    (∫ z, (meshBrownian T N z)^2 ∂gaussianProduct (Fin (N+1)))=T := by
-  rw [←(coupledCoordinates_hasLaw hB hT hN).integral_comp
-    ((meshBrownian_measurable T N).pow_const 2).aestronglyMeasurable]
-  have heq : (fun ω ↦ (meshBrownian T N (coupledCoordinates T N B ω))^2)=ᵐ[μ]
-      (fun ω ↦ (B (Real.toNNReal T) ω)^2) := by
-    filter_upwards [hB.eval_zero_ae_eq_zero] with ω hω
-    simp only [coupled_meshBrownian B hT hN ω, hω, sub_zero]
-  exact (integral_congr_ae heq).trans ((brownian_eval_second_moment hB _).trans
-    (Real.coe_toNNReal _ hT.le))
-
-lemma mesh_terminal_brownian_error {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0<T) (h : GradientStrip T γ u K L)
-    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D) {N : ℕ} (hN : 0<N) :
-    |(∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
-      stepSize T N * ∑ i : Fin N, ∫ z, meshRaw T N γ u i z ∂gaussianProduct (Fin (N+1))| ≤
-    Real.sqrt (gradientMeshError T N γ u)*Real.sqrt T := by
-  have hm (i : Fin N) := mesh_raw_measurable hT.le h hN i.isLt.le
-  have hp (i : Fin N) := mesh_raw_depends T i.isLt.le γ u
-  have hH (i : Fin N) := mesh_raw_memLp hT.le h hD hN i.isLt.le
-  have hu := mesh_terminal_memLp hT.le h hN
-  have hr := finiteRaw_memLp (δ := stepSize T N) hm hp hH
-  have hb := meshBrownian_memLp T N
-  rw [←finiteRaw_brownian_covariance (δ := stepSize T N) (div_nonneg hT.le (Nat.cast_nonneg N)) hm hp hH]
-  change |(∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
-      (∫ z, finiteRawMartingale (stepSize T N) (fun i : Fin N ↦ meshRaw T N γ u i) z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))|≤_
-  rw [←integral_sub (f := fun z ↦ meshTerminal T N γ u z * meshBrownian T N z)
-    (g := fun z ↦ finiteRawMartingale (stepSize T N) (fun i : Fin N ↦ meshRaw T N γ u i) z * meshBrownian T N z)
-    (hu.integrable_mul hb) (hr.integrable_mul hb)]
-  simp_rw [←sub_mul]
-  have hcs := integral_product_CS (hu.sub hr) hb
-  rw [meshBrownian_second_moment hB hT hN] at hcs
-  exact hcs
-
-lemma coupled_terminal_brownian_error {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    [IsProbabilityMeasure μ] {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    {X : ℝ → Ω → ℝ} {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0<T) (h : GradientStrip T γ u K L) (hLu : 0≤Lu)
-    (huLip : ∀ x y, |u T x-u T y|≤Lu*|x-y|)
-    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
-    (hpaths : ∀ᵐ ω ∂μ,
-      (∀ t∈Icc (0 : ℝ) T, X t ω=B (Real.toNNReal t) ω+
-        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0)
-    {N : ℕ} (hN : 0<N) :
-    |(∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
-      (∫ ω, u T (X T ω)*B (Real.toNNReal T) ω ∂μ)| ≤
-      Lu*Real.sqrt (∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)*Real.sqrt T := by
-  let F (ω : Ω) := u T (coupledEuler T γ u B N N ω)
-  let G (ω : Ω) := u T (X T ω)
-  let E := meshMaxError T X (coupledEuler T γ u B) N
-  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
-  have hd := (h.smooth T ht).continuous
-  have hFM : MemLp F 2 μ := MemLp.of_bound
-    (hd.measurable.comp_aemeasurable (coupled_euler_measurable hB hT h N N le_rfl).aemeasurable).aestronglyMeasurable 1
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, F] using h.bounded T ht _))
-  have hGM : MemLp G 2 μ := MemLp.of_bound
-    (hd.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable 1
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, G] using h.bounded T ht _))
-  have hEM : MemLp E 2 μ := coupled_mesh_error_memLp hB hT h hXM hpaths hN
-  have hlast : meshTime T N N = T := by dsimp [meshTime, stepSize]; field_simp
-  have hFG (ω : Ω) : |F ω-G ω|≤Lu*|E ω| := by
-    have he : |coupledEuler T γ u B N N ω-X T ω|≤E ω := by
-      change |coupledEuler T γ u B N N ω-X T ω| ≤ initialMax _ N
-      have hm := le_initialMax (f := fun k ↦ |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
-      rw [hlast] at hm
-      exact hm
-    exact (huLip _ _).trans (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) hLu)
-  have hlaw := coupledCoordinates_hasLaw hB hT hN
-  have heq : (∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))=
-      ∫ ω, F ω*B (Real.toNNReal T) ω ∂μ := by
-    have hm : AEStronglyMeasurable (fun z ↦ meshTerminal T N γ u z * meshBrownian T N z) (gaussianProduct (Fin (N+1))) :=
-      ((mesh_terminal_measurable hT.le h hN).mul (meshBrownian_measurable T N)).aestronglyMeasurable
-    rw [←hlaw.integral_comp hm]
-    apply integral_congr_ae
-    filter_upwards [hB.eval_zero_ae_eq_zero] with ω hω
-    simp only [Function.comp_apply, F, meshTerminal, coupledEuler, ite_eq_right (Nat.ne_of_gt hN),
-      coupled_meshBrownian B hT hN ω, hω, sub_zero]
-  rw [heq]
-  have hh := weighted_first_moment_perturbation hFM hGM hEM (hB.hasLaw_eval T.toNNReal).hasGaussianLaw.memLp_two hLu
-    (Eventually.of_forall hFG)
-  rw [brownian_eval_second_moment hB,Real.coe_toNNReal _ hT.le] at hh
-  exact hh
-
-theorem diffusion_gradient_brownian_covariance
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
-    {X : ℝ → Ω → ℝ} {T K L D Lu La : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0<T) (hT1 : T≤1) (h : GradientStrip T γ u K L)
-    (hD0 : 0≤D) (hLu : 0≤Lu) (hLa : 0≤La)
-    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
-    (huLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (haLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
-      |deriv (u s) x-deriv (u t) y|≤La*(|s-t|+|x-y|))
-    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
-    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      (∀ t∈Icc (0 : ℝ) T, X t ω = B (Real.toNNReal t) ω+
-        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0) :
-    (∫ ω, u T (X T ω)*B (Real.toNNReal T) ω ∂μ) =
-      ∫ t in (0 : ℝ)..T, ∫ ω, deriv (u t) (X t ω) ∂μ := by
-  let E (N : ℕ) := ∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ
-  have hE : Tendsto E atTop (𝓝 0) := coupled_euler_L2_convergence hB hT h hLu huLip hXM hpaths
-  have hEs : Tendsto (fun N ↦ Real.sqrt (E N)) atTop (𝓝 0) := by
-    simpa only [Real.sqrt_zero, Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp hE
-  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
-  have hcovariance : Tendsto (fun N ↦ ∫ z, meshTerminal T N γ u z * meshBrownian T N z
-      ∂gaussianProduct (Fin (N+1))) atTop
-      (𝓝 (∫ ω, u T (X T ω)*B T.toNNReal ω ∂μ)) := by
-    apply tendsto_iff_norm_sub_tendsto_zero.mpr
-    apply squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) ?_
-      (by simpa only [mul_zero, zero_mul] using (hEs.const_mul Lu).mul_const (Real.sqrt T))
-    filter_upwards [eventually_gt_atTop 0] with N hN
-    rw [Real.norm_eq_abs]
-    exact coupled_terminal_brownian_error hB hT h hLu
-      (fun x y ↦ by simpa only [sub_self, abs_zero, zero_add] using huLip T ht T ht x y)
-      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN
-  have hmean := mean_derivative_continuous hLa
-    (fun t ht ↦ (h.smooth t ht).continuous_deriv (by norm_num)) hD haLip hXM
-    (hpaths.mono (fun _ hω ↦ hω.1))
-  have hquadrature : Tendsto (fun N ↦ stepSize T N * ∑ i : Fin N,
-      ∫ z, meshRaw T N γ u i z ∂gaussianProduct (Fin (N+1))) atTop
-      (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, deriv (u t) (X t ω) ∂μ)) := by
-    apply perturbed_grid_riemann_tendsto (r := fun N ↦ La*Real.sqrt (E N)) hT hmean
-      (by simpa only [mul_zero] using hEs.const_mul La)
-    filter_upwards [eventually_gt_atTop 0] with N hN i
-    exact (coupled_mesh_general_moment_bounds hB hT h hD0 hLa hD
-      (fun t ht x y ↦ by simpa only [sub_self, abs_zero, zero_add] using haLip t ht t ht x y)
-      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN i).2
-  have herror : Tendsto (fun N ↦
-      (∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
-      stepSize T N * ∑ i : Fin N, ∫ z, meshRaw T N γ u i z ∂gaussianProduct (Fin (N+1)))
-      atTop (𝓝 0) := by
-    apply squeeze_zero_norm' (a := fun N ↦
-      Real.sqrt (gradientMeshError T N γ u) * Real.sqrt T)
-    · filter_upwards [eventually_gt_atTop 0] with N hN
-      rw [Real.norm_eq_abs]
-      exact mesh_terminal_brownian_error hB hT h hD hN
-    · simpa only [Real.sqrt_zero, zero_mul, Function.comp_def] using
-        ((Real.continuous_sqrt.tendsto 0).comp (gradientMeshError_tendsto_zero hT hT1 h)).mul_const
-          (Real.sqrt T)
-  have hlimit := herror.add hquadrature
-  simp only [sub_add_cancel, zero_add] at hlimit
-  exact tendsto_nhds_unique hcovariance hlimit
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Evolution.MeanDerivativeContinuous
+import OAI.Probability.SKValue.Equations.RawIncrement
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology BigOperators
+
+lemma coupled_terminal_second_moment_bound {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    [IsProbabilityMeasure μ] {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    {X : ℝ → Ω → ℝ} {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0<T) (h : GradientStrip T γ u K L) (hLu : 0≤Lu)
+    (huLip : ∀ x y, |u T x-u T y|≤Lu*|x-y|)
+    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
+    (hpaths : ∀ᵐ ω ∂μ,
+      (∀ t∈Icc (0 : ℝ) T, X t ω=B (Real.toNNReal t) ω+
+        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0)
+    {N : ℕ} (hN : 0<N) :
+    |(∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))-
+      (∫ ω, (u T (X T ω))^2 ∂μ)| ≤
+      (2*Lu)*Real.sqrt (∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ) := by
+  let F (ω : Ω) := u T (coupledEuler T γ u B N N ω)
+  let G (ω : Ω) := u T (X T ω)
+  let E := meshMaxError T X (coupledEuler T γ u B) N
+  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
+  have hd := (h.smooth T ht).continuous
+  have hFM : MemLp F 2 μ := MemLp.of_bound
+    (hd.measurable.comp_aemeasurable (coupled_euler_measurable hB hT h N N le_rfl).aemeasurable).aestronglyMeasurable 1
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, F] using h.bounded T ht _))
+  have hGM : MemLp G 2 μ := MemLp.of_bound
+    (hd.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable 1
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, G] using h.bounded T ht _))
+  have hEM : MemLp E 2 μ := coupled_mesh_error_memLp hB hT h hXM hpaths hN
+  have hlast : meshTime T N N = T := by dsimp [meshTime, stepSize]; field_simp
+  have hFG (ω : Ω) : |F ω-G ω|≤Lu*|E ω| := by
+    have he : |coupledEuler T γ u B N N ω-X T ω|≤E ω := by
+      change |coupledEuler T γ u B N N ω-X T ω| ≤ initialMax _ N
+      have hm := le_initialMax (f := fun k ↦ |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
+      change |coupledEuler T γ u B N N ω-X (meshTime T N N) ω| ≤ initialMax _ N at hm
+      rw [hlast] at hm
+      exact hm
+
+    exact (huLip _ _).trans (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) hLu)
+  have hlaw := coupledCoordinates_hasLaw hB hT hN
+  have heq : (∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))=∫ ω, (F ω)^2 ∂μ := by
+    rw [←hlaw.integral_comp ((mesh_terminal_measurable hT.le h hN).pow_const 2).aestronglyMeasurable]
+    simp only [Function.comp_apply, F, meshTerminal, coupledEuler, ite_eq_right (Nat.ne_of_gt hN)]
+  rw [heq]
+  simpa only [mul_one] using second_moment_perturbation hFM hGM hEM (by norm_num : (0 : ℝ)≤1) hLu
+    (Eventually.of_forall (fun ω ↦ ⟨h.bounded T ht _, h.bounded T ht _, hFG ω⟩))
+
+theorem diffusion_gradient_isometry
+    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    {X : ℝ → Ω → ℝ} {T K L D Lu La : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0<T) (hT1 : T≤1) (h : GradientStrip T γ u K L)
+    (hD0 : 0≤D) (hLu : 0≤Lu) (hLa : 0≤La)
+    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
+    (huLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (haLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |deriv (u s) x-deriv (u t) y|≤La*(|s-t|+|x-y|))
+    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
+    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      (∀ t∈Icc (0 : ℝ) T, X t ω = B (Real.toNNReal t) ω+
+        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0) :
+    (∫ ω, (u T (X T ω))^2 ∂μ) =
+      ∫ t in (0 : ℝ)..T, ∫ ω, (deriv (u t) (X t ω))^2 ∂μ := by
+  let E (N : ℕ) := ∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ
+  have hE : Tendsto E atTop (𝓝 0) := coupled_euler_L2_convergence hB hT h hLu huLip hXM hpaths
+  have hEs : Tendsto (fun N ↦ Real.sqrt (E N)) atTop (𝓝 0) := by
+    simpa only [Real.sqrt_zero, Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp hE
+  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
+  have hmoment : Tendsto (fun N ↦ ∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))
+      atTop (𝓝 (∫ ω, (u T (X T ω))^2 ∂μ)) := by
+    apply tendsto_iff_norm_sub_tendsto_zero.mpr
+    apply squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) ?_
+      (by simpa only [mul_zero] using hEs.const_mul (2*Lu))
+    filter_upwards [eventually_gt_atTop 0] with N hN
+    rw [Real.norm_eq_abs]
+    exact coupled_terminal_second_moment_bound hB hT h hLu
+      (fun x y ↦ by simpa only [sub_self, abs_zero, zero_add] using huLip T ht T ht x y)
+      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN
+  have hmean := mean_derivative_square_continuous hD0 hLa
+    (fun t ht ↦ (h.smooth t ht).continuous_deriv (by norm_num)) hD haLip hXM
+    (hpaths.mono (fun _ hω ↦ hω.1))
+  have hquadrature : Tendsto (fun N ↦ stepSize T N * ∑ i : Fin N,
+      ∫ z, (meshRaw T N γ u i z)^2 ∂gaussianProduct (Fin (N+1))) atTop
+      (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, (deriv (u t) (X t ω))^2 ∂μ)) := by
+    apply perturbed_grid_riemann_tendsto (r := fun N ↦ (2*D*La)*Real.sqrt (E N)) hT hmean
+      (by simpa only [mul_zero] using hEs.const_mul (2*D*La))
+    filter_upwards [eventually_gt_atTop 0] with N hN i
+    exact (coupled_mesh_general_moment_bounds hB hT h hD0 hLa hD
+      (fun t ht x y ↦ by simpa only [sub_self, abs_zero, zero_add] using haLip t ht t ht x y)
+      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN i).1
+  have herror : Tendsto (fun N ↦
+      (∫ z, (meshTerminal T N γ u z)^2 ∂gaussianProduct (Fin (N+1)))-
+      stepSize T N * ∑ i : Fin N, ∫ z, (meshRaw T N γ u i z)^2 ∂gaussianProduct (Fin (N+1)))
+      atTop (𝓝 0) := by
+    apply squeeze_zero_norm' (a := fun N ↦
+      Real.sqrt (gradientMeshError T N γ u) * Real.sqrt (2*(1+T*D^2)))
+    · filter_upwards [eventually_gt_atTop 0] with N hN
+      rw [Real.norm_eq_abs]
+      exact mesh_terminal_stochastic_second_moment_error hT h hD0 hD hN
+    · simpa only [Real.sqrt_zero, zero_mul, Function.comp_def] using
+        ((Real.continuous_sqrt.tendsto 0).comp (gradientMeshError_tendsto_zero hT hT1 h)).mul_const
+          (Real.sqrt (2*(1+T*D^2)))
+  have hlimit := herror.add hquadrature
+  simp only [sub_add_cancel, zero_add] at hlimit
+  exact tendsto_nhds_unique hmoment hlimit
+
+lemma weighted_first_moment_perturbation {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    {f g e b : Ω → ℝ} {L : ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ)
+    (he : MemLp e 2 μ) (hb : MemLp b 2 μ) (hL : 0≤L)
+    (hfg : ∀ᵐ ω ∂μ, |f ω-g ω|≤L*|e ω|) :
+    |(∫ ω, f ω*b ω ∂μ)-(∫ ω, g ω*b ω ∂μ)| ≤
+      L*Real.sqrt (∫ ω, (e ω)^2 ∂μ)*Real.sqrt (∫ ω, (b ω)^2 ∂μ) := by
+  rw [←integral_sub (f := fun ω ↦ f ω*b ω) (g := fun ω ↦ g ω*b ω) (hf.integrable_mul hb) (hg.integrable_mul hb)]
+  have heq : (fun ω ↦ f ω*b ω-g ω*b ω)=(fun ω ↦ (f ω-g ω)*b ω) := by funext ω; ring
+  rw [heq]
+  calc
+    _ ≤ ∫ ω, |(f ω-g ω)*b ω| ∂μ := abs_integral_le_integral_abs
+    _ ≤ ∫ ω, L*(|e ω| * |b ω|) ∂μ := by
+      apply integral_mono_ae ((hf.sub hg).integrable_mul hb).abs ((he.abs.integrable_mul hb.abs).const_mul L)
+      filter_upwards [hfg] with ω hω
+      change |(f ω-g ω)*b ω| ≤ L*(|e ω| * |b ω|)
+      rw [abs_mul]
+      exact (mul_le_mul_of_nonneg_right hω (abs_nonneg _)).trans_eq (mul_assoc _ _ _)
+    _ = L*(∫ ω, |e ω| * |b ω| ∂μ) := integral_const_mul _ _
+    _ ≤ L*(Real.sqrt (∫ ω, (e ω)^2 ∂μ)*Real.sqrt (∫ ω, (b ω)^2 ∂μ)) := by
+      have hcs := integral_product_CS he.abs hb.abs
+      change |∫ ω, |e ω| * |b ω| ∂μ|≤Real.sqrt (∫ ω, |e ω|^2 ∂μ)*Real.sqrt (∫ ω, |b ω|^2 ∂μ) at hcs
+      simp only [sq_abs] at hcs
+      exact mul_le_mul_of_nonneg_left ((le_abs_self _).trans hcs) hL
+    _ = _ := by ring
+
+lemma brownian_eval_second_moment {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ) (t : ℝ≥0) :
+    (∫ ω, (B t ω)^2 ∂μ)=(t : ℝ) := by
+  have hv := (hB.hasLaw_eval t).variance_eq
+  rw [variance_eq_integral (hB.aemeasurable t), hB.integral_eval t] at hv
+  simpa using hv
+
+noncomputable def meshBrownian (T : ℝ) (N : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
+  ∑ j : Fin N, Real.sqrt (stepSize T N)*coordinate N j z
+
+lemma meshBrownian_measurable (T : ℝ) (N : ℕ) : Measurable (meshBrownian T N) := by
+  apply Finset.measurable_sum
+  intro i _
+  exact measurable_const.mul (measurable_coordinate N i)
+
+lemma meshBrownian_memLp (T : ℝ) (N : ℕ) :
+    MemLp (meshBrownian T N) 2 (gaussianProduct (Fin (N+1))) := by
+  convert memLp_finsetSum' Finset.univ (fun i : Fin N ↦ fun _ ↦
+    (gaussian_memLp (coordinate_hasLaw N i) 2 (by norm_num)).const_mul (Real.sqrt (stepSize T N))) using 1
+  funext z
+  simp only [meshBrownian, Finset.sum_apply]
+
+lemma coupled_meshBrownian {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) {T : ℝ} (hT : 0<T)
+    {N : ℕ} (hN : 0<N) (ω : Ω) :
+    meshBrownian T N (coupledCoordinates T N B ω)=B (Real.toNNReal T) ω-B 0 ω := by
+  have hlast : (N : ℝ)*(T/N)=T := by field_simp
+  unfold meshBrownian
+  have heq (j : Fin N) := coupled_coordinate_exact B hT hN (show (j : ℕ)<N+1 by omega) ω
+  simp_rw [stepSize, heq]
+  rw [Fin.sum_univ_eq_sum_range (fun j ↦ B (Real.toNNReal (((j+1 : ℕ) : ℝ)*(T/N))) ω-
+    B (Real.toNNReal ((j : ℝ)*(T/N))) ω), Finset.sum_range_sub (fun j : ℕ ↦ B (Real.toNNReal ((j : ℝ)*(T/N))) ω) N]
+  simp only [Nat.cast_zero, zero_mul, Real.toNNReal_zero, hlast]
+
+lemma meshBrownian_second_moment {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ) {T : ℝ} (hT : 0<T)
+    {N : ℕ} (hN : 0<N) :
+    (∫ z, (meshBrownian T N z)^2 ∂gaussianProduct (Fin (N+1)))=T := by
+  rw [←(coupledCoordinates_hasLaw hB hT hN).integral_comp
+    ((meshBrownian_measurable T N).pow_const 2).aestronglyMeasurable]
+  have heq : (fun ω ↦ (meshBrownian T N (coupledCoordinates T N B ω))^2)=ᵐ[μ]
+      (fun ω ↦ (B (Real.toNNReal T) ω)^2) := by
+    filter_upwards [hB.eval_zero_ae_eq_zero] with ω hω
+    simp only [coupled_meshBrownian B hT hN ω, hω, sub_zero]
+  exact (integral_congr_ae heq).trans ((brownian_eval_second_moment hB _).trans
+    (Real.coe_toNNReal _ hT.le))
+
+lemma mesh_terminal_brownian_error {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0<T) (h : GradientStrip T γ u K L)
+    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D) {N : ℕ} (hN : 0<N) :
+    |(∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
+      stepSize T N * ∑ i : Fin N, ∫ z, meshRaw T N γ u i z ∂gaussianProduct (Fin (N+1))| ≤
+    Real.sqrt (gradientMeshError T N γ u)*Real.sqrt T := by
+  have hm (i : Fin N) := mesh_raw_measurable hT.le h hN i.isLt.le
+  have hp (i : Fin N) := mesh_raw_depends T i.isLt.le γ u
+  have hH (i : Fin N) := mesh_raw_memLp hT.le h hD hN i.isLt.le
+  have hu := mesh_terminal_memLp hT.le h hN
+  have hr := finiteRaw_memLp (δ := stepSize T N) hm hp hH
+  have hb := meshBrownian_memLp T N
+  rw [←finiteRaw_brownian_covariance (δ := stepSize T N) (div_nonneg hT.le (Nat.cast_nonneg N)) hm hp hH]
+  change |(∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
+      (∫ z, finiteRawMartingale (stepSize T N) (fun i : Fin N ↦ meshRaw T N γ u i) z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))|≤_
+  rw [←integral_sub (f := fun z ↦ meshTerminal T N γ u z * meshBrownian T N z)
+    (g := fun z ↦ finiteRawMartingale (stepSize T N) (fun i : Fin N ↦ meshRaw T N γ u i) z * meshBrownian T N z)
+    (hu.integrable_mul hb) (hr.integrable_mul hb)]
+  simp_rw [←sub_mul]
+  have hcs := integral_product_CS (hu.sub hr) hb
+  rw [meshBrownian_second_moment hB hT hN] at hcs
+  exact hcs
+
+lemma coupled_terminal_brownian_error {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    [IsProbabilityMeasure μ] {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    {X : ℝ → Ω → ℝ} {T K L Lu : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0<T) (h : GradientStrip T γ u K L) (hLu : 0≤Lu)
+    (huLip : ∀ x y, |u T x-u T y|≤Lu*|x-y|)
+    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
+    (hpaths : ∀ᵐ ω ∂μ,
+      (∀ t∈Icc (0 : ℝ) T, X t ω=B (Real.toNNReal t) ω+
+        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0)
+    {N : ℕ} (hN : 0<N) :
+    |(∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
+      (∫ ω, u T (X T ω)*B (Real.toNNReal T) ω ∂μ)| ≤
+      Lu*Real.sqrt (∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)*Real.sqrt T := by
+  let F (ω : Ω) := u T (coupledEuler T γ u B N N ω)
+  let G (ω : Ω) := u T (X T ω)
+  let E := meshMaxError T X (coupledEuler T γ u B) N
+  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
+  have hd := (h.smooth T ht).continuous
+  have hFM : MemLp F 2 μ := MemLp.of_bound
+    (hd.measurable.comp_aemeasurable (coupled_euler_measurable hB hT h N N le_rfl).aemeasurable).aestronglyMeasurable 1
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, F] using h.bounded T ht _))
+  have hGM : MemLp G 2 μ := MemLp.of_bound
+    (hd.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable 1
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs, G] using h.bounded T ht _))
+  have hEM : MemLp E 2 μ := coupled_mesh_error_memLp hB hT h hXM hpaths hN
+  have hlast : meshTime T N N = T := by dsimp [meshTime, stepSize]; field_simp
+  have hFG (ω : Ω) : |F ω-G ω|≤Lu*|E ω| := by
+    have he : |coupledEuler T γ u B N N ω-X T ω|≤E ω := by
+      change |coupledEuler T γ u B N N ω-X T ω| ≤ initialMax _ N
+      have hm := le_initialMax (f := fun k ↦ |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
+      rw [hlast] at hm
+      exact hm
+    exact (huLip _ _).trans (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) hLu)
+  have hlaw := coupledCoordinates_hasLaw hB hT hN
+  have heq : (∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))=
+      ∫ ω, F ω*B (Real.toNNReal T) ω ∂μ := by
+    have hm : AEStronglyMeasurable (fun z ↦ meshTerminal T N γ u z * meshBrownian T N z) (gaussianProduct (Fin (N+1))) :=
+      ((mesh_terminal_measurable hT.le h hN).mul (meshBrownian_measurable T N)).aestronglyMeasurable
+    rw [←hlaw.integral_comp hm]
+    apply integral_congr_ae
+    filter_upwards [hB.eval_zero_ae_eq_zero] with ω hω
+    simp only [Function.comp_apply, F, meshTerminal, coupledEuler, ite_eq_right (Nat.ne_of_gt hN),
+      coupled_meshBrownian B hT hN ω, hω, sub_zero]
+  rw [heq]
+  have hh := weighted_first_moment_perturbation hFM hGM hEM (hB.hasLaw_eval T.toNNReal).hasGaussianLaw.memLp_two hLu
+    (Eventually.of_forall hFG)
+  rw [brownian_eval_second_moment hB,Real.coe_toNNReal _ hT.le] at hh
+  exact hh
+
+theorem diffusion_gradient_brownian_covariance
+    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B μ)
+    {X : ℝ → Ω → ℝ} {T K L D Lu La : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0<T) (hT1 : T≤1) (h : GradientStrip T γ u K L)
+    (hD0 : 0≤D) (hLu : 0≤Lu) (hLa : 0≤La)
+    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
+    (huLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (haLip : ∀ s∈Icc (0 : ℝ) T, ∀ t∈Icc (0 : ℝ) T, ∀ x y,
+      |deriv (u s) x-deriv (u t) y|≤La*(|s-t|+|x-y|))
+    (hXM : ∀ t∈Icc (0 : ℝ) T, AEStronglyMeasurable (X t) μ)
+    (hpaths : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      (∀ t∈Icc (0 : ℝ) T, X t ω = B (Real.toNNReal t) ω+
+        ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω=0) :
+    (∫ ω, u T (X T ω)*B (Real.toNNReal T) ω ∂μ) =
+      ∫ t in (0 : ℝ)..T, ∫ ω, deriv (u t) (X t ω) ∂μ := by
+  let E (N : ℕ) := ∫ ω, (meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ
+  have hE : Tendsto E atTop (𝓝 0) := coupled_euler_L2_convergence hB hT h hLu huLip hXM hpaths
+  have hEs : Tendsto (fun N ↦ Real.sqrt (E N)) atTop (𝓝 0) := by
+    simpa only [Real.sqrt_zero, Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp hE
+  have ht : T∈Icc (0 : ℝ) T := ⟨hT.le,le_rfl⟩
+  have hcovariance : Tendsto (fun N ↦ ∫ z, meshTerminal T N γ u z * meshBrownian T N z
+      ∂gaussianProduct (Fin (N+1))) atTop
+      (𝓝 (∫ ω, u T (X T ω)*B T.toNNReal ω ∂μ)) := by
+    apply tendsto_iff_norm_sub_tendsto_zero.mpr
+    apply squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) ?_
+      (by simpa only [mul_zero, zero_mul] using (hEs.const_mul Lu).mul_const (Real.sqrt T))
+    filter_upwards [eventually_gt_atTop 0] with N hN
+    rw [Real.norm_eq_abs]
+    exact coupled_terminal_brownian_error hB hT h hLu
+      (fun x y ↦ by simpa only [sub_self, abs_zero, zero_add] using huLip T ht T ht x y)
+      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN
+  have hmean := mean_derivative_continuous hLa
+    (fun t ht ↦ (h.smooth t ht).continuous_deriv (by norm_num)) hD haLip hXM
+    (hpaths.mono (fun _ hω ↦ hω.1))
+  have hquadrature : Tendsto (fun N ↦ stepSize T N * ∑ i : Fin N,
+      ∫ z, meshRaw T N γ u i z ∂gaussianProduct (Fin (N+1))) atTop
+      (𝓝 (∫ t in (0 : ℝ)..T, ∫ ω, deriv (u t) (X t ω) ∂μ)) := by
+    apply perturbed_grid_riemann_tendsto (r := fun N ↦ La*Real.sqrt (E N)) hT hmean
+      (by simpa only [mul_zero] using hEs.const_mul La)
+    filter_upwards [eventually_gt_atTop 0] with N hN i
+    exact (coupled_mesh_general_moment_bounds hB hT h hD0 hLa hD
+      (fun t ht x y ↦ by simpa only [sub_self, abs_zero, zero_add] using haLip t ht t ht x y)
+      hXM (hpaths.mono (fun _ hω ↦ hω.2.2)) hN i).2
+  have herror : Tendsto (fun N ↦
+      (∫ z, meshTerminal T N γ u z * meshBrownian T N z ∂gaussianProduct (Fin (N+1)))-
+      stepSize T N * ∑ i : Fin N, ∫ z, meshRaw T N γ u i z ∂gaussianProduct (Fin (N+1)))
+      atTop (𝓝 0) := by
+    apply squeeze_zero_norm' (a := fun N ↦
+      Real.sqrt (gradientMeshError T N γ u) * Real.sqrt T)
+    · filter_upwards [eventually_gt_atTop 0] with N hN
+      rw [Real.norm_eq_abs]
+      exact mesh_terminal_brownian_error hB hT h hD hN
+    · simpa only [Real.sqrt_zero, zero_mul, Function.comp_def] using
+        ((Real.continuous_sqrt.tendsto 0).comp (gradientMeshError_tendsto_zero hT hT1 h)).mul_const
+          (Real.sqrt T)
+  have hlimit := herror.add hquadrature
+  simp only [sub_add_cancel, zero_add] at hlimit
+  exact tendsto_nhds_unique hcovariance hlimit
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/MeshMaxError.lean b/lean/OAI/Probability/SKValue/Processes/MeshMaxError.lean
index 8dccc9388f759bcb407b7d8a4bedc8c976d3e14d..02b5878eea6bdbec3f3e80c2bb230f784a44d077 100644
--- a/lean/OAI/Probability/SKValue/Processes/MeshMaxError.lean
+++ b/lean/OAI/Probability/SKValue/Processes/MeshMaxError.lean
@@ -1,87 +1,87 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.InitialMax
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-
-noncomputable def meshMaxError {Ω : Type*} (T : ℝ) (X : ℝ → Ω → ℝ)
-    (Y : ℕ → ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : ℝ :=
-  initialMax (fun j ↦ |Y N j ω-X ((j : ℝ)*(T/N)) ω|) N
-
-lemma meshMaxError_nonneg {Ω : Type*} (T : ℝ) (X : ℝ → Ω → ℝ)
-    (Y : ℕ → ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : 0 ≤ meshMaxError T X Y N ω :=
-  (abs_nonneg _).trans (le_initialMax
-    (f := fun j ↦ |Y N j ω-X ((j : ℝ)*(T/N)) ω|) (Nat.zero_le N))
-
-lemma driven_euler_L2_convergence {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    [IsFiniteMeasure μ] {T L : ℝ} {u : ℝ → ℝ → ℝ} {γ : ℝ → ℝ}
-    {W X : ℝ → Ω → ℝ} {Y : ℕ → ℕ → Ω → ℝ}
-    (hT : 0 ≤ T) (hL : 0 ≤ L)
-    (hγ : MonotoneOn γ (Icc (0 : ℝ) T)) (hγ0 : 0 ≤ γ 0)
-    (hu : ∀ t ∈ Icc (0 : ℝ) T, ∀ x, |u t x| ≤ 1)
-    (hLip : ∀ s ∈ Icc (0 : ℝ) T, ∀ t ∈ Icc (0 : ℝ) T, ∀ x y,
-      |u s x-u t y| ≤ L*(|s-t|+|x-y|))
-    (hXM : ∀ t, AEStronglyMeasurable (X t) μ)
-    (hYM : ∀ N j, AEStronglyMeasurable (Y N j) μ)
-    (hpaths : ∀ᵐ ω ∂μ,
-      ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      (∀ t ∈ Icc (0 : ℝ) T,
-        X t ω = W t ω + ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω = 0)
-    (hY0 : ∀ N ω, Y N 0 ω = 0)
-    (hY : ∀ N > 0, ∀ j < N, ∀ ω,
-      Y N (j+1) ω = Y N j ω + W (((j+1 : ℕ) : ℝ)*(T/N)) ω-W ((j : ℝ)*(T/N)) ω +
-        (T/N)*γ ((j : ℝ)*(T/N))*u ((j : ℝ)*(T/N)) (Y N j ω)) :
-    Tendsto (fun N ↦ ∫ ω, (meshMaxError T X Y N ω)^2 ∂μ) atTop (𝓝 (0 : ℝ)) := by
-  have hmeas : ∀ N, AEStronglyMeasurable (fun ω ↦ (meshMaxError T X Y N ω)^2) μ := by
-    intro N
-    exact (aestronglyMeasurable_initialMax (fun j ↦ ((hYM N j).sub (hXM _)).norm) N).fun_pow 2
-  have hbound : ∀ᶠ N in atTop, ∀ᵐ ω ∂μ,
-      ‖(meshMaxError T X Y N ω)^2‖ ≤ (2*T*γ T)^2 := by
-    filter_upwards [eventually_gt_atTop 0] with N hN
-    filter_upwards [hpaths] with ω hω
-    have hδ : 0 ≤ T/(N : ℝ) := div_nonneg hT (Nat.cast_nonneg _)
-    have horizon : (N : ℝ)*(T/N) = T := by field_simp
-    have H := driven_euler_uniform_bound (Y := fun j ↦ Y N j ω) hδ hT horizon hγ hγ0 hu hω.2.2.1 hω.2.2.2
-      (hY0 N ω) (fun j hj ↦ hY N hN j hj ω)
-    have he : meshMaxError T X Y N ω ≤ 2*T*γ T := initialMax_le_iff.mpr H
-    rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
-    exact pow_le_pow_left₀ (meshMaxError_nonneg T X Y N ω) he 2
-  have hlim : ∀ᵐ ω ∂μ, Tendsto (fun N ↦ (meshMaxError T X Y N ω)^2) atTop (𝓝 (0 : ℝ)) := by
-    filter_upwards [hpaths] with ω hω
-    have H := driven_euler_uniform_convergence (Y := fun N j ↦ Y N j ω) hT hL hγ hγ0 hu hLip hω.1 hω.2.1 hω.2.2.1
-      hω.2.2.2 (fun N ↦ hY0 N ω) (fun N hN j hj ↦ hY N hN j hj ω)
-    have hzero : Tendsto (fun N ↦ meshMaxError T X Y N ω) atTop (𝓝 (0 : ℝ)) := by
-      rw [Metric.tendsto_nhds]
-      intro ε hε
-      filter_upwards [H ε hε] with N hN
-      rw [Real.dist_eq,sub_zero,abs_of_nonneg (meshMaxError_nonneg T X Y N ω)]
-      exact initialMax_lt_iff.mpr hN
-    simpa using hzero.pow 2
-  have H := tendsto_integral_filter_of_dominated_convergence (fun _ : Ω ↦ (2*T*γ T)^2)
-    (Eventually.of_forall hmeas) hbound (integrable_const _) hlim
-  simpa using H
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.InitialMax
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+
+noncomputable def meshMaxError {Ω : Type*} (T : ℝ) (X : ℝ → Ω → ℝ)
+    (Y : ℕ → ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : ℝ :=
+  initialMax (fun j ↦ |Y N j ω-X ((j : ℝ)*(T/N)) ω|) N
+
+lemma meshMaxError_nonneg {Ω : Type*} (T : ℝ) (X : ℝ → Ω → ℝ)
+    (Y : ℕ → ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : 0 ≤ meshMaxError T X Y N ω :=
+  (abs_nonneg _).trans (le_initialMax
+    (f := fun j ↦ |Y N j ω-X ((j : ℝ)*(T/N)) ω|) (Nat.zero_le N))
+
+lemma driven_euler_L2_convergence {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    [IsFiniteMeasure μ] {T L : ℝ} {u : ℝ → ℝ → ℝ} {γ : ℝ → ℝ}
+    {W X : ℝ → Ω → ℝ} {Y : ℕ → ℕ → Ω → ℝ}
+    (hT : 0 ≤ T) (hL : 0 ≤ L)
+    (hγ : MonotoneOn γ (Icc (0 : ℝ) T)) (hγ0 : 0 ≤ γ 0)
+    (hu : ∀ t ∈ Icc (0 : ℝ) T, ∀ x, |u t x| ≤ 1)
+    (hLip : ∀ s ∈ Icc (0 : ℝ) T, ∀ t ∈ Icc (0 : ℝ) T, ∀ x y,
+      |u s x-u t y| ≤ L*(|s-t|+|x-y|))
+    (hXM : ∀ t, AEStronglyMeasurable (X t) μ)
+    (hYM : ∀ N j, AEStronglyMeasurable (Y N j) μ)
+    (hpaths : ∀ᵐ ω ∂μ,
+      ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      (∀ t ∈ Icc (0 : ℝ) T,
+        X t ω = W t ω + ∫ s in (0 : ℝ)..t, γ s*u s (X s ω)) ∧ X 0 ω = 0)
+    (hY0 : ∀ N ω, Y N 0 ω = 0)
+    (hY : ∀ N > 0, ∀ j < N, ∀ ω,
+      Y N (j+1) ω = Y N j ω + W (((j+1 : ℕ) : ℝ)*(T/N)) ω-W ((j : ℝ)*(T/N)) ω +
+        (T/N)*γ ((j : ℝ)*(T/N))*u ((j : ℝ)*(T/N)) (Y N j ω)) :
+    Tendsto (fun N ↦ ∫ ω, (meshMaxError T X Y N ω)^2 ∂μ) atTop (𝓝 (0 : ℝ)) := by
+  have hmeas : ∀ N, AEStronglyMeasurable (fun ω ↦ (meshMaxError T X Y N ω)^2) μ := by
+    intro N
+    exact (aestronglyMeasurable_initialMax (fun j ↦ ((hYM N j).sub (hXM _)).norm) N).fun_pow 2
+  have hbound : ∀ᶠ N in atTop, ∀ᵐ ω ∂μ,
+      ‖(meshMaxError T X Y N ω)^2‖ ≤ (2*T*γ T)^2 := by
+    filter_upwards [eventually_gt_atTop 0] with N hN
+    filter_upwards [hpaths] with ω hω
+    have hδ : 0 ≤ T/(N : ℝ) := div_nonneg hT (Nat.cast_nonneg _)
+    have horizon : (N : ℝ)*(T/N) = T := by field_simp
+    have H := driven_euler_uniform_bound (Y := fun j ↦ Y N j ω) hδ hT horizon hγ hγ0 hu hω.2.2.1 hω.2.2.2
+      (hY0 N ω) (fun j hj ↦ hY N hN j hj ω)
+    have he : meshMaxError T X Y N ω ≤ 2*T*γ T := initialMax_le_iff.mpr H
+    rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
+    exact pow_le_pow_left₀ (meshMaxError_nonneg T X Y N ω) he 2
+  have hlim : ∀ᵐ ω ∂μ, Tendsto (fun N ↦ (meshMaxError T X Y N ω)^2) atTop (𝓝 (0 : ℝ)) := by
+    filter_upwards [hpaths] with ω hω
+    have H := driven_euler_uniform_convergence (Y := fun N j ↦ Y N j ω) hT hL hγ hγ0 hu hLip hω.1 hω.2.1 hω.2.2.1
+      hω.2.2.2 (fun N ↦ hY0 N ω) (fun N hN j hj ↦ hY N hN j hj ω)
+    have hzero : Tendsto (fun N ↦ meshMaxError T X Y N ω) atTop (𝓝 (0 : ℝ)) := by
+      rw [Metric.tendsto_nhds]
+      intro ε hε
+      filter_upwards [H ε hε] with N hN
+      rw [Real.dist_eq,sub_zero,abs_of_nonneg (meshMaxError_nonneg T X Y N ω)]
+      exact initialMax_lt_iff.mpr hN
+    simpa using hzero.pow 2
+  have H := tendsto_integral_filter_of_dominated_convergence (fun _ : Ω ↦ (2*T*γ T)^2)
+    (Eventually.of_forall hmeas) hbound (integrable_const _) hlim
+  simpa using H
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/MeshMeanBound.lean b/lean/OAI/Probability/SKValue/Processes/MeshMeanBound.lean
index a83f5a6b2a41138925ca1ecbd9c11fec9bf248c8..e13f7024a3e52cfcb652df87812509325b14e86f 100644
--- a/lean/OAI/Probability/SKValue/Processes/MeshMeanBound.lean
+++ b/lean/OAI/Probability/SKValue/Processes/MeshMeanBound.lean
@@ -1,145 +1,145 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.ShiftedPairing
-import OAI.Probability.SKValue.Evolution.UniformGridRiemannBound
-import OAI.Probability.SKValue.Processes.MeshRaw
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-
-lemma mesh_mean_bound {T D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0≤T) (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
-    {N : ℕ} (hN : 0<N) {j : ℕ} (hj : j≤N) :
-    |∫ z, meshRaw T N γ u j z ∂gaussianProduct (Fin (N+1))|≤D := by
-  have := gaussianProduct_probability (Fin (N+1))
-  simpa [Real.norm_eq_abs] using
-    (norm_integral_le_of_norm_le_const (μ := gaussianProduct (Fin (N+1)))
-      (Eventually.of_forall (fun z ↦ show ‖meshRaw T N γ u j z‖≤D from by
-        simpa only [Real.norm_eq_abs, meshRaw] using hD _ (mesh_time_mem hT hN hj) _)))
-
-lemma product_perturbation_bound {a b f D r : ℝ} (hD : 0≤D) (_ : 0≤r)
-    (ha : |a|≤D) (hb : |b-1|≤r) (hf : |a-f|≤r) : |a*b-f|≤(D+1)*r := by
-  calc
-    |a*b-f| = |a*(b-1)+(a-f)| := by congr 1; ring
-    _ ≤ |a*(b-1)|+|a-f| := abs_add_le _ _
-    _ ≤ D*r+r := by
-      rw [abs_mul]
-      exact add_le_add (mul_le_mul ha hb (abs_nonneg _) hD) hf
-    _ = _ := by ring
-
-theorem finiteGaussian_from_strip_and_moments
-    {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {f : ℝ → ℝ} {r : ℕ → ℝ}
-    (hT : 0<T) (hT1 : T≤1) (h : GradientStrip T γ u K L) (hD0 : 0≤D)
-    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
-    (hf : ContinuousOn f (Icc (0 : ℝ) T)) (hr : ∀ N, 0≤r N)
-    (hr0 : Tendsto r atTop (𝓝 0))
-    (hq : ∀ N, 0<N → ∀ j : Fin N,
-      |(∫ z, (meshRaw T N γ u j z)^2 ∂gaussianProduct (Fin (N+1)))-1|≤r N)
-    (hm : ∀ N, 0<N → ∀ j : Fin N,
-      |(∫ z, meshRaw T N γ u j z ∂gaussianProduct (Fin (N+1)))-f (meshTime T N j)|≤r N) :
-    (∀ᶠ N in atTop, ∀ j<N, 0<predictableNormalizer (meshRaw T N γ u j)
-      (gaussianProduct (Fin (N+1)))) ∧
-    Tendsto (fun N ↦ finiteShiftedValue T N γ u) atTop (𝓝 (∫ t in (0 : ℝ)..T, f t)) := by
-  let q (N j : ℕ) := ∫ z, (meshRaw T N γ u j z)^2 ∂gaussianProduct (Fin (N+1))
-  have hqpos : ∀ᶠ N in atTop, ∀ j<N, 0<q N j :=
-    uniform_moment_positive hr0 (fun N j hj ↦ hq N (by omega) ⟨j,hj⟩)
-  constructor
-  · filter_upwards [hqpos] with N hN j hj
-    exact predictableNormalizer_pos (hN j hj)
-  · let A (N : ℕ) (i : Fin (N+1)) := meshA T (N+1) γ u i
-    let B (N : ℕ) (i : Fin (N+1)) := meshB T (N+1) γ u i
-    let α (N : ℕ) (i : Fin (N+1)) := Real.sqrt (stepSize T (N+1))/
-      predictableNormalizer (meshRaw T (N+1) γ u i) (gaussianProduct (Fin ((N+1)+1)))
-    let β (N : ℕ) (i : Fin (N+1)) := Real.sqrt (stepSize T (N+1))*
-      ∫ z, meshRaw T (N+1) γ u i z ∂gaussianProduct (Fin ((N+1)+1))
-    let bstar (N : ℕ) := meshB T (N+1) γ u (N+1)
-    let e (N : ℕ) := gradientMeshError T (N+1) γ u
-    let rend (N : ℕ) := Real.sqrt (stepSize T (N+1))*D
-    have hproj : ∀ᶠ N in atTop,
-        (∑ i, (A N i-α N i)^2)≤e N ∧ (∑ i, (B N i-β N i)^2)≤e N := by
-      filter_upwards [(tendsto_add_atTop_nat 1).eventually hqpos] with N hN
-      exact mesh_coefficient_errors hT.le h hD (Nat.succ_pos N) (fun i ↦ hN i i.isLt)
-    have hnorm (N : ℕ) : (∑ i, (B N i)^2)≤1 ∧ |bstar N|≤1 :=
-      mesh_coefficients_norms hT.le h (Nat.succ_pos N)
-    have hαnorm (N : ℕ) : (∑ i, (α N i)^2)≤(Real.sqrt T*D)^2 :=
-      mesh_leading_norm hT.le h hD0 hD (Nat.succ_pos N)
-    have hend (N : ℕ) : |α N (Fin.last N)|≤rend N := by
-      have := gaussianProduct_probability (Fin ((N+1)+1))
-      exact sqrt_div_normalizer_bound
-        (mesh_raw_memLp hT.le h hD (Nat.succ_pos N) (Nat.le_succ N)) hD0
-        (Eventually.of_forall (fun z ↦ hD _ (mesh_time_mem hT.le (Nat.succ_pos N) (Nat.le_succ N)) _))
-    have he0 : Tendsto e atTop (𝓝 0) :=
-      (gradientMeshError_tendsto_zero hT hT1 h).comp (tendsto_add_atTop_nat 1)
-    have hδ0 : Tendsto (fun N : ℕ ↦ stepSize T (N+1)) atTop (𝓝 0) :=
-      (tendsto_const_div_atTop_nhds_zero_nat T).comp (tendsto_add_atTop_nat 1)
-    have hrend0 : Tendsto rend atTop (𝓝 0) := by
-      simpa only [rend, Function.comp_def, Real.sqrt_zero, zero_mul] using
-        ((Real.continuous_sqrt.tendsto 0).comp hδ0).mul_const D
-    have herr := shiftedPairing_tendsto_error_zero
-      (fun N ↦ gradientMeshError_nonneg T (N+1) γ u) (mul_nonneg (Real.sqrt_nonneg T) hD0)
-      (hproj.mono (fun _ hN ↦ hN.1)) (hproj.mono (fun _ hN ↦ hN.2))
-      (Eventually.of_forall (fun N ↦ (hnorm N).1)) (Eventually.of_forall hαnorm)
-      (Eventually.of_forall (fun N ↦ (hnorm N).2)) (Eventually.of_forall hend) he0 hrend0
-    let v (N : ℕ) (i : Fin N) :=
-      (∫ z, meshRaw T (N+1) γ u (i+1) z ∂gaussianProduct (Fin ((N+1)+1)))*Real.sqrt (q (N+1) i)
-    have hv (N : ℕ) (i : Fin N) :
-        |v N i-f ((i+1 : ℕ)*(T/(N+1)))|≤(D+1)*r (N+1) := by
-      apply product_perturbation_bound hD0 (hr (N+1))
-      · exact mesh_mean_bound hT.le hD (Nat.succ_pos N) (by omega)
-      · exact (abs_sqrt_sub_one_le (integral_nonneg (fun z ↦ sq_nonneg _))).trans
-          (hq (N+1) (Nat.succ_pos N) i.castSucc)
-      · simpa only [meshTime, stepSize, Fin.val_succ, Nat.cast_add, Nat.cast_one] using
-          hm (N+1) (Nat.succ_pos N) i.succ
-    have hv0 : Tendsto (fun N ↦ (D+1)*r (N+1)) atTop (𝓝 0) := by
-      simpa only [mul_zero, Function.comp_def] using (hr0.comp (tendsto_add_atTop_nat 1)).const_mul (D+1)
-    have hvlim := perturbed_shifted_grid_tendsto hT hf
-      (fun N ↦ mul_nonneg (by positivity : 0≤D+1) (hr (N+1))) hv0 (Eventually.of_forall hv)
-    have hsum (N : ℕ) : (∑ i : Fin N, α N i.castSucc*β N i.succ)=
-        (T/(N+1))*∑ i : Fin N, v N i := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro i _
-      have hδ : 0 ≤ stepSize T (N+1) := div_nonneg hT.le (Nat.cast_nonneg _)
-      dsimp only [α, β, v, q, predictableNormalizer]
-      rw [div_inv_eq_mul]
-      calc
-        _ = (Real.sqrt (stepSize T (N+1)))^2 *
-          ((∫ z, meshRaw T (N+1) γ u (i+1) z ∂gaussianProduct (Fin ((N+1)+1)))*
-            Real.sqrt (∫ z, (meshRaw T (N+1) γ u i z)^2 ∂gaussianProduct (Fin ((N+1)+1)))) := by
-              simp only [Fin.val_castSucc, Fin.val_succ]
-              ring
-        _ = _ := by rw [Real.sq_sqrt hδ]; simp only [stepSize, Nat.cast_add, Nat.cast_one]
-    have hp (N : ℕ) : shiftedPairing (A N) (B N) (bstar N)=finiteShiftedValue T (N+1) γ u := by
-      simp only [shiftedPairing, finiteShiftedValue, A, B, bstar]
-      rw [Fin.sum_univ_castSucc]
-      simp only [Fin.val_castSucc, Fin.val_succ, Fin.val_last]
-    simp_rw [hp, hsum] at herr
-    have hlim := herr.add hvlim
-    simp only [sub_add_cancel, zero_add] at hlim
-    exact (tendsto_add_atTop_iff_nat 1).mp hlim
-
-end SKValue
-
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.ShiftedPairing
+import OAI.Probability.SKValue.Evolution.UniformGridRiemannBound
+import OAI.Probability.SKValue.Processes.MeshRaw
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+
+lemma mesh_mean_bound {T D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0≤T) (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
+    {N : ℕ} (hN : 0<N) {j : ℕ} (hj : j≤N) :
+    |∫ z, meshRaw T N γ u j z ∂gaussianProduct (Fin (N+1))|≤D := by
+  have := gaussianProduct_probability (Fin (N+1))
+  simpa [Real.norm_eq_abs] using
+    (norm_integral_le_of_norm_le_const (μ := gaussianProduct (Fin (N+1)))
+      (Eventually.of_forall (fun z ↦ show ‖meshRaw T N γ u j z‖≤D from by
+        simpa only [Real.norm_eq_abs, meshRaw] using hD _ (mesh_time_mem hT hN hj) _)))
+
+lemma product_perturbation_bound {a b f D r : ℝ} (hD : 0≤D) (_ : 0≤r)
+    (ha : |a|≤D) (hb : |b-1|≤r) (hf : |a-f|≤r) : |a*b-f|≤(D+1)*r := by
+  calc
+    |a*b-f| = |a*(b-1)+(a-f)| := by congr 1; ring
+    _ ≤ |a*(b-1)|+|a-f| := abs_add_le _ _
+    _ ≤ D*r+r := by
+      rw [abs_mul]
+      exact add_le_add (mul_le_mul ha hb (abs_nonneg _) hD) hf
+    _ = _ := by ring
+
+theorem finiteGaussian_from_strip_and_moments
+    {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ} {f : ℝ → ℝ} {r : ℕ → ℝ}
+    (hT : 0<T) (hT1 : T≤1) (h : GradientStrip T γ u K L) (hD0 : 0≤D)
+    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
+    (hf : ContinuousOn f (Icc (0 : ℝ) T)) (hr : ∀ N, 0≤r N)
+    (hr0 : Tendsto r atTop (𝓝 0))
+    (hq : ∀ N, 0<N → ∀ j : Fin N,
+      |(∫ z, (meshRaw T N γ u j z)^2 ∂gaussianProduct (Fin (N+1)))-1|≤r N)
+    (hm : ∀ N, 0<N → ∀ j : Fin N,
+      |(∫ z, meshRaw T N γ u j z ∂gaussianProduct (Fin (N+1)))-f (meshTime T N j)|≤r N) :
+    (∀ᶠ N in atTop, ∀ j<N, 0<predictableNormalizer (meshRaw T N γ u j)
+      (gaussianProduct (Fin (N+1)))) ∧
+    Tendsto (fun N ↦ finiteShiftedValue T N γ u) atTop (𝓝 (∫ t in (0 : ℝ)..T, f t)) := by
+  let q (N j : ℕ) := ∫ z, (meshRaw T N γ u j z)^2 ∂gaussianProduct (Fin (N+1))
+  have hqpos : ∀ᶠ N in atTop, ∀ j<N, 0<q N j :=
+    uniform_moment_positive hr0 (fun N j hj ↦ hq N (by omega) ⟨j,hj⟩)
+  constructor
+  · filter_upwards [hqpos] with N hN j hj
+    exact predictableNormalizer_pos (hN j hj)
+  · let A (N : ℕ) (i : Fin (N+1)) := meshA T (N+1) γ u i
+    let B (N : ℕ) (i : Fin (N+1)) := meshB T (N+1) γ u i
+    let α (N : ℕ) (i : Fin (N+1)) := Real.sqrt (stepSize T (N+1))/
+      predictableNormalizer (meshRaw T (N+1) γ u i) (gaussianProduct (Fin ((N+1)+1)))
+    let β (N : ℕ) (i : Fin (N+1)) := Real.sqrt (stepSize T (N+1))*
+      ∫ z, meshRaw T (N+1) γ u i z ∂gaussianProduct (Fin ((N+1)+1))
+    let bstar (N : ℕ) := meshB T (N+1) γ u (N+1)
+    let e (N : ℕ) := gradientMeshError T (N+1) γ u
+    let rend (N : ℕ) := Real.sqrt (stepSize T (N+1))*D
+    have hproj : ∀ᶠ N in atTop,
+        (∑ i, (A N i-α N i)^2)≤e N ∧ (∑ i, (B N i-β N i)^2)≤e N := by
+      filter_upwards [(tendsto_add_atTop_nat 1).eventually hqpos] with N hN
+      exact mesh_coefficient_errors hT.le h hD (Nat.succ_pos N) (fun i ↦ hN i i.isLt)
+    have hnorm (N : ℕ) : (∑ i, (B N i)^2)≤1 ∧ |bstar N|≤1 :=
+      mesh_coefficients_norms hT.le h (Nat.succ_pos N)
+    have hαnorm (N : ℕ) : (∑ i, (α N i)^2)≤(Real.sqrt T*D)^2 :=
+      mesh_leading_norm hT.le h hD0 hD (Nat.succ_pos N)
+    have hend (N : ℕ) : |α N (Fin.last N)|≤rend N := by
+      have := gaussianProduct_probability (Fin ((N+1)+1))
+      exact sqrt_div_normalizer_bound
+        (mesh_raw_memLp hT.le h hD (Nat.succ_pos N) (Nat.le_succ N)) hD0
+        (Eventually.of_forall (fun z ↦ hD _ (mesh_time_mem hT.le (Nat.succ_pos N) (Nat.le_succ N)) _))
+    have he0 : Tendsto e atTop (𝓝 0) :=
+      (gradientMeshError_tendsto_zero hT hT1 h).comp (tendsto_add_atTop_nat 1)
+    have hδ0 : Tendsto (fun N : ℕ ↦ stepSize T (N+1)) atTop (𝓝 0) :=
+      (tendsto_const_div_atTop_nhds_zero_nat T).comp (tendsto_add_atTop_nat 1)
+    have hrend0 : Tendsto rend atTop (𝓝 0) := by
+      simpa only [rend, Function.comp_def, Real.sqrt_zero, zero_mul] using
+        ((Real.continuous_sqrt.tendsto 0).comp hδ0).mul_const D
+    have herr := shiftedPairing_tendsto_error_zero
+      (fun N ↦ gradientMeshError_nonneg T (N+1) γ u) (mul_nonneg (Real.sqrt_nonneg T) hD0)
+      (hproj.mono (fun _ hN ↦ hN.1)) (hproj.mono (fun _ hN ↦ hN.2))
+      (Eventually.of_forall (fun N ↦ (hnorm N).1)) (Eventually.of_forall hαnorm)
+      (Eventually.of_forall (fun N ↦ (hnorm N).2)) (Eventually.of_forall hend) he0 hrend0
+    let v (N : ℕ) (i : Fin N) :=
+      (∫ z, meshRaw T (N+1) γ u (i+1) z ∂gaussianProduct (Fin ((N+1)+1)))*Real.sqrt (q (N+1) i)
+    have hv (N : ℕ) (i : Fin N) :
+        |v N i-f ((i+1 : ℕ)*(T/(N+1)))|≤(D+1)*r (N+1) := by
+      apply product_perturbation_bound hD0 (hr (N+1))
+      · exact mesh_mean_bound hT.le hD (Nat.succ_pos N) (by omega)
+      · exact (abs_sqrt_sub_one_le (integral_nonneg (fun z ↦ sq_nonneg _))).trans
+          (hq (N+1) (Nat.succ_pos N) i.castSucc)
+      · simpa only [meshTime, stepSize, Fin.val_succ, Nat.cast_add, Nat.cast_one] using
+          hm (N+1) (Nat.succ_pos N) i.succ
+    have hv0 : Tendsto (fun N ↦ (D+1)*r (N+1)) atTop (𝓝 0) := by
+      simpa only [mul_zero, Function.comp_def] using (hr0.comp (tendsto_add_atTop_nat 1)).const_mul (D+1)
+    have hvlim := perturbed_shifted_grid_tendsto hT hf
+      (fun N ↦ mul_nonneg (by positivity : 0≤D+1) (hr (N+1))) hv0 (Eventually.of_forall hv)
+    have hsum (N : ℕ) : (∑ i : Fin N, α N i.castSucc*β N i.succ)=
+        (T/(N+1))*∑ i : Fin N, v N i := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro i _
+      have hδ : 0 ≤ stepSize T (N+1) := div_nonneg hT.le (Nat.cast_nonneg _)
+      dsimp only [α, β, v, q, predictableNormalizer]
+      rw [div_inv_eq_mul]
+      calc
+        _ = (Real.sqrt (stepSize T (N+1)))^2 *
+          ((∫ z, meshRaw T (N+1) γ u (i+1) z ∂gaussianProduct (Fin ((N+1)+1)))*
+            Real.sqrt (∫ z, (meshRaw T (N+1) γ u i z)^2 ∂gaussianProduct (Fin ((N+1)+1)))) := by
+              simp only [Fin.val_castSucc, Fin.val_succ]
+              ring
+        _ = _ := by rw [Real.sq_sqrt hδ]; simp only [stepSize, Nat.cast_add, Nat.cast_one]
+    have hp (N : ℕ) : shiftedPairing (A N) (B N) (bstar N)=finiteShiftedValue T (N+1) γ u := by
+      simp only [shiftedPairing, finiteShiftedValue, A, B, bstar]
+      rw [Fin.sum_univ_castSucc]
+      simp only [Fin.val_castSucc, Fin.val_succ, Fin.val_last]
+    simp_rw [hp, hsum] at herr
+    have hlim := herr.add hvlim
+    simp only [sub_add_cancel, zero_add] at hlim
+    exact (tendsto_add_atTop_iff_nat 1).mp hlim
+
+end SKValue
+
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/MeshRaw.lean b/lean/OAI/Probability/SKValue/Processes/MeshRaw.lean
index c5c337eaba3e049b3291980436df28b74f9fd6a8..95db26407243507496ffcafee9abaacbcd514ea2 100644
--- a/lean/OAI/Probability/SKValue/Processes/MeshRaw.lean
+++ b/lean/OAI/Probability/SKValue/Processes/MeshRaw.lean
@@ -1,143 +1,143 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.FiniteRounded
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-
-noncomputable def meshRaw (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
-  deriv (u (meshTime T N j)) (euler T N γ u z j)
-
-noncomputable def meshTerminal (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (z : Fin (N+1) → ℝ) : ℝ := u T (euler T N γ u z N)
-
-noncomputable def meshA (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (j : ℕ) : ℝ :=
-  ∫ z, finiteRounded (meshTerminal T N γ u) z *
-    normalizedIncrement (meshRaw T N γ u j) (coordinate N j) (gaussianProduct (Fin (N+1))) z
-    ∂gaussianProduct (Fin (N+1))
-
-noncomputable def meshB (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
-    (j : ℕ) : ℝ :=
-  ∫ z, finiteRounded (meshTerminal T N γ u) z * coordinate N j z ∂gaussianProduct (Fin (N+1))
-
-noncomputable def finiteShiftedValue (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) : ℝ :=
-  ∑ i : Fin N, meshA T N γ u i * meshB T N γ u (i+1)
-
-lemma mesh_time_mem {T : ℝ} (hT : 0≤T) {N j : ℕ} (hN : 0<N) (hj : j≤N) :
-    meshTime T N j∈Icc (0 : ℝ) T := by
-  apply mesh_mem_strip_ito (δ := stepSize T N) (div_nonneg hT (Nat.cast_nonneg N)) _ hj
-  have hNr : (N : ℝ)≠0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hN)
-  unfold stepSize
-  field_simp
-
-lemma mesh_terminal_measurable {T K L : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0≤T) (h : GradientStrip T γ u K L) {N : ℕ} (hN : 0<N) :
-    Measurable (meshTerminal T N γ u) :=
-  (h.smooth T ⟨hT,le_rfl⟩).continuous.measurable.comp
-    (measurable_euler T N γ u (fun _ hj ↦
-      (h.smooth _ (mesh_time_mem hT hN hj.le)).continuous.measurable) N le_rfl)
-
-lemma mesh_raw_measurable {T K L : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0≤T) (h : GradientStrip T γ u K L) {N : ℕ} (hN : 0<N) {j : ℕ} (hj : j≤N) :
-    Measurable (meshRaw T N γ u j) :=
-  ((h.smooth _ (mesh_time_mem hT hN hj)).continuous_deriv (by norm_num)).measurable.comp
-    (measurable_euler T N γ u (fun k hk ↦
-      (h.smooth _ (mesh_time_mem hT hN hk.le)).continuous.measurable) j hj)
-
-lemma mesh_raw_depends (T : ℝ) {N j : ℕ} (hj : j≤N) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) :
-    DependsBefore j (meshRaw T N γ u j) :=
-  eulerCoefficient_dependsBefore T hj γ u (fun t ↦ deriv (u t))
-
-lemma mesh_terminal_depends (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) :
-    DependsBefore N (meshTerminal T N γ u) :=
-  eulerCoefficient_dependsBefore T le_rfl γ u (fun _ ↦ u T)
-
-lemma mesh_raw_memLp {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0≤T) (h : GradientStrip T γ u K L)
-    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
-    {N : ℕ} (hN : 0<N) {j : ℕ} (hj : j≤N) :
-    MemLp (meshRaw T N γ u j) 2 (gaussianProduct (Fin (N+1))) := by
-  have := gaussianProduct_probability (Fin (N+1))
-  apply MemLp.of_bound (mesh_raw_measurable hT h hN hj).aestronglyMeasurable D
-  exact Eventually.of_forall (fun z ↦ by
-    simpa only [meshRaw, Real.norm_eq_abs] using hD _ (mesh_time_mem hT hN hj) (euler T N γ u z j))
-
-lemma mesh_coefficient_errors {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0≤T) (h : GradientStrip T γ u K L)
-    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
-    {N : ℕ} (hN : 0<N)
-    (hpos : ∀ j : Fin N, 0<∫ z, (meshRaw T N γ u j z)^2 ∂gaussianProduct (Fin (N+1))) :
-    (∑ j : Fin N, (meshA T N γ u j-Real.sqrt (stepSize T N)/
-      predictableNormalizer (meshRaw T N γ u j) (gaussianProduct (Fin (N+1))))^2)≤
-      gradientMeshError T N γ u ∧
-    (∑ j : Fin N, (meshB T N γ u j-Real.sqrt (stepSize T N)*
-      ∫ z, meshRaw T N γ u j z ∂gaussianProduct (Fin (N+1)))^2)≤
-      gradientMeshError T N γ u := by
-  exact finite_coefficient_L2_errors
-    (fun j ↦ mesh_raw_measurable hT h hN j.isLt.le)
-    (fun j ↦ mesh_raw_depends T j.isLt.le γ u)
-    (fun j ↦ mesh_raw_memLp hT h hD hN j.isLt.le) hpos
-    (mesh_terminal_measurable hT h hN) (mesh_terminal_depends T N γ u)
-    (fun z ↦ h.bounded T ⟨hT,le_rfl⟩ _)
-
-lemma mesh_coefficients_norms {T K L : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0≤T) (h : GradientStrip T γ u K L) {N : ℕ} (hN : 0<N) :
-    (∑ j : Fin N, (meshB T N γ u j)^2)≤1 ∧ |meshB T N γ u N|≤1 :=
-  finiteRounded_coordinate_norms (mesh_terminal_measurable hT h hN)
-
-lemma mesh_leading_norm {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
-    (hT : 0≤T) (h : GradientStrip T γ u K L) (hD0 : 0≤D)
-    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
-    {N : ℕ} (hN : 0<N) :
-    (∑ j : Fin N, (Real.sqrt (stepSize T N)/
-      predictableNormalizer (meshRaw T N γ u j) (gaussianProduct (Fin (N+1))))^2)≤
-      (Real.sqrt T*D)^2 := by
-  have := gaussianProduct_probability (Fin (N+1))
-  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
-  calc
-    _ ≤ ∑ j : Fin N, (Real.sqrt (stepSize T N)*D)^2 := by
-      apply Finset.sum_le_sum
-      intro j _
-      rw [←sq_abs (Real.sqrt _ / _)]
-      apply (sq_le_sq₀ (abs_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) hD0)).2
-      exact sqrt_div_normalizer_bound (mesh_raw_memLp hT h hD hN j.isLt.le) hD0
-        (Eventually.of_forall (fun z ↦ hD _ (mesh_time_mem hT hN j.isLt.le) _))
-    _ = (Real.sqrt T*D)^2 := by
-      simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, mul_pow]
-      rw [Real.sq_sqrt hδ, Real.sq_sqrt hT]
-      unfold stepSize
-      have hNr : (N : ℝ)≠0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hN)
-      field_simp
-
-lemma abs_sqrt_sub_one_le {q : ℝ} (hq : 0≤q) : |Real.sqrt q-1|≤|q-1| := by
-  have heq : |q-1|=|Real.sqrt q-1| * (Real.sqrt q+1) := by
-    rw [←abs_of_nonneg (by positivity : 0≤Real.sqrt q+1), ←abs_mul]
-    congr 1
-    nlinarith [Real.sq_sqrt hq]
-  rw [heq]
-  nlinarith [Real.sqrt_nonneg q, abs_nonneg (Real.sqrt q-1)]
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.FiniteRounded
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+
+noncomputable def meshRaw (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (j : ℕ) (z : Fin (N+1) → ℝ) : ℝ :=
+  deriv (u (meshTime T N j)) (euler T N γ u z j)
+
+noncomputable def meshTerminal (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (z : Fin (N+1) → ℝ) : ℝ := u T (euler T N γ u z N)
+
+noncomputable def meshA (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (j : ℕ) : ℝ :=
+  ∫ z, finiteRounded (meshTerminal T N γ u) z *
+    normalizedIncrement (meshRaw T N γ u j) (coordinate N j) (gaussianProduct (Fin (N+1))) z
+    ∂gaussianProduct (Fin (N+1))
+
+noncomputable def meshB (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ)
+    (j : ℕ) : ℝ :=
+  ∫ z, finiteRounded (meshTerminal T N γ u) z * coordinate N j z ∂gaussianProduct (Fin (N+1))
+
+noncomputable def finiteShiftedValue (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) : ℝ :=
+  ∑ i : Fin N, meshA T N γ u i * meshB T N γ u (i+1)
+
+lemma mesh_time_mem {T : ℝ} (hT : 0≤T) {N j : ℕ} (hN : 0<N) (hj : j≤N) :
+    meshTime T N j∈Icc (0 : ℝ) T := by
+  apply mesh_mem_strip_ito (δ := stepSize T N) (div_nonneg hT (Nat.cast_nonneg N)) _ hj
+  have hNr : (N : ℝ)≠0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hN)
+  unfold stepSize
+  field_simp
+
+lemma mesh_terminal_measurable {T K L : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0≤T) (h : GradientStrip T γ u K L) {N : ℕ} (hN : 0<N) :
+    Measurable (meshTerminal T N γ u) :=
+  (h.smooth T ⟨hT,le_rfl⟩).continuous.measurable.comp
+    (measurable_euler T N γ u (fun _ hj ↦
+      (h.smooth _ (mesh_time_mem hT hN hj.le)).continuous.measurable) N le_rfl)
+
+lemma mesh_raw_measurable {T K L : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0≤T) (h : GradientStrip T γ u K L) {N : ℕ} (hN : 0<N) {j : ℕ} (hj : j≤N) :
+    Measurable (meshRaw T N γ u j) :=
+  ((h.smooth _ (mesh_time_mem hT hN hj)).continuous_deriv (by norm_num)).measurable.comp
+    (measurable_euler T N γ u (fun k hk ↦
+      (h.smooth _ (mesh_time_mem hT hN hk.le)).continuous.measurable) j hj)
+
+lemma mesh_raw_depends (T : ℝ) {N j : ℕ} (hj : j≤N) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) :
+    DependsBefore j (meshRaw T N γ u j) :=
+  eulerCoefficient_dependsBefore T hj γ u (fun t ↦ deriv (u t))
+
+lemma mesh_terminal_depends (T : ℝ) (N : ℕ) (γ : ℝ → ℝ) (u : ℝ → ℝ → ℝ) :
+    DependsBefore N (meshTerminal T N γ u) :=
+  eulerCoefficient_dependsBefore T le_rfl γ u (fun _ ↦ u T)
+
+lemma mesh_raw_memLp {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0≤T) (h : GradientStrip T γ u K L)
+    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
+    {N : ℕ} (hN : 0<N) {j : ℕ} (hj : j≤N) :
+    MemLp (meshRaw T N γ u j) 2 (gaussianProduct (Fin (N+1))) := by
+  have := gaussianProduct_probability (Fin (N+1))
+  apply MemLp.of_bound (mesh_raw_measurable hT h hN hj).aestronglyMeasurable D
+  exact Eventually.of_forall (fun z ↦ by
+    simpa only [meshRaw, Real.norm_eq_abs] using hD _ (mesh_time_mem hT hN hj) (euler T N γ u z j))
+
+lemma mesh_coefficient_errors {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0≤T) (h : GradientStrip T γ u K L)
+    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
+    {N : ℕ} (hN : 0<N)
+    (hpos : ∀ j : Fin N, 0<∫ z, (meshRaw T N γ u j z)^2 ∂gaussianProduct (Fin (N+1))) :
+    (∑ j : Fin N, (meshA T N γ u j-Real.sqrt (stepSize T N)/
+      predictableNormalizer (meshRaw T N γ u j) (gaussianProduct (Fin (N+1))))^2)≤
+      gradientMeshError T N γ u ∧
+    (∑ j : Fin N, (meshB T N γ u j-Real.sqrt (stepSize T N)*
+      ∫ z, meshRaw T N γ u j z ∂gaussianProduct (Fin (N+1)))^2)≤
+      gradientMeshError T N γ u := by
+  exact finite_coefficient_L2_errors
+    (fun j ↦ mesh_raw_measurable hT h hN j.isLt.le)
+    (fun j ↦ mesh_raw_depends T j.isLt.le γ u)
+    (fun j ↦ mesh_raw_memLp hT h hD hN j.isLt.le) hpos
+    (mesh_terminal_measurable hT h hN) (mesh_terminal_depends T N γ u)
+    (fun z ↦ h.bounded T ⟨hT,le_rfl⟩ _)
+
+lemma mesh_coefficients_norms {T K L : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0≤T) (h : GradientStrip T γ u K L) {N : ℕ} (hN : 0<N) :
+    (∑ j : Fin N, (meshB T N γ u j)^2)≤1 ∧ |meshB T N γ u N|≤1 :=
+  finiteRounded_coordinate_norms (mesh_terminal_measurable hT h hN)
+
+lemma mesh_leading_norm {T K L D : ℝ} {γ : ℝ → ℝ} {u : ℝ → ℝ → ℝ}
+    (hT : 0≤T) (h : GradientStrip T γ u K L) (hD0 : 0≤D)
+    (hD : ∀ t∈Icc (0 : ℝ) T, ∀ x, |deriv (u t) x|≤D)
+    {N : ℕ} (hN : 0<N) :
+    (∑ j : Fin N, (Real.sqrt (stepSize T N)/
+      predictableNormalizer (meshRaw T N γ u j) (gaussianProduct (Fin (N+1))))^2)≤
+      (Real.sqrt T*D)^2 := by
+  have := gaussianProduct_probability (Fin (N+1))
+  have hδ : 0 ≤ stepSize T N := div_nonneg hT (Nat.cast_nonneg N)
+  calc
+    _ ≤ ∑ j : Fin N, (Real.sqrt (stepSize T N)*D)^2 := by
+      apply Finset.sum_le_sum
+      intro j _
+      rw [←sq_abs (Real.sqrt _ / _)]
+      apply (sq_le_sq₀ (abs_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) hD0)).2
+      exact sqrt_div_normalizer_bound (mesh_raw_memLp hT h hD hN j.isLt.le) hD0
+        (Eventually.of_forall (fun z ↦ hD _ (mesh_time_mem hT hN j.isLt.le) _))
+    _ = (Real.sqrt T*D)^2 := by
+      simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, mul_pow]
+      rw [Real.sq_sqrt hδ, Real.sq_sqrt hT]
+      unfold stepSize
+      have hNr : (N : ℝ)≠0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hN)
+      field_simp
+
+lemma abs_sqrt_sub_one_le {q : ℝ} (hq : 0≤q) : |Real.sqrt q-1|≤|q-1| := by
+  have heq : |q-1|=|Real.sqrt q-1| * (Real.sqrt q+1) := by
+    rw [←abs_of_nonneg (by positivity : 0≤Real.sqrt q+1), ←abs_mul]
+    congr 1
+    nlinarith [Real.sq_sqrt hq]
+  rw [heq]
+  nlinarith [Real.sqrt_nonneg q, abs_nonneg (Real.sqrt q-1)]
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/StripPaths.lean b/lean/OAI/Probability/SKValue/Processes/StripPaths.lean
index 2426599ee635f77db09287796f52c008335ef79d..560b8aadc4f70519ea14996a785aa7bd343a53c6 100644
--- a/lean/OAI/Probability/SKValue/Processes/StripPaths.lean
+++ b/lean/OAI/Probability/SKValue/Processes/StripPaths.lean
@@ -1,78 +1,78 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.Cutoff
-import OAI.Probability.SKValue.Equations.LipschitzCompMemLpFinite
-import OAI.Probability.SKValue.Processes.Measurable
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology BigOperators
-
-lemma IsDiffusion.strip_paths {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T Lu : ℝ}
-    (hT : 0≤T) (hT1 : T<1) (hLu : 0≤Lu)
-    (huLip : ∀ t∈Icc (0 : ℝ) T, ∀ s∈Icc (0 : ℝ) T, ∀ x y,
-      |gradient W γ s y-gradient W γ t x|≤Lu*(|s-t|+|y-x|)) :
-    ∀ᵐ ω ∂W.μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s ω)) volume 0 T ∧
-      (∀ t∈Icc (0 : ℝ) T, X t ω = W.B (Real.toNNReal t) ω+
-        ∫ s in (0 : ℝ)..t, γ.coeff s*gradient W γ s (X s ω)) ∧ X 0 ω=0 := by
-  filter_upwards [hX.2, W.brownian.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hω h0
-  have hsub : Icc (0 : ℝ) T ⊆ Icc (0 : ℝ) 1 := Icc_subset_Icc le_rfl hT1.le
-  refine ⟨hω.1.mono hsub, ?_, fun t ht ↦ hω.2 t (hsub ht), ?_⟩
-  · have hj : ContinuousOn (fun p : ℝ×ℝ ↦ gradient W γ p.1 p.2)
-        {p | p.1∈Icc (0 : ℝ) T} :=
-      derivative_joint_continuous hLu (fun s hs t ht x y ↦ huLip t ht s hs y x)
-    have hp : ContinuousOn (fun t : ℝ ↦ (t, X t ω)) (Icc (0 : ℝ) T) :=
-      continuousOn_id.prodMk (hω.1.mono hsub)
-    have hc : ContinuousOn (fun t ↦ gradient W γ t (X t ω)) (Icc (0 : ℝ) T) := by
-      exact ContinuousOn.comp (g := fun p : ℝ×ℝ ↦ gradient W γ p.1 p.2)
-        (f := fun t : ℝ ↦ (t, X t ω)) hj hp (fun t ht ↦ ht)
-    have hgi : IntervalIntegrable γ.coeff volume 0 T :=
-      γ.intervalIntegrable.mono_set (by
-        simpa only [uIcc_of_le hT, uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using
-          (Icc_subset_Icc le_rfl hT1.le))
-    exact hgi.mul_continuousOn (by simpa only [uIcc_of_le hT] using hc)
-  · simpa only [intervalIntegral.integral_same, add_zero, Real.toNNReal_zero, h0]
-      using hω.2 0 (by norm_num)
-
-lemma IsDiffusion.value_expectation {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T K L Kg Lg Lu : ℝ}
-    (hT : 0<T) (hT1 : T<1)
-    (h : ValueStrip T γ.coeff (phi W γ) K L)
-    (hg : GradientStrip T γ.coeff (gradient W γ) Kg Lg)
-    (hLu : 0≤Lu)
-    (huLip : ∀ t∈Icc (0 : ℝ) T, ∀ s∈Icc (0 : ℝ) T, ∀ x y,
-      |gradient W γ s y-gradient W γ t x|≤Lu*(|s-t|+|y-x|)) :
-    (∫ ω, phi W γ T (X T ω) ∂W.μ) = phi W γ 0 0+
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..T, γ.coeff t*(∫ ω, (gradient W γ t (X t ω))^2 ∂W.μ)) := by
-  apply diffusion_value_expectation (μ := W.μ) (B := W.B) (X := X)
-    W.brownian.toIsPreBrownianReal hT hT1.le h hg hLu
-    (fun s hs t ht x y ↦ huLip t ht s hs y x)
-  · intro t ht
-    exact hX.memLp ⟨ht.1,ht.2.trans hT1.le⟩ (by norm_num)
-  · exact hX.strip_paths hT.le hT1 hLu huLip
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.Cutoff
+import OAI.Probability.SKValue.Equations.LipschitzCompMemLpFinite
+import OAI.Probability.SKValue.Processes.Measurable
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology BigOperators
+
+lemma IsDiffusion.strip_paths {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T Lu : ℝ}
+    (hT : 0≤T) (hT1 : T<1) (hLu : 0≤Lu)
+    (huLip : ∀ t∈Icc (0 : ℝ) T, ∀ s∈Icc (0 : ℝ) T, ∀ x y,
+      |gradient W γ s y-gradient W γ t x|≤Lu*(|s-t|+|y-x|)) :
+    ∀ᵐ ω ∂W.μ, ContinuousOn (fun t ↦ X t ω) (Icc (0 : ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s ω)) volume 0 T ∧
+      (∀ t∈Icc (0 : ℝ) T, X t ω = W.B (Real.toNNReal t) ω+
+        ∫ s in (0 : ℝ)..t, γ.coeff s*gradient W γ s (X s ω)) ∧ X 0 ω=0 := by
+  filter_upwards [hX.2, W.brownian.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hω h0
+  have hsub : Icc (0 : ℝ) T ⊆ Icc (0 : ℝ) 1 := Icc_subset_Icc le_rfl hT1.le
+  refine ⟨hω.1.mono hsub, ?_, fun t ht ↦ hω.2 t (hsub ht), ?_⟩
+  · have hj : ContinuousOn (fun p : ℝ×ℝ ↦ gradient W γ p.1 p.2)
+        {p | p.1∈Icc (0 : ℝ) T} :=
+      derivative_joint_continuous hLu (fun s hs t ht x y ↦ huLip t ht s hs y x)
+    have hp : ContinuousOn (fun t : ℝ ↦ (t, X t ω)) (Icc (0 : ℝ) T) :=
+      continuousOn_id.prodMk (hω.1.mono hsub)
+    have hc : ContinuousOn (fun t ↦ gradient W γ t (X t ω)) (Icc (0 : ℝ) T) := by
+      exact ContinuousOn.comp (g := fun p : ℝ×ℝ ↦ gradient W γ p.1 p.2)
+        (f := fun t : ℝ ↦ (t, X t ω)) hj hp (fun t ht ↦ ht)
+    have hgi : IntervalIntegrable γ.coeff volume 0 T :=
+      γ.intervalIntegrable.mono_set (by
+        simpa only [uIcc_of_le hT, uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using
+          (Icc_subset_Icc le_rfl hT1.le))
+    exact hgi.mul_continuousOn (by simpa only [uIcc_of_le hT] using hc)
+  · simpa only [intervalIntegral.integral_same, add_zero, Real.toNNReal_zero, h0]
+      using hω.2 0 (by norm_num)
+
+lemma IsDiffusion.value_expectation {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T K L Kg Lg Lu : ℝ}
+    (hT : 0<T) (hT1 : T<1)
+    (h : ValueStrip T γ.coeff (phi W γ) K L)
+    (hg : GradientStrip T γ.coeff (gradient W γ) Kg Lg)
+    (hLu : 0≤Lu)
+    (huLip : ∀ t∈Icc (0 : ℝ) T, ∀ s∈Icc (0 : ℝ) T, ∀ x y,
+      |gradient W γ s y-gradient W γ t x|≤Lu*(|s-t|+|y-x|)) :
+    (∫ ω, phi W γ T (X T ω) ∂W.μ) = phi W γ 0 0+
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..T, γ.coeff t*(∫ ω, (gradient W γ t (X t ω))^2 ∂W.μ)) := by
+  apply diffusion_value_expectation (μ := W.μ) (B := W.B) (X := X)
+    W.brownian.toIsPreBrownianReal hT hT1.le h hg hLu
+    (fun s hs t ht x y ↦ huLip t ht s hs y x)
+  · intro t ht
+    exact hX.memLp ⟨ht.1,ht.2.trans hT1.le⟩ (by norm_num)
+  · exact hX.strip_paths hT.le hT1 hLu huLip
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/TerminalCondExp.lean b/lean/OAI/Probability/SKValue/Processes/TerminalCondExp.lean
index 3a6f225744f80eb7ce9e0d47fb7cc62f5867149f..aa0727e7055e207299b5b331f35e30df2bb84dea 100644
--- a/lean/OAI/Probability/SKValue/Processes/TerminalCondExp.lean
+++ b/lean/OAI/Probability/SKValue/Processes/TerminalCondExp.lean
@@ -1,105 +1,105 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.UnitMomentMartingale
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-variable {Ω : Type*} {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
-variable {ℱ : Filtration ℝ m} {f : ℝ → Ω → ℝ}
-
-lemma UnitMomentMartingale.terminal_condExp
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
-    (hUm : StronglyMeasurable[ℱ 1] U)
-    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
-      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω)))
-    {s : ℝ} (hs : s ∈ Ico (0 : ℝ) 1) : μ[U | ℱ s] =ᵐ[μ] f s := by
-  have hUi : Integrable U μ := Integrable.of_bound (hUm.mono (ℱ.le _)).aestronglyMeasurable 1
-    (hU.mono (fun ω hω ↦ (Real.norm_eq_abs _).trans_le hω.1))
-  apply Filter.EventuallyEq.symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq (ℱ.le s) hUi
-    (fun S _ _ ↦ (h.integrable hs).integrableOn) _ (h.measurable s hs).aestronglyMeasurable
-  intro S hS _
-  have hlim : Tendsto (fun n ↦ ∫ ω in S, f (terminalTime n) ω ∂μ) atTop (𝓝 (∫ ω in S, U ω ∂μ)) :=
-    tendsto_integral_of_dominated_convergence (fun _ : Ω ↦ (1 : ℝ))
-      (fun n ↦ ((h.measurable _ (terminalTime_mem n)).mono (ℱ.le _)).aestronglyMeasurable.restrict)
-      (integrable_const 1)
-      (fun n ↦ Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le
-        (h.bounded _ (terminalTime_mem n) ω)))
-      (ae_restrict_of_ae (hU.mono (fun _ hω ↦ hω.2)))
-  have he : ∀ᶠ n in atTop, (∫ ω in S, f (terminalTime n) ω ∂μ) = ∫ ω in S, f s ω ∂μ := by
-    have hsn : ∀ᶠ n in atTop, s ≤ terminalTime n :=
-      ((tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1.eventually (Ioi_mem_nhds hs.2)).mono
-        (fun _ hn ↦ hn.le)
-    filter_upwards [hsn] with n hn
-    calc
-      _ = ∫ ω in S, μ[f (terminalTime n) | ℱ s] ω ∂μ :=
-        (setIntegral_condExp (ℱ.le s) (h.integrable (terminalTime_mem n)) hS).symm
-      _ = _ := integral_congr_ae (ae_restrict_of_ae (h.condExp s hs _ (terminalTime_mem n) hn))
-  exact tendsto_nhds_unique (tendsto_const_nhds.congr' (Filter.EventuallyEq.symm he)) hlim
-
-lemma UnitMomentMartingale.terminal_stopped_martingale
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
-    (hUm : StronglyMeasurable[ℱ 1] U)
-    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
-      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω))) :
-    Martingale (fun t : ℝ≥0 ↦ fun ω ↦ if (t : ℝ)<1 then f t ω else U ω)
-      (⟨fun t : ℝ≥0 ↦ ℱ t,fun _ _ hij ↦ ℱ.mono hij,fun t ↦ ℱ.le t⟩ :
-        Filtration ℝ≥0 m) μ := by
-  have hUi : Integrable U μ := Integrable.of_bound (hUm.mono (ℱ.le _)).aestronglyMeasurable 1
-    (hU.mono (fun ω hω ↦ (Real.norm_eq_abs _).trans_le hω.1))
-  constructor
-  · intro t
-    by_cases ht : (t : ℝ)<1
-    · simpa only [ht,ite_true,ite_false] using h.measurable t ⟨t.coe_nonneg,ht⟩
-    · simpa only [ht,ite_true,ite_false] using hUm.mono (ℱ.mono (not_lt.mp ht))
-  · intro s t hst
-    by_cases hs : (s : ℝ)<1
-    · by_cases ht : (t : ℝ)<1
-      · simpa only [hs,ht,ite_true,ite_false] using
-          h.condExp s ⟨s.coe_nonneg,hs⟩ t ⟨t.coe_nonneg,ht⟩ hst
-      · simpa only [hs,ht,ite_true,ite_false] using h.terminal_condExp hUm hU ⟨s.coe_nonneg,hs⟩
-    · have ht : ¬(t : ℝ)<1 := not_lt.mpr ((not_lt.mp hs).trans hst)
-      have he := condExp_of_stronglyMeasurable (ℱ.le s)
-        (hUm.mono (ℱ.mono (not_lt.mp hs))) hUi
-      simpa only [hs,ht,ite_true,ite_false] using Filter.EventuallyEq.of_eq he
-
-lemma UnitMomentMartingale.exists_terminal
-    (h : UnitMomentMartingale (μ := μ) ℱ f)
-    (hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ f t ω) (Ico (0 : ℝ) 1)) :
-    ∃ U : Ω → ℝ, StronglyMeasurable[ℱ 1] U ∧
-      Martingale (fun t : ℝ≥0 ↦ fun ω ↦ if (t : ℝ)<1 then f t ω else U ω)
-        (⟨fun t : ℝ≥0 ↦ ℱ t,fun _ _ hij ↦ ℱ.mono hij,fun t ↦ ℱ.le t⟩ :
-          Filtration ℝ≥0 m) μ ∧
-      (∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧ (U ω)^2=1 ∧
-        Tendsto (fun t ↦ f t ω) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) ∧
-      Tendsto (fun t ↦ ∫ ω, (f t ω-U ω)^2 ∂μ) (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
-  obtain ⟨U,hUm,hU⟩ := h.exists_continuous_terminal hc
-  have hseq : ∀ᵐ ω ∂μ, |U ω|≤1 ∧
-      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω)) :=
-    hU.mono (fun ω hω ↦ ⟨hω.1,hω.2.comp terminalTime_tendsto⟩)
-  refine ⟨U,hUm,h.terminal_stopped_martingale hUm hseq,?_,h.terminal_mean_square hUm hU⟩
-  filter_upwards [hU,h.terminal_square hUm hseq] with ω hω hsq
-  exact ⟨hω.1,hsq,hω.2⟩
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.UnitMomentMartingale
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+variable {Ω : Type*} {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+variable {ℱ : Filtration ℝ m} {f : ℝ → Ω → ℝ}
+
+lemma UnitMomentMartingale.terminal_condExp
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
+    (hUm : StronglyMeasurable[ℱ 1] U)
+    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
+      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω)))
+    {s : ℝ} (hs : s ∈ Ico (0 : ℝ) 1) : μ[U | ℱ s] =ᵐ[μ] f s := by
+  have hUi : Integrable U μ := Integrable.of_bound (hUm.mono (ℱ.le _)).aestronglyMeasurable 1
+    (hU.mono (fun ω hω ↦ (Real.norm_eq_abs _).trans_le hω.1))
+  apply Filter.EventuallyEq.symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq (ℱ.le s) hUi
+    (fun S _ _ ↦ (h.integrable hs).integrableOn) _ (h.measurable s hs).aestronglyMeasurable
+  intro S hS _
+  have hlim : Tendsto (fun n ↦ ∫ ω in S, f (terminalTime n) ω ∂μ) atTop (𝓝 (∫ ω in S, U ω ∂μ)) :=
+    tendsto_integral_of_dominated_convergence (fun _ : Ω ↦ (1 : ℝ))
+      (fun n ↦ ((h.measurable _ (terminalTime_mem n)).mono (ℱ.le _)).aestronglyMeasurable.restrict)
+      (integrable_const 1)
+      (fun n ↦ Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le
+        (h.bounded _ (terminalTime_mem n) ω)))
+      (ae_restrict_of_ae (hU.mono (fun _ hω ↦ hω.2)))
+  have he : ∀ᶠ n in atTop, (∫ ω in S, f (terminalTime n) ω ∂μ) = ∫ ω in S, f s ω ∂μ := by
+    have hsn : ∀ᶠ n in atTop, s ≤ terminalTime n :=
+      ((tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1.eventually (Ioi_mem_nhds hs.2)).mono
+        (fun _ hn ↦ hn.le)
+    filter_upwards [hsn] with n hn
+    calc
+      _ = ∫ ω in S, μ[f (terminalTime n) | ℱ s] ω ∂μ :=
+        (setIntegral_condExp (ℱ.le s) (h.integrable (terminalTime_mem n)) hS).symm
+      _ = _ := integral_congr_ae (ae_restrict_of_ae (h.condExp s hs _ (terminalTime_mem n) hn))
+  exact tendsto_nhds_unique (tendsto_const_nhds.congr' (Filter.EventuallyEq.symm he)) hlim
+
+lemma UnitMomentMartingale.terminal_stopped_martingale
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
+    (hUm : StronglyMeasurable[ℱ 1] U)
+    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
+      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω))) :
+    Martingale (fun t : ℝ≥0 ↦ fun ω ↦ if (t : ℝ)<1 then f t ω else U ω)
+      (⟨fun t : ℝ≥0 ↦ ℱ t,fun _ _ hij ↦ ℱ.mono hij,fun t ↦ ℱ.le t⟩ :
+        Filtration ℝ≥0 m) μ := by
+  have hUi : Integrable U μ := Integrable.of_bound (hUm.mono (ℱ.le _)).aestronglyMeasurable 1
+    (hU.mono (fun ω hω ↦ (Real.norm_eq_abs _).trans_le hω.1))
+  constructor
+  · intro t
+    by_cases ht : (t : ℝ)<1
+    · simpa only [ht,ite_true,ite_false] using h.measurable t ⟨t.coe_nonneg,ht⟩
+    · simpa only [ht,ite_true,ite_false] using hUm.mono (ℱ.mono (not_lt.mp ht))
+  · intro s t hst
+    by_cases hs : (s : ℝ)<1
+    · by_cases ht : (t : ℝ)<1
+      · simpa only [hs,ht,ite_true,ite_false] using
+          h.condExp s ⟨s.coe_nonneg,hs⟩ t ⟨t.coe_nonneg,ht⟩ hst
+      · simpa only [hs,ht,ite_true,ite_false] using h.terminal_condExp hUm hU ⟨s.coe_nonneg,hs⟩
+    · have ht : ¬(t : ℝ)<1 := not_lt.mpr ((not_lt.mp hs).trans hst)
+      have he := condExp_of_stronglyMeasurable (ℱ.le s)
+        (hUm.mono (ℱ.mono (not_lt.mp hs))) hUi
+      simpa only [hs,ht,ite_true,ite_false] using Filter.EventuallyEq.of_eq he
+
+lemma UnitMomentMartingale.exists_terminal
+    (h : UnitMomentMartingale (μ := μ) ℱ f)
+    (hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ f t ω) (Ico (0 : ℝ) 1)) :
+    ∃ U : Ω → ℝ, StronglyMeasurable[ℱ 1] U ∧
+      Martingale (fun t : ℝ≥0 ↦ fun ω ↦ if (t : ℝ)<1 then f t ω else U ω)
+        (⟨fun t : ℝ≥0 ↦ ℱ t,fun _ _ hij ↦ ℱ.mono hij,fun t ↦ ℱ.le t⟩ :
+          Filtration ℝ≥0 m) μ ∧
+      (∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧ (U ω)^2=1 ∧
+        Tendsto (fun t ↦ f t ω) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) ∧
+      Tendsto (fun t ↦ ∫ ω, (f t ω-U ω)^2 ∂μ) (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
+  obtain ⟨U,hUm,hU⟩ := h.exists_continuous_terminal hc
+  have hseq : ∀ᵐ ω ∂μ, |U ω|≤1 ∧
+      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω)) :=
+    hU.mono (fun ω hω ↦ ⟨hω.1,hω.2.comp terminalTime_tendsto⟩)
+  refine ⟨U,hUm,h.terminal_stopped_martingale hUm hseq,?_,h.terminal_mean_square hUm hU⟩
+  filter_upwards [hU,h.terminal_square hUm hseq] with ω hω hsq
+  exact ⟨hω.1,hsq,hω.2⟩
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/TestDiffusion.lean b/lean/OAI/Probability/SKValue/Processes/TestDiffusion.lean
index 7fb9ba0a3fb23c0dbfef8884ac58373c8053ff5d..87866979bfd1a19e4f20bbc4a3aa46007fb219a1 100644
--- a/lean/OAI/Probability/SKValue/Processes/TestDiffusion.lean
+++ b/lean/OAI/Probability/SKValue/Processes/TestDiffusion.lean
@@ -1,57 +1,57 @@
-import Mathlib
-import OAI.Probability.SKValue.Gaussian.CoupledCoordinates
-import OAI.Probability.SKValue.Evolution.BoundedSmooth
-import OAI.Probability.SKValue.Equations.TestGenerator
-import OAI.Probability.SKValue.Processes.TestExpectation
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal
-namespace SKValue
-lemma BackwardTest.diffusion_expectation {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
-    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
-    {X:ℝ → Ω → ℝ} {T K L K' L' Lu:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
-    (hT:0<T) (hT1:T≤1) (hu:GradientStrip T γ u K L) (h:BackwardTest T γ u V K' L')
-    (hLu:0≤Lu) (hLip:∀ s∈Icc (0:ℝ) T,∀ t∈Icc (0:ℝ) T,∀ x y,
-      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
-    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
-    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
-      (∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) :
-    (∫ ω,V T (X T ω) ∂μ)=V 0 0 := by
-  let A := fun N ↦ ∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))
-  let E := fun N ↦ T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L')+
-    stepSize T N*K'*(γ T-γ 0)
-  let F := fun N ↦ K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)
-  have hδ:Tendsto (fun N:ℕ ↦ stepSize T N) atTop (𝓝 (0:ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
-  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
-  have hE:Tendsto E atTop (𝓝 0) := by
-    simpa only [E,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,add_zero] using
-      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L'))).add
-        ((hδ.mul_const K').mul_const (γ T-γ 0)))
-  have hF:Tendsto F atTop (𝓝 0) := by
-    simpa only [F,Function.comp_def,Real.sqrt_zero,mul_zero] using
-      (((Real.continuous_sqrt.tendsto 0).comp
-        (coupled_euler_L2_convergence hB hT hu hLu hLip hXM hpaths)).const_mul K')
-  have hbound:∀ᶠ N in atTop,|(∫ ω,V T (X T ω) ∂μ)-V 0 0|≤E N+F N := by
-    filter_upwards [eventually_gt_atTop 0] with N hN
-    have he := h.mesh_expectation hT hT1 hN
-    have hf := h.coupled_terminal_error hB hT hu hXM
-      (by filter_upwards [hpaths] with ω hω using hω.2.2) hN
-    have ht := abs_sub_le (∫ ω,V T (X T ω) ∂μ) (A N) (V 0 0)
-    change |A N-V 0 0|≤E N at he
-    change |A N-(∫ ω,V T (X T ω) ∂μ)|≤F N at hf
-    have hf' : |(∫ ω,V T (X T ω) ∂μ)-A N|≤F N := by
-      rw [abs_sub_comm];exact hf
-    linarith
-  have hz := le_of_tendsto_of_tendsto tendsto_const_nhds (hE.add hF) hbound
-  have habs:|(∫ ω,V T (X T ω) ∂μ)-V 0 0|=0 := le_antisymm (by simpa using hz) (abs_nonneg _)
-  exact sub_eq_zero.mp (abs_eq_zero.mp habs)
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Gaussian.CoupledCoordinates
+import OAI.Probability.SKValue.Evolution.BoundedSmooth
+import OAI.Probability.SKValue.Equations.TestGenerator
+import OAI.Probability.SKValue.Processes.TestExpectation
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal
+namespace SKValue
+lemma BackwardTest.diffusion_expectation {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
+    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
+    {X:ℝ → Ω → ℝ} {T K L K' L' Lu:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
+    (hT:0<T) (hT1:T≤1) (hu:GradientStrip T γ u K L) (h:BackwardTest T γ u V K' L')
+    (hLu:0≤Lu) (hLip:∀ s∈Icc (0:ℝ) T,∀ t∈Icc (0:ℝ) T,∀ x y,
+      |u s x-u t y|≤Lu*(|s-t|+|x-y|))
+    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
+    (hpaths:∀ᵐ ω ∂μ,ContinuousOn (fun t ↦ X t ω) (Icc (0:ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ s*u s (X s ω)) volume 0 T ∧
+      (∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) :
+    (∫ ω,V T (X T ω) ∂μ)=V 0 0 := by
+  let A := fun N ↦ ∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))
+  let E := fun N ↦ T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L')+
+    stepSize T N*K'*(γ T-γ 0)
+  let F := fun N ↦ K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ)
+  have hδ:Tendsto (fun N:ℕ ↦ stepSize T N) atTop (𝓝 (0:ℝ)) := tendsto_const_div_atTop_nhds_zero_nat T
+  have hsqrt := (Real.continuous_sqrt.tendsto 0).comp hδ
+  have hE:Tendsto E atTop (𝓝 0) := by
+    simpa only [E,Function.comp_def,Real.sqrt_zero,mul_zero,zero_mul,add_zero] using
+      (((hsqrt.const_mul T).mul_const (cubicEnvelopeMean (γ T) K' ((1/2+γ T)*L'))).add
+        ((hδ.mul_const K').mul_const (γ T-γ 0)))
+  have hF:Tendsto F atTop (𝓝 0) := by
+    simpa only [F,Function.comp_def,Real.sqrt_zero,mul_zero] using
+      (((Real.continuous_sqrt.tendsto 0).comp
+        (coupled_euler_L2_convergence hB hT hu hLu hLip hXM hpaths)).const_mul K')
+  have hbound:∀ᶠ N in atTop,|(∫ ω,V T (X T ω) ∂μ)-V 0 0|≤E N+F N := by
+    filter_upwards [eventually_gt_atTop 0] with N hN
+    have he := h.mesh_expectation hT hT1 hN
+    have hf := h.coupled_terminal_error hB hT hu hXM
+      (by filter_upwards [hpaths] with ω hω using hω.2.2) hN
+    have ht := abs_sub_le (∫ ω,V T (X T ω) ∂μ) (A N) (V 0 0)
+    change |A N-V 0 0|≤E N at he
+    change |A N-(∫ ω,V T (X T ω) ∂μ)|≤F N at hf
+    have hf' : |(∫ ω,V T (X T ω) ∂μ)-A N|≤F N := by
+      rw [abs_sub_comm];exact hf
+    linarith
+  have hz := le_of_tendsto_of_tendsto tendsto_const_nhds (hE.add hF) hbound
+  have habs:|(∫ ω,V T (X T ω) ∂μ)-V 0 0|=0 := le_antisymm (by simpa using hz) (abs_nonneg _)
+  exact sub_eq_zero.mp (abs_eq_zero.mp habs)
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/TestExpectation.lean b/lean/OAI/Probability/SKValue/Processes/TestExpectation.lean
index 0251cf68b209546d1b17f8af39e56a7473c8bd77..2dbf22f2d0809df42c55a42e9103986f5a789f46 100644
--- a/lean/OAI/Probability/SKValue/Processes/TestExpectation.lean
+++ b/lean/OAI/Probability/SKValue/Processes/TestExpectation.lean
@@ -1,191 +1,191 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.CrossMesh
-import OAI.Probability.SKValue.Equations.TestGenerator
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators NNReal
-namespace SKValue
-lemma BackwardTest.expectation_step {T K L t δ:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
-    (h:BackwardTest T γ u V K L) (ht:0≤t) (hδ:0≤δ) (hδ1:δ≤1) (htδ:t+δ≤T)
-    {N j:ℕ} (hj:j<N+1) {Y:(Fin (N+1) → ℝ) → ℝ}
-    (hYm:Measurable Y) (hYp:DependsBefore j Y) :
-    let μ := gaussianProduct (Fin (N+1))
-    let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
-    |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)|≤
-      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
-  let μ := gaussianProduct (Fin (N+1))
-  let p := fun z ↦ deriv (V t) (Y z)
-  let a := fun z ↦ deriv (deriv (V t)) (Y z)
-  let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
-  let P := fun z ↦ Real.sqrt δ*rawIncrement p j z
-  let Q := fun z ↦ (δ/2)*(a z*((coordinate N j z)^2-1))
-  let D := fun z ↦ V (t+δ) (Y' z)-V t (Y z)
-  let R := fun z ↦ D z-P z-Q z
-  let E := fun z ↦ δ*Real.sqrt δ*cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z)+δ*K*(γ (t+δ)-γ t)
-  have htT:t≤T := (le_add_of_nonneg_right hδ).trans htδ
-  have htm:t∈Icc (0:ℝ) T := ⟨ht,htT⟩
-  have hem:t+δ∈Icc (0:ℝ) T := ⟨by linarith,htδ⟩
-  have hpm:Measurable p := ((h.smooth t htm).continuous_deriv (by norm_num)).measurable.comp hYm
-  have ham:Measurable a := by
-    have hh := (h.smooth t htm).continuous_iteratedDeriv 2 (by norm_num)
-    simp only [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ] at hh
-    exact hh.measurable.comp hYm
-  have hpM:MemLp p 2 μ := MemLp.of_bound hpm.aestronglyMeasurable K
-    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound t htm (Y z)))
-  have haM:MemLp a 2 μ := MemLp.of_bound ham.aestronglyMeasurable K
-    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.second_bound t htm (Y z)))
-  have hpp:DependsBefore j p := fun z z' hz ↦ by dsimp only [p];rw [hYp z z' hz]
-  have hap:DependsBefore j a := fun z z' hz ↦ by dsimp only [a];rw [hYp z z' hz]
-  have hPi:Integrable P μ := (((rawIncrement_memLp hj hpm hpp hpM).const_mul _).integrable (by norm_num))
-  have hQi:Integrable Q μ := (((predictableCenteredSquare_memLp ham (fun z ↦ h.second_bound t htm (Y z))).const_mul _).integrable (by norm_num))
-  have hP0:(∫ z,P z ∂μ)=0 := by
-    dsimp only [P]
-    rw [integral_const_mul,rawIncrement_mean hj hpm hpp,mul_zero]
-  have hQ0:(∫ z,Q z ∂μ)=0 := by
-    dsimp only [Q]
-    rw [integral_const_mul,predictableCenteredSquare_mean hj ham hap,mul_zero]
-  have hY'm:Measurable Y' := (hYm.add ((measurable_coordinate N j).const_mul _)).add
-    (((h.drift_measurable t htm).comp hYm).const_mul _)
-  have hcur:Integrable (fun z ↦ V t (Y z)) μ :=
-    (MemLp.of_bound ((h.smooth t htm).continuous.measurable.comp hYm).aestronglyMeasurable K
-      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound t htm (Y z))) :
-        MemLp (fun z ↦ V t (Y z)) 2 μ).integrable (by norm_num)
-  have hnext:Integrable (fun z ↦ V (t+δ) (Y' z)) μ :=
-    (MemLp.of_bound ((h.smooth (t+δ) hem).continuous.measurable.comp hY'm).aestronglyMeasurable K
-      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound (t+δ) hem (Y' z))) :
-        MemLp (fun z ↦ V (t+δ) (Y' z)) 2 μ).integrable (by norm_num)
-  have hDi:Integrable D μ := hnext.sub hcur
-  have hRi:Integrable R μ := (hDi.sub hPi).sub hQi
-  have hEi0 := (cubicEnvelope_memLp_two (coordinate_hasLaw N j)
-    (h.gamma_nonneg T ⟨ht.trans htT,le_rfl⟩) K ((1/2+γ T)*L)).integrable (by norm_num)
-  have hEi:Integrable E μ := (hEi0.const_mul _).add (integrable_const _)
-  have hRE (z) : |R z|≤E z := by
-    have hh := h.euler_one_step ht hδ hδ1 htδ (x:=Y z) (z:=coordinate N j z)
-    convert! hh using 1
-    dsimp only [R,D,P,Q,p,a,Y',E,rawIncrement]
-    congr 1
-    ring_nf
-  have hRmean:(∫ z,R z ∂μ)=(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ) := by
-    have hs := integral_sub (hDi.sub hPi) hQi
-    change (∫ z,D z-P z-Q z ∂μ)=(∫ z,D z-P z ∂μ)-(∫ z,Q z ∂μ) at hs
-    have hs1 := integral_sub hDi hPi
-    change (∫ z,D z-P z ∂μ)=(∫ z,D z ∂μ)-(∫ z,P z ∂μ) at hs1
-    change (∫ z,D z-P z-Q z ∂μ)=_
-    rw [hs,hs1,hP0,hQ0,sub_zero,sub_zero]
-    exact integral_sub hnext hcur
-  have hEmean:(∫ z,E z ∂μ)=
-      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
-    dsimp only [E]
-    rw [integral_add (hEi0.const_mul _) (integrable_const _),integral_const_mul]
-    simp only [integral_const,probReal_univ,smul_eq_mul,one_mul]
-    have he:(∫ z,cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z) ∂μ)=
-        cubicEnvelopeMean (γ T) K ((1/2+γ T)*L) := by
-      simpa only [Function.comp_def,cubicEnvelopeMean,standardGaussian] using
-        (coordinate_hasLaw N j).integral_comp (measurable_cubicEnvelope (γ T) K ((1/2+γ T)*L)).aestronglyMeasurable
-    rw [he]
-  change |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)|≤_
-  rw [←hRmean,←hEmean]
-  exact abs_integral_le_integral_abs.trans (integral_mono hRi.norm hEi hRE)
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators NNReal
-namespace SKValue
-lemma BackwardTest.mesh_expectation {T K L:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
-    (h:BackwardTest T γ u V K L) (hT:0<T) (hT1:T≤1) {N:ℕ} (hN:0<N) :
-    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-V 0 0|≤
-      T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
-        stepSize T N*K*(γ T-γ 0) := by
-  let δ := stepSize T N
-  let A := fun j ↦ ∫ z,V (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))
-  have hNr:0<(N:ℝ) := by exact_mod_cast hN
-  have hδ:0≤δ := div_nonneg hT.le hNr.le
-  have hδ1:δ≤1 := by
-    apply (div_le_iff₀ hNr).mpr
-    have hn:(1:ℝ)≤N := by exact_mod_cast hN
-    simpa only [one_mul] using hT1.trans hn
-  have htime (j:ℕ) (hj:j≤N) := mesh_time_mem hT.le hN hj
-  have hs (j:ℕ) : meshTime T N j+δ=meshTime T N (j+1) := by dsimp [meshTime];push_cast;ring
-  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
-  have hzero:meshTime T N 0=0 := by simp [meshTime]
-  have hum (j:ℕ) (hj:j<N) : Measurable (u (meshTime T N j)) := h.drift_measurable _ (htime j hj.le)
-  have hstep (j:ℕ) (hj:j<N) : |A (j+1)-A j+0|≤
-      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
-        δ*K*(γ (meshTime T N (j+1))-γ (meshTime T N j))+0*((0:ℝ)-0) := by
-    have hYp:DependsBefore j (fun z ↦ euler T N γ u z j) :=
-      eulerCoefficient_dependsBefore T hj.le γ u (fun _ y ↦ y)
-    have he := h.expectation_step (htime j hj.le).1 hδ hδ1
-      (by rw [hs];exact (htime (j+1) (by omega)).2) (by omega:j<N+1)
-      (measurable_euler T N γ u hum j hj.le) hYp
-    dsimp only at he
-    rw [hs] at he
-    simpa only [A,euler,δ,mul_assoc,add_zero,sub_self,mul_zero] using he
-  have hh := abs_telescope_source (A:=A) (b:=fun _ ↦ 0) (Γ:=fun j ↦ γ (meshTime T N j))
-    (Δ:=fun _ ↦ 0) (e:=0) hstep
-  have hA0:A 0=V 0 0 := by simp [A,meshTime,euler]
-  rw [hA0,show A N=(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))) by
-    dsimp only [A];rw [hlast],hlast,hzero] at hh
-  have hNd:(N:ℝ)*δ=T := by dsimp [δ,stepSize];field_simp
-  simpa only [Finset.sum_const_zero,add_zero,sub_self,mul_zero,←mul_assoc,hNd] using hh
-
-lemma BackwardTest.terminal_lipschitz {T K L:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
-    (h:BackwardTest T γ u V K L) (hT:0≤T) (x y:ℝ) : |V T x-V T y|≤K*|x-y| := by
-  have hd := (h.smooth T ⟨hT,le_rfl⟩).differentiable (by norm_num)
-  have hb:∀ z,‖deriv (V T) z‖≤K := fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound T ⟨hT,le_rfl⟩ z
-  have hl : LipschitzWith ⟨K,h.K_nonneg⟩ (V T) := by
-    apply lipschitzWith_of_nnnorm_deriv_le hd
-    intro z
-    exact_mod_cast hb z
-  have hh := hl.dist_le_mul x y
-  change |V T x-V T y|≤K*|x-y| at hh
-  exact hh
-
-lemma BackwardTest.coupled_terminal_error {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
-    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
-    {X:ℝ → Ω → ℝ} {T K L K' L':ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
-    (hT:0<T) (hu:GradientStrip T γ u K L) (h:BackwardTest T γ u V K' L')
-    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
-    (hpaths:∀ᵐ ω ∂μ,(∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+
-      ∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) {N:ℕ} (hN:0<N) :
-    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-
-      (∫ ω,V T (X T ω) ∂μ)|≤K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ) := by
-  let F := fun ω ↦ V T (coupledEuler T γ u B N N ω)
-  let G := fun ω ↦ V T (X T ω)
-  let E := meshMaxError T X (coupledEuler T γ u B) N
-  have ht:T∈Icc (0:ℝ) T := ⟨hT.le,le_rfl⟩
-  have hm:Continuous (V T) := (h.smooth T ht).continuous
-  have hFM:MemLp F 2 μ := MemLp.of_bound
-    (hm.measurable.comp_aemeasurable (coupled_euler_measurable hB hT hu N N le_rfl).aemeasurable).aestronglyMeasurable K'
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,F] using h.bound T ht _))
-  have hGM:MemLp G 2 μ := MemLp.of_bound
-    (hm.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable K'
-      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,G] using h.bound T ht _))
-  have hEM:MemLp E 2 μ := coupled_mesh_error_memLp hB hT hu hXM hpaths hN
-  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
-  have hFG (ω:Ω) : |F ω-G ω|≤K'*|E ω| := by
-    have he:|coupledEuler T γ u B N N ω-X T ω|≤E ω := by
-      have hh := le_initialMax (f:=fun k ↦
-        |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
-      change |coupledEuler T γ u B N N ω-X (meshTime T N N) ω|≤E ω at hh
-      rwa [hlast] at hh
-    exact (h.terminal_lipschitz hT.le _ _).trans
-      (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) h.K_nonneg)
-  have heq:(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))=(∫ ω,F ω ∂μ) := by
-    have heum := measurable_euler T N γ u (fun j hj ↦ (hu.smooth _ (mesh_time_mem hT.le hN hj.le)).continuous.measurable) N le_rfl
-    have hh := (coupledCoordinates_hasLaw hB hT hN).integral_comp (hm.measurable.comp heum).aestronglyMeasurable
-    simpa only [Function.comp_apply,F,coupledEuler,ite_eq_right (Nat.ne_of_gt hN)] using hh.symm
-  rw [heq]
-  exact first_moment_perturbation hFM hGM hEM h.K_nonneg (Eventually.of_forall hFG)
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.CrossMesh
+import OAI.Probability.SKValue.Equations.TestGenerator
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators NNReal
+namespace SKValue
+lemma BackwardTest.expectation_step {T K L t δ:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
+    (h:BackwardTest T γ u V K L) (ht:0≤t) (hδ:0≤δ) (hδ1:δ≤1) (htδ:t+δ≤T)
+    {N j:ℕ} (hj:j<N+1) {Y:(Fin (N+1) → ℝ) → ℝ}
+    (hYm:Measurable Y) (hYp:DependsBefore j Y) :
+    let μ := gaussianProduct (Fin (N+1))
+    let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
+    |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)|≤
+      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
+  let μ := gaussianProduct (Fin (N+1))
+  let p := fun z ↦ deriv (V t) (Y z)
+  let a := fun z ↦ deriv (deriv (V t)) (Y z)
+  let Y' := fun z ↦ Y z+Real.sqrt δ*coordinate N j z+δ*γ t*u t (Y z)
+  let P := fun z ↦ Real.sqrt δ*rawIncrement p j z
+  let Q := fun z ↦ (δ/2)*(a z*((coordinate N j z)^2-1))
+  let D := fun z ↦ V (t+δ) (Y' z)-V t (Y z)
+  let R := fun z ↦ D z-P z-Q z
+  let E := fun z ↦ δ*Real.sqrt δ*cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z)+δ*K*(γ (t+δ)-γ t)
+  have htT:t≤T := (le_add_of_nonneg_right hδ).trans htδ
+  have htm:t∈Icc (0:ℝ) T := ⟨ht,htT⟩
+  have hem:t+δ∈Icc (0:ℝ) T := ⟨by linarith,htδ⟩
+  have hpm:Measurable p := ((h.smooth t htm).continuous_deriv (by norm_num)).measurable.comp hYm
+  have ham:Measurable a := by
+    have hh := (h.smooth t htm).continuous_iteratedDeriv 2 (by norm_num)
+    simp only [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ] at hh
+    exact hh.measurable.comp hYm
+  have hpM:MemLp p 2 μ := MemLp.of_bound hpm.aestronglyMeasurable K
+    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound t htm (Y z)))
+  have haM:MemLp a 2 μ := MemLp.of_bound ham.aestronglyMeasurable K
+    (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.second_bound t htm (Y z)))
+  have hpp:DependsBefore j p := fun z z' hz ↦ by dsimp only [p];rw [hYp z z' hz]
+  have hap:DependsBefore j a := fun z z' hz ↦ by dsimp only [a];rw [hYp z z' hz]
+  have hPi:Integrable P μ := (((rawIncrement_memLp hj hpm hpp hpM).const_mul _).integrable (by norm_num))
+  have hQi:Integrable Q μ := (((predictableCenteredSquare_memLp ham (fun z ↦ h.second_bound t htm (Y z))).const_mul _).integrable (by norm_num))
+  have hP0:(∫ z,P z ∂μ)=0 := by
+    dsimp only [P]
+    rw [integral_const_mul,rawIncrement_mean hj hpm hpp,mul_zero]
+  have hQ0:(∫ z,Q z ∂μ)=0 := by
+    dsimp only [Q]
+    rw [integral_const_mul,predictableCenteredSquare_mean hj ham hap,mul_zero]
+  have hY'm:Measurable Y' := (hYm.add ((measurable_coordinate N j).const_mul _)).add
+    (((h.drift_measurable t htm).comp hYm).const_mul _)
+  have hcur:Integrable (fun z ↦ V t (Y z)) μ :=
+    (MemLp.of_bound ((h.smooth t htm).continuous.measurable.comp hYm).aestronglyMeasurable K
+      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound t htm (Y z))) :
+        MemLp (fun z ↦ V t (Y z)) 2 μ).integrable (by norm_num)
+  have hnext:Integrable (fun z ↦ V (t+δ) (Y' z)) μ :=
+    (MemLp.of_bound ((h.smooth (t+δ) hem).continuous.measurable.comp hY'm).aestronglyMeasurable K
+      (Eventually.of_forall (fun z ↦ by simpa only [Real.norm_eq_abs] using h.bound (t+δ) hem (Y' z))) :
+        MemLp (fun z ↦ V (t+δ) (Y' z)) 2 μ).integrable (by norm_num)
+  have hDi:Integrable D μ := hnext.sub hcur
+  have hRi:Integrable R μ := (hDi.sub hPi).sub hQi
+  have hEi0 := (cubicEnvelope_memLp_two (coordinate_hasLaw N j)
+    (h.gamma_nonneg T ⟨ht.trans htT,le_rfl⟩) K ((1/2+γ T)*L)).integrable (by norm_num)
+  have hEi:Integrable E μ := (hEi0.const_mul _).add (integrable_const _)
+  have hRE (z) : |R z|≤E z := by
+    have hh := h.euler_one_step ht hδ hδ1 htδ (x:=Y z) (z:=coordinate N j z)
+    convert! hh using 1
+    dsimp only [R,D,P,Q,p,a,Y',E,rawIncrement]
+    congr 1
+    ring_nf
+  have hRmean:(∫ z,R z ∂μ)=(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ) := by
+    have hs := integral_sub (hDi.sub hPi) hQi
+    change (∫ z,D z-P z-Q z ∂μ)=(∫ z,D z-P z ∂μ)-(∫ z,Q z ∂μ) at hs
+    have hs1 := integral_sub hDi hPi
+    change (∫ z,D z-P z ∂μ)=(∫ z,D z ∂μ)-(∫ z,P z ∂μ) at hs1
+    change (∫ z,D z-P z-Q z ∂μ)=_
+    rw [hs,hs1,hP0,hQ0,sub_zero,sub_zero]
+    exact integral_sub hnext hcur
+  have hEmean:(∫ z,E z ∂μ)=
+      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+δ*K*(γ (t+δ)-γ t) := by
+    dsimp only [E]
+    rw [integral_add (hEi0.const_mul _) (integrable_const _),integral_const_mul]
+    simp only [integral_const,probReal_univ,smul_eq_mul,one_mul]
+    have he:(∫ z,cubicEnvelope (γ T) K ((1/2+γ T)*L) (coordinate N j z) ∂μ)=
+        cubicEnvelopeMean (γ T) K ((1/2+γ T)*L) := by
+      simpa only [Function.comp_def,cubicEnvelopeMean,standardGaussian] using
+        (coordinate_hasLaw N j).integral_comp (measurable_cubicEnvelope (γ T) K ((1/2+γ T)*L)).aestronglyMeasurable
+    rw [he]
+  change |(∫ z,V (t+δ) (Y' z) ∂μ)-(∫ z,V t (Y z) ∂μ)|≤_
+  rw [←hRmean,←hEmean]
+  exact abs_integral_le_integral_abs.trans (integral_mono hRi.norm hEi hRE)
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators NNReal
+namespace SKValue
+lemma BackwardTest.mesh_expectation {T K L:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
+    (h:BackwardTest T γ u V K L) (hT:0<T) (hT1:T≤1) {N:ℕ} (hN:0<N) :
+    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-V 0 0|≤
+      T*Real.sqrt (stepSize T N)*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
+        stepSize T N*K*(γ T-γ 0) := by
+  let δ := stepSize T N
+  let A := fun j ↦ ∫ z,V (meshTime T N j) (euler T N γ u z j) ∂gaussianProduct (Fin (N+1))
+  have hNr:0<(N:ℝ) := by exact_mod_cast hN
+  have hδ:0≤δ := div_nonneg hT.le hNr.le
+  have hδ1:δ≤1 := by
+    apply (div_le_iff₀ hNr).mpr
+    have hn:(1:ℝ)≤N := by exact_mod_cast hN
+    simpa only [one_mul] using hT1.trans hn
+  have htime (j:ℕ) (hj:j≤N) := mesh_time_mem hT.le hN hj
+  have hs (j:ℕ) : meshTime T N j+δ=meshTime T N (j+1) := by dsimp [meshTime];push_cast;ring
+  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
+  have hzero:meshTime T N 0=0 := by simp [meshTime]
+  have hum (j:ℕ) (hj:j<N) : Measurable (u (meshTime T N j)) := h.drift_measurable _ (htime j hj.le)
+  have hstep (j:ℕ) (hj:j<N) : |A (j+1)-A j+0|≤
+      δ*Real.sqrt δ*cubicEnvelopeMean (γ T) K ((1/2+γ T)*L)+
+        δ*K*(γ (meshTime T N (j+1))-γ (meshTime T N j))+0*((0:ℝ)-0) := by
+    have hYp:DependsBefore j (fun z ↦ euler T N γ u z j) :=
+      eulerCoefficient_dependsBefore T hj.le γ u (fun _ y ↦ y)
+    have he := h.expectation_step (htime j hj.le).1 hδ hδ1
+      (by rw [hs];exact (htime (j+1) (by omega)).2) (by omega:j<N+1)
+      (measurable_euler T N γ u hum j hj.le) hYp
+    dsimp only at he
+    rw [hs] at he
+    simpa only [A,euler,δ,mul_assoc,add_zero,sub_self,mul_zero] using he
+  have hh := abs_telescope_source (A:=A) (b:=fun _ ↦ 0) (Γ:=fun j ↦ γ (meshTime T N j))
+    (Δ:=fun _ ↦ 0) (e:=0) hstep
+  have hA0:A 0=V 0 0 := by simp [A,meshTime,euler]
+  rw [hA0,show A N=(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1))) by
+    dsimp only [A];rw [hlast],hlast,hzero] at hh
+  have hNd:(N:ℝ)*δ=T := by dsimp [δ,stepSize];field_simp
+  simpa only [Finset.sum_const_zero,add_zero,sub_self,mul_zero,←mul_assoc,hNd] using hh
+
+lemma BackwardTest.terminal_lipschitz {T K L:ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
+    (h:BackwardTest T γ u V K L) (hT:0≤T) (x y:ℝ) : |V T x-V T y|≤K*|x-y| := by
+  have hd := (h.smooth T ⟨hT,le_rfl⟩).differentiable (by norm_num)
+  have hb:∀ z,‖deriv (V T) z‖≤K := fun z ↦ by simpa only [Real.norm_eq_abs] using h.first_bound T ⟨hT,le_rfl⟩ z
+  have hl : LipschitzWith ⟨K,h.K_nonneg⟩ (V T) := by
+    apply lipschitzWith_of_nnnorm_deriv_le hd
+    intro z
+    exact_mod_cast hb z
+  have hh := hl.dist_le_mul x y
+  change |V T x-V T y|≤K*|x-y| at hh
+  exact hh
+
+lemma BackwardTest.coupled_terminal_error {Ω:Type*} [MeasurableSpace Ω] {μ:Measure Ω}
+    [IsProbabilityMeasure μ] {B:ℝ≥0 → Ω → ℝ} (hB:IsPreBrownianReal B μ)
+    {X:ℝ → Ω → ℝ} {T K L K' L':ℝ} {γ:ℝ → ℝ} {u V:ℝ → ℝ → ℝ}
+    (hT:0<T) (hu:GradientStrip T γ u K L) (h:BackwardTest T γ u V K' L')
+    (hXM:∀ t∈Icc (0:ℝ) T,AEStronglyMeasurable (X t) μ)
+    (hpaths:∀ᵐ ω ∂μ,(∀ t∈Icc (0:ℝ) T,X t ω=B t.toNNReal ω+
+      ∫ s in (0:ℝ)..t,γ s*u s (X s ω)) ∧ X 0 ω=0) {N:ℕ} (hN:0<N) :
+    |(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))-
+      (∫ ω,V T (X T ω) ∂μ)|≤K'*Real.sqrt (∫ ω,(meshMaxError T X (coupledEuler T γ u B) N ω)^2 ∂μ) := by
+  let F := fun ω ↦ V T (coupledEuler T γ u B N N ω)
+  let G := fun ω ↦ V T (X T ω)
+  let E := meshMaxError T X (coupledEuler T γ u B) N
+  have ht:T∈Icc (0:ℝ) T := ⟨hT.le,le_rfl⟩
+  have hm:Continuous (V T) := (h.smooth T ht).continuous
+  have hFM:MemLp F 2 μ := MemLp.of_bound
+    (hm.measurable.comp_aemeasurable (coupled_euler_measurable hB hT hu N N le_rfl).aemeasurable).aestronglyMeasurable K'
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,F] using h.bound T ht _))
+  have hGM:MemLp G 2 μ := MemLp.of_bound
+    (hm.measurable.comp_aemeasurable (hXM T ht).aemeasurable).aestronglyMeasurable K'
+      (Eventually.of_forall (fun ω ↦ by simpa only [Real.norm_eq_abs,G] using h.bound T ht _))
+  have hEM:MemLp E 2 μ := coupled_mesh_error_memLp hB hT hu hXM hpaths hN
+  have hlast:meshTime T N N=T := by dsimp [meshTime,stepSize];field_simp
+  have hFG (ω:Ω) : |F ω-G ω|≤K'*|E ω| := by
+    have he:|coupledEuler T γ u B N N ω-X T ω|≤E ω := by
+      have hh := le_initialMax (f:=fun k ↦
+        |coupledEuler T γ u B N k ω-X (meshTime T N k) ω|) (le_refl N)
+      change |coupledEuler T γ u B N N ω-X (meshTime T N N) ω|≤E ω at hh
+      rwa [hlast] at hh
+    exact (h.terminal_lipschitz hT.le _ _).trans
+      (mul_le_mul_of_nonneg_left (he.trans (le_abs_self _)) h.K_nonneg)
+  have heq:(∫ z,V T (euler T N γ u z N) ∂gaussianProduct (Fin (N+1)))=(∫ ω,F ω ∂μ) := by
+    have heum := measurable_euler T N γ u (fun j hj ↦ (hu.smooth _ (mesh_time_mem hT.le hN hj.le)).continuous.measurable) N le_rfl
+    have hh := (coupledCoordinates_hasLaw hB hT hN).integral_comp (hm.measurable.comp heum).aestronglyMeasurable
+    simpa only [Function.comp_apply,F,coupledEuler,ite_eq_right (Nat.ne_of_gt hN)] using hh.symm
+  rw [heq]
+  exact first_moment_perturbation hFM hGM hEM h.K_nonneg (Eventually.of_forall hFG)
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/UnitMomentMartingale.lean b/lean/OAI/Probability/SKValue/Processes/UnitMomentMartingale.lean
index e1ab08b0c23eb8034d342f9eebd984187e738218..0432cf970f8a9baab24622b7f6d706d98fe4534d 100644
--- a/lean/OAI/Probability/SKValue/Processes/UnitMomentMartingale.lean
+++ b/lean/OAI/Probability/SKValue/Processes/UnitMomentMartingale.lean
@@ -1,396 +1,396 @@
-import Mathlib
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-
-variable {Ω : Type*} {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
-
-structure UnitMomentMartingale (ℱ : Filtration ℝ m) (f : ℝ → Ω → ℝ) : Prop where
-  measurable : ∀ t ∈ Ico (0 : ℝ) 1, StronglyMeasurable[ℱ t] (f t)
-  bounded : ∀ t ∈ Ico (0 : ℝ) 1, ∀ ω, |f t ω| ≤ 1
-  condExp : ∀ s ∈ Ico (0 : ℝ) 1, ∀ t ∈ Ico (0 : ℝ) 1, s ≤ t →
-    μ[f t | ℱ s] =ᵐ[μ] f s
-  secondMoment : ∀ t ∈ Ico (0 : ℝ) 1, ∫ ω, (f t ω)^2 ∂μ = t
-
-variable {ℱ : Filtration ℝ m} {f : ℝ → Ω → ℝ}
-
-lemma UnitMomentMartingale.integrable (h : UnitMomentMartingale (μ := μ) ℱ f)
-    {t : ℝ} (ht : t ∈ Ico (0 : ℝ) 1) : Integrable (f t) μ :=
-  Integrable.of_bound ((h.measurable t ht).mono (ℱ.le t)).aestronglyMeasurable 1
-    (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le (h.bounded t ht ω)))
-
-lemma UnitMomentMartingale.integrable_mul (h : UnitMomentMartingale (μ := μ) ℱ f)
-    {s t : ℝ} (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) :
-    Integrable (fun ω ↦ f s ω * f t ω) μ :=
-  Integrable.of_bound (((h.measurable s hs).mono (ℱ.le s)).mul
-    ((h.measurable t ht).mono (ℱ.le t))).aestronglyMeasurable 1
-    (Eventually.of_forall (fun ω ↦ by
-      rw [Real.norm_eq_abs,abs_mul]
-      nlinarith [h.bounded s hs ω,h.bounded t ht ω,abs_nonneg (f s ω),abs_nonneg (f t ω)]))
-
-lemma UnitMomentMartingale.integral_past_mul (h : UnitMomentMartingale (μ := μ) ℱ f)
-    {s t : ℝ} (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) (hst : s ≤ t) :
-    ∫ ω, f s ω * f t ω ∂μ = s := by
-  have hc := condExp_mul_of_stronglyMeasurable_left (h.measurable s hs)
-    (h.integrable_mul hs ht) (h.integrable ht)
-  have he : μ[(f s) * (f t) | ℱ s] =ᵐ[μ] fun ω ↦ (f s ω)^2 := by
-    filter_upwards [hc,h.condExp s hs t ht hst] with ω hω hω'
-    simpa only [Pi.mul_apply,hω',pow_two] using hω
-  calc
-    _ = ∫ ω, μ[(f s) * (f t) | ℱ s] ω ∂μ := (integral_condExp (ℱ.le s)).symm
-    _ = ∫ ω, (f s ω)^2 ∂μ := integral_congr_ae he
-    _ = s := h.secondMoment s hs
-
-lemma UnitMomentMartingale.sampled_gap_martingale
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {s : ℝ}
-    (hs : s ∈ Ico (0 : ℝ) 1) (r : ℕ → ℝ) (hr : Monotone r)
-    (hrmem : ∀ n, r n ∈ Ico s 1) :
-    Martingale (fun n ω ↦ 1-f s ω*f (r n) ω)
-      (⟨fun n ↦ ℱ (r n),fun _ _ hij ↦ ℱ.mono (hr hij),fun n ↦ ℱ.le (r n)⟩ :
-        Filtration ℕ m) μ := by
-  have ht (n : ℕ) : r n ∈ Ico (0 : ℝ) 1 := ⟨hs.1.trans (hrmem n).1,(hrmem n).2⟩
-  refine ⟨fun n ↦ stronglyMeasurable_const.sub
-    (((h.measurable s hs).mono (ℱ.mono (hrmem n).1)).mul (h.measurable (r n) (ht n))),?_⟩
-  intro i j hij
-  have hc := condExp_mul_of_stronglyMeasurable_left
-    ((h.measurable s hs).mono (ℱ.mono (hrmem i).1))
-    (h.integrable_mul hs (ht j)) (h.integrable (ht j))
-  have hc1 := condExp_of_stronglyMeasurable (ℱ.le (r i))
-    (stronglyMeasurable_const (b := (1 : ℝ))) (integrable_const (μ := μ) 1)
-  filter_upwards [condExp_sub (integrable_const (1 : ℝ))
-      (h.integrable_mul hs (ht j)) (ℱ (r i)),hc,
-      h.condExp (r i) (ht i) (r j) (ht j) (hr hij)] with ω hsub hmul he
-  simp only [Pi.sub_apply] at hsub
-  change μ[(fun ω ↦ f s ω*f (r j) ω) | ℱ (r i)] ω =
-    f s ω*μ[f (r j) | ℱ (r i)] ω at hmul
-  rw [hc1,hmul] at hsub
-  simpa only [Pi.sub_def,Pi.mul_def,Pi.mul_apply,he] using hsub
-
-omit [IsProbabilityMeasure μ] in
-lemma UnitMomentMartingale.sampled_gap_nonneg
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {s t : ℝ}
-    (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) (ω : Ω) :
-    0 ≤ 1-f s ω*f t ω := by
-  have hh : f s ω*f t ω ≤ 1 := calc
-    _ ≤ |f s ω*f t ω| := le_abs_self _
-    _ = |f s ω| * |f t ω| := abs_mul _ _
-    _ ≤ 1 := by nlinarith [h.bounded s hs ω,h.bounded t ht ω,
-      abs_nonneg (f s ω),abs_nonneg (f t ω)]
-  linarith
-
-lemma UnitMomentMartingale.sampled_gap_integral
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {s t : ℝ}
-    (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) (hst : s ≤ t) :
-    ∫ ω, 1-f s ω*f t ω ∂μ = 1-s := by
-  rw [integral_sub (integrable_const 1) (h.integrable_mul hs ht),
-    h.integral_past_mul hs ht hst]
-  simp
-
-lemma UnitMomentMartingale.finite_gap_maximal
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {s : ℝ}
-    (hs : s ∈ Ico (0 : ℝ) 1) (S : Finset ℝ)
-    (hS : ∀ t ∈ S, t ∈ Ico s 1) (ε : ℝ≥0) :
-    (ε : ℝ≥0∞) * μ {ω | ∃ t ∈ S, (ε : ℝ) ≤ 1-f s ω*f t ω} ≤
-      ENNReal.ofReal (1-s) := by
-  classical
-  by_cases hne : S.Nonempty
-  · have hcard : 0 < S.card := Finset.card_pos.mpr hne
-    let e := S.orderEmbOfFin rfl
-    let r : ℕ → ℝ := fun n ↦ e ⟨min n (S.card-1),by omega⟩
-    have hr : Monotone r := by
-      intro i j hij
-      exact e.monotone (show min i (S.card-1) ≤ min j (S.card-1) from min_le_min_right _ hij)
-    have hrmem (n : ℕ) : r n ∈ Ico s 1 := hS _ (S.orderEmbOfFin_mem rfl _)
-    have hm := h.sampled_gap_martingale hs r hr hrmem
-    have hn : 0 ≤ (fun n ω ↦ 1-f s ω*f (r n) ω) :=
-      fun n ω ↦ h.sampled_gap_nonneg hs ⟨hs.1.trans (hrmem n).1,(hrmem n).2⟩ ω
-    have hb := maximal_ineq hm.submartingale hn (ε := ε) (S.card-1)
-    have hset : {ω | ∃ t ∈ S, (ε : ℝ) ≤ 1-f s ω*f t ω} ⊆
-        {ω | (ε : ℝ) ≤ (Finset.range ((S.card-1)+1)).sup' Finset.nonempty_range_add_one
-          (fun k ↦ 1-f s ω*f (r k) ω)} := by
-      rintro ω ⟨t,ht,heps⟩
-      let i := (S.orderIsoOfFin rfl).symm ⟨t,ht⟩
-      have her : r i.val = t := by
-        dsimp only [r]
-        have hh : (⟨min i.val (S.card-1),by omega⟩ : Fin S.card) = i :=
-          Fin.ext (min_eq_left (by omega))
-        rw [hh]
-        change ((S.orderIsoOfFin rfl) i).val = t
-        simp [i]
-      refine heps.trans ?_
-      rw [← her]
-      exact Finset.le_sup' (f := fun k ↦ 1-f s ω*f (r k) ω)
-        (Finset.mem_range.mpr (by omega : i.val < (S.card-1)+1))
-    calc
-      _ ≤ (ε : ℝ≥0∞) * μ {ω | (ε : ℝ) ≤
-          (Finset.range ((S.card-1)+1)).sup' Finset.nonempty_range_add_one
-            (fun k ↦ 1-f s ω*f (r k) ω)} := by gcongr
-      _ ≤ _ := hb
-      _ ≤ ENNReal.ofReal (∫ ω, 1-f s ω*f (r (S.card-1)) ω ∂μ) := by
-        apply ENNReal.ofReal_le_ofReal
-        exact setIntegral_le_integral (integrable_const 1 |>.sub
-          (h.integrable_mul hs ⟨hs.1.trans (hrmem _).1,(hrmem _).2⟩))
-          (Eventually.of_forall (hn (S.card-1)))
-      _ = _ := congrArg ENNReal.ofReal (h.sampled_gap_integral hs
-        ⟨hs.1.trans (hrmem _).1,(hrmem _).2⟩ (hrmem _).1)
-  · simp [Finset.not_nonempty_iff_eq_empty.mp hne]
-
-lemma UnitMomentMartingale.rational_gap_maximal
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {s : ℝ}
-    (hs : s ∈ Ico (0 : ℝ) 1) (ε : ℝ≥0) :
-    (ε : ℝ≥0∞) * μ {ω | ∃ q : ℚ, (q : ℝ) ∈ Ico s 1 ∧
-      (ε : ℝ) ≤ 1-f s ω*f q ω} ≤ ENNReal.ofReal (1-s) := by
-  classical
-  let E : Finset ℚ → Set Ω := fun S ↦ {ω | ∃ q ∈ S, (q : ℝ) ∈ Ico s 1 ∧
-    (ε : ℝ) ≤ 1-f s ω*f q ω}
-  have hd : Directed (· ⊆ ·) E := by
-    intro S T
-    refine ⟨S∪T, ?_, ?_⟩
-    · rintro ω ⟨q,hq,hmem,he⟩
-      exact ⟨q,Finset.mem_union_left _ hq,hmem,he⟩
-    · rintro ω ⟨q,hq,hmem,he⟩
-      exact ⟨q,Finset.mem_union_right _ hq,hmem,he⟩
-  have heq : {ω | ∃ q : ℚ, (q : ℝ) ∈ Ico s 1 ∧ (ε : ℝ) ≤ 1-f s ω*f q ω} =
-      ⋃ S : Finset ℚ, E S := by
-    ext ω
-    constructor
-    · rintro ⟨q,hmem,he⟩
-      exact mem_iUnion.mpr ⟨{q},q,Finset.mem_singleton_self _,hmem,he⟩
-    · intro hω
-      obtain ⟨S,q,hq,hmem,he⟩ := mem_iUnion.mp hω
-      exact ⟨q,hmem,he⟩
-  rw [heq,hd.measure_iUnion,ENNReal.mul_iSup]
-  refine iSup_le fun S ↦ ?_
-  let R : Finset ℝ := (S.filter (fun q : ℚ ↦ (q : ℝ) ∈ Ico s 1)).image (fun q : ℚ ↦ (q : ℝ))
-  have hR : ∀ t ∈ R, t ∈ Ico s 1 := by
-    intro t ht
-    obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp ht
-    exact (Finset.mem_filter.mp hq).2
-  have hE : E S ⊆ {ω | ∃ t ∈ R, (ε : ℝ) ≤ 1-f s ω*f t ω} := by
-    rintro ω ⟨q,hq,hmem,he⟩
-    exact ⟨(q : ℝ),Finset.mem_image.mpr ⟨q,Finset.mem_filter.mpr ⟨hq,hmem⟩,rfl⟩,he⟩
-  calc
-    _ ≤ (ε : ℝ≥0∞) * μ {ω | ∃ t ∈ R, (ε : ℝ) ≤ 1-f s ω*f t ω} := by gcongr
-    _ ≤ _ := h.finite_gap_maximal hs R hR ε
-
-noncomputable def terminalTime (n : ℕ) : ℝ := 1-1/(n+1)
-
-lemma terminalTime_mem (n : ℕ) : terminalTime n ∈ Ico (0 : ℝ) 1 := by
-  have hn : (1 : ℝ) ≤ n+1 := by linarith [Nat.cast_nonneg (α := ℝ) n]
-  have hp : (0 : ℝ) < n+1 := by positivity
-  simp only [terminalTime]
-  constructor
-  · have hh : (1 : ℝ)/(n+1) ≤ 1 := (div_le_one hp).mpr hn
-    linarith
-  · have hh : (0 : ℝ) < 1/(n+1) := by positivity
-    linarith
-
-lemma terminalTime_monotone : Monotone terminalTime := by
-  intro i j hij
-  simp only [terminalTime]
-  have hh : (1 : ℝ)/(j+1) ≤ 1/(i+1) := one_div_le_one_div_of_le
-    (by positivity) (by exact_mod_cast Nat.add_le_add_right hij 1)
-  linarith
-
-lemma terminalTime_tendsto : Tendsto terminalTime atTop (𝓝[<] (1 : ℝ)) := by
-  refine tendsto_nhdsWithin_iff.mpr ⟨?_,Eventually.of_forall (fun n ↦ (terminalTime_mem n).2)⟩
-  change Tendsto (fun n : ℕ ↦ (1 : ℝ)-1/((n : ℝ)+1)) atTop (𝓝 (1 : ℝ))
-  simpa only [sub_zero] using (tendsto_const_nhds (x := (1 : ℝ))).sub
-    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
-
-lemma UnitMomentMartingale.ae_exists_rational_gap_bound
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {ε : ℝ} (hε : 0 < ε) :
-    ∀ᵐ ω ∂μ, ∃ n, ∀ q : ℚ, (q : ℝ) ∈ Ico (terminalTime n) 1 →
-      1-f (terminalTime n) ω*f q ω < ε := by
-  let E : ℕ → Set Ω := fun n ↦ {ω | ∃ q : ℚ, (q : ℝ) ∈ Ico (terminalTime n) 1 ∧
-    ε ≤ 1-f (terminalTime n) ω*f q ω}
-  have hbound (n : ℕ) : ENNReal.ofReal ε * μ (⋂ n, E n) ≤
-      ENNReal.ofReal (1-terminalTime n) := calc
-    _ ≤ ENNReal.ofReal ε * μ (E n) := by gcongr; exact iInter_subset E n
-    _ ≤ _ := by
-      have hh := h.rational_gap_maximal (terminalTime_mem n) (NNReal.mk ε hε.le)
-      simpa only [E,NNReal.coe_mk,ENNReal.coe_nnreal_eq] using hh
-  have hzero : ENNReal.ofReal ε * μ (⋂ n, E n) = 0 := by
-    apply le_antisymm _ bot_le
-    have hlim : Tendsto (fun n ↦ ENNReal.ofReal (1-terminalTime n)) atTop (𝓝 0) := by
-      have hh : Tendsto (fun n ↦ 1-terminalTime n) atTop (𝓝 (0 : ℝ)) := by
-        simpa only [sub_self] using ((tendsto_const_nhds (x := (1 : ℝ))).sub
-          (tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1)
-      simpa only [ENNReal.ofReal_zero,Function.comp_def] using
-        ENNReal.continuous_ofReal.continuousAt.tendsto.comp hh
-    exact ge_of_tendsto' hlim hbound
-  have hn : μ (⋂ n, E n) = 0 := (mul_eq_zero.mp hzero).resolve_left
-    (ne_of_gt (ENNReal.ofReal_pos.mpr hε))
-  have ha : ∀ᵐ ω ∂μ, ω ∉ ⋂ n, E n := by
-    rw [ae_iff]
-    convert hn using 1
-    simp only [not_not,Set.ofPred_mem_eq]
-  filter_upwards [ha] with ω hω
-  simp only [mem_iInter,E,mem_ofPred_eq,not_forall,not_exists,not_and,not_le] at hω
-  exact hω
-
-lemma continuous_gap_bound_of_rat {v : ℝ → ℝ}
-    (hv : ContinuousOn v (Ico (0 : ℝ) 1)) {s ε : ℝ}
-    (hs : s ∈ Ico (0 : ℝ) 1)
-    (hq : ∀ q : ℚ, (q : ℝ) ∈ Ico s 1 → 1-v s*v q < ε) :
-    ∀ t ∈ Ioo s 1, 1-v s*v t ≤ ε := by
-  intro t ht
-  by_contra! he
-  have hvt : ContinuousAt v t := (hv t ⟨hs.1.trans ht.1.le,ht.2⟩).continuousAt
-    (Ico_mem_nhds_iff.mpr ⟨hs.1.trans_lt ht.1,ht.2⟩)
-  have hgt : ContinuousAt (fun r ↦ 1-v s*v r) t := continuousAt_const.sub
-    (continuousAt_const.mul hvt)
-  have hevent : {r | ε < 1-v s*v r} ∈ 𝓝 t := hgt (Ioi_mem_nhds he)
-  obtain ⟨q,hq'⟩ := Rat.denseRange_cast.mem_nhds (inter_mem hevent (Ioo_mem_nhds ht.1 ht.2))
-  exact (hq q ⟨hq'.2.1.le,hq'.2.2⟩).not_gt hq'.1
-
-lemma UnitMomentMartingale.exists_sequence_terminal
-    (h : UnitMomentMartingale (μ := μ) ℱ f) :
-    ∃ U : Ω → ℝ, StronglyMeasurable[ℱ 1] U ∧
-      (∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
-        Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω))) := by
-  let 𝒢 : Filtration ℕ m := ⟨fun n ↦ ℱ (terminalTime n),
-    fun _ _ hij ↦ ℱ.mono (terminalTime_monotone hij),fun n ↦ ℱ.le _⟩
-  have hm : Martingale (fun n ↦ f (terminalTime n)) 𝒢 μ :=
-    ⟨fun n ↦ h.measurable _ (terminalTime_mem n),fun i j hij ↦
-      h.condExp _ (terminalTime_mem i) _ (terminalTime_mem j) (terminalTime_monotone hij)⟩
-  have hb (n : ℕ) : eLpNorm (f (terminalTime n)) 1 μ ≤ (1 : ℝ≥0) := by
-    have hh := eLpNorm_le_of_ae_bound (μ := μ) (p := (1 : ℝ≥0∞))
-      (h.integrable (terminalTime_mem n)).aestronglyMeasurable
-      (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le
-        (h.bounded _ (terminalTime_mem n) ω)))
-    simpa using hh
-  let U := 𝒢.limitProcess (fun n ↦ f (terminalTime n)) μ
-  have hU : StronglyMeasurable[ℱ 1] U :=
-    Filtration.stronglyMeasurable_limitProcess.mono (iSup_le fun n ↦ ℱ.mono (terminalTime_mem n).2.le)
-  refine ⟨U,hU,?_⟩
-  filter_upwards [hm.submartingale.ae_tendsto_limitProcess hb] with ω hω
-  exact ⟨le_of_tendsto' hω.abs (fun n ↦ h.bounded _ (terminalTime_mem n) ω),hω⟩
-
-lemma UnitMomentMartingale.exists_continuous_terminal
-    (h : UnitMomentMartingale (μ := μ) ℱ f)
-    (hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ f t ω) (Ico (0 : ℝ) 1)) :
-    ∃ U : Ω → ℝ, StronglyMeasurable[ℱ 1] U ∧
-      (∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
-        Tendsto (fun t ↦ f t ω) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) := by
-  obtain ⟨U,hU,hseq⟩ := h.exists_sequence_terminal
-  have hg : ∀ᵐ ω ∂μ, ∀ k : ℕ, ∃ n, ∀ q : ℚ,
-      (q : ℝ) ∈ Ico (terminalTime n) 1 →
-        1-f (terminalTime n) ω*f q ω < 1/(k+1) := by
-    apply ae_all_iff.mpr
-    intro k
-    exact h.ae_exists_rational_gap_bound (by positivity)
-  refine ⟨U,hU,?_⟩
-  filter_upwards [hc,hseq,hg] with ω hcont hseqω hgap
-  refine ⟨hseqω.1,Metric.tendsto_nhds.mpr (fun δ hδ ↦ ?_)⟩
-  obtain ⟨k,hk⟩ : ∃ k : ℕ, (1 : ℝ)/(k+1) < δ^2/32 := by
-    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
-      (Iio_mem_nhds (show (0 : ℝ) < δ^2/32 by positivity))
-    exact hh.exists
-  obtain ⟨n,hn⟩ := hgap k
-  have hreal := continuous_gap_bound_of_rat hcont (terminalTime_mem n) hn
-  have hbound (t : ℝ) (ht : t ∈ Ioo (terminalTime n) 1) :
-      |f t ω-f (terminalTime n) ω| ≤ δ/4 := by
-    have hbt := h.bounded t ⟨(terminalTime_mem n).1.trans ht.1.le,ht.2⟩ ω
-    have hbs := h.bounded _ (terminalTime_mem n) ω
-    have he := hreal t ht
-    have hsq1 : (f t ω)^2 ≤ 1 := by nlinarith [abs_le.mp hbt]
-    have hsq2 : (f (terminalTime n) ω)^2 ≤ 1 := by nlinarith [abs_le.mp hbs]
-    have habs : |f t ω-f (terminalTime n) ω|^2 < (δ/4)^2 := by
-      rw [sq_abs]
-      nlinarith
-    nlinarith [abs_nonneg (f t ω-f (terminalTime n) ω)]
-  have hUb : |U ω-f (terminalTime n) ω| ≤ δ/4 := by
-    apply le_of_tendsto ((hseqω.2.sub_const _).abs)
-    have hev : ∀ᶠ k in atTop, terminalTime k ∈ Ioo (terminalTime n) 1 :=
-      terminalTime_tendsto.eventually (Ioo_mem_nhdsLT (terminalTime_mem n).2)
-    exact hev.mono (fun k hk ↦ hbound _ hk)
-  filter_upwards [Ioo_mem_nhdsLT (terminalTime_mem n).2] with t ht
-  rw [Real.dist_eq]
-  have hh := abs_add_le (f t ω-f (terminalTime n) ω) (f (terminalTime n) ω-U ω)
-  have hh' := hbound t ht
-  have hUb' : |f (terminalTime n) ω-U ω| ≤ δ/4 := by simpa only [abs_sub_comm] using hUb
-  have heq : f t ω-U ω = (f t ω-f (terminalTime n) ω)+(f (terminalTime n) ω-U ω) := by ring
-  rw [heq]
-  linarith
-
-lemma UnitMomentMartingale.terminal_square
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
-    (hUm : StronglyMeasurable[ℱ 1] U)
-    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
-      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω))) :
-    ∀ᵐ ω ∂μ, (U ω)^2 = 1 := by
-  have hU2i : Integrable (fun ω ↦ (U ω)^2) μ :=
-    Integrable.of_bound ((hUm.mono (ℱ.le 1)).pow 2).aestronglyMeasurable 1
-      (hU.mono (fun ω hω ↦ by
-        rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
-        nlinarith [abs_le.mp hω.1]))
-  have ht := tendsto_integral_of_dominated_convergence (μ := μ)
-    (F := fun n ω ↦ (f (terminalTime n) ω)^2) (fun _ : Ω ↦ (1 : ℝ))
-    (fun n ↦ (((h.measurable _ (terminalTime_mem n)).mono (ℱ.le _)).pow 2).aestronglyMeasurable)
-    (integrable_const 1)
-    (fun n ↦ Eventually.of_forall (fun ω ↦ by
-      rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
-      nlinarith [abs_le.mp (h.bounded _ (terminalTime_mem n) ω)]))
-    (hU.mono (fun ω hω ↦ hω.2.pow 2))
-  have hi : ∫ ω, (U ω)^2 ∂μ = 1 := by
-    have hh : (fun n ↦ ∫ ω, (f (terminalTime n) ω)^2 ∂μ) = terminalTime :=
-      funext (fun n ↦ h.secondMoment _ (terminalTime_mem n))
-    rw [hh] at ht
-    exact tendsto_nhds_unique ht (tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1
-  have hnonneg : 0 ≤ᵐ[μ] (fun ω ↦ 1-(U ω)^2) := hU.mono (fun ω hω ↦ by
-    dsimp
-    nlinarith [abs_le.mp hω.1])
-  have hzero : ∫ ω, 1-(U ω)^2 ∂μ = 0 := by
-    rw [integral_sub (integrable_const 1) hU2i,hi]
-    simp
-  have he := (integral_eq_zero_iff_of_nonneg_ae hnonneg ((integrable_const 1).sub hU2i)).mp hzero
-  filter_upwards [he] with ω hω
-  change 1-(U ω)^2=0 at hω
-  linarith
-
-lemma UnitMomentMartingale.terminal_mean_square
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
-    (hUm : StronglyMeasurable[ℱ 1] U)
-    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
-      Tendsto (fun t ↦ f t ω) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) :
-    Tendsto (fun t ↦ ∫ ω, (f t ω-U ω)^2 ∂μ) (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
-  have hev : ∀ᶠ t in 𝓝[<] (1 : ℝ), t ∈ Ico (0 : ℝ) 1 := by
-    filter_upwards [Ioo_mem_nhdsLT (show (0 : ℝ)<1 by norm_num)] with t ht
-    exact ⟨ht.1.le,ht.2⟩
-  have ht := tendsto_integral_filter_of_dominated_convergence (μ := μ)
-    (F := fun t ω ↦ (f t ω-U ω)^2) (f := fun _ ↦ (0 : ℝ)) (fun _ : Ω ↦ (4 : ℝ))
-    (hev.mono (fun t ht ↦ ((((h.measurable t ht).mono (ℱ.le t)).sub
-      (hUm.mono (ℱ.le 1))).pow 2).aestronglyMeasurable))
-    (hev.mono (fun t ht ↦ hU.mono (fun ω hω ↦ by
-      rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
-      have hb := h.bounded t ht ω
-      have hd : |f t ω-U ω| ≤ 2 := (abs_sub _ _).trans (by linarith [hω.1])
-      nlinarith [sq_abs (f t ω-U ω),abs_nonneg (f t ω-U ω)])))
-    (integrable_const 4)
-    (hU.mono (fun ω hω ↦ by
-      simpa only [sub_self,zero_pow (by decide : (2 : ℕ) ≠ 0)] using (hω.2.sub_const (U ω)).pow 2))
-  simpa only [integral_zero] using ht
-
-end SKValue
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+
+variable {Ω : Type*} {m : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+
+structure UnitMomentMartingale (ℱ : Filtration ℝ m) (f : ℝ → Ω → ℝ) : Prop where
+  measurable : ∀ t ∈ Ico (0 : ℝ) 1, StronglyMeasurable[ℱ t] (f t)
+  bounded : ∀ t ∈ Ico (0 : ℝ) 1, ∀ ω, |f t ω| ≤ 1
+  condExp : ∀ s ∈ Ico (0 : ℝ) 1, ∀ t ∈ Ico (0 : ℝ) 1, s ≤ t →
+    μ[f t | ℱ s] =ᵐ[μ] f s
+  secondMoment : ∀ t ∈ Ico (0 : ℝ) 1, ∫ ω, (f t ω)^2 ∂μ = t
+
+variable {ℱ : Filtration ℝ m} {f : ℝ → Ω → ℝ}
+
+lemma UnitMomentMartingale.integrable (h : UnitMomentMartingale (μ := μ) ℱ f)
+    {t : ℝ} (ht : t ∈ Ico (0 : ℝ) 1) : Integrable (f t) μ :=
+  Integrable.of_bound ((h.measurable t ht).mono (ℱ.le t)).aestronglyMeasurable 1
+    (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le (h.bounded t ht ω)))
+
+lemma UnitMomentMartingale.integrable_mul (h : UnitMomentMartingale (μ := μ) ℱ f)
+    {s t : ℝ} (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) :
+    Integrable (fun ω ↦ f s ω * f t ω) μ :=
+  Integrable.of_bound (((h.measurable s hs).mono (ℱ.le s)).mul
+    ((h.measurable t ht).mono (ℱ.le t))).aestronglyMeasurable 1
+    (Eventually.of_forall (fun ω ↦ by
+      rw [Real.norm_eq_abs,abs_mul]
+      nlinarith [h.bounded s hs ω,h.bounded t ht ω,abs_nonneg (f s ω),abs_nonneg (f t ω)]))
+
+lemma UnitMomentMartingale.integral_past_mul (h : UnitMomentMartingale (μ := μ) ℱ f)
+    {s t : ℝ} (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) (hst : s ≤ t) :
+    ∫ ω, f s ω * f t ω ∂μ = s := by
+  have hc := condExp_mul_of_stronglyMeasurable_left (h.measurable s hs)
+    (h.integrable_mul hs ht) (h.integrable ht)
+  have he : μ[(f s) * (f t) | ℱ s] =ᵐ[μ] fun ω ↦ (f s ω)^2 := by
+    filter_upwards [hc,h.condExp s hs t ht hst] with ω hω hω'
+    simpa only [Pi.mul_apply,hω',pow_two] using hω
+  calc
+    _ = ∫ ω, μ[(f s) * (f t) | ℱ s] ω ∂μ := (integral_condExp (ℱ.le s)).symm
+    _ = ∫ ω, (f s ω)^2 ∂μ := integral_congr_ae he
+    _ = s := h.secondMoment s hs
+
+lemma UnitMomentMartingale.sampled_gap_martingale
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {s : ℝ}
+    (hs : s ∈ Ico (0 : ℝ) 1) (r : ℕ → ℝ) (hr : Monotone r)
+    (hrmem : ∀ n, r n ∈ Ico s 1) :
+    Martingale (fun n ω ↦ 1-f s ω*f (r n) ω)
+      (⟨fun n ↦ ℱ (r n),fun _ _ hij ↦ ℱ.mono (hr hij),fun n ↦ ℱ.le (r n)⟩ :
+        Filtration ℕ m) μ := by
+  have ht (n : ℕ) : r n ∈ Ico (0 : ℝ) 1 := ⟨hs.1.trans (hrmem n).1,(hrmem n).2⟩
+  refine ⟨fun n ↦ stronglyMeasurable_const.sub
+    (((h.measurable s hs).mono (ℱ.mono (hrmem n).1)).mul (h.measurable (r n) (ht n))),?_⟩
+  intro i j hij
+  have hc := condExp_mul_of_stronglyMeasurable_left
+    ((h.measurable s hs).mono (ℱ.mono (hrmem i).1))
+    (h.integrable_mul hs (ht j)) (h.integrable (ht j))
+  have hc1 := condExp_of_stronglyMeasurable (ℱ.le (r i))
+    (stronglyMeasurable_const (b := (1 : ℝ))) (integrable_const (μ := μ) 1)
+  filter_upwards [condExp_sub (integrable_const (1 : ℝ))
+      (h.integrable_mul hs (ht j)) (ℱ (r i)),hc,
+      h.condExp (r i) (ht i) (r j) (ht j) (hr hij)] with ω hsub hmul he
+  simp only [Pi.sub_apply] at hsub
+  change μ[(fun ω ↦ f s ω*f (r j) ω) | ℱ (r i)] ω =
+    f s ω*μ[f (r j) | ℱ (r i)] ω at hmul
+  rw [hc1,hmul] at hsub
+  simpa only [Pi.sub_def,Pi.mul_def,Pi.mul_apply,he] using hsub
+
+omit [IsProbabilityMeasure μ] in
+lemma UnitMomentMartingale.sampled_gap_nonneg
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {s t : ℝ}
+    (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) (ω : Ω) :
+    0 ≤ 1-f s ω*f t ω := by
+  have hh : f s ω*f t ω ≤ 1 := calc
+    _ ≤ |f s ω*f t ω| := le_abs_self _
+    _ = |f s ω| * |f t ω| := abs_mul _ _
+    _ ≤ 1 := by nlinarith [h.bounded s hs ω,h.bounded t ht ω,
+      abs_nonneg (f s ω),abs_nonneg (f t ω)]
+  linarith
+
+lemma UnitMomentMartingale.sampled_gap_integral
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {s t : ℝ}
+    (hs : s ∈ Ico (0 : ℝ) 1) (ht : t ∈ Ico (0 : ℝ) 1) (hst : s ≤ t) :
+    ∫ ω, 1-f s ω*f t ω ∂μ = 1-s := by
+  rw [integral_sub (integrable_const 1) (h.integrable_mul hs ht),
+    h.integral_past_mul hs ht hst]
+  simp
+
+lemma UnitMomentMartingale.finite_gap_maximal
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {s : ℝ}
+    (hs : s ∈ Ico (0 : ℝ) 1) (S : Finset ℝ)
+    (hS : ∀ t ∈ S, t ∈ Ico s 1) (ε : ℝ≥0) :
+    (ε : ℝ≥0∞) * μ {ω | ∃ t ∈ S, (ε : ℝ) ≤ 1-f s ω*f t ω} ≤
+      ENNReal.ofReal (1-s) := by
+  classical
+  by_cases hne : S.Nonempty
+  · have hcard : 0 < S.card := Finset.card_pos.mpr hne
+    let e := S.orderEmbOfFin rfl
+    let r : ℕ → ℝ := fun n ↦ e ⟨min n (S.card-1),by omega⟩
+    have hr : Monotone r := by
+      intro i j hij
+      exact e.monotone (show min i (S.card-1) ≤ min j (S.card-1) from min_le_min_right _ hij)
+    have hrmem (n : ℕ) : r n ∈ Ico s 1 := hS _ (S.orderEmbOfFin_mem rfl _)
+    have hm := h.sampled_gap_martingale hs r hr hrmem
+    have hn : 0 ≤ (fun n ω ↦ 1-f s ω*f (r n) ω) :=
+      fun n ω ↦ h.sampled_gap_nonneg hs ⟨hs.1.trans (hrmem n).1,(hrmem n).2⟩ ω
+    have hb := maximal_ineq hm.submartingale hn (ε := ε) (S.card-1)
+    have hset : {ω | ∃ t ∈ S, (ε : ℝ) ≤ 1-f s ω*f t ω} ⊆
+        {ω | (ε : ℝ) ≤ (Finset.range ((S.card-1)+1)).sup' Finset.nonempty_range_add_one
+          (fun k ↦ 1-f s ω*f (r k) ω)} := by
+      rintro ω ⟨t,ht,heps⟩
+      let i := (S.orderIsoOfFin rfl).symm ⟨t,ht⟩
+      have her : r i.val = t := by
+        dsimp only [r]
+        have hh : (⟨min i.val (S.card-1),by omega⟩ : Fin S.card) = i :=
+          Fin.ext (min_eq_left (by omega))
+        rw [hh]
+        change ((S.orderIsoOfFin rfl) i).val = t
+        simp [i]
+      refine heps.trans ?_
+      rw [← her]
+      exact Finset.le_sup' (f := fun k ↦ 1-f s ω*f (r k) ω)
+        (Finset.mem_range.mpr (by omega : i.val < (S.card-1)+1))
+    calc
+      _ ≤ (ε : ℝ≥0∞) * μ {ω | (ε : ℝ) ≤
+          (Finset.range ((S.card-1)+1)).sup' Finset.nonempty_range_add_one
+            (fun k ↦ 1-f s ω*f (r k) ω)} := by gcongr
+      _ ≤ _ := hb
+      _ ≤ ENNReal.ofReal (∫ ω, 1-f s ω*f (r (S.card-1)) ω ∂μ) := by
+        apply ENNReal.ofReal_le_ofReal
+        exact setIntegral_le_integral (integrable_const 1 |>.sub
+          (h.integrable_mul hs ⟨hs.1.trans (hrmem _).1,(hrmem _).2⟩))
+          (Eventually.of_forall (hn (S.card-1)))
+      _ = _ := congrArg ENNReal.ofReal (h.sampled_gap_integral hs
+        ⟨hs.1.trans (hrmem _).1,(hrmem _).2⟩ (hrmem _).1)
+  · simp [Finset.not_nonempty_iff_eq_empty.mp hne]
+
+lemma UnitMomentMartingale.rational_gap_maximal
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {s : ℝ}
+    (hs : s ∈ Ico (0 : ℝ) 1) (ε : ℝ≥0) :
+    (ε : ℝ≥0∞) * μ {ω | ∃ q : ℚ, (q : ℝ) ∈ Ico s 1 ∧
+      (ε : ℝ) ≤ 1-f s ω*f q ω} ≤ ENNReal.ofReal (1-s) := by
+  classical
+  let E : Finset ℚ → Set Ω := fun S ↦ {ω | ∃ q ∈ S, (q : ℝ) ∈ Ico s 1 ∧
+    (ε : ℝ) ≤ 1-f s ω*f q ω}
+  have hd : Directed (· ⊆ ·) E := by
+    intro S T
+    refine ⟨S∪T, ?_, ?_⟩
+    · rintro ω ⟨q,hq,hmem,he⟩
+      exact ⟨q,Finset.mem_union_left _ hq,hmem,he⟩
+    · rintro ω ⟨q,hq,hmem,he⟩
+      exact ⟨q,Finset.mem_union_right _ hq,hmem,he⟩
+  have heq : {ω | ∃ q : ℚ, (q : ℝ) ∈ Ico s 1 ∧ (ε : ℝ) ≤ 1-f s ω*f q ω} =
+      ⋃ S : Finset ℚ, E S := by
+    ext ω
+    constructor
+    · rintro ⟨q,hmem,he⟩
+      exact mem_iUnion.mpr ⟨{q},q,Finset.mem_singleton_self _,hmem,he⟩
+    · intro hω
+      obtain ⟨S,q,hq,hmem,he⟩ := mem_iUnion.mp hω
+      exact ⟨q,hmem,he⟩
+  rw [heq,hd.measure_iUnion,ENNReal.mul_iSup]
+  refine iSup_le fun S ↦ ?_
+  let R : Finset ℝ := (S.filter (fun q : ℚ ↦ (q : ℝ) ∈ Ico s 1)).image (fun q : ℚ ↦ (q : ℝ))
+  have hR : ∀ t ∈ R, t ∈ Ico s 1 := by
+    intro t ht
+    obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp ht
+    exact (Finset.mem_filter.mp hq).2
+  have hE : E S ⊆ {ω | ∃ t ∈ R, (ε : ℝ) ≤ 1-f s ω*f t ω} := by
+    rintro ω ⟨q,hq,hmem,he⟩
+    exact ⟨(q : ℝ),Finset.mem_image.mpr ⟨q,Finset.mem_filter.mpr ⟨hq,hmem⟩,rfl⟩,he⟩
+  calc
+    _ ≤ (ε : ℝ≥0∞) * μ {ω | ∃ t ∈ R, (ε : ℝ) ≤ 1-f s ω*f t ω} := by gcongr
+    _ ≤ _ := h.finite_gap_maximal hs R hR ε
+
+noncomputable def terminalTime (n : ℕ) : ℝ := 1-1/(n+1)
+
+lemma terminalTime_mem (n : ℕ) : terminalTime n ∈ Ico (0 : ℝ) 1 := by
+  have hn : (1 : ℝ) ≤ n+1 := by linarith [Nat.cast_nonneg (α := ℝ) n]
+  have hp : (0 : ℝ) < n+1 := by positivity
+  simp only [terminalTime]
+  constructor
+  · have hh : (1 : ℝ)/(n+1) ≤ 1 := (div_le_one hp).mpr hn
+    linarith
+  · have hh : (0 : ℝ) < 1/(n+1) := by positivity
+    linarith
+
+lemma terminalTime_monotone : Monotone terminalTime := by
+  intro i j hij
+  simp only [terminalTime]
+  have hh : (1 : ℝ)/(j+1) ≤ 1/(i+1) := one_div_le_one_div_of_le
+    (by positivity) (by exact_mod_cast Nat.add_le_add_right hij 1)
+  linarith
+
+lemma terminalTime_tendsto : Tendsto terminalTime atTop (𝓝[<] (1 : ℝ)) := by
+  refine tendsto_nhdsWithin_iff.mpr ⟨?_,Eventually.of_forall (fun n ↦ (terminalTime_mem n).2)⟩
+  change Tendsto (fun n : ℕ ↦ (1 : ℝ)-1/((n : ℝ)+1)) atTop (𝓝 (1 : ℝ))
+  simpa only [sub_zero] using (tendsto_const_nhds (x := (1 : ℝ))).sub
+    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
+
+lemma UnitMomentMartingale.ae_exists_rational_gap_bound
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᵐ ω ∂μ, ∃ n, ∀ q : ℚ, (q : ℝ) ∈ Ico (terminalTime n) 1 →
+      1-f (terminalTime n) ω*f q ω < ε := by
+  let E : ℕ → Set Ω := fun n ↦ {ω | ∃ q : ℚ, (q : ℝ) ∈ Ico (terminalTime n) 1 ∧
+    ε ≤ 1-f (terminalTime n) ω*f q ω}
+  have hbound (n : ℕ) : ENNReal.ofReal ε * μ (⋂ n, E n) ≤
+      ENNReal.ofReal (1-terminalTime n) := calc
+    _ ≤ ENNReal.ofReal ε * μ (E n) := by gcongr; exact iInter_subset E n
+    _ ≤ _ := by
+      have hh := h.rational_gap_maximal (terminalTime_mem n) (NNReal.mk ε hε.le)
+      simpa only [E,NNReal.coe_mk,ENNReal.coe_nnreal_eq] using hh
+  have hzero : ENNReal.ofReal ε * μ (⋂ n, E n) = 0 := by
+    apply le_antisymm _ bot_le
+    have hlim : Tendsto (fun n ↦ ENNReal.ofReal (1-terminalTime n)) atTop (𝓝 0) := by
+      have hh : Tendsto (fun n ↦ 1-terminalTime n) atTop (𝓝 (0 : ℝ)) := by
+        simpa only [sub_self] using ((tendsto_const_nhds (x := (1 : ℝ))).sub
+          (tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1)
+      simpa only [ENNReal.ofReal_zero,Function.comp_def] using
+        ENNReal.continuous_ofReal.continuousAt.tendsto.comp hh
+    exact ge_of_tendsto' hlim hbound
+  have hn : μ (⋂ n, E n) = 0 := (mul_eq_zero.mp hzero).resolve_left
+    (ne_of_gt (ENNReal.ofReal_pos.mpr hε))
+  have ha : ∀ᵐ ω ∂μ, ω ∉ ⋂ n, E n := by
+    rw [ae_iff]
+    convert hn using 1
+    simp only [not_not,Set.ofPred_mem_eq]
+  filter_upwards [ha] with ω hω
+  simp only [mem_iInter,E,mem_ofPred_eq,not_forall,not_exists,not_and,not_le] at hω
+  exact hω
+
+lemma continuous_gap_bound_of_rat {v : ℝ → ℝ}
+    (hv : ContinuousOn v (Ico (0 : ℝ) 1)) {s ε : ℝ}
+    (hs : s ∈ Ico (0 : ℝ) 1)
+    (hq : ∀ q : ℚ, (q : ℝ) ∈ Ico s 1 → 1-v s*v q < ε) :
+    ∀ t ∈ Ioo s 1, 1-v s*v t ≤ ε := by
+  intro t ht
+  by_contra! he
+  have hvt : ContinuousAt v t := (hv t ⟨hs.1.trans ht.1.le,ht.2⟩).continuousAt
+    (Ico_mem_nhds_iff.mpr ⟨hs.1.trans_lt ht.1,ht.2⟩)
+  have hgt : ContinuousAt (fun r ↦ 1-v s*v r) t := continuousAt_const.sub
+    (continuousAt_const.mul hvt)
+  have hevent : {r | ε < 1-v s*v r} ∈ 𝓝 t := hgt (Ioi_mem_nhds he)
+  obtain ⟨q,hq'⟩ := Rat.denseRange_cast.mem_nhds (inter_mem hevent (Ioo_mem_nhds ht.1 ht.2))
+  exact (hq q ⟨hq'.2.1.le,hq'.2.2⟩).not_gt hq'.1
+
+lemma UnitMomentMartingale.exists_sequence_terminal
+    (h : UnitMomentMartingale (μ := μ) ℱ f) :
+    ∃ U : Ω → ℝ, StronglyMeasurable[ℱ 1] U ∧
+      (∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
+        Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω))) := by
+  let 𝒢 : Filtration ℕ m := ⟨fun n ↦ ℱ (terminalTime n),
+    fun _ _ hij ↦ ℱ.mono (terminalTime_monotone hij),fun n ↦ ℱ.le _⟩
+  have hm : Martingale (fun n ↦ f (terminalTime n)) 𝒢 μ :=
+    ⟨fun n ↦ h.measurable _ (terminalTime_mem n),fun i j hij ↦
+      h.condExp _ (terminalTime_mem i) _ (terminalTime_mem j) (terminalTime_monotone hij)⟩
+  have hb (n : ℕ) : eLpNorm (f (terminalTime n)) 1 μ ≤ (1 : ℝ≥0) := by
+    have hh := eLpNorm_le_of_ae_bound (μ := μ) (p := (1 : ℝ≥0∞))
+      (h.integrable (terminalTime_mem n)).aestronglyMeasurable
+      (Eventually.of_forall (fun ω ↦ (Real.norm_eq_abs _).trans_le
+        (h.bounded _ (terminalTime_mem n) ω)))
+    simpa using hh
+  let U := 𝒢.limitProcess (fun n ↦ f (terminalTime n)) μ
+  have hU : StronglyMeasurable[ℱ 1] U :=
+    Filtration.stronglyMeasurable_limitProcess.mono (iSup_le fun n ↦ ℱ.mono (terminalTime_mem n).2.le)
+  refine ⟨U,hU,?_⟩
+  filter_upwards [hm.submartingale.ae_tendsto_limitProcess hb] with ω hω
+  exact ⟨le_of_tendsto' hω.abs (fun n ↦ h.bounded _ (terminalTime_mem n) ω),hω⟩
+
+lemma UnitMomentMartingale.exists_continuous_terminal
+    (h : UnitMomentMartingale (μ := μ) ℱ f)
+    (hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ f t ω) (Ico (0 : ℝ) 1)) :
+    ∃ U : Ω → ℝ, StronglyMeasurable[ℱ 1] U ∧
+      (∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
+        Tendsto (fun t ↦ f t ω) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) := by
+  obtain ⟨U,hU,hseq⟩ := h.exists_sequence_terminal
+  have hg : ∀ᵐ ω ∂μ, ∀ k : ℕ, ∃ n, ∀ q : ℚ,
+      (q : ℝ) ∈ Ico (terminalTime n) 1 →
+        1-f (terminalTime n) ω*f q ω < 1/(k+1) := by
+    apply ae_all_iff.mpr
+    intro k
+    exact h.ae_exists_rational_gap_bound (by positivity)
+  refine ⟨U,hU,?_⟩
+  filter_upwards [hc,hseq,hg] with ω hcont hseqω hgap
+  refine ⟨hseqω.1,Metric.tendsto_nhds.mpr (fun δ hδ ↦ ?_)⟩
+  obtain ⟨k,hk⟩ : ∃ k : ℕ, (1 : ℝ)/(k+1) < δ^2/32 := by
+    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
+      (Iio_mem_nhds (show (0 : ℝ) < δ^2/32 by positivity))
+    exact hh.exists
+  obtain ⟨n,hn⟩ := hgap k
+  have hreal := continuous_gap_bound_of_rat hcont (terminalTime_mem n) hn
+  have hbound (t : ℝ) (ht : t ∈ Ioo (terminalTime n) 1) :
+      |f t ω-f (terminalTime n) ω| ≤ δ/4 := by
+    have hbt := h.bounded t ⟨(terminalTime_mem n).1.trans ht.1.le,ht.2⟩ ω
+    have hbs := h.bounded _ (terminalTime_mem n) ω
+    have he := hreal t ht
+    have hsq1 : (f t ω)^2 ≤ 1 := by nlinarith [abs_le.mp hbt]
+    have hsq2 : (f (terminalTime n) ω)^2 ≤ 1 := by nlinarith [abs_le.mp hbs]
+    have habs : |f t ω-f (terminalTime n) ω|^2 < (δ/4)^2 := by
+      rw [sq_abs]
+      nlinarith
+    nlinarith [abs_nonneg (f t ω-f (terminalTime n) ω)]
+  have hUb : |U ω-f (terminalTime n) ω| ≤ δ/4 := by
+    apply le_of_tendsto ((hseqω.2.sub_const _).abs)
+    have hev : ∀ᶠ k in atTop, terminalTime k ∈ Ioo (terminalTime n) 1 :=
+      terminalTime_tendsto.eventually (Ioo_mem_nhdsLT (terminalTime_mem n).2)
+    exact hev.mono (fun k hk ↦ hbound _ hk)
+  filter_upwards [Ioo_mem_nhdsLT (terminalTime_mem n).2] with t ht
+  rw [Real.dist_eq]
+  have hh := abs_add_le (f t ω-f (terminalTime n) ω) (f (terminalTime n) ω-U ω)
+  have hh' := hbound t ht
+  have hUb' : |f (terminalTime n) ω-U ω| ≤ δ/4 := by simpa only [abs_sub_comm] using hUb
+  have heq : f t ω-U ω = (f t ω-f (terminalTime n) ω)+(f (terminalTime n) ω-U ω) := by ring
+  rw [heq]
+  linarith
+
+lemma UnitMomentMartingale.terminal_square
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
+    (hUm : StronglyMeasurable[ℱ 1] U)
+    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
+      Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω))) :
+    ∀ᵐ ω ∂μ, (U ω)^2 = 1 := by
+  have hU2i : Integrable (fun ω ↦ (U ω)^2) μ :=
+    Integrable.of_bound ((hUm.mono (ℱ.le 1)).pow 2).aestronglyMeasurable 1
+      (hU.mono (fun ω hω ↦ by
+        rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+        nlinarith [abs_le.mp hω.1]))
+  have ht := tendsto_integral_of_dominated_convergence (μ := μ)
+    (F := fun n ω ↦ (f (terminalTime n) ω)^2) (fun _ : Ω ↦ (1 : ℝ))
+    (fun n ↦ (((h.measurable _ (terminalTime_mem n)).mono (ℱ.le _)).pow 2).aestronglyMeasurable)
+    (integrable_const 1)
+    (fun n ↦ Eventually.of_forall (fun ω ↦ by
+      rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+      nlinarith [abs_le.mp (h.bounded _ (terminalTime_mem n) ω)]))
+    (hU.mono (fun ω hω ↦ hω.2.pow 2))
+  have hi : ∫ ω, (U ω)^2 ∂μ = 1 := by
+    have hh : (fun n ↦ ∫ ω, (f (terminalTime n) ω)^2 ∂μ) = terminalTime :=
+      funext (fun n ↦ h.secondMoment _ (terminalTime_mem n))
+    rw [hh] at ht
+    exact tendsto_nhds_unique ht (tendsto_nhdsWithin_iff.mp terminalTime_tendsto).1
+  have hnonneg : 0 ≤ᵐ[μ] (fun ω ↦ 1-(U ω)^2) := hU.mono (fun ω hω ↦ by
+    dsimp
+    nlinarith [abs_le.mp hω.1])
+  have hzero : ∫ ω, 1-(U ω)^2 ∂μ = 0 := by
+    rw [integral_sub (integrable_const 1) hU2i,hi]
+    simp
+  have he := (integral_eq_zero_iff_of_nonneg_ae hnonneg ((integrable_const 1).sub hU2i)).mp hzero
+  filter_upwards [he] with ω hω
+  change 1-(U ω)^2=0 at hω
+  linarith
+
+lemma UnitMomentMartingale.terminal_mean_square
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
+    (hUm : StronglyMeasurable[ℱ 1] U)
+    (hU : ∀ᵐ ω ∂μ, |U ω| ≤ 1 ∧
+      Tendsto (fun t ↦ f t ω) (𝓝[<] (1 : ℝ)) (𝓝 (U ω))) :
+    Tendsto (fun t ↦ ∫ ω, (f t ω-U ω)^2 ∂μ) (𝓝[<] (1 : ℝ)) (𝓝 (0 : ℝ)) := by
+  have hev : ∀ᶠ t in 𝓝[<] (1 : ℝ), t ∈ Ico (0 : ℝ) 1 := by
+    filter_upwards [Ioo_mem_nhdsLT (show (0 : ℝ)<1 by norm_num)] with t ht
+    exact ⟨ht.1.le,ht.2⟩
+  have ht := tendsto_integral_filter_of_dominated_convergence (μ := μ)
+    (F := fun t ω ↦ (f t ω-U ω)^2) (f := fun _ ↦ (0 : ℝ)) (fun _ : Ω ↦ (4 : ℝ))
+    (hev.mono (fun t ht ↦ ((((h.measurable t ht).mono (ℱ.le t)).sub
+      (hUm.mono (ℱ.le 1))).pow 2).aestronglyMeasurable))
+    (hev.mono (fun t ht ↦ hU.mono (fun ω hω ↦ by
+      rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+      have hb := h.bounded t ht ω
+      have hd : |f t ω-U ω| ≤ 2 := (abs_sub _ _).trans (by linarith [hω.1])
+      nlinarith [sq_abs (f t ω-U ω),abs_nonneg (f t ω-U ω)])))
+    (integrable_const 4)
+    (hU.mono (fun ω hω ↦ by
+      simpa only [sub_self,zero_pow (by decide : (2 : ℕ) ≠ 0)] using (hω.2.sub_const (U ω)).pow 2))
+  simpa only [integral_zero] using ht
+
+end SKValue
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/ValueEulerOneStep.lean b/lean/OAI/Probability/SKValue/Processes/ValueEulerOneStep.lean
index c38e7fd1274d7117ead57beea33440c4c902daa5..8692957e01f0859b41bdbe4b9a2be6fc03edbc46 100644
--- a/lean/OAI/Probability/SKValue/Processes/ValueEulerOneStep.lean
+++ b/lean/OAI/Probability/SKValue/Processes/ValueEulerOneStep.lean
@@ -1,91 +1,91 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.ThirdOrderRemainderBound
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-
-lemma value_euler_one_step {V : ℝ → ℝ → ℝ} {γ : ℝ → ℝ}
-    {t δ x z G K L : ℝ} (hδ : 0≤δ) (hδ1 : δ≤1)
-    (hG : 0≤G) (hK : 0≤K) (hL : 0≤L)
-    (hγ : MonotoneOn γ (Icc t (t+δ))) (hγ0 : 0≤γ t) (hγG : γ t≤G)
-    (hu : |deriv (V t) x|≤1) (hsmooth : ContDiff ℝ 3 (V t))
-    (ha : |deriv (deriv (V t)) x|≤K)
-    (hthird : ∀ y, |iteratedDeriv 3 (V t) y|≤K)
-    (hp : ∀ s ∈ Icc t (t+δ), ∀ y, |(1/2 : ℝ)*(deriv (V s) y)^2|≤K)
-    (haLip : ∀ s ∈ Icc t (t+δ), ∀ y,
-      |deriv (deriv (V s)) y-deriv (deriv (V t)) x|≤L*(|s-t|+|y-x|))
-    (hpLip : ∀ s ∈ Icc t (t+δ), ∀ y,
-      |(1/2 : ℝ)*(deriv (V s) y)^2-(1/2 : ℝ)*(deriv (V t) x)^2|≤L*(|s-t|+|y-x|))
-    (haint : ∀ y, IntervalIntegrable (fun s ↦ deriv (deriv (V s)) y) volume t (t+δ))
-    (hpint : ∀ y, IntervalIntegrable (fun s ↦ γ s*((1/2 : ℝ)*(deriv (V s) y)^2)) volume t (t+δ))
-    (hPDE : ∀ y, V (t+δ) y-V t y =
-      -(∫ s in t..(t+δ), (1/2 : ℝ)*deriv (deriv (V s)) y+γ s*((1/2 : ℝ)*(deriv (V s) y)^2))) :
-    let d := Real.sqrt δ*z+δ*γ t*deriv (V t) x
-    |V (t+δ) (x+d)-V t x-deriv (V t) x*Real.sqrt δ*z-
-      (1/2 : ℝ)*δ*deriv (deriv (V t)) x*(z^2-1)-
-      (1/2 : ℝ)*δ*γ t*(deriv (V t) x)^2| ≤
-    δ*Real.sqrt δ*(K*(|z|+G)^3+(1/2+G)*L*(1+|z|+G)+K*G*(|z|+G/2))+
-      δ*K*(γ (t+δ)-γ t) := by
-  dsimp only
-  let d := Real.sqrt δ*z+δ*γ t*deriv (V t) x
-  have htm := gradient_generator_quadrature
-    (b := fun s y ↦ deriv (deriv (V s)) y)
-    (p := fun s y ↦ (1/2 : ℝ)*(deriv (V s) y)^2) (y := x+d)
-    hδ hG hK hL hγ hγ0 hγG
-    (fun s hs ↦ hp s hs _) (fun s hs ↦ haLip s hs _) (fun s hs ↦ hpLip s hs _)
-    (haint _) (hpint _)
-  have ht : |(V (t+δ) (x+d)-V t x+deriv (V t) x-
-        (1/2 : ℝ)*δ*γ t*(deriv (V t) x)^2)-
-      (V t (x+d)-V t x+deriv (V t) x)+
-      δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*deriv (V t) x*deriv (V t) x)| ≤
-      δ*(((1/2 : ℝ)+G)*L*(δ+|d|)+K*(γ (t+δ)-γ t)) := by
-    have heq : (V (t+δ) (x+d)-V t x+deriv (V t) x-
-        (1/2 : ℝ)*δ*γ t*(deriv (V t) x)^2)-
-      (V t (x+d)-V t x+deriv (V t) x)+
-      δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*deriv (V t) x*deriv (V t) x) =
-      (V (t+δ) (x+d)-V t (x+d))+
-      δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*((1/2 : ℝ)*(deriv (V t) x)^2)) := by ring
-    rw [heq,hPDE]
-    rw [show -(∫ s in t..(t+δ), (1/2 : ℝ)*deriv (deriv (V s)) (x+d)+
-        γ s*((1/2 : ℝ)*(deriv (V s) (x+d))^2))+
-        δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*((1/2 : ℝ)*(deriv (V t) x)^2)) =
-        -((∫ s in t..(t+δ), (1/2 : ℝ)*deriv (deriv (V s)) (x+d)+
-        γ s*((1/2 : ℝ)*(deriv (V s) (x+d))^2))-
-        δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*((1/2 : ℝ)*(deriv (V t) x)^2))) by ring,
-      abs_neg]
-    simpa only [add_sub_cancel_left] using htm
-  have hs : |(V t (x+d)-V t x+deriv (V t) x)-deriv (V t) x-
-      deriv (V t) x*d-(1/2 : ℝ)*deriv (deriv (V t)) x*d^2| ≤ K*|d|^3 := by
-    simpa only [add_sub_cancel_right] using third_order_remainder_bound hsmooth hK hthird x d
-  have hh := euler_cubic_remainder hδ hδ1 hG hK (by positivity : 0≤(1/2+G)*L)
-    (by rwa [abs_of_nonneg hγ0]) hu ha hs ht
-  convert hh using 1
-  congr 1
-  dsimp only [d]
-  ring
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.ThirdOrderRemainderBound
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+
+lemma value_euler_one_step {V : ℝ → ℝ → ℝ} {γ : ℝ → ℝ}
+    {t δ x z G K L : ℝ} (hδ : 0≤δ) (hδ1 : δ≤1)
+    (hG : 0≤G) (hK : 0≤K) (hL : 0≤L)
+    (hγ : MonotoneOn γ (Icc t (t+δ))) (hγ0 : 0≤γ t) (hγG : γ t≤G)
+    (hu : |deriv (V t) x|≤1) (hsmooth : ContDiff ℝ 3 (V t))
+    (ha : |deriv (deriv (V t)) x|≤K)
+    (hthird : ∀ y, |iteratedDeriv 3 (V t) y|≤K)
+    (hp : ∀ s ∈ Icc t (t+δ), ∀ y, |(1/2 : ℝ)*(deriv (V s) y)^2|≤K)
+    (haLip : ∀ s ∈ Icc t (t+δ), ∀ y,
+      |deriv (deriv (V s)) y-deriv (deriv (V t)) x|≤L*(|s-t|+|y-x|))
+    (hpLip : ∀ s ∈ Icc t (t+δ), ∀ y,
+      |(1/2 : ℝ)*(deriv (V s) y)^2-(1/2 : ℝ)*(deriv (V t) x)^2|≤L*(|s-t|+|y-x|))
+    (haint : ∀ y, IntervalIntegrable (fun s ↦ deriv (deriv (V s)) y) volume t (t+δ))
+    (hpint : ∀ y, IntervalIntegrable (fun s ↦ γ s*((1/2 : ℝ)*(deriv (V s) y)^2)) volume t (t+δ))
+    (hPDE : ∀ y, V (t+δ) y-V t y =
+      -(∫ s in t..(t+δ), (1/2 : ℝ)*deriv (deriv (V s)) y+γ s*((1/2 : ℝ)*(deriv (V s) y)^2))) :
+    let d := Real.sqrt δ*z+δ*γ t*deriv (V t) x
+    |V (t+δ) (x+d)-V t x-deriv (V t) x*Real.sqrt δ*z-
+      (1/2 : ℝ)*δ*deriv (deriv (V t)) x*(z^2-1)-
+      (1/2 : ℝ)*δ*γ t*(deriv (V t) x)^2| ≤
+    δ*Real.sqrt δ*(K*(|z|+G)^3+(1/2+G)*L*(1+|z|+G)+K*G*(|z|+G/2))+
+      δ*K*(γ (t+δ)-γ t) := by
+  dsimp only
+  let d := Real.sqrt δ*z+δ*γ t*deriv (V t) x
+  have htm := gradient_generator_quadrature
+    (b := fun s y ↦ deriv (deriv (V s)) y)
+    (p := fun s y ↦ (1/2 : ℝ)*(deriv (V s) y)^2) (y := x+d)
+    hδ hG hK hL hγ hγ0 hγG
+    (fun s hs ↦ hp s hs _) (fun s hs ↦ haLip s hs _) (fun s hs ↦ hpLip s hs _)
+    (haint _) (hpint _)
+  have ht : |(V (t+δ) (x+d)-V t x+deriv (V t) x-
+        (1/2 : ℝ)*δ*γ t*(deriv (V t) x)^2)-
+      (V t (x+d)-V t x+deriv (V t) x)+
+      δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*deriv (V t) x*deriv (V t) x)| ≤
+      δ*(((1/2 : ℝ)+G)*L*(δ+|d|)+K*(γ (t+δ)-γ t)) := by
+    have heq : (V (t+δ) (x+d)-V t x+deriv (V t) x-
+        (1/2 : ℝ)*δ*γ t*(deriv (V t) x)^2)-
+      (V t (x+d)-V t x+deriv (V t) x)+
+      δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*deriv (V t) x*deriv (V t) x) =
+      (V (t+δ) (x+d)-V t (x+d))+
+      δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*((1/2 : ℝ)*(deriv (V t) x)^2)) := by ring
+    rw [heq,hPDE]
+    rw [show -(∫ s in t..(t+δ), (1/2 : ℝ)*deriv (deriv (V s)) (x+d)+
+        γ s*((1/2 : ℝ)*(deriv (V s) (x+d))^2))+
+        δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*((1/2 : ℝ)*(deriv (V t) x)^2)) =
+        -((∫ s in t..(t+δ), (1/2 : ℝ)*deriv (deriv (V s)) (x+d)+
+        γ s*((1/2 : ℝ)*(deriv (V s) (x+d))^2))-
+        δ*((1/2 : ℝ)*deriv (deriv (V t)) x+γ t*((1/2 : ℝ)*(deriv (V t) x)^2))) by ring,
+      abs_neg]
+    simpa only [add_sub_cancel_left] using htm
+  have hs : |(V t (x+d)-V t x+deriv (V t) x)-deriv (V t) x-
+      deriv (V t) x*d-(1/2 : ℝ)*deriv (deriv (V t)) x*d^2| ≤ K*|d|^3 := by
+    simpa only [add_sub_cancel_right] using third_order_remainder_bound hsmooth hK hthird x d
+  have hh := euler_cubic_remainder hδ hδ1 hG hK (by positivity : 0≤(1/2+G)*L)
+    (by rwa [abs_of_nonneg hγ0]) hu ha hs ht
+  convert hh using 1
+  congr 1
+  dsimp only [d]
+  ring
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/VariableTestLaw.lean b/lean/OAI/Probability/SKValue/Processes/VariableTestLaw.lean
index 67b31ad4655cfadabac4208bcb7a170f12b8ee5c..152cb8df2a3e46aa700867971b27b1f918e33d75 100644
--- a/lean/OAI/Probability/SKValue/Processes/VariableTestLaw.lean
+++ b/lean/OAI/Probability/SKValue/Processes/VariableTestLaw.lean
@@ -1,32 +1,32 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.ResponseContraction
-import OAI.Probability.SKValue.Gaussian.UniformError
-import OAI.Probability.SKValue.Equations.UniformPolynomial
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal
-namespace SKValue
-lemma IsDiffusion.splitResponse_varying {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
-    (hX:IsDiffusion W γ X) (t:Ioo (0:ℝ) 1) {F:ℕ → ℝ → ℝ} {f:ℝ → ℝ}
-    (hF:∀ n,BoundedSmooth (F n)) (hf:BoundedSmooth f) (hl:BoundedUniformLimit F f) :
-    Tendsto (fun n ↦ profileResponse (splitTerminal γ t n) (F n) (γ.splitSegment 0 t t.property.1.le n) 0 0)
-      atTop (𝓝 (∫ z, f (X t z) ∂W.μ)) := by
-  obtain ⟨C,hC,hb,hfb⟩:=hl.bound
-  have he:∀ n,∃ D:ℝ,∀ x,|F n x-f x| ≤ D :=
-    fun n ↦ ⟨C+C,fun x ↦ (abs_sub _ _).trans (add_le_add (hb n x) (hfb x))⟩
-  have hel:=uniformDistance_tendsto hl.tendsto he
-  have hd:Tendsto (fun n ↦ profileResponse (splitTerminal γ t n) (F n) (γ.splitSegment 0 t t.property.1.le n) 0 0-
-      profileResponse (splitTerminal γ t n) f (γ.splitSegment 0 t t.property.1.le n) 0 0) atTop (𝓝 0) := by
-    apply squeeze_zero_norm (fun n ↦ ?_) hel
-    rw [Real.norm_eq_abs]
-    exact (splitTerminal_smooth γ t n).profile_response_contraction (hF n) hf (le_uniformDistance (he n)) _ _ _
-  have ht:=hd.add (hX.splitResponse_tendsto t hf)
-  simpa only [sub_add_cancel,zero_add] using ht
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.ResponseContraction
+import OAI.Probability.SKValue.Gaussian.UniformError
+import OAI.Probability.SKValue.Equations.UniformPolynomial
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal
+namespace SKValue
+lemma IsDiffusion.splitResponse_varying {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
+    (hX:IsDiffusion W γ X) (t:Ioo (0:ℝ) 1) {F:ℕ → ℝ → ℝ} {f:ℝ → ℝ}
+    (hF:∀ n,BoundedSmooth (F n)) (hf:BoundedSmooth f) (hl:BoundedUniformLimit F f) :
+    Tendsto (fun n ↦ profileResponse (splitTerminal γ t n) (F n) (γ.splitSegment 0 t t.property.1.le n) 0 0)
+      atTop (𝓝 (∫ z, f (X t z) ∂W.μ)) := by
+  obtain ⟨C,hC,hb,hfb⟩:=hl.bound
+  have he:∀ n,∃ D:ℝ,∀ x,|F n x-f x| ≤ D :=
+    fun n ↦ ⟨C+C,fun x ↦ (abs_sub _ _).trans (add_le_add (hb n x) (hfb x))⟩
+  have hel:=uniformDistance_tendsto hl.tendsto he
+  have hd:Tendsto (fun n ↦ profileResponse (splitTerminal γ t n) (F n) (γ.splitSegment 0 t t.property.1.le n) 0 0-
+      profileResponse (splitTerminal γ t n) f (γ.splitSegment 0 t t.property.1.le n) 0 0) atTop (𝓝 0) := by
+    apply squeeze_zero_norm (fun n ↦ ?_) hel
+    rw [Real.norm_eq_abs]
+    exact (splitTerminal_smooth γ t n).profile_response_contraction (hF n) hf (le_uniformDistance (he n)) _ _ _
+  have ht:=hd.add (hX.splitResponse_tendsto t hf)
+  simpa only [sub_add_cancel,zero_add] using ht
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/Volterra.lean b/lean/OAI/Probability/SKValue/Processes/Volterra.lean
index ef76d08d1008b00c8905a20a1219a515722d75c8..45c60a7060a99b6e6dbfb6f2eaa09c7712413749 100644
--- a/lean/OAI/Probability/SKValue/Processes/Volterra.lean
+++ b/lean/OAI/Probability/SKValue/Processes/Volterra.lean
@@ -1,131 +1,131 @@
-import Mathlib
-import OAI.Probability.SKValue.Control.Regularity
-
-namespace OAI
-
-section
-open MeasureTheory Set Filter Function
-open scoped Topology NNReal Nat
-namespace SKValue
-structure VolterraData (T:ℝ) (hT:0≤T) where
-  field:ℝ → ℝ → ℝ
-  lip:ℝ≥0
-  bound:∀t∈Icc (0:ℝ) T,∀x y,dist (field t x) (field t y)≤lip*dist x y
-  integ:∀Y:C(Icc (0:ℝ) T,ℝ),IntervalIntegrable (fun s ↦ field s (Y (projIcc 0 T hT s))) volume 0 T
-namespace VolterraData
-variable {T:ℝ} {hT:0≤T} (D:VolterraData T hT)
-noncomputable def next (Y:C(Icc (0:ℝ) T,ℝ)):C(Icc (0:ℝ) T,ℝ) where
-  toFun t:=∫s in (0:ℝ)..(t:ℝ),D.field s (Y (projIcc 0 T hT s))
-  continuous_toFun:=by
-    have hc:=intervalIntegral.continuousOn_primitive_interval' (D.integ Y) left_mem_uIcc
-    rw [uIcc_of_le hT] at hc
-    exact hc.domRestrict
-lemma next_apply (Y:C(Icc (0:ℝ) T,ℝ)) (t:Icc (0:ℝ) T):
-    D.next Y t=∫s in (0:ℝ)..(t:ℝ),D.field s (Y (projIcc 0 T hT s)) :=rfl
-lemma integ_to (Y:C(Icc (0:ℝ) T,ℝ)) (t:Icc (0:ℝ) T):
-    IntervalIntegrable (fun s ↦ D.field s (Y (projIcc 0 T hT s))) volume 0 t := by
-  apply (D.integ Y).mono_set
-  rw [uIcc_of_le t.2.1,uIcc_of_le hT]
-  exact Icc_subset_Icc le_rfl t.2.2
-lemma iterate_bound (Y Z:C(Icc (0:ℝ) T,ℝ)) (n:ℕ) (t:Icc (0:ℝ) T):
-    dist ((D.next)^[n] Y t) ((D.next)^[n] Z t)≤
-      (D.lip*|((t:ℝ)-0)|)^n/n.factorial*dist Y Z := by
-  induction n generalizing t with
-  | zero=>simpa using ContinuousMap.dist_apply_le_dist (f:=Y) (g:=Z) t
-  | succ n hn=>
-    rw [iterate_succ_apply',iterate_succ_apply',dist_eq_norm,next_apply,next_apply,
-      ←intervalIntegral.integral_sub (D.integ_to _ t) (D.integ_to _ t)]
-    calc
-      _≤∫s in uIoc (0:ℝ) (t:ℝ),(D.lip:ℝ)^(n+1)*|s-0|^n/n.factorial*dist Y Z := by
-        rw [intervalIntegral.norm_intervalIntegral_eq]
-        apply norm_integral_le_of_norm_le ((by fun_prop:Continuous (fun s:ℝ ↦
-          (D.lip:ℝ)^(n+1)*|s-0|^n/n.factorial*dist Y Z)).integrableOn_uIoc)
-        filter_upwards [ae_restrict_mem measurableSet_uIoc] with s hs
-        rw [uIoc_of_le t.2.1] at hs
-        have hsT:s∈Icc (0:ℝ) T:=⟨hs.1.le,hs.2.trans t.2.2⟩
-        rw [←dist_eq_norm]
-        have hp:=D.bound s hsT ((D.next)^[n] Y (projIcc 0 T hT s)) ((D.next)^[n] Z (projIcc 0 T hT s))
-        have hh:=hn (⟨s,hsT⟩:Icc (0:ℝ) T)
-        have he:projIcc 0 T hT s=(⟨s,hsT⟩:Icc (0:ℝ) T):=projIcc_of_mem hT hsT
-        rw [he] at hp ⊢
-        apply hp.trans
-        calc
-          _≤(D.lip:ℝ)*((D.lip*|s-0|)^n/n.factorial*dist Y Z):=mul_le_mul_of_nonneg_left hh D.lip.coe_nonneg
-          _=_:=by rw [mul_pow,pow_succ];ring
-      _≤_:=by
-        apply le_of_abs_le
-        rw [←intervalIntegral.abs_intervalIntegral_eq,intervalIntegral.integral_mul_const,
-          intervalIntegral.integral_div,intervalIntegral.integral_const_mul,abs_mul,abs_div,
-          abs_mul,intervalIntegral.abs_intervalIntegral_eq,integral_pow_abs_sub_uIoc,abs_div,
-          abs_pow,abs_pow,abs_dist,NNReal.abs_eq,abs_abs,mul_div,div_div,←abs_mul,
-          ←Nat.cast_succ,←Nat.cast_mul,←Nat.factorial_succ,Nat.abs_cast,←mul_pow]
-lemma exists_fixedPoint:∃Y:C(Icc (0:ℝ) T,ℝ),D.next Y=Y := by
-  obtain ⟨n,hn⟩:= (FloorSemiring.tendsto_pow_div_factorial_atTop ((D.lip:ℝ)*T)).eventually
-    (gt_mem_nhds zero_lt_one) |>.exists
-  have hn0:0≤((D.lip:ℝ)*T)^n/n.factorial:=by positivity
-  have hc:ContractingWith ⟨((D.lip:ℝ)*T)^n/n.factorial,hn0⟩ (D.next)^[n] := by
-    refine ⟨hn,LipschitzWith.of_dist_le_mul (fun Y Z ↦ ?_)⟩
-    rw [ContinuousMap.dist_le]
-    · intro t
-      apply (D.iterate_bound Y Z n t).trans
-      simp only [sub_zero,abs_of_nonneg t.2.1]
-      change (D.lip*↑t)^n/n.factorial*dist Y Z ≤ ((D.lip:ℝ)*T)^n/n.factorial*dist Y Z
-      gcongr
-      · exact mul_nonneg D.lip.coe_nonneg t.2.1
-      · exact t.2.2
-    · positivity
-  exact ⟨_,hc.isFixedPt_fixedPoint_iterate⟩
-end VolterraData
-end SKValue
-
-end
-
-section
-open MeasureTheory ProbabilityTheory Set Filter Function
-open scoped Topology NNReal
-namespace SKValue
-lemma exists_local_diffusion_path (W:BrownianSpace) (γ:OrderParameter) {T:ℝ}
-    (hT:T∈Ico (0:ℝ) 1) (b:ℝ → ℝ) (hb:ContinuousOn b (Icc (0:ℝ) T)):
-    ∃X:ℝ → ℝ,ContinuousOn X (Icc (0:ℝ) T) ∧
-      IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s)) volume 0 T ∧
-      ∀t∈Icc (0:ℝ) T,X t=b t+∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (X s) := by
-  obtain ⟨K,L,Lu,hLu,hG,hu⟩:=(sourceStripRegularity W γ).core T hT
-  have hg0:0≤γ.coeff T:=γ.nonneg T hT
-  have hi:IntervalIntegrable γ.coeff volume 0 T:=γ.intervalIntegrable.mono_set (by
-    rw [uIcc_of_le hT.1,uIcc_of_le (by norm_num:(0:ℝ)≤1)];exact Icc_subset_Icc le_rfl hT.2.le)
-  have hj:=derivative_joint_continuous hLu hu
-  let D:VolterraData T hT.1:={
-    field:=fun t y ↦ γ.coeff t*gradient W γ t (b t+y)
-    lip:=⟨γ.coeff T*Lu,mul_nonneg hg0 hLu⟩
-    bound:=by
-      intro t ht x y
-      have hg:0≤γ.coeff t:=γ.nonneg t ⟨ht.1,ht.2.trans_lt hT.2⟩
-      have hgt:γ.coeff t≤γ.coeff T:=γ.monotone ⟨ht.1,ht.2.trans_lt hT.2⟩ hT ht.2
-      simp only [Real.dist_eq,←mul_sub,abs_mul,abs_of_nonneg hg]
-      have hh:=hu t ht t ht (b t+x) (b t+y)
-      simp only [sub_self,abs_zero,zero_add,add_sub_add_left_eq_sub] at hh
-      change _≤(γ.coeff T*Lu)*|x-y|
-      calc
-        _≤γ.coeff t*(Lu*|x-y|):=mul_le_mul_of_nonneg_left hh hg
-        _≤γ.coeff T*(Lu*|x-y|):=mul_le_mul_of_nonneg_right hgt (by positivity)
-        _=_:=by ring
-    integ:=by
-      intro Y
-      apply hi.mul_continuousOn
-      rw [uIcc_of_le hT.1]
-      apply hj.comp (continuousOn_id.prodMk (hb.add ((Y.continuous.comp continuous_projIcc).continuousOn)))
-      exact fun t ht ↦ ht }
-  obtain ⟨Y,hY⟩:=D.exists_fixedPoint
-  let X:=fun t ↦ b t+Y (projIcc 0 T hT.1 t)
-  have hXcont:ContinuousOn X (Icc (0:ℝ) T):=hb.add ((Y.continuous.comp continuous_projIcc).continuousOn)
-  refine ⟨X,hXcont,D.integ Y,?_⟩
-  intro t ht
-  have he:=congrArg (fun f:C(Icc (0:ℝ) T,ℝ) ↦ f ⟨t,ht⟩) hY
-  change (∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (X s))=Y ⟨t,ht⟩ at he
-  dsimp only [X]
-  rw [projIcc_of_mem hT.1 ht,he]
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Control.Regularity
+
+namespace OAI
+
+section
+open MeasureTheory Set Filter Function
+open scoped Topology NNReal Nat
+namespace SKValue
+structure VolterraData (T:ℝ) (hT:0≤T) where
+  field:ℝ → ℝ → ℝ
+  lip:ℝ≥0
+  bound:∀t∈Icc (0:ℝ) T,∀x y,dist (field t x) (field t y)≤lip*dist x y
+  integ:∀Y:C(Icc (0:ℝ) T,ℝ),IntervalIntegrable (fun s ↦ field s (Y (projIcc 0 T hT s))) volume 0 T
+namespace VolterraData
+variable {T:ℝ} {hT:0≤T} (D:VolterraData T hT)
+noncomputable def next (Y:C(Icc (0:ℝ) T,ℝ)):C(Icc (0:ℝ) T,ℝ) where
+  toFun t:=∫s in (0:ℝ)..(t:ℝ),D.field s (Y (projIcc 0 T hT s))
+  continuous_toFun:=by
+    have hc:=intervalIntegral.continuousOn_primitive_interval' (D.integ Y) left_mem_uIcc
+    rw [uIcc_of_le hT] at hc
+    exact hc.domRestrict
+lemma next_apply (Y:C(Icc (0:ℝ) T,ℝ)) (t:Icc (0:ℝ) T):
+    D.next Y t=∫s in (0:ℝ)..(t:ℝ),D.field s (Y (projIcc 0 T hT s)) :=rfl
+lemma integ_to (Y:C(Icc (0:ℝ) T,ℝ)) (t:Icc (0:ℝ) T):
+    IntervalIntegrable (fun s ↦ D.field s (Y (projIcc 0 T hT s))) volume 0 t := by
+  apply (D.integ Y).mono_set
+  rw [uIcc_of_le t.2.1,uIcc_of_le hT]
+  exact Icc_subset_Icc le_rfl t.2.2
+lemma iterate_bound (Y Z:C(Icc (0:ℝ) T,ℝ)) (n:ℕ) (t:Icc (0:ℝ) T):
+    dist ((D.next)^[n] Y t) ((D.next)^[n] Z t)≤
+      (D.lip*|((t:ℝ)-0)|)^n/n.factorial*dist Y Z := by
+  induction n generalizing t with
+  | zero=>simpa using ContinuousMap.dist_apply_le_dist (f:=Y) (g:=Z) t
+  | succ n hn=>
+    rw [iterate_succ_apply',iterate_succ_apply',dist_eq_norm,next_apply,next_apply,
+      ←intervalIntegral.integral_sub (D.integ_to _ t) (D.integ_to _ t)]
+    calc
+      _≤∫s in uIoc (0:ℝ) (t:ℝ),(D.lip:ℝ)^(n+1)*|s-0|^n/n.factorial*dist Y Z := by
+        rw [intervalIntegral.norm_intervalIntegral_eq]
+        apply norm_integral_le_of_norm_le ((by fun_prop:Continuous (fun s:ℝ ↦
+          (D.lip:ℝ)^(n+1)*|s-0|^n/n.factorial*dist Y Z)).integrableOn_uIoc)
+        filter_upwards [ae_restrict_mem measurableSet_uIoc] with s hs
+        rw [uIoc_of_le t.2.1] at hs
+        have hsT:s∈Icc (0:ℝ) T:=⟨hs.1.le,hs.2.trans t.2.2⟩
+        rw [←dist_eq_norm]
+        have hp:=D.bound s hsT ((D.next)^[n] Y (projIcc 0 T hT s)) ((D.next)^[n] Z (projIcc 0 T hT s))
+        have hh:=hn (⟨s,hsT⟩:Icc (0:ℝ) T)
+        have he:projIcc 0 T hT s=(⟨s,hsT⟩:Icc (0:ℝ) T):=projIcc_of_mem hT hsT
+        rw [he] at hp ⊢
+        apply hp.trans
+        calc
+          _≤(D.lip:ℝ)*((D.lip*|s-0|)^n/n.factorial*dist Y Z):=mul_le_mul_of_nonneg_left hh D.lip.coe_nonneg
+          _=_:=by rw [mul_pow,pow_succ];ring
+      _≤_:=by
+        apply le_of_abs_le
+        rw [←intervalIntegral.abs_intervalIntegral_eq,intervalIntegral.integral_mul_const,
+          intervalIntegral.integral_div,intervalIntegral.integral_const_mul,abs_mul,abs_div,
+          abs_mul,intervalIntegral.abs_intervalIntegral_eq,integral_pow_abs_sub_uIoc,abs_div,
+          abs_pow,abs_pow,abs_dist,NNReal.abs_eq,abs_abs,mul_div,div_div,←abs_mul,
+          ←Nat.cast_succ,←Nat.cast_mul,←Nat.factorial_succ,Nat.abs_cast,←mul_pow]
+lemma exists_fixedPoint:∃Y:C(Icc (0:ℝ) T,ℝ),D.next Y=Y := by
+  obtain ⟨n,hn⟩:= (FloorSemiring.tendsto_pow_div_factorial_atTop ((D.lip:ℝ)*T)).eventually
+    (gt_mem_nhds zero_lt_one) |>.exists
+  have hn0:0≤((D.lip:ℝ)*T)^n/n.factorial:=by positivity
+  have hc:ContractingWith ⟨((D.lip:ℝ)*T)^n/n.factorial,hn0⟩ (D.next)^[n] := by
+    refine ⟨hn,LipschitzWith.of_dist_le_mul (fun Y Z ↦ ?_)⟩
+    rw [ContinuousMap.dist_le]
+    · intro t
+      apply (D.iterate_bound Y Z n t).trans
+      simp only [sub_zero,abs_of_nonneg t.2.1]
+      change (D.lip*↑t)^n/n.factorial*dist Y Z ≤ ((D.lip:ℝ)*T)^n/n.factorial*dist Y Z
+      gcongr
+      · exact mul_nonneg D.lip.coe_nonneg t.2.1
+      · exact t.2.2
+    · positivity
+  exact ⟨_,hc.isFixedPt_fixedPoint_iterate⟩
+end VolterraData
+end SKValue
+
+end
+
+section
+open MeasureTheory ProbabilityTheory Set Filter Function
+open scoped Topology NNReal
+namespace SKValue
+lemma exists_local_diffusion_path (W:BrownianSpace) (γ:OrderParameter) {T:ℝ}
+    (hT:T∈Ico (0:ℝ) 1) (b:ℝ → ℝ) (hb:ContinuousOn b (Icc (0:ℝ) T)):
+    ∃X:ℝ → ℝ,ContinuousOn X (Icc (0:ℝ) T) ∧
+      IntervalIntegrable (fun s ↦ γ.coeff s*gradient W γ s (X s)) volume 0 T ∧
+      ∀t∈Icc (0:ℝ) T,X t=b t+∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (X s) := by
+  obtain ⟨K,L,Lu,hLu,hG,hu⟩:=(sourceStripRegularity W γ).core T hT
+  have hg0:0≤γ.coeff T:=γ.nonneg T hT
+  have hi:IntervalIntegrable γ.coeff volume 0 T:=γ.intervalIntegrable.mono_set (by
+    rw [uIcc_of_le hT.1,uIcc_of_le (by norm_num:(0:ℝ)≤1)];exact Icc_subset_Icc le_rfl hT.2.le)
+  have hj:=derivative_joint_continuous hLu hu
+  let D:VolterraData T hT.1:={
+    field:=fun t y ↦ γ.coeff t*gradient W γ t (b t+y)
+    lip:=⟨γ.coeff T*Lu,mul_nonneg hg0 hLu⟩
+    bound:=by
+      intro t ht x y
+      have hg:0≤γ.coeff t:=γ.nonneg t ⟨ht.1,ht.2.trans_lt hT.2⟩
+      have hgt:γ.coeff t≤γ.coeff T:=γ.monotone ⟨ht.1,ht.2.trans_lt hT.2⟩ hT ht.2
+      simp only [Real.dist_eq,←mul_sub,abs_mul,abs_of_nonneg hg]
+      have hh:=hu t ht t ht (b t+x) (b t+y)
+      simp only [sub_self,abs_zero,zero_add,add_sub_add_left_eq_sub] at hh
+      change _≤(γ.coeff T*Lu)*|x-y|
+      calc
+        _≤γ.coeff t*(Lu*|x-y|):=mul_le_mul_of_nonneg_left hh hg
+        _≤γ.coeff T*(Lu*|x-y|):=mul_le_mul_of_nonneg_right hgt (by positivity)
+        _=_:=by ring
+    integ:=by
+      intro Y
+      apply hi.mul_continuousOn
+      rw [uIcc_of_le hT.1]
+      apply hj.comp (continuousOn_id.prodMk (hb.add ((Y.continuous.comp continuous_projIcc).continuousOn)))
+      exact fun t ht ↦ ht }
+  obtain ⟨Y,hY⟩:=D.exists_fixedPoint
+  let X:=fun t ↦ b t+Y (projIcc 0 T hT.1 t)
+  have hXcont:ContinuousOn X (Icc (0:ℝ) T):=hb.add ((Y.continuous.comp continuous_projIcc).continuousOn)
+  refine ⟨X,hXcont,D.integ Y,?_⟩
+  intro t ht
+  have he:=congrArg (fun f:C(Icc (0:ℝ) T,ℝ) ↦ f ⟨t,ht⟩) hY
+  change (∫s in (0:ℝ)..t,γ.coeff s*gradient W γ s (X s))=Y ⟨t,ht⟩ at he
+  dsimp only [X]
+  rw [projIcc_of_mem hT.1 ht,he]
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Processes/WeightedTerminalExpectation.lean b/lean/OAI/Probability/SKValue/Processes/WeightedTerminalExpectation.lean
index c67ba66823030b79a26c89254b6972dbeb90e049..177465827f830cd41a7e7039ec10127f73451126 100644
--- a/lean/OAI/Probability/SKValue/Processes/WeightedTerminalExpectation.lean
+++ b/lean/OAI/Probability/SKValue/Processes/WeightedTerminalExpectation.lean
@@ -1,78 +1,78 @@
-import Mathlib
-import OAI.Probability.SKValue.Equations.RealFiltration
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set
-open scoped ENNReal NNReal BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped BigOperators Topology
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology ENNReal NNReal
-open Filter Set
-open scoped Topology BigOperators
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-open MeasureTheory Set Filter
-open scoped Topology BigOperators
-open MeasureTheory Set Filter Finset
-open scoped Topology BigOperators
-namespace SKValue
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology
-
-lemma UnitMomentMartingale.weighted_terminal_expectation
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
-    {ℱ : Filtration ℝ ‹MeasurableSpace Ω›} {f : ℝ → Ω → ℝ}
-    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
-    (hUm : StronglyMeasurable[ℱ 1] U)
-    (hU : ∀ᵐ ω ∂μ, |U ω|≤1 ∧ Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω)))
-    {γ : ℝ → ℝ} {T : ℝ} (hT : 0≤T) (hT1 : T<1)
-    (hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ f t ω) (Icc (0 : ℝ) T))
-    (hγ : IntervalIntegrable γ volume 0 T) :
-    (∫ ω, U ω*(∫ t in (0 : ℝ)..T, γ t*f t ω) ∂μ)=∫ t in (0 : ℝ)..T, t*γ t := by
-  have hm (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : t∈Ico (0 : ℝ) 1 := ⟨ht.1,ht.2.trans_lt hT1⟩
-  rw [compact_weighted_fubini hT (fun t ht ↦ (h.measurable t (hm t ht)).mono (ℱ.le t))
-    hc (fun t ht ↦ h.bounded t (hm t ht)) (hUm.mono (ℱ.le 1)).aestronglyMeasurable
-    (hU.mono (fun _ hω ↦ hω.1)) hγ]
-  apply intervalIntegral.integral_congr
-  intro t ht
-  rw [uIcc_of_le hT] at ht
-  change γ t * (∫ ω, U ω * f t ω ∂μ) = t * γ t
-  rw [h.terminal_past_product hUm hU (hm t ht),mul_comm]
-
-lemma bounded_multiplier_integral_difference {Ω : Type*} [MeasurableSpace Ω]
-    {μ : Measure Ω} {U X Y : Ω → ℝ}
-    (hU : AEStronglyMeasurable U μ) (hb : ∀ᵐ ω ∂μ, |U ω|≤1)
-    (hX : Integrable X μ) (hY : Integrable Y μ) :
-    |(∫ ω, U ω*X ω ∂μ)-(∫ ω, U ω*Y ω ∂μ)|≤∫ ω, |X ω-Y ω| ∂μ := by
-  have hUb : ∀ᵐ ω ∂μ, ‖U ω‖≤1 := hb.mono (fun ω hω ↦ by simpa only [Real.norm_eq_abs] using hω)
-  have hUX := hX.bdd_mul hU hUb
-  have hUY := hY.bdd_mul hU hUb
-  rw [←integral_sub hUX hUY]
-  apply abs_integral_le_integral_abs.trans
-  apply integral_mono_ae (hUX.sub hUY).abs (hX.sub hY).abs
-  filter_upwards [hb] with ω hω
-  dsimp only [Pi.sub_apply]
-  rw [←mul_sub,abs_mul]
-  exact (mul_le_mul_of_nonneg_right hω (abs_nonneg _)).trans_eq (one_mul _)
-
-lemma bounded_multiplier_integral_tendsto {Ω I : Type*} [MeasurableSpace Ω]
-    {μ : Measure Ω} {l : Filter I} {U : Ω → ℝ} {X : I → Ω → ℝ} {Y : Ω → ℝ}
-    (hU : AEStronglyMeasurable U μ) (hb : ∀ᵐ ω ∂μ, |U ω|≤1)
-    (hX : ∀ᶠ t in l, Integrable (X t) μ) (hY : Integrable Y μ)
-    (hL1 : Tendsto (fun t ↦ ∫ ω, |X t ω-Y ω| ∂μ) l (𝓝 (0 : ℝ))) :
-    Tendsto (fun t ↦ ∫ ω, U ω*X t ω ∂μ) l (𝓝 (∫ ω, U ω*Y ω ∂μ)) := by
-  apply tendsto_iff_norm_sub_tendsto_zero.mpr
-  apply squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) ?_ hL1
-  filter_upwards [hX] with t ht
-  simpa only [Real.norm_eq_abs] using bounded_multiplier_integral_difference hU hb ht hY
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Equations.RealFiltration
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped BigOperators Topology
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology ENNReal NNReal
+open Filter Set
+open scoped Topology BigOperators
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+open MeasureTheory Set Filter
+open scoped Topology BigOperators
+open MeasureTheory Set Filter Finset
+open scoped Topology BigOperators
+namespace SKValue
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology
+
+lemma UnitMomentMartingale.weighted_terminal_expectation
+    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
+    {ℱ : Filtration ℝ ‹MeasurableSpace Ω›} {f : ℝ → Ω → ℝ}
+    (h : UnitMomentMartingale (μ := μ) ℱ f) {U : Ω → ℝ}
+    (hUm : StronglyMeasurable[ℱ 1] U)
+    (hU : ∀ᵐ ω ∂μ, |U ω|≤1 ∧ Tendsto (fun n ↦ f (terminalTime n) ω) atTop (𝓝 (U ω)))
+    {γ : ℝ → ℝ} {T : ℝ} (hT : 0≤T) (hT1 : T<1)
+    (hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t ↦ f t ω) (Icc (0 : ℝ) T))
+    (hγ : IntervalIntegrable γ volume 0 T) :
+    (∫ ω, U ω*(∫ t in (0 : ℝ)..T, γ t*f t ω) ∂μ)=∫ t in (0 : ℝ)..T, t*γ t := by
+  have hm (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : t∈Ico (0 : ℝ) 1 := ⟨ht.1,ht.2.trans_lt hT1⟩
+  rw [compact_weighted_fubini hT (fun t ht ↦ (h.measurable t (hm t ht)).mono (ℱ.le t))
+    hc (fun t ht ↦ h.bounded t (hm t ht)) (hUm.mono (ℱ.le 1)).aestronglyMeasurable
+    (hU.mono (fun _ hω ↦ hω.1)) hγ]
+  apply intervalIntegral.integral_congr
+  intro t ht
+  rw [uIcc_of_le hT] at ht
+  change γ t * (∫ ω, U ω * f t ω ∂μ) = t * γ t
+  rw [h.terminal_past_product hUm hU (hm t ht),mul_comm]
+
+lemma bounded_multiplier_integral_difference {Ω : Type*} [MeasurableSpace Ω]
+    {μ : Measure Ω} {U X Y : Ω → ℝ}
+    (hU : AEStronglyMeasurable U μ) (hb : ∀ᵐ ω ∂μ, |U ω|≤1)
+    (hX : Integrable X μ) (hY : Integrable Y μ) :
+    |(∫ ω, U ω*X ω ∂μ)-(∫ ω, U ω*Y ω ∂μ)|≤∫ ω, |X ω-Y ω| ∂μ := by
+  have hUb : ∀ᵐ ω ∂μ, ‖U ω‖≤1 := hb.mono (fun ω hω ↦ by simpa only [Real.norm_eq_abs] using hω)
+  have hUX := hX.bdd_mul hU hUb
+  have hUY := hY.bdd_mul hU hUb
+  rw [←integral_sub hUX hUY]
+  apply abs_integral_le_integral_abs.trans
+  apply integral_mono_ae (hUX.sub hUY).abs (hX.sub hY).abs
+  filter_upwards [hb] with ω hω
+  dsimp only [Pi.sub_apply]
+  rw [←mul_sub,abs_mul]
+  exact (mul_le_mul_of_nonneg_right hω (abs_nonneg _)).trans_eq (one_mul _)
+
+lemma bounded_multiplier_integral_tendsto {Ω I : Type*} [MeasurableSpace Ω]
+    {μ : Measure Ω} {l : Filter I} {U : Ω → ℝ} {X : I → Ω → ℝ} {Y : Ω → ℝ}
+    (hU : AEStronglyMeasurable U μ) (hb : ∀ᵐ ω ∂μ, |U ω|≤1)
+    (hX : ∀ᶠ t in l, Integrable (X t) μ) (hY : Integrable Y μ)
+    (hL1 : Tendsto (fun t ↦ ∫ ω, |X t ω-Y ω| ∂μ) l (𝓝 (0 : ℝ))) :
+    Tendsto (fun t ↦ ∫ ω, U ω*X t ω ∂μ) l (𝓝 (∫ ω, U ω*Y ω ∂μ)) := by
+  apply tendsto_iff_norm_sub_tendsto_zero.mpr
+  apply squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) ?_ hL1
+  filter_upwards [hX] with t ht
+  simpa only [Real.norm_eq_abs] using bounded_multiplier_integral_difference hU hb ht hY
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/ContactAssembly.lean b/lean/OAI/Probability/SKValue/Variation/ContactAssembly.lean
index 9dafafbc5528304f446e368857060367aa3390eb..91b908ec7db0219fc719e9f5f4a9cea044a8cde0 100644
--- a/lean/OAI/Probability/SKValue/Variation/ContactAssembly.lean
+++ b/lean/OAI/Probability/SKValue/Variation/ContactAssembly.lean
@@ -1,78 +1,78 @@
-import Mathlib
-import OAI.Probability.SKValue.Processes.FiniteGaussianFixedStrip
-import OAI.Probability.SKValue.Evolution.IntegralPrimitiveTerminal
-import OAI.Probability.SKValue.Processes.GradientSquareIntegral
-import OAI.Probability.SKValue.GroundState.WeakGuerra
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma normalizer_pos_iff (W : BrownianSpace) (γ : OrderParameter) (T : ℝ) (N j : ℕ) :
-    0<normalizer W γ T N j ↔
-    0<∫ z,(rawCoefficient W γ T N j z)^2 ∂gaussianProduct (Fin (N+1)) := by
-  unfold normalizer
-  rw [inv_pos, Real.sqrt_pos]
-
-lemma IsDiffusion.strip_value_le_groundState {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
-    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t)
-    {T : ℝ} (hT : 0<T) (hT1 : T<1) :
-    (∫ t in (0 : ℝ)..T,∫ ω,curvature W γ t (X t ω) ∂W.μ)≤groundStateValue := by
-  have hreg := sourceStripRegularity W γ
-  have hamom := hX.curvature_unit_moment_of_contact hreg hmom
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨hT.le,hT1⟩
-  obtain ⟨hp,hl⟩ := hX.finite_gaussian_fixed_strip hT hT1 hG hD hLu hLa hDb hu ha
-    (fun t ht ↦ hamom t ⟨ht.1,ht.2.trans_lt hT1⟩)
-  apply le_of_tendsto hl
-  filter_upwards [hp,eventually_gt_atTop (0 : ℕ)] with N hNpos hN
-  exact shiftedCoefficientSum_le_groundState W γ hT.le hT1 hN
-    (fun j hj ↦ (normalizer_pos_iff W γ T N j).mp (hNpos j hj))
-
-theorem IsDiffusion.groundState_eq_parisi_of_contact {W : BrownianSpace}
-    {γ : OrderParameter} {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
-    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t) :
-    groundStateValue=parisi W γ := by
-  have hreg := sourceStripRegularity W γ
-  obtain ⟨U,hmart,hU,hU2,hP,hB,hA,hi,hl⟩ :=
-    hX.parisi_value_assembly hreg hmom (hX.curvature_unit_moment_of_contact hreg hmom)
-  refine le_antisymm (groundStateValue_le_parisi W γ) ?_
-  apply le_of_tendsto hl
-  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds
-    (Ioi_mem_nhds (by norm_num : (0 : ℝ)<1))] with T hT1 hT
-  exact hX.strip_value_le_groundState hmom hT hT1
-
-theorem IsDiffusion.scalar_value_of_contact {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
-    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t) :
-    ScalarValueConclusion W γ X := by
-  have hreg := sourceStripRegularity W γ
-  obtain ⟨U,hmart,hU,hU2,hP,hB,hA,hi,hl⟩ :=
-    hX.parisi_value_assembly hreg hmom (hX.curvature_unit_moment_of_contact hreg hmom)
-  refine ⟨SKValueG.groundStateSequence_tendsto,U,hmart,hU,hU2,?_,?_,?_,hi⟩
-  · exact (hX.groundState_eq_parisi_of_contact hmom).trans hP
-  · exact (hX.groundState_eq_parisi_of_contact hmom).trans hB
-  · exact (hX.groundState_eq_parisi_of_contact hmom).trans hA
-
-theorem IsDiffusion.finite_gaussian_of_contact {W : BrownianSpace}
-    {γ : OrderParameter} {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
-    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t)
-    {T : ℝ} (hT : 0<T) (hT1 : T<1) : FiniteGaussianConclusion W γ X T := by
-  have hreg := sourceStripRegularity W γ
-  have hamom := hX.curvature_unit_moment_of_contact hreg hmom
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨hT.le,hT1⟩
-  obtain ⟨hp,hl⟩ := hX.finite_gaussian_fixed_strip hT hT1 hG hD hLu hLa hDb hu ha
-    (fun t ht ↦ hamom t ⟨ht.1,ht.2.trans_lt hT1⟩)
-  obtain ⟨U,hmart,hU,hU2,hP,hB,hA,hi,hend⟩ := hX.parisi_value_assembly hreg hmom hamom
-  refine ⟨hp,hl,?_⟩
-  rwa [hX.groundState_eq_parisi_of_contact hmom]
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Processes.FiniteGaussianFixedStrip
+import OAI.Probability.SKValue.Evolution.IntegralPrimitiveTerminal
+import OAI.Probability.SKValue.Processes.GradientSquareIntegral
+import OAI.Probability.SKValue.GroundState.WeakGuerra
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma normalizer_pos_iff (W : BrownianSpace) (γ : OrderParameter) (T : ℝ) (N j : ℕ) :
+    0<normalizer W γ T N j ↔
+    0<∫ z,(rawCoefficient W γ T N j z)^2 ∂gaussianProduct (Fin (N+1)) := by
+  unfold normalizer
+  rw [inv_pos, Real.sqrt_pos]
+
+lemma IsDiffusion.strip_value_le_groundState {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
+    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t)
+    {T : ℝ} (hT : 0<T) (hT1 : T<1) :
+    (∫ t in (0 : ℝ)..T,∫ ω,curvature W γ t (X t ω) ∂W.μ)≤groundStateValue := by
+  have hreg := sourceStripRegularity W γ
+  have hamom := hX.curvature_unit_moment_of_contact hreg hmom
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨hT.le,hT1⟩
+  obtain ⟨hp,hl⟩ := hX.finite_gaussian_fixed_strip hT hT1 hG hD hLu hLa hDb hu ha
+    (fun t ht ↦ hamom t ⟨ht.1,ht.2.trans_lt hT1⟩)
+  apply le_of_tendsto hl
+  filter_upwards [hp,eventually_gt_atTop (0 : ℕ)] with N hNpos hN
+  exact shiftedCoefficientSum_le_groundState W γ hT.le hT1 hN
+    (fun j hj ↦ (normalizer_pos_iff W γ T N j).mp (hNpos j hj))
+
+theorem IsDiffusion.groundState_eq_parisi_of_contact {W : BrownianSpace}
+    {γ : OrderParameter} {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
+    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t) :
+    groundStateValue=parisi W γ := by
+  have hreg := sourceStripRegularity W γ
+  obtain ⟨U,hmart,hU,hU2,hP,hB,hA,hi,hl⟩ :=
+    hX.parisi_value_assembly hreg hmom (hX.curvature_unit_moment_of_contact hreg hmom)
+  refine le_antisymm (groundStateValue_le_parisi W γ) ?_
+  apply le_of_tendsto hl
+  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds
+    (Ioi_mem_nhds (by norm_num : (0 : ℝ)<1))] with T hT1 hT
+  exact hX.strip_value_le_groundState hmom hT hT1
+
+theorem IsDiffusion.scalar_value_of_contact {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
+    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t) :
+    ScalarValueConclusion W γ X := by
+  have hreg := sourceStripRegularity W γ
+  obtain ⟨U,hmart,hU,hU2,hP,hB,hA,hi,hl⟩ :=
+    hX.parisi_value_assembly hreg hmom (hX.curvature_unit_moment_of_contact hreg hmom)
+  refine ⟨SKValueG.groundStateSequence_tendsto,U,hmart,hU,hU2,?_,?_,?_,hi⟩
+  · exact (hX.groundState_eq_parisi_of_contact hmom).trans hP
+  · exact (hX.groundState_eq_parisi_of_contact hmom).trans hB
+  · exact (hX.groundState_eq_parisi_of_contact hmom).trans hA
+
+theorem IsDiffusion.finite_gaussian_of_contact {W : BrownianSpace}
+    {γ : OrderParameter} {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X)
+    (hmom : ∀ t∈Ico (0 : ℝ) 1, (∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ)=t)
+    {T : ℝ} (hT : 0<T) (hT1 : T<1) : FiniteGaussianConclusion W γ X T := by
+  have hreg := sourceStripRegularity W γ
+  have hamom := hX.curvature_unit_moment_of_contact hreg hmom
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := hreg T ⟨hT.le,hT1⟩
+  obtain ⟨hp,hl⟩ := hX.finite_gaussian_fixed_strip hT hT1 hG hD hLu hLa hDb hu ha
+    (fun t ht ↦ hamom t ⟨ht.1,ht.2.trans_lt hT1⟩)
+  obtain ⟨U,hmart,hU,hU2,hP,hB,hA,hi,hend⟩ := hX.parisi_value_assembly hreg hmom hamom
+  refine ⟨hp,hl,?_⟩
+  rwa [hX.groundState_eq_parisi_of_contact hmom]
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/ContactIntegral.lean b/lean/OAI/Probability/SKValue/Variation/ContactIntegral.lean
index ae77838d111da6f634c5578c29f63d1b64f4be4b..7b5801abcc213605a918184c62e3cf2758b5f1d9 100644
--- a/lean/OAI/Probability/SKValue/Variation/ContactIntegral.lean
+++ b/lean/OAI/Probability/SKValue/Variation/ContactIntegral.lean
@@ -1,131 +1,131 @@
-import Mathlib
-import OAI.Probability.SKValue.Variation.ContactVariations
-import OAI.Probability.SKValue.Equations.TruncatedStieltjes
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma IsDiffusion.contact_fubini {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T : ℝ}
-    (hT : T∈Ico (0 : ℝ) 1) :
-    Integrable (contactPotential W γ X) (γ.truncatedMass T hT) ∧
-    (∫ s,contactPotential W γ X s ∂γ.truncatedMass T hT)=
-      ∫ t in (0 : ℝ)..1,min (γ.coeff t) (γ.coeff T)*(gradientMoment W γ X t-t) := by
-  let μ := γ.truncatedMass T hT
-  let ν : Measure ℝ := volume.restrict (Ioc 0 1)
-  let F : ℝ×ℝ → ℝ := {p : ℝ×ℝ | p.1≤p.2}.indicator (fun p ↦ gradientMoment W γ X p.2-p.2)
-  have hi : Integrable (fun t ↦ gradientMoment W γ X t-t) ν :=
-    (intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num : (0 : ℝ)≤1)).mp hX.moment_difference_integrable
-  have hF : Integrable F (μ.prod ν) := by
-    apply Integrable.of_bound (hi.aestronglyMeasurable.comp_snd.indicator
-      (measurableSet_le measurable_fst measurable_snd)) 1
-    filter_upwards [Measure.quasiMeasurePreserving_snd.ae (ae_restrict_mem (μ := volume) measurableSet_Ioc)] with p hp
-    by_cases hs : p.1≤p.2
-    · rw [indicator_of_mem (show p∈{a : ℝ×ℝ | a.1≤a.2} from hs),Real.norm_eq_abs]
-      apply abs_le.mpr
-      have h0 := gradientMoment_nonneg W γ X p.2
-      have h1 := (abs_le.mp (gradientMoment_bound W γ X ⟨hp.1.le,hp.2⟩)).2
-      constructor <;> linarith [hp.1,hp.2]
-    · rw [indicator_of_notMem (show p∉{a : ℝ×ℝ | a.1≤a.2} from hs),norm_zero]; norm_num
-  have hleft : (fun s ↦ ∫ t,F (s,t) ∂ν)=ᵐ[μ] contactPotential W γ X := by
-    filter_upwards [γ.truncatedMass_ae_mem hT] with s hs
-    have hs' : s∈Icc (0 : ℝ) 1 := ⟨hs.1,hs.2.trans hT.2.le⟩
-    change (∫ t in Ioc (0 : ℝ) 1,F (s,t))=contactPotential W γ X s
-    rw [←intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1)]
-    calc
-      (∫ t in (0 : ℝ)..1,F (s,t))=
-          ∫ t in (0 : ℝ)..1,(OrderParameter.atom s).coeff t*(gradientMoment W γ X t-t) := by
-        apply intervalIntegral.integral_congr
-        intro t _
-        dsimp [F,OrderParameter.atom,indicator]
-        split_ifs <;> simp
-      _=contactPotential W γ X s := integral_atom_mul s hs' _
-  have hright : (fun t ↦ ∫ s,F (s,t) ∂μ)=ᵐ[ν]
-      (fun t ↦ min (γ.coeff t) (γ.coeff T)*(gradientMoment W γ X t-t)) := by
-    have hν : ∀ᵐ t ∂ν,t∈Ico (0 : ℝ) 1 := by
-      dsimp [ν]
-      rw [←restrict_Ico_eq_restrict_Ioc]
-      exact ae_restrict_mem measurableSet_Ico
-    filter_upwards [hν] with t ht
-    have he : (fun s ↦ F (s,t))=(Iic t).indicator (fun _ ↦ gradientMoment W γ X t-t) := rfl
-    rw [he,integral_indicator measurableSet_Iic,integral_const]
-    simp only [Measure.real,Measure.restrict_apply_univ,μ,γ.truncatedMass_cdf hT ht,
-      ENNReal.toReal_ofReal (le_min (γ.nonneg t ht) (γ.nonneg T hT)),smul_eq_mul]
-  refine ⟨hF.integral_prod_left.congr hleft,?_⟩
-  rw [←integral_congr_ae hleft]
-  have hswap := integral_integral_swap (f := fun s t ↦ F (s,t)) hF
-  rw [hswap,integral_congr_ae hright]
-  exact (intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1)).symm
-
-lemma IsMinimizer.contactPotential_ae_zero {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
-    {T : ℝ} (hT : T∈Ico (0 : ℝ) 1) :
-    ∀ᵐ s ∂γ.truncatedMass T hT,contactPotential W γ X s=0 := by
-  have hh := hγ.variation_inequality hX (γ.removeBelow (γ.coeff T))
-  simp_rw [γ.removeBelow_sub,neg_mul] at hh
-  rw [intervalIntegral.integral_neg] at hh
-  obtain ⟨hi,he⟩ := hX.contact_fubini hT
-  have hnn : 0≤ᵐ[γ.truncatedMass T hT] contactPotential W γ X := by
-    filter_upwards [γ.truncatedMass_ae_mem hT] with s hs
-    exact hγ.contactPotential_nonneg hX ⟨hs.1,hs.2.trans hT.2.le⟩
-  apply (integral_eq_zero_iff_of_nonneg_ae hnn hi).mp
-  have hpos := integral_nonneg_of_ae hnn
-  rw [he] at hpos ⊢
-  linarith
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma OrderParameter.truncatedMass_Ioc (γ : OrderParameter) {a b : ℝ}
-    (ha : 0≤a) (hab : a≤b) (hb : b<1) :
-    γ.truncatedMass b ⟨ha.trans hab,hb⟩ (Ioc a b)=ENNReal.ofReal (γ.coeff b-γ.coeff a) := by
-  have he : Ioc a b∩Ioc (0 : ℝ) b=Ioc a b := inter_eq_left.mpr (Ioc_subset_Ioc ha le_rfl)
-  have hst : γ.truncatedStieltjes b ⟨ha.trans hab,hb⟩ a=γ.coeff a := by
-    simp [OrderParameter.truncatedStieltjes,max_eq_right ha,min_eq_right hab]
-  simp only [OrderParameter.truncatedMass,Measure.add_apply,
-    Measure.restrict_apply measurableSet_Ioc,he,StieltjesFunction.measure_Ioc,
-    hst,Measure.smul_apply,
-    Measure.dirac_apply' _ measurableSet_Ioc]
-  rw [indicator_of_notMem (show (0 : ℝ)∉Ioc a b from fun h ↦ not_lt_of_ge ha h.1)]
-  simp
-  simp [OrderParameter.truncatedStieltjes,max_eq_right (ha.trans hab)]
-
-lemma IsMinimizer.coeff_eq_of_contactPositive {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
-    {a b : ℝ} (ha : 0≤a) (hab : a≤b) (hb : b<1)
-    (hG : ∀ s∈Ioc a b,0<contactPotential W γ X s) : γ.coeff a=γ.coeff b := by
-  have hz := hγ.contactPotential_ae_zero hX (T := b) ⟨ha.trans hab,hb⟩
-  rw [ae_iff] at hz
-  have hI : γ.truncatedMass b ⟨ha.trans hab,hb⟩ (Ioc a b)=0 :=
-    measure_mono_null (fun s hs ↦ ne_of_gt (hG s hs)) hz
-  rw [γ.truncatedMass_Ioc ha hab hb,ENNReal.ofReal_eq_zero] at hI
-  exact le_antisymm (γ.monotone ⟨ha,hab.trans_lt hb⟩ ⟨ha.trans hab,hb⟩ hab) (by linarith)
-
-lemma IsMinimizer.coeff_constant_of_contactPositive {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
-    {a b : ℝ} (ha : 0≤a) (hb : b≤1)
-    (hG : ∀ s∈Ioo a b,0<contactPotential W γ X s) :
-    ∀ s∈Ioo a b,∀ t∈Ioo a b,γ.coeff s=γ.coeff t := by
-  intro s hs t ht
-  wlog hst : s≤t generalizing s t
-  · exact (this t ht s hs (le_of_not_ge hst)).symm
-  exact hγ.coeff_eq_of_contactPositive hX (ha.trans hs.1.le) hst (ht.2.trans_le hb)
-    (fun u hu ↦ hG u ⟨hs.1.trans hu.1,hu.2.trans_lt ht.2⟩)
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Variation.ContactVariations
+import OAI.Probability.SKValue.Equations.TruncatedStieltjes
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma IsDiffusion.contact_fubini {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {T : ℝ}
+    (hT : T∈Ico (0 : ℝ) 1) :
+    Integrable (contactPotential W γ X) (γ.truncatedMass T hT) ∧
+    (∫ s,contactPotential W γ X s ∂γ.truncatedMass T hT)=
+      ∫ t in (0 : ℝ)..1,min (γ.coeff t) (γ.coeff T)*(gradientMoment W γ X t-t) := by
+  let μ := γ.truncatedMass T hT
+  let ν : Measure ℝ := volume.restrict (Ioc 0 1)
+  let F : ℝ×ℝ → ℝ := {p : ℝ×ℝ | p.1≤p.2}.indicator (fun p ↦ gradientMoment W γ X p.2-p.2)
+  have hi : Integrable (fun t ↦ gradientMoment W γ X t-t) ν :=
+    (intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num : (0 : ℝ)≤1)).mp hX.moment_difference_integrable
+  have hF : Integrable F (μ.prod ν) := by
+    apply Integrable.of_bound (hi.aestronglyMeasurable.comp_snd.indicator
+      (measurableSet_le measurable_fst measurable_snd)) 1
+    filter_upwards [Measure.quasiMeasurePreserving_snd.ae (ae_restrict_mem (μ := volume) measurableSet_Ioc)] with p hp
+    by_cases hs : p.1≤p.2
+    · rw [indicator_of_mem (show p∈{a : ℝ×ℝ | a.1≤a.2} from hs),Real.norm_eq_abs]
+      apply abs_le.mpr
+      have h0 := gradientMoment_nonneg W γ X p.2
+      have h1 := (abs_le.mp (gradientMoment_bound W γ X ⟨hp.1.le,hp.2⟩)).2
+      constructor <;> linarith [hp.1,hp.2]
+    · rw [indicator_of_notMem (show p∉{a : ℝ×ℝ | a.1≤a.2} from hs),norm_zero]; norm_num
+  have hleft : (fun s ↦ ∫ t,F (s,t) ∂ν)=ᵐ[μ] contactPotential W γ X := by
+    filter_upwards [γ.truncatedMass_ae_mem hT] with s hs
+    have hs' : s∈Icc (0 : ℝ) 1 := ⟨hs.1,hs.2.trans hT.2.le⟩
+    change (∫ t in Ioc (0 : ℝ) 1,F (s,t))=contactPotential W γ X s
+    rw [←intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1)]
+    calc
+      (∫ t in (0 : ℝ)..1,F (s,t))=
+          ∫ t in (0 : ℝ)..1,(OrderParameter.atom s).coeff t*(gradientMoment W γ X t-t) := by
+        apply intervalIntegral.integral_congr
+        intro t _
+        dsimp [F,OrderParameter.atom,indicator]
+        split_ifs <;> simp
+      _=contactPotential W γ X s := integral_atom_mul s hs' _
+  have hright : (fun t ↦ ∫ s,F (s,t) ∂μ)=ᵐ[ν]
+      (fun t ↦ min (γ.coeff t) (γ.coeff T)*(gradientMoment W γ X t-t)) := by
+    have hν : ∀ᵐ t ∂ν,t∈Ico (0 : ℝ) 1 := by
+      dsimp [ν]
+      rw [←restrict_Ico_eq_restrict_Ioc]
+      exact ae_restrict_mem measurableSet_Ico
+    filter_upwards [hν] with t ht
+    have he : (fun s ↦ F (s,t))=(Iic t).indicator (fun _ ↦ gradientMoment W γ X t-t) := rfl
+    rw [he,integral_indicator measurableSet_Iic,integral_const]
+    simp only [Measure.real,Measure.restrict_apply_univ,μ,γ.truncatedMass_cdf hT ht,
+      ENNReal.toReal_ofReal (le_min (γ.nonneg t ht) (γ.nonneg T hT)),smul_eq_mul]
+  refine ⟨hF.integral_prod_left.congr hleft,?_⟩
+  rw [←integral_congr_ae hleft]
+  have hswap := integral_integral_swap (f := fun s t ↦ F (s,t)) hF
+  rw [hswap,integral_congr_ae hright]
+  exact (intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1)).symm
+
+lemma IsMinimizer.contactPotential_ae_zero {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
+    {T : ℝ} (hT : T∈Ico (0 : ℝ) 1) :
+    ∀ᵐ s ∂γ.truncatedMass T hT,contactPotential W γ X s=0 := by
+  have hh := hγ.variation_inequality hX (γ.removeBelow (γ.coeff T))
+  simp_rw [γ.removeBelow_sub,neg_mul] at hh
+  rw [intervalIntegral.integral_neg] at hh
+  obtain ⟨hi,he⟩ := hX.contact_fubini hT
+  have hnn : 0≤ᵐ[γ.truncatedMass T hT] contactPotential W γ X := by
+    filter_upwards [γ.truncatedMass_ae_mem hT] with s hs
+    exact hγ.contactPotential_nonneg hX ⟨hs.1,hs.2.trans hT.2.le⟩
+  apply (integral_eq_zero_iff_of_nonneg_ae hnn hi).mp
+  have hpos := integral_nonneg_of_ae hnn
+  rw [he] at hpos ⊢
+  linarith
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma OrderParameter.truncatedMass_Ioc (γ : OrderParameter) {a b : ℝ}
+    (ha : 0≤a) (hab : a≤b) (hb : b<1) :
+    γ.truncatedMass b ⟨ha.trans hab,hb⟩ (Ioc a b)=ENNReal.ofReal (γ.coeff b-γ.coeff a) := by
+  have he : Ioc a b∩Ioc (0 : ℝ) b=Ioc a b := inter_eq_left.mpr (Ioc_subset_Ioc ha le_rfl)
+  have hst : γ.truncatedStieltjes b ⟨ha.trans hab,hb⟩ a=γ.coeff a := by
+    simp [OrderParameter.truncatedStieltjes,max_eq_right ha,min_eq_right hab]
+  simp only [OrderParameter.truncatedMass,Measure.add_apply,
+    Measure.restrict_apply measurableSet_Ioc,he,StieltjesFunction.measure_Ioc,
+    hst,Measure.smul_apply,
+    Measure.dirac_apply' _ measurableSet_Ioc]
+  rw [indicator_of_notMem (show (0 : ℝ)∉Ioc a b from fun h ↦ not_lt_of_ge ha h.1)]
+  simp
+  simp [OrderParameter.truncatedStieltjes,max_eq_right (ha.trans hab)]
+
+lemma IsMinimizer.coeff_eq_of_contactPositive {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
+    {a b : ℝ} (ha : 0≤a) (hab : a≤b) (hb : b<1)
+    (hG : ∀ s∈Ioc a b,0<contactPotential W γ X s) : γ.coeff a=γ.coeff b := by
+  have hz := hγ.contactPotential_ae_zero hX (T := b) ⟨ha.trans hab,hb⟩
+  rw [ae_iff] at hz
+  have hI : γ.truncatedMass b ⟨ha.trans hab,hb⟩ (Ioc a b)=0 :=
+    measure_mono_null (fun s hs ↦ ne_of_gt (hG s hs)) hz
+  rw [γ.truncatedMass_Ioc ha hab hb,ENNReal.ofReal_eq_zero] at hI
+  exact le_antisymm (γ.monotone ⟨ha,hab.trans_lt hb⟩ ⟨ha.trans hab,hb⟩ hab) (by linarith)
+
+lemma IsMinimizer.coeff_constant_of_contactPositive {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
+    {a b : ℝ} (ha : 0≤a) (hb : b≤1)
+    (hG : ∀ s∈Ioo a b,0<contactPotential W γ X s) :
+    ∀ s∈Ioo a b,∀ t∈Ioo a b,γ.coeff s=γ.coeff t := by
+  intro s hs t ht
+  wlog hst : s≤t generalizing s t
+  · exact (this t ht s hs (le_of_not_ge hst)).symm
+  exact hγ.coeff_eq_of_contactPositive hX (ha.trans hs.1.le) hst (ht.2.trans_le hb)
+    (fun u hu ↦ hG u ⟨hs.1.trans hu.1,hu.2.trans_lt ht.2⟩)
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/ContactOrigin.lean b/lean/OAI/Probability/SKValue/Variation/ContactOrigin.lean
index a26be6d55e2475b41b6fa4ae8a0614885adb532a..823d7272f5b5f24ea031e3e7cea03f65bac06872 100644
--- a/lean/OAI/Probability/SKValue/Variation/ContactOrigin.lean
+++ b/lean/OAI/Probability/SKValue/Variation/ContactOrigin.lean
@@ -1,22 +1,22 @@
-import Mathlib
-import OAI.Probability.SKValue.Variation.ContactIntegral
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal
-namespace SKValue
-lemma IsMinimizer.contact_origin_of_positive {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
-    (hγ:IsMinimizer W γ) (hX:IsDiffusion W γ X) (hc:0<γ.coeff 0):contactPotential W γ X 0=0 := by
-  have hz:=hγ.contactPotential_ae_zero hX (T:=0) (by constructor <;> norm_num)
-  have hh:∀ᵐ s ∂(ENNReal.ofReal (γ.coeff 0) • Measure.dirac (0:ℝ)),contactPotential W γ X s=0 := by
-    simpa only [OrderParameter.truncatedMass,Ioc_self,Measure.restrict_empty,zero_add] using hz
-  have ha:∀ᵐ s ∂Measure.dirac (0:ℝ),contactPotential W γ X s=0 := by
-    exact (Measure.ae_ennreal_smul_measure_iff (ne_of_gt (ENNReal.ofReal_pos.mpr hc))).mp hh
-  simpa only [ae_dirac_eq, Filter.eventually_pure] using ha
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Variation.ContactIntegral
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal
+namespace SKValue
+lemma IsMinimizer.contact_origin_of_positive {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
+    (hγ:IsMinimizer W γ) (hX:IsDiffusion W γ X) (hc:0<γ.coeff 0):contactPotential W γ X 0=0 := by
+  have hz:=hγ.contactPotential_ae_zero hX (T:=0) (by constructor <;> norm_num)
+  have hh:∀ᵐ s ∂(ENNReal.ofReal (γ.coeff 0) • Measure.dirac (0:ℝ)),contactPotential W γ X s=0 := by
+    simpa only [OrderParameter.truncatedMass,Ioc_self,Measure.restrict_empty,zero_add] using hz
+  have ha:∀ᵐ s ∂Measure.dirac (0:ℝ),contactPotential W γ X s=0 := by
+    exact (Measure.ae_ennreal_smul_measure_iff (ne_of_gt (ENNReal.ofReal_pos.mpr hc))).mp hh
+  simpa only [ae_dirac_eq, Filter.eventually_pure] using ha
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/ContactVariations.lean b/lean/OAI/Probability/SKValue/Variation/ContactVariations.lean
index 306aa054ee22230555328f53b31ffed0512d90dc..6fb31633f30f0c6395f88372f2c746e55a0991ee 100644
--- a/lean/OAI/Probability/SKValue/Variation/ContactVariations.lean
+++ b/lean/OAI/Probability/SKValue/Variation/ContactVariations.lean
@@ -1,156 +1,156 @@
-import Mathlib
-import OAI.Probability.SKValue.Variation.FirstVariationCompact
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def OrderParameter.zero : OrderParameter where
-  coeff := 0
-  nonneg _ _ := le_rfl
-  monotone _ _ _ _ _ := le_rfl
-  rightContinuous _ _ := continuousWithinAt_const
-  integrable := integrableOn_zero
-
-noncomputable def OrderParameter.add (γ η : OrderParameter) : OrderParameter where
-  coeff t := γ.coeff t+η.coeff t
-  nonneg t ht := add_nonneg (γ.nonneg t ht) (η.nonneg t ht)
-  monotone _ hs _ ht hst := add_le_add (γ.monotone hs ht hst) (η.monotone hs ht hst)
-  rightContinuous t ht := (γ.rightContinuous t ht).add (η.rightContinuous t ht)
-  integrable := γ.integrable.add η.integrable
-
-noncomputable def OrderParameter.atom (s : ℝ) : OrderParameter where
-  coeff t := if s≤t then 1 else 0
-  nonneg _ _ := by split_ifs <;> norm_num
-  monotone t _ u _ htu := by
-    by_cases hst : s≤t
-    · simp only [ite_eq_left hst,ite_eq_left (hst.trans htu),le_refl]
-    · simp only [ite_eq_right hst]; split_ifs <;> norm_num
-  rightContinuous t _ := by
-    by_cases hst : s≤t
-    · apply (continuousWithinAt_const (b := (1 : ℝ))).congr
-      · intro u hu
-        exact ite_eq_left (hst.trans hu)
-      · exact ite_eq_left hst
-    · apply (continuousWithinAt_const (b := (0 : ℝ))).congr_of_eventuallyEq
-      · filter_upwards [mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds (lt_of_not_ge hst))] with u hu
-        exact ite_eq_right (not_le.mpr hu)
-      · exact ite_eq_right hst
-  integrable := by
-    apply Integrable.of_bound
-      (StronglyMeasurable.ite measurableSet_Ici stronglyMeasurable_const stronglyMeasurable_const).aestronglyMeasurable 1
-    exact Eventually.of_forall (fun t ↦ by split_ifs <;> norm_num)
-
-noncomputable def OrderParameter.removeBelow (γ : OrderParameter) (c : ℝ) : OrderParameter where
-  coeff t := max (γ.coeff t-c) 0
-  nonneg _ _ := le_max_right _ _
-  monotone _ hs _ ht hst := max_le_max (sub_le_sub_right (γ.monotone hs ht hst) c) le_rfl
-  rightContinuous t ht := ((γ.rightContinuous t ht).sub continuousWithinAt_const).sup continuousWithinAt_const
-  integrable := (γ.integrable.sub (integrableOn_const (μ := volume) (s := Ioc (0 : ℝ) 1) (C := c) (by simp))).sup integrableOn_zero
-
-lemma OrderParameter.removeBelow_sub (γ : OrderParameter) (c t : ℝ) :
-    (γ.removeBelow c).coeff t-γ.coeff t= -min (γ.coeff t) c := by
-  dsimp only [OrderParameter.removeBelow]
-  by_cases h : γ.coeff t≤c
-  · rw [max_eq_right (sub_nonpos.mpr h),min_eq_left h]; ring
-  · rw [max_eq_left (sub_nonneg.mpr (le_of_not_ge h)),min_eq_right (le_of_not_ge h)]; ring
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def contactPotential (W : BrownianSpace) (γ : OrderParameter)
-    (X : ℝ → W.Ω → ℝ) (s : ℝ) : ℝ :=
-  ∫ t in s..1,gradientMoment W γ X t-t
-
-lemma IsDiffusion.moment_difference_integrable {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
-    IntervalIntegrable (fun t ↦ gradientMoment W γ X t-t) volume 0 1 := by
-  have hm := hX.gradientMoment_continuous.aestronglyMeasurable (μ := volume) measurableSet_Ico
-  rw [restrict_Ico_eq_restrict_Ioc] at hm
-  have hi : IntegrableOn (gradientMoment W γ X) (Ioc (0 : ℝ) 1) :=
-    Integrable.of_bound hm 1 (by
-      filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
-      exact gradientMoment_bound W γ X ⟨ht.1.le,ht.2⟩)
-  exact ((intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num : (0 : ℝ)≤1)).mpr hi).sub
-    (continuous_id.intervalIntegrable 0 1)
-
-lemma integral_atom_mul (s : ℝ) (hs : s∈Icc (0 : ℝ) 1) (f : ℝ → ℝ) :
-    (∫ t in (0 : ℝ)..1,(OrderParameter.atom s).coeff t*f t)=∫ t in s..1,f t := by
-  have he : (fun t ↦ (OrderParameter.atom s).coeff t*f t)=(Ici s).indicator f := by
-    funext t
-    simp only [OrderParameter.atom,indicator,mem_Ici]
-    split_ifs <;> simp
-  have hs' : Ici s∩Icc (0 : ℝ) 1=Icc s 1 := by
-    ext t
-    simp only [mem_inter_iff,mem_Ici,mem_Icc]
-    constructor
-    · rintro ⟨hst,_,ht1⟩; exact ⟨hst,ht1⟩
-    · rintro ⟨hst,ht1⟩; exact ⟨hst,hs.1.trans hst,ht1⟩
-  rw [he,intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1),
-    ←integral_Icc_eq_integral_Ioc,integral_indicator measurableSet_Ici,
-    Measure.restrict_restrict measurableSet_Ici,hs',
-    intervalIntegral.integral_of_le hs.2,integral_Icc_eq_integral_Ioc]
-
-lemma IsMinimizer.contactPotential_nonneg {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
-    {s : ℝ} (hs : s∈Icc (0 : ℝ) 1) : 0≤contactPotential W γ X s := by
-  have hh := hγ.variation_inequality hX (γ.add (OrderParameter.atom s))
-  simp only [OrderParameter.add,add_sub_cancel_left] at hh
-  rwa [integral_atom_mul s hs] at hh
-
-lemma IsDiffusion.contactPotential_continuous {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
-    ContinuousOn (contactPotential W γ X) (Icc (0 : ℝ) 1) := by
-  change ContinuousOn (fun s : ℝ ↦ ∫ t in s..1,gradientMoment W γ X t-t) (Icc (0 : ℝ) 1)
-  have hi : IntegrableOn (fun t ↦ gradientMoment W γ X t-t) (uIcc (0 : ℝ) 1) volume := by
-    rw [uIcc_of_le (by norm_num : (0 : ℝ)≤1)]
-    exact (intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num : (0 : ℝ)≤1)).mp
-      hX.moment_difference_integrable
-  have hc : ContinuousOn (fun s : ℝ ↦ ∫ t in s..1,gradientMoment W γ X t-t) (uIcc (0 : ℝ) 1) :=
-    intervalIntegral.continuousOn_primitive_interval_left hi
-  simpa only [uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using hc
-
-lemma IsDiffusion.contactPotential_hasDerivAt {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {s : ℝ} (hs : s∈Ioo (0 : ℝ) 1) :
-    HasDerivAt (contactPotential W γ X) (s-gradientMoment W γ X s) s := by
-  change HasDerivAt (fun u : ℝ ↦ ∫ t in u..1,gradientMoment W γ X t-t) _ s
-  have hc : ContinuousOn (fun t ↦ gradientMoment W γ X t-t) (Ico (0 : ℝ) 1) :=
-    hX.gradientMoment_continuous.sub continuousOn_id
-  have hns : Ico (0 : ℝ) 1∈𝓝 s := mem_of_superset (Ioo_mem_nhds hs.1 hs.2) Ioo_subset_Ico_self
-  have hi : IntervalIntegrable (fun t ↦ gradientMoment W γ X t-t) volume s 1 :=
-    hX.moment_difference_integrable.mono_set (by
-      rw [uIcc_of_le hs.2.le,uIcc_of_le (by norm_num : (0 : ℝ)≤1)]
-      exact Icc_subset_Icc hs.1.le le_rfl)
-  have hh : HasDerivAt (fun u : ℝ ↦ ∫ t in u..1,gradientMoment W γ X t-t)
-      (-(gradientMoment W γ X s-s)) s := intervalIntegral.integral_hasDerivAt_left hi
-        (ContinuousOn.stronglyMeasurableAtFilter isOpen_Ioo (hc.mono Ioo_subset_Ico_self) s hs)
-        (hc.continuousAt hns)
-  simpa only [neg_sub] using hh
-
-lemma IsMinimizer.moment_eq_at_contact {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
-    {s : ℝ} (hs : s∈Ioo (0 : ℝ) 1) (hG : contactPotential W γ X s=0) :
-    gradientMoment W γ X s=s := by
-  have hm : IsLocalMin (contactPotential W γ X) s := by
-    filter_upwards [Ioo_mem_nhds hs.1 hs.2] with t ht
-    rw [hG]
-    exact hγ.contactPotential_nonneg hX ⟨ht.1.le,ht.2.le⟩
-  have hz := hm.hasDerivAt_eq_zero (hX.contactPotential_hasDerivAt hs)
-  linarith
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Variation.FirstVariationCompact
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def OrderParameter.zero : OrderParameter where
+  coeff := 0
+  nonneg _ _ := le_rfl
+  monotone _ _ _ _ _ := le_rfl
+  rightContinuous _ _ := continuousWithinAt_const
+  integrable := integrableOn_zero
+
+noncomputable def OrderParameter.add (γ η : OrderParameter) : OrderParameter where
+  coeff t := γ.coeff t+η.coeff t
+  nonneg t ht := add_nonneg (γ.nonneg t ht) (η.nonneg t ht)
+  monotone _ hs _ ht hst := add_le_add (γ.monotone hs ht hst) (η.monotone hs ht hst)
+  rightContinuous t ht := (γ.rightContinuous t ht).add (η.rightContinuous t ht)
+  integrable := γ.integrable.add η.integrable
+
+noncomputable def OrderParameter.atom (s : ℝ) : OrderParameter where
+  coeff t := if s≤t then 1 else 0
+  nonneg _ _ := by split_ifs <;> norm_num
+  monotone t _ u _ htu := by
+    by_cases hst : s≤t
+    · simp only [ite_eq_left hst,ite_eq_left (hst.trans htu),le_refl]
+    · simp only [ite_eq_right hst]; split_ifs <;> norm_num
+  rightContinuous t _ := by
+    by_cases hst : s≤t
+    · apply (continuousWithinAt_const (b := (1 : ℝ))).congr
+      · intro u hu
+        exact ite_eq_left (hst.trans hu)
+      · exact ite_eq_left hst
+    · apply (continuousWithinAt_const (b := (0 : ℝ))).congr_of_eventuallyEq
+      · filter_upwards [mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds (lt_of_not_ge hst))] with u hu
+        exact ite_eq_right (not_le.mpr hu)
+      · exact ite_eq_right hst
+  integrable := by
+    apply Integrable.of_bound
+      (StronglyMeasurable.ite measurableSet_Ici stronglyMeasurable_const stronglyMeasurable_const).aestronglyMeasurable 1
+    exact Eventually.of_forall (fun t ↦ by split_ifs <;> norm_num)
+
+noncomputable def OrderParameter.removeBelow (γ : OrderParameter) (c : ℝ) : OrderParameter where
+  coeff t := max (γ.coeff t-c) 0
+  nonneg _ _ := le_max_right _ _
+  monotone _ hs _ ht hst := max_le_max (sub_le_sub_right (γ.monotone hs ht hst) c) le_rfl
+  rightContinuous t ht := ((γ.rightContinuous t ht).sub continuousWithinAt_const).sup continuousWithinAt_const
+  integrable := (γ.integrable.sub (integrableOn_const (μ := volume) (s := Ioc (0 : ℝ) 1) (C := c) (by simp))).sup integrableOn_zero
+
+lemma OrderParameter.removeBelow_sub (γ : OrderParameter) (c t : ℝ) :
+    (γ.removeBelow c).coeff t-γ.coeff t= -min (γ.coeff t) c := by
+  dsimp only [OrderParameter.removeBelow]
+  by_cases h : γ.coeff t≤c
+  · rw [max_eq_right (sub_nonpos.mpr h),min_eq_left h]; ring
+  · rw [max_eq_left (sub_nonneg.mpr (le_of_not_ge h)),min_eq_right (le_of_not_ge h)]; ring
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def contactPotential (W : BrownianSpace) (γ : OrderParameter)
+    (X : ℝ → W.Ω → ℝ) (s : ℝ) : ℝ :=
+  ∫ t in s..1,gradientMoment W γ X t-t
+
+lemma IsDiffusion.moment_difference_integrable {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
+    IntervalIntegrable (fun t ↦ gradientMoment W γ X t-t) volume 0 1 := by
+  have hm := hX.gradientMoment_continuous.aestronglyMeasurable (μ := volume) measurableSet_Ico
+  rw [restrict_Ico_eq_restrict_Ioc] at hm
+  have hi : IntegrableOn (gradientMoment W γ X) (Ioc (0 : ℝ) 1) :=
+    Integrable.of_bound hm 1 (by
+      filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
+      exact gradientMoment_bound W γ X ⟨ht.1.le,ht.2⟩)
+  exact ((intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num : (0 : ℝ)≤1)).mpr hi).sub
+    (continuous_id.intervalIntegrable 0 1)
+
+lemma integral_atom_mul (s : ℝ) (hs : s∈Icc (0 : ℝ) 1) (f : ℝ → ℝ) :
+    (∫ t in (0 : ℝ)..1,(OrderParameter.atom s).coeff t*f t)=∫ t in s..1,f t := by
+  have he : (fun t ↦ (OrderParameter.atom s).coeff t*f t)=(Ici s).indicator f := by
+    funext t
+    simp only [OrderParameter.atom,indicator,mem_Ici]
+    split_ifs <;> simp
+  have hs' : Ici s∩Icc (0 : ℝ) 1=Icc s 1 := by
+    ext t
+    simp only [mem_inter_iff,mem_Ici,mem_Icc]
+    constructor
+    · rintro ⟨hst,_,ht1⟩; exact ⟨hst,ht1⟩
+    · rintro ⟨hst,ht1⟩; exact ⟨hst,hs.1.trans hst,ht1⟩
+  rw [he,intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1),
+    ←integral_Icc_eq_integral_Ioc,integral_indicator measurableSet_Ici,
+    Measure.restrict_restrict measurableSet_Ici,hs',
+    intervalIntegral.integral_of_le hs.2,integral_Icc_eq_integral_Ioc]
+
+lemma IsMinimizer.contactPotential_nonneg {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
+    {s : ℝ} (hs : s∈Icc (0 : ℝ) 1) : 0≤contactPotential W γ X s := by
+  have hh := hγ.variation_inequality hX (γ.add (OrderParameter.atom s))
+  simp only [OrderParameter.add,add_sub_cancel_left] at hh
+  rwa [integral_atom_mul s hs] at hh
+
+lemma IsDiffusion.contactPotential_continuous {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
+    ContinuousOn (contactPotential W γ X) (Icc (0 : ℝ) 1) := by
+  change ContinuousOn (fun s : ℝ ↦ ∫ t in s..1,gradientMoment W γ X t-t) (Icc (0 : ℝ) 1)
+  have hi : IntegrableOn (fun t ↦ gradientMoment W γ X t-t) (uIcc (0 : ℝ) 1) volume := by
+    rw [uIcc_of_le (by norm_num : (0 : ℝ)≤1)]
+    exact (intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num : (0 : ℝ)≤1)).mp
+      hX.moment_difference_integrable
+  have hc : ContinuousOn (fun s : ℝ ↦ ∫ t in s..1,gradientMoment W γ X t-t) (uIcc (0 : ℝ) 1) :=
+    intervalIntegral.continuousOn_primitive_interval_left hi
+  simpa only [uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using hc
+
+lemma IsDiffusion.contactPotential_hasDerivAt {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) {s : ℝ} (hs : s∈Ioo (0 : ℝ) 1) :
+    HasDerivAt (contactPotential W γ X) (s-gradientMoment W γ X s) s := by
+  change HasDerivAt (fun u : ℝ ↦ ∫ t in u..1,gradientMoment W γ X t-t) _ s
+  have hc : ContinuousOn (fun t ↦ gradientMoment W γ X t-t) (Ico (0 : ℝ) 1) :=
+    hX.gradientMoment_continuous.sub continuousOn_id
+  have hns : Ico (0 : ℝ) 1∈𝓝 s := mem_of_superset (Ioo_mem_nhds hs.1 hs.2) Ioo_subset_Ico_self
+  have hi : IntervalIntegrable (fun t ↦ gradientMoment W γ X t-t) volume s 1 :=
+    hX.moment_difference_integrable.mono_set (by
+      rw [uIcc_of_le hs.2.le,uIcc_of_le (by norm_num : (0 : ℝ)≤1)]
+      exact Icc_subset_Icc hs.1.le le_rfl)
+  have hh : HasDerivAt (fun u : ℝ ↦ ∫ t in u..1,gradientMoment W γ X t-t)
+      (-(gradientMoment W γ X s-s)) s := intervalIntegral.integral_hasDerivAt_left hi
+        (ContinuousOn.stronglyMeasurableAtFilter isOpen_Ioo (hc.mono Ioo_subset_Ico_self) s hs)
+        (hc.continuousAt hns)
+  simpa only [neg_sub] using hh
+
+lemma IsMinimizer.moment_eq_at_contact {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
+    {s : ℝ} (hs : s∈Ioo (0 : ℝ) 1) (hG : contactPotential W γ X s=0) :
+    gradientMoment W γ X s=s := by
+  have hm : IsLocalMin (contactPotential W γ X) s := by
+    filter_upwards [Ioo_mem_nhds hs.1 hs.2] with t ht
+    rw [hG]
+    exact hγ.contactPotential_nonneg hX ⟨ht.1.le,ht.2.le⟩
+  have hz := hm.hasDerivAt_eq_zero (hX.contactPotential_hasDerivAt hs)
+  linarith
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/FirstVariationCompact.lean b/lean/OAI/Probability/SKValue/Variation/FirstVariationCompact.lean
index 09e0e997e471dea56f5f78f254649bcc80afdbe6..c396db9a4c233d8d32b263fc59e7774a9a5da1a7 100644
--- a/lean/OAI/Probability/SKValue/Variation/FirstVariationCompact.lean
+++ b/lean/OAI/Probability/SKValue/Variation/FirstVariationCompact.lean
@@ -1,126 +1,126 @@
-import Mathlib
-import OAI.Probability.SKValue.Evolution.IntegralPrimitiveTerminal
-import OAI.Probability.SKValue.Processes.CrossValueBound
-import OAI.Probability.SKValue.Variation.OrderVariations
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma IsMinimizer.compact_variation_inequality {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
-    (η : OrderParameter) {T : ℝ} (hT : 0<T) (hT1 : T<1) :
-    (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤
-      (3/2 : ℝ)*(∫ t in T..1,|η.cutoff t-γ.cutoff t|)+
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..T,(η.coeff t-γ.coeff t)*gradientMoment W γ X t) := by
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨hT.le,hT1⟩
-  let A := ∫ t in (0 : ℝ)..1,|η.cutoff t-γ.cutoff t|
-  let C := η.coeff T+γ.coeff T
-  let R (h : ℝ) := T*(h^2*C+γ.coeff T*(h*(Lu+3*A)))*Real.exp (T*γ.coeff T*Lu)
-  let B := (3/2 : ℝ)*(∫ t in T..1,|η.cutoff t-γ.cutoff t|)+
-    (1/2 : ℝ)*(∫ t in (0 : ℝ)..T,(η.coeff t-γ.coeff t)*gradientMoment W γ X t)
-  have hA : 0≤A := intervalIntegral.integral_nonneg (by norm_num) (fun t ht ↦ abs_nonneg _)
-  have hC : 0≤C := add_nonneg (η.nonneg T ⟨hT.le,hT1⟩) (γ.nonneg T ⟨hT.le,hT1⟩)
-  have hAi : IntervalIntegrable (fun t ↦ |η.cutoff t-γ.cutoff t|) volume 0 1 :=
-    (η.cutoff_integrable.sub γ.cutoff_integrable).norm.intervalIntegrable
-  have htA (t : ℝ) (ht : t∈Icc (0 : ℝ) T) :
-      (∫ s in t..1,|η.cutoff s-γ.cutoff s|)≤A := by
-    exact intervalIntegral.integral_mono_interval ht.1 (ht.2.trans hT1.le) le_rfl
-      (Eventually.of_forall (fun s ↦ abs_nonneg _)) hAi
-  have hul (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x y : ℝ) :
-      |gradient W γ t x-gradient W γ t y|≤Lu*|x-y| := by
-    simpa only [sub_self,abs_zero,zero_add] using hu t ht t ht x y
-  have hh (h : ℝ) (hh0 : 0<h) (hh1 : h≤1) :
-      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤B+T*C*Lv*R h := by
-    have hε : h^2∈Icc (0 : ℝ) 1 := ⟨sq_nonneg _,(sq_le_one_iff_abs_le_one _).mpr (by rw [abs_of_pos hh0]; exact hh1)⟩
-    let β := γ.mix η (h^2) hε
-    have hph (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
-        |phi W β t x-phi W γ t x|≤(3/2 : ℝ)*h^2*A := by
-      have hb := phi_difference_bound W γ β ⟨ht.1,ht.2.trans hT1.le⟩ x
-      rw [γ.mix_difference_integral η hε] at hb
-      exact hb.trans (by nlinarith [mul_le_mul_of_nonneg_left (htA t ht) (sq_nonneg h)])
-    have hgs (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
-        |gradient W β t x-gradient W γ t x|≤h*(Lu+3*A) := by
-      have hb := phi_gradient_difference_bound W γ β ⟨hT.le,hT1⟩ hul hph hh0 ht x
-      have heq : Lu*h+2*((3/2 : ℝ)*h^2*A)/h=h*(Lu+3*A) := by field_simp
-      rwa [heq] at hb
-    have hv := hX.cross_value_bound β hT hT1 hV hLu
-      (mul_nonneg (sq_nonneg h) hC) (mul_nonneg hh0.le (by positivity : 0≤Lu+3*A))
-      (fun t ht ↦ γ.mix_strip_difference_bound η hε ⟨hT.le,hT1⟩ ht) hul hgs
-    rw [γ.mix_difference_integral η hε,hX.mix_moment_integral η hε ⟨hT.le,hT1.le⟩] at hv
-    have hm := hγ.mix_value_inequality η hε
-    have hv' : (h^2/2)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤
-        h^2*(B+T*C*Lv*R h) := by
-      calc
-        _≤phi W β 0 0-phi W γ 0 0 := hm
-        _≤_ := hv
-        _=h^2*(B+T*C*Lv*R h) := by dsimp only [B,R,C] at *; ring
-    have heq : (h^2/2)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))=
-        h^2*((1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))) := by ring
-    rw [heq] at hv'
-    exact (mul_le_mul_iff_right₀ (sq_pos_of_pos hh0)).mp hv'
-  have hzero : Tendsto (fun h : ℝ ↦ h) (𝓝[>] (0 : ℝ)) (𝓝 0) := nhdsWithin_le_nhds
-  have hRlim : Tendsto R (𝓝[>] (0 : ℝ)) (𝓝 0) := by
-    simpa only [R,zero_pow (by norm_num : (2 : ℕ)≠0),zero_mul,mul_zero,add_zero] using
-      (((hzero.pow 2).mul_const C).add ((hzero.mul_const (Lu+3*A)).const_mul (γ.coeff T))).const_mul T |>.mul_const (Real.exp (T*γ.coeff T*Lu))
-  have hlim := (hRlim.const_mul (T*C*Lv)).const_add B
-  have hb := ge_of_tendsto hlim (by
-    filter_upwards [Ioo_mem_nhdsGT (by norm_num : (0 : ℝ)<1)] with h hhs
-    exact hh h hhs.1 hhs.2.le)
-  simpa only [mul_zero,add_zero,B] using hb
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma tail_integral_tendsto_zero {f : ℝ → ℝ} (hf : IntervalIntegrable f volume 0 1) :
-    Tendsto (fun T ↦ ∫ t in T..1,f t) (𝓝[<] (1 : ℝ)) (𝓝 0) := by
-  have hlim := (integral_primitive_terminal hf).const_sub (∫ t in (0 : ℝ)..1,f t)
-  have hlim' : Tendsto (fun T ↦ (∫ t in (0 : ℝ)..1,f t)-(∫ t in (0 : ℝ)..T,f t))
-      (𝓝[<] (1 : ℝ)) (𝓝 0) := by simpa only [sub_self] using hlim
-  apply hlim'.congr'
-  filter_upwards [Ioo_mem_nhdsLT (by norm_num : (0 : ℝ)<1)] with T hT
-  have hi : IntervalIntegrable f volume 0 T := hf.mono_set (by simpa only [uIcc_of_le hT.1.le,uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using Icc_subset_Icc le_rfl hT.2.le)
-  have hi' : IntervalIntegrable f volume T 1 := hf.mono_set (by simpa only [uIcc_of_le hT.2.le,uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using Icc_subset_Icc hT.1.le le_rfl)
-  linarith [intervalIntegral.integral_add_adjacent_intervals hi hi']
-
-lemma IsMinimizer.variation_inequality {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
-    (η : OrderParameter) :
-    0≤∫ t in (0 : ℝ)..1,(η.coeff t-γ.coeff t)*(gradientMoment W γ X t-t) := by
-  have hAi : IntervalIntegrable (fun t ↦ |η.cutoff t-γ.cutoff t|) volume 0 1 :=
-    (η.cutoff_integrable.sub γ.cutoff_integrable).norm.intervalIntegrable
-  have hqi : IntervalIntegrable (fun t ↦ (η.coeff t-γ.coeff t)*gradientMoment W γ X t) volume 0 1 := by
-    simpa only [sub_mul] using (hX.weightedMoment_integrable η).sub (hX.weightedMoment_integrable γ)
-  have hti : IntervalIntegrable (fun t ↦ t*(η.coeff t-γ.coeff t)) volume 0 1 := by
-    simpa only [mul_sub] using η.weighted_intervalIntegrable.sub γ.weighted_intervalIntegrable
-  have hlim := ((tail_integral_tendsto_zero hAi).const_mul (3/2 : ℝ)).add
-    ((integral_primitive_terminal hqi).const_mul (1/2 : ℝ))
-  have hh := ge_of_tendsto hlim (by
-    filter_upwards [Ioo_mem_nhdsLT (by norm_num : (0 : ℝ)<1)] with T hT
-    exact hγ.compact_variation_inequality hX η hT.1 hT.2)
-  have heq : (∫ t in (0 : ℝ)..1,(η.coeff t-γ.coeff t)*(gradientMoment W γ X t-t))=
-      (∫ t in (0 : ℝ)..1,(η.coeff t-γ.coeff t)*gradientMoment W γ X t)-
-        (∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t)) := by
-    rw [←intervalIntegral.integral_sub hqi hti]
-    apply intervalIntegral.integral_congr
-    intro t ht
-    ring
-  rw [heq]
-  linarith
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Evolution.IntegralPrimitiveTerminal
+import OAI.Probability.SKValue.Processes.CrossValueBound
+import OAI.Probability.SKValue.Variation.OrderVariations
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma IsMinimizer.compact_variation_inequality {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
+    (η : OrderParameter) {T : ℝ} (hT : 0<T) (hT1 : T<1) :
+    (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤
+      (3/2 : ℝ)*(∫ t in T..1,|η.cutoff t-γ.cutoff t|)+
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..T,(η.coeff t-γ.coeff t)*gradientMoment W γ X t) := by
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨hT.le,hT1⟩
+  let A := ∫ t in (0 : ℝ)..1,|η.cutoff t-γ.cutoff t|
+  let C := η.coeff T+γ.coeff T
+  let R (h : ℝ) := T*(h^2*C+γ.coeff T*(h*(Lu+3*A)))*Real.exp (T*γ.coeff T*Lu)
+  let B := (3/2 : ℝ)*(∫ t in T..1,|η.cutoff t-γ.cutoff t|)+
+    (1/2 : ℝ)*(∫ t in (0 : ℝ)..T,(η.coeff t-γ.coeff t)*gradientMoment W γ X t)
+  have hA : 0≤A := intervalIntegral.integral_nonneg (by norm_num) (fun t ht ↦ abs_nonneg _)
+  have hC : 0≤C := add_nonneg (η.nonneg T ⟨hT.le,hT1⟩) (γ.nonneg T ⟨hT.le,hT1⟩)
+  have hAi : IntervalIntegrable (fun t ↦ |η.cutoff t-γ.cutoff t|) volume 0 1 :=
+    (η.cutoff_integrable.sub γ.cutoff_integrable).norm.intervalIntegrable
+  have htA (t : ℝ) (ht : t∈Icc (0 : ℝ) T) :
+      (∫ s in t..1,|η.cutoff s-γ.cutoff s|)≤A := by
+    exact intervalIntegral.integral_mono_interval ht.1 (ht.2.trans hT1.le) le_rfl
+      (Eventually.of_forall (fun s ↦ abs_nonneg _)) hAi
+  have hul (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x y : ℝ) :
+      |gradient W γ t x-gradient W γ t y|≤Lu*|x-y| := by
+    simpa only [sub_self,abs_zero,zero_add] using hu t ht t ht x y
+  have hh (h : ℝ) (hh0 : 0<h) (hh1 : h≤1) :
+      (1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤B+T*C*Lv*R h := by
+    have hε : h^2∈Icc (0 : ℝ) 1 := ⟨sq_nonneg _,(sq_le_one_iff_abs_le_one _).mpr (by rw [abs_of_pos hh0]; exact hh1)⟩
+    let β := γ.mix η (h^2) hε
+    have hph (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
+        |phi W β t x-phi W γ t x|≤(3/2 : ℝ)*h^2*A := by
+      have hb := phi_difference_bound W γ β ⟨ht.1,ht.2.trans hT1.le⟩ x
+      rw [γ.mix_difference_integral η hε] at hb
+      exact hb.trans (by nlinarith [mul_le_mul_of_nonneg_left (htA t ht) (sq_nonneg h)])
+    have hgs (t : ℝ) (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
+        |gradient W β t x-gradient W γ t x|≤h*(Lu+3*A) := by
+      have hb := phi_gradient_difference_bound W γ β ⟨hT.le,hT1⟩ hul hph hh0 ht x
+      have heq : Lu*h+2*((3/2 : ℝ)*h^2*A)/h=h*(Lu+3*A) := by field_simp
+      rwa [heq] at hb
+    have hv := hX.cross_value_bound β hT hT1 hV hLu
+      (mul_nonneg (sq_nonneg h) hC) (mul_nonneg hh0.le (by positivity : 0≤Lu+3*A))
+      (fun t ht ↦ γ.mix_strip_difference_bound η hε ⟨hT.le,hT1⟩ ht) hul hgs
+    rw [γ.mix_difference_integral η hε,hX.mix_moment_integral η hε ⟨hT.le,hT1.le⟩] at hv
+    have hm := hγ.mix_value_inequality η hε
+    have hv' : (h^2/2)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤
+        h^2*(B+T*C*Lv*R h) := by
+      calc
+        _≤phi W β 0 0-phi W γ 0 0 := hm
+        _≤_ := hv
+        _=h^2*(B+T*C*Lv*R h) := by dsimp only [B,R,C] at *; ring
+    have heq : (h^2/2)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))=
+        h^2*((1/2 : ℝ)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))) := by ring
+    rw [heq] at hv'
+    exact (mul_le_mul_iff_right₀ (sq_pos_of_pos hh0)).mp hv'
+  have hzero : Tendsto (fun h : ℝ ↦ h) (𝓝[>] (0 : ℝ)) (𝓝 0) := nhdsWithin_le_nhds
+  have hRlim : Tendsto R (𝓝[>] (0 : ℝ)) (𝓝 0) := by
+    simpa only [R,zero_pow (by norm_num : (2 : ℕ)≠0),zero_mul,mul_zero,add_zero] using
+      (((hzero.pow 2).mul_const C).add ((hzero.mul_const (Lu+3*A)).const_mul (γ.coeff T))).const_mul T |>.mul_const (Real.exp (T*γ.coeff T*Lu))
+  have hlim := (hRlim.const_mul (T*C*Lv)).const_add B
+  have hb := ge_of_tendsto hlim (by
+    filter_upwards [Ioo_mem_nhdsGT (by norm_num : (0 : ℝ)<1)] with h hhs
+    exact hh h hhs.1 hhs.2.le)
+  simpa only [mul_zero,add_zero,B] using hb
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma tail_integral_tendsto_zero {f : ℝ → ℝ} (hf : IntervalIntegrable f volume 0 1) :
+    Tendsto (fun T ↦ ∫ t in T..1,f t) (𝓝[<] (1 : ℝ)) (𝓝 0) := by
+  have hlim := (integral_primitive_terminal hf).const_sub (∫ t in (0 : ℝ)..1,f t)
+  have hlim' : Tendsto (fun T ↦ (∫ t in (0 : ℝ)..1,f t)-(∫ t in (0 : ℝ)..T,f t))
+      (𝓝[<] (1 : ℝ)) (𝓝 0) := by simpa only [sub_self] using hlim
+  apply hlim'.congr'
+  filter_upwards [Ioo_mem_nhdsLT (by norm_num : (0 : ℝ)<1)] with T hT
+  have hi : IntervalIntegrable f volume 0 T := hf.mono_set (by simpa only [uIcc_of_le hT.1.le,uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using Icc_subset_Icc le_rfl hT.2.le)
+  have hi' : IntervalIntegrable f volume T 1 := hf.mono_set (by simpa only [uIcc_of_le hT.2.le,uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using Icc_subset_Icc hT.1.le le_rfl)
+  linarith [intervalIntegral.integral_add_adjacent_intervals hi hi']
+
+lemma IsMinimizer.variation_inequality {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hγ : IsMinimizer W γ) (hX : IsDiffusion W γ X)
+    (η : OrderParameter) :
+    0≤∫ t in (0 : ℝ)..1,(η.coeff t-γ.coeff t)*(gradientMoment W γ X t-t) := by
+  have hAi : IntervalIntegrable (fun t ↦ |η.cutoff t-γ.cutoff t|) volume 0 1 :=
+    (η.cutoff_integrable.sub γ.cutoff_integrable).norm.intervalIntegrable
+  have hqi : IntervalIntegrable (fun t ↦ (η.coeff t-γ.coeff t)*gradientMoment W γ X t) volume 0 1 := by
+    simpa only [sub_mul] using (hX.weightedMoment_integrable η).sub (hX.weightedMoment_integrable γ)
+  have hti : IntervalIntegrable (fun t ↦ t*(η.coeff t-γ.coeff t)) volume 0 1 := by
+    simpa only [mul_sub] using η.weighted_intervalIntegrable.sub γ.weighted_intervalIntegrable
+  have hlim := ((tail_integral_tendsto_zero hAi).const_mul (3/2 : ℝ)).add
+    ((integral_primitive_terminal hqi).const_mul (1/2 : ℝ))
+  have hh := ge_of_tendsto hlim (by
+    filter_upwards [Ioo_mem_nhdsLT (by norm_num : (0 : ℝ)<1)] with T hT
+    exact hγ.compact_variation_inequality hX η hT.1 hT.2)
+  have heq : (∫ t in (0 : ℝ)..1,(η.coeff t-γ.coeff t)*(gradientMoment W γ X t-t))=
+      (∫ t in (0 : ℝ)..1,(η.coeff t-γ.coeff t)*gradientMoment W γ X t)-
+        (∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t)) := by
+    rw [←intervalIntegral.integral_sub hqi hti]
+    apply intervalIntegral.integral_congr
+    intro t ht
+    ring
+  rw [heq]
+  linarith
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/MinimizerContact.lean b/lean/OAI/Probability/SKValue/Variation/MinimizerContact.lean
index 1b3ff57091cf9e8d86cc93432c235ece985c2ae1..cd0c36489325a6ad49ccf955470bed1cc54acc34 100644
--- a/lean/OAI/Probability/SKValue/Variation/MinimizerContact.lean
+++ b/lean/OAI/Probability/SKValue/Variation/MinimizerContact.lean
@@ -1,87 +1,87 @@
-import Mathlib
-import OAI.Probability.SKValue.Variation.ContactOrigin
-import OAI.Probability.SKValue.Coercivity.CurvatureMoment
-import OAI.Probability.SKValue.Coercivity.GapBoundary
-import OAI.Probability.SKValue.Coercivity.ZeroGapHump
-
-namespace OAI
-
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal
-namespace SKValue
-
-lemma IsMinimizer.contactPotential_zero {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
-    (hγ:IsMinimizer W γ) (hX:IsDiffusion W γ X) {x:ℝ} (hx:x∈Ioo (0:ℝ) 1):
-    contactPotential W γ X x=0 := by
-  by_contra hne
-  have hxp:0<contactPotential W γ X x:=lt_of_le_of_ne
-    (hγ.contactPotential_nonneg hX ⟨hx.1.le,hx.2.le⟩) (Ne.symm hne)
-  obtain ⟨a,b,ha,hax,hxb,hb,haG,hGb,hpos⟩:=positive_gap_endpoints
-    hX.contactPotential_continuous (fun t ht ↦ hγ.contactPotential_nonneg hX ht)
-    (by simp only [contactPotential,intervalIntegral.integral_same]) hx hxp
-  have hab:a<b:=hax.trans hxb
-  have ha1:a<1:=hax.trans hx.2
-  have hb0:0<b:=hx.1.trans hxb
-  let c:=γ.coeff x
-  have he:∀t∈Ioo a b,γ.coeff t=c := fun t ht ↦
-    hγ.coeff_constant_of_contactPositive hX ha hb hpos t ht x ⟨hax,hxb⟩
-  let f:=fun t ↦ gradientMoment W γ X t-t
-  have hf0:f a=0 := by
-    rcases haG with ha0|haG
-    · simp only [f,ha0,hX.moment_origin,sub_self]
-    · by_cases ha0:a=0
-      · simp only [f,ha0,hX.moment_origin,sub_self]
-      · exact sub_eq_zero.mpr (hγ.moment_eq_at_contact hX ⟨lt_of_le_of_ne ha (Ne.symm ha0),ha1⟩ haG)
-  have hGc:ContinuousOn (contactPotential W γ X) (Icc a b):=
-    hX.contactPotential_continuous.mono (Icc_subset_Icc ha hb)
-  have hGd:∀t∈Ioo a b,HasDerivAt (contactPotential W γ X) (-f t) t := by
-    intro t ht
-    simpa only [f,neg_sub] using hX.contactPotential_hasDerivAt ⟨ha.trans_lt ht.1,ht.2.trans_le hb⟩
-  have hfc:ContinuousOn f (Ico a 1):=
-    (hX.gradientMoment_continuous.sub continuousOn_id).mono (Ico_subset_Ico_left ha)
-  have hend:∀s∈Ioo a b,0<f s → ∃z∈Ioo s 1,z≤b ∧ f z≤f s := by
-    apply endpoint_condition hb
-    · intro hb1
-      exact sub_eq_zero.mpr (hγ.moment_eq_at_contact hX ⟨hb0,hb1⟩ hGb)
-    · intro t ht
-      have hh:gradientMoment W γ X t≤1:=(le_abs_self _).trans
-        (gradientMoment_bound W γ X ⟨(ha.trans_lt ht.1).le,ht.2.le⟩)
-      dsimp only [f];linarith
-  by_cases hc:c=0
-  · apply positive_right_hump_impossible hax hxb hGc hGd hGb hxp hf0 ?_ hend
-    intro z hz hzb
-    have hv:=hX.moment_convexOn_zero_closed ha hz.2
-      (fun t ht ↦ (he t ⟨ht.1,ht.2.trans_le hzb⟩).trans hc)
-    exact hv.sub (concaveOn_id (convex_Icc a z))
-  · have hcpos:0<c:=lt_of_le_of_ne (γ.nonneg x ⟨hx.1.le,hx.2⟩) (Ne.symm hc)
-    have hGa:contactPotential W γ X a=0 := by
-      rcases haG with ha0|haG
-      · subst a
-        have h0:=γ.origin_eq_of_constant hb0 he
-        exact hγ.contact_origin_of_positive hX (by simpa only [h0] using hcpos)
-      · exact haG
-    let f':=fun t ↦ jetMoment W γ X JetExpr.curvatureExpr t-1
-    have hfd:∀t∈Ioo a b,HasDerivAt f (f' t) t := by
-      intro t ht
-      have hd:=hX.momentExpr_hasDerivAt ⟨ha.trans_lt ht.1,ht.2.trans_le hb⟩
-        (continuousAt_of_constant_interval ht he)
-      rw [momentExpr_eq_gradientMoment] at hd
-      exact hd.sub (hasDerivAt_id t)
-    have hcv:ConvexOn ℝ (Ioo a b) f':=
-      (hX.curvature_convexOn ha hb he).sub (concaveOn_const 1 (convex_Ioo a b))
-    exact positive_hump_impossible hax hxb hb hGc hGd hGa hGb hxp hfc hfd hf0 hcv hend
-
-theorem IsMinimizer.gradient_moment_contact {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
-    (hγ:IsMinimizer W γ) (hX:IsDiffusion W γ X):
-    ∀t∈Ico (0:ℝ) 1,(∫ z,(gradient W γ t (X t z))^2 ∂W.μ)=t := by
-  intro t ht
-  change gradientMoment W γ X t=t
-  rcases ht.1.eq_or_lt with he|he
-  · rw [←he];exact hX.moment_origin
-  · exact hγ.moment_eq_at_contact hX ⟨he,ht.2⟩ (hγ.contactPotential_zero hX ⟨he,ht.2⟩)
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Variation.ContactOrigin
+import OAI.Probability.SKValue.Coercivity.CurvatureMoment
+import OAI.Probability.SKValue.Coercivity.GapBoundary
+import OAI.Probability.SKValue.Coercivity.ZeroGapHump
+
+namespace OAI
+
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal
+namespace SKValue
+
+lemma IsMinimizer.contactPotential_zero {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
+    (hγ:IsMinimizer W γ) (hX:IsDiffusion W γ X) {x:ℝ} (hx:x∈Ioo (0:ℝ) 1):
+    contactPotential W γ X x=0 := by
+  by_contra hne
+  have hxp:0<contactPotential W γ X x:=lt_of_le_of_ne
+    (hγ.contactPotential_nonneg hX ⟨hx.1.le,hx.2.le⟩) (Ne.symm hne)
+  obtain ⟨a,b,ha,hax,hxb,hb,haG,hGb,hpos⟩:=positive_gap_endpoints
+    hX.contactPotential_continuous (fun t ht ↦ hγ.contactPotential_nonneg hX ht)
+    (by simp only [contactPotential,intervalIntegral.integral_same]) hx hxp
+  have hab:a<b:=hax.trans hxb
+  have ha1:a<1:=hax.trans hx.2
+  have hb0:0<b:=hx.1.trans hxb
+  let c:=γ.coeff x
+  have he:∀t∈Ioo a b,γ.coeff t=c := fun t ht ↦
+    hγ.coeff_constant_of_contactPositive hX ha hb hpos t ht x ⟨hax,hxb⟩
+  let f:=fun t ↦ gradientMoment W γ X t-t
+  have hf0:f a=0 := by
+    rcases haG with ha0|haG
+    · simp only [f,ha0,hX.moment_origin,sub_self]
+    · by_cases ha0:a=0
+      · simp only [f,ha0,hX.moment_origin,sub_self]
+      · exact sub_eq_zero.mpr (hγ.moment_eq_at_contact hX ⟨lt_of_le_of_ne ha (Ne.symm ha0),ha1⟩ haG)
+  have hGc:ContinuousOn (contactPotential W γ X) (Icc a b):=
+    hX.contactPotential_continuous.mono (Icc_subset_Icc ha hb)
+  have hGd:∀t∈Ioo a b,HasDerivAt (contactPotential W γ X) (-f t) t := by
+    intro t ht
+    simpa only [f,neg_sub] using hX.contactPotential_hasDerivAt ⟨ha.trans_lt ht.1,ht.2.trans_le hb⟩
+  have hfc:ContinuousOn f (Ico a 1):=
+    (hX.gradientMoment_continuous.sub continuousOn_id).mono (Ico_subset_Ico_left ha)
+  have hend:∀s∈Ioo a b,0<f s → ∃z∈Ioo s 1,z≤b ∧ f z≤f s := by
+    apply endpoint_condition hb
+    · intro hb1
+      exact sub_eq_zero.mpr (hγ.moment_eq_at_contact hX ⟨hb0,hb1⟩ hGb)
+    · intro t ht
+      have hh:gradientMoment W γ X t≤1:=(le_abs_self _).trans
+        (gradientMoment_bound W γ X ⟨(ha.trans_lt ht.1).le,ht.2.le⟩)
+      dsimp only [f];linarith
+  by_cases hc:c=0
+  · apply positive_right_hump_impossible hax hxb hGc hGd hGb hxp hf0 ?_ hend
+    intro z hz hzb
+    have hv:=hX.moment_convexOn_zero_closed ha hz.2
+      (fun t ht ↦ (he t ⟨ht.1,ht.2.trans_le hzb⟩).trans hc)
+    exact hv.sub (concaveOn_id (convex_Icc a z))
+  · have hcpos:0<c:=lt_of_le_of_ne (γ.nonneg x ⟨hx.1.le,hx.2⟩) (Ne.symm hc)
+    have hGa:contactPotential W γ X a=0 := by
+      rcases haG with ha0|haG
+      · subst a
+        have h0:=γ.origin_eq_of_constant hb0 he
+        exact hγ.contact_origin_of_positive hX (by simpa only [h0] using hcpos)
+      · exact haG
+    let f':=fun t ↦ jetMoment W γ X JetExpr.curvatureExpr t-1
+    have hfd:∀t∈Ioo a b,HasDerivAt f (f' t) t := by
+      intro t ht
+      have hd:=hX.momentExpr_hasDerivAt ⟨ha.trans_lt ht.1,ht.2.trans_le hb⟩
+        (continuousAt_of_constant_interval ht he)
+      rw [momentExpr_eq_gradientMoment] at hd
+      exact hd.sub (hasDerivAt_id t)
+    have hcv:ConvexOn ℝ (Ioo a b) f':=
+      (hX.curvature_convexOn ha hb he).sub (concaveOn_const 1 (convex_Ioo a b))
+    exact positive_hump_impossible hax hxb hb hGc hGd hGa hGb hxp hfc hfd hf0 hcv hend
+
+theorem IsMinimizer.gradient_moment_contact {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
+    (hγ:IsMinimizer W γ) (hX:IsDiffusion W γ X):
+    ∀t∈Ico (0:ℝ) 1,(∫ z,(gradient W γ t (X t z))^2 ∂W.μ)=t := by
+  intro t ht
+  change gradientMoment W γ X t=t
+  rcases ht.1.eq_or_lt with he|he
+  · rw [←he];exact hX.moment_origin
+  · exact hγ.moment_eq_at_contact hX ⟨he,ht.2⟩ (hγ.contactPotential_zero hX ⟨he,ht.2⟩)
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/OrderVariations.lean b/lean/OAI/Probability/SKValue/Variation/OrderVariations.lean
index 28d1b23c9b0f29bb58e2d153a8849d84f6279539..8475c21a2917e9cf03ac4417d137fb6a1508d3fb 100644
--- a/lean/OAI/Probability/SKValue/Variation/OrderVariations.lean
+++ b/lean/OAI/Probability/SKValue/Variation/OrderVariations.lean
@@ -1,147 +1,147 @@
-import Mathlib
-import OAI.Probability.SKValue.Control.Regularity
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def OrderParameter.mix (γ η : OrderParameter) (ε : ℝ)
-    (hε : ε∈Icc (0 : ℝ) 1) : OrderParameter where
-  coeff t := (1-ε)*γ.coeff t+ε*η.coeff t
-  nonneg t ht := add_nonneg (mul_nonneg (sub_nonneg.mpr hε.2) (γ.nonneg t ht))
-    (mul_nonneg hε.1 (η.nonneg t ht))
-  monotone _s hs _t ht hst := add_le_add
-    (mul_le_mul_of_nonneg_left (γ.monotone hs ht hst) (sub_nonneg.mpr hε.2))
-    (mul_le_mul_of_nonneg_left (η.monotone hs ht hst) hε.1)
-  rightContinuous t ht := ((γ.rightContinuous t ht).const_mul _).add ((η.rightContinuous t ht).const_mul _)
-  integrable := (γ.integrable.const_mul _).add (η.integrable.const_mul _)
-
-lemma OrderParameter.mix_coeff_sub (γ η : OrderParameter) {ε : ℝ} (hε : ε∈Icc (0 : ℝ) 1) (t : ℝ) :
-    (γ.mix η ε hε).coeff t-γ.coeff t=ε*(η.coeff t-γ.coeff t) := by dsimp [OrderParameter.mix]; ring
-
-lemma OrderParameter.mix_cutoff_sub (γ η : OrderParameter) {ε : ℝ} (hε : ε∈Icc (0 : ℝ) 1) (t : ℝ) :
-    (γ.mix η ε hε).cutoff t-γ.cutoff t=ε*(η.cutoff t-γ.cutoff t) := by
-  by_cases ht : t∈Ico (0 : ℝ) 1
-  · simp only [OrderParameter.cutoff,indicator_of_mem ht]
-    exact γ.mix_coeff_sub η hε t
-  · simp only [OrderParameter.cutoff,indicator_of_notMem ht,sub_self,mul_zero]
-
-lemma OrderParameter.mix_difference_integral (γ η : OrderParameter) {ε : ℝ}
-    (hε : ε∈Icc (0 : ℝ) 1) (a b : ℝ) :
-    (∫ t in a..b,|(γ.mix η ε hε).cutoff t-γ.cutoff t|)=
-      ε*(∫ t in a..b,|η.cutoff t-γ.cutoff t|) := by
-  simp_rw [γ.mix_cutoff_sub η hε,abs_mul,abs_of_nonneg hε.1]
-  exact intervalIntegral.integral_const_mul ε _
-
-lemma OrderParameter.mix_strip_difference_bound (γ η : OrderParameter) {ε T : ℝ}
-    (hε : ε∈Icc (0 : ℝ) 1) (hT : T∈Ico (0 : ℝ) 1) {t : ℝ} (ht : t∈Icc (0 : ℝ) T) :
-    |(γ.mix η ε hε).coeff t-γ.coeff t|≤ε*(η.coeff T+γ.coeff T) := by
-  rw [γ.mix_coeff_sub η hε,abs_mul,abs_of_nonneg hε.1]
-  apply mul_le_mul_of_nonneg_left _ hε.1
-  have ht' : t∈Ico (0 : ℝ) 1 := ⟨ht.1,ht.2.trans_lt hT.2⟩
-  calc |η.coeff t-γ.coeff t|≤|η.coeff t|+|γ.coeff t| := abs_sub _ _
-    _=η.coeff t+γ.coeff t := by rw [abs_of_nonneg (η.nonneg t ht'),abs_of_nonneg (γ.nonneg t ht')]
-    _≤η.coeff T+γ.coeff T := add_le_add (η.monotone ht' hT ht.2) (γ.monotone ht' hT ht.2)
-
-lemma OrderParameter.weighted_intervalIntegrable (γ : OrderParameter) :
-    IntervalIntegrable (fun t ↦ t*γ.coeff t) volume 0 1 :=
-  γ.intervalIntegrable.continuousOn_mul continuousOn_id
-
-lemma IsMinimizer.mix_value_inequality {W : BrownianSpace} {γ : OrderParameter}
-    (hγ : IsMinimizer W γ) (η : OrderParameter) {ε : ℝ} (hε : ε∈Icc (0 : ℝ) 1) :
-    (ε/2)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤
-      phi W (γ.mix η ε hε) 0 0-phi W γ 0 0 := by
-  have hh := hγ (γ.mix η ε hε)
-  have hi : (∫ t in (0 : ℝ)..1,t*(γ.mix η ε hε).coeff t)-
-      (∫ t in (0 : ℝ)..1,t*γ.coeff t)=ε*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t)) := by
-    rw [←intervalIntegral.integral_sub (γ.mix η ε hε).weighted_intervalIntegrable γ.weighted_intervalIntegrable,
-      ←intervalIntegral.integral_const_mul]
-    apply intervalIntegral.integral_congr
-    intro t ht
-    dsimp [OrderParameter.mix]
-    ring
-  dsimp only [parisi] at hh
-  linarith
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def gradientMoment (W : BrownianSpace) (γ : OrderParameter)
-    (X : ℝ → W.Ω → ℝ) (t : ℝ) : ℝ := ∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ
-
-lemma gradientMoment_nonneg (W : BrownianSpace) (γ : OrderParameter)
-    (X : ℝ → W.Ω → ℝ) (t : ℝ) : 0≤gradientMoment W γ X t :=
-  integral_nonneg (fun _ ↦ sq_nonneg _)
-
-lemma gradientMoment_bound (W : BrownianSpace) (γ : OrderParameter)
-    (X : ℝ → W.Ω → ℝ) {t : ℝ} (ht : t∈Icc (0 : ℝ) 1) :
-    |gradientMoment W γ X t|≤1 := by
-  simpa only [gradientMoment,Real.norm_eq_abs,probReal_univ,mul_one] using
-    (norm_integral_le_of_norm_le_const (μ := W.μ) (C := (1 : ℝ))
-      (f := fun ω ↦ (gradient W γ t (X t ω))^2)
-      (Eventually.of_forall (fun ω ↦ by
-        rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
-        exact (sq_le_one_iff_abs_le_one _).mpr (gradient_bound W γ ht _))))
-
-lemma IsDiffusion.gradientMoment_continuous {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
-    ContinuousOn (gradientMoment W γ X) (Ico (0 : ℝ) 1) := by
-  intro t ht
-  obtain ⟨T,htT,hT1⟩ := exists_between ht.2
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨ht.1.trans htT.le,hT1⟩
-  have hc := mean_derivative_square_continuous (by norm_num : (0 : ℝ)≤1) hLu
-    (fun s hs ↦ (hV.smooth s hs).continuous_deriv (by norm_num)) hV.bounded hu
-    (fun s hs ↦ (hX.measurable ⟨hs.1,hs.2.trans hT1.le⟩).aestronglyMeasurable)
-    (hX.2.mono (fun _ hω ↦ hω.1.mono (Icc_subset_Icc le_rfl hT1.le)))
-  apply (hc t ⟨ht.1,htT.le⟩).mono_of_mem_nhdsWithin
-  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds htT)] with s hs hsT
-  exact ⟨hs.1,hsT.le⟩
-
-lemma IsDiffusion.weightedMoment_integrable {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter) :
-    IntervalIntegrable (fun t ↦ η.coeff t*gradientMoment W γ X t) volume 0 1 := by
-  apply (intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num)).mpr
-  have hm := hX.gradientMoment_continuous.aestronglyMeasurable (μ := volume) measurableSet_Ico
-  rw [restrict_Ico_eq_restrict_Ioc] at hm
-  exact η.integrable.mul_bdd hm (by
-    filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
-    simpa only [Real.norm_eq_abs] using gradientMoment_bound W γ X ⟨ht.1.le,ht.2⟩)
-
-lemma IsDiffusion.weightedMean_integrable {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter) :
-    IntervalIntegrable (fun t ↦ η.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ)) volume 0 1 := by
-  have hi := (hX.weightedMoment_integrable η).const_mul (1/2 : ℝ)
-  simpa only [integral_const_mul,mul_left_comm,gradientMoment] using hi
-
-lemma IsDiffusion.mix_moment_integral {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter)
-    {ε T : ℝ} (hε : ε∈Icc (0 : ℝ) 1) (hT : T∈Icc (0 : ℝ) 1) :
-    (∫ t in (0 : ℝ)..T,(γ.mix η ε hε).coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))-
-    (∫ t in (0 : ℝ)..T,γ.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))=
-    (ε/2)*(∫ t in (0 : ℝ)..T,(η.coeff t-γ.coeff t)*gradientMoment W γ X t) := by
-  have ht : uIcc (0 : ℝ) T⊆uIcc (0 : ℝ) 1 := by
-    simpa only [uIcc_of_le hT.1,uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using Icc_subset_Icc le_rfl hT.2
-  rw [←intervalIntegral.integral_sub ((hX.weightedMean_integrable _).mono_set ht)
-    ((hX.weightedMean_integrable _).mono_set ht),←intervalIntegral.integral_const_mul]
-  apply intervalIntegral.integral_congr
-  intro t ht
-  simp only [integral_const_mul]
-  dsimp only [OrderParameter.mix,gradientMoment]
-  ring
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Control.Regularity
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def OrderParameter.mix (γ η : OrderParameter) (ε : ℝ)
+    (hε : ε∈Icc (0 : ℝ) 1) : OrderParameter where
+  coeff t := (1-ε)*γ.coeff t+ε*η.coeff t
+  nonneg t ht := add_nonneg (mul_nonneg (sub_nonneg.mpr hε.2) (γ.nonneg t ht))
+    (mul_nonneg hε.1 (η.nonneg t ht))
+  monotone _s hs _t ht hst := add_le_add
+    (mul_le_mul_of_nonneg_left (γ.monotone hs ht hst) (sub_nonneg.mpr hε.2))
+    (mul_le_mul_of_nonneg_left (η.monotone hs ht hst) hε.1)
+  rightContinuous t ht := ((γ.rightContinuous t ht).const_mul _).add ((η.rightContinuous t ht).const_mul _)
+  integrable := (γ.integrable.const_mul _).add (η.integrable.const_mul _)
+
+lemma OrderParameter.mix_coeff_sub (γ η : OrderParameter) {ε : ℝ} (hε : ε∈Icc (0 : ℝ) 1) (t : ℝ) :
+    (γ.mix η ε hε).coeff t-γ.coeff t=ε*(η.coeff t-γ.coeff t) := by dsimp [OrderParameter.mix]; ring
+
+lemma OrderParameter.mix_cutoff_sub (γ η : OrderParameter) {ε : ℝ} (hε : ε∈Icc (0 : ℝ) 1) (t : ℝ) :
+    (γ.mix η ε hε).cutoff t-γ.cutoff t=ε*(η.cutoff t-γ.cutoff t) := by
+  by_cases ht : t∈Ico (0 : ℝ) 1
+  · simp only [OrderParameter.cutoff,indicator_of_mem ht]
+    exact γ.mix_coeff_sub η hε t
+  · simp only [OrderParameter.cutoff,indicator_of_notMem ht,sub_self,mul_zero]
+
+lemma OrderParameter.mix_difference_integral (γ η : OrderParameter) {ε : ℝ}
+    (hε : ε∈Icc (0 : ℝ) 1) (a b : ℝ) :
+    (∫ t in a..b,|(γ.mix η ε hε).cutoff t-γ.cutoff t|)=
+      ε*(∫ t in a..b,|η.cutoff t-γ.cutoff t|) := by
+  simp_rw [γ.mix_cutoff_sub η hε,abs_mul,abs_of_nonneg hε.1]
+  exact intervalIntegral.integral_const_mul ε _
+
+lemma OrderParameter.mix_strip_difference_bound (γ η : OrderParameter) {ε T : ℝ}
+    (hε : ε∈Icc (0 : ℝ) 1) (hT : T∈Ico (0 : ℝ) 1) {t : ℝ} (ht : t∈Icc (0 : ℝ) T) :
+    |(γ.mix η ε hε).coeff t-γ.coeff t|≤ε*(η.coeff T+γ.coeff T) := by
+  rw [γ.mix_coeff_sub η hε,abs_mul,abs_of_nonneg hε.1]
+  apply mul_le_mul_of_nonneg_left _ hε.1
+  have ht' : t∈Ico (0 : ℝ) 1 := ⟨ht.1,ht.2.trans_lt hT.2⟩
+  calc |η.coeff t-γ.coeff t|≤|η.coeff t|+|γ.coeff t| := abs_sub _ _
+    _=η.coeff t+γ.coeff t := by rw [abs_of_nonneg (η.nonneg t ht'),abs_of_nonneg (γ.nonneg t ht')]
+    _≤η.coeff T+γ.coeff T := add_le_add (η.monotone ht' hT ht.2) (γ.monotone ht' hT ht.2)
+
+lemma OrderParameter.weighted_intervalIntegrable (γ : OrderParameter) :
+    IntervalIntegrable (fun t ↦ t*γ.coeff t) volume 0 1 :=
+  γ.intervalIntegrable.continuousOn_mul continuousOn_id
+
+lemma IsMinimizer.mix_value_inequality {W : BrownianSpace} {γ : OrderParameter}
+    (hγ : IsMinimizer W γ) (η : OrderParameter) {ε : ℝ} (hε : ε∈Icc (0 : ℝ) 1) :
+    (ε/2)*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t))≤
+      phi W (γ.mix η ε hε) 0 0-phi W γ 0 0 := by
+  have hh := hγ (γ.mix η ε hε)
+  have hi : (∫ t in (0 : ℝ)..1,t*(γ.mix η ε hε).coeff t)-
+      (∫ t in (0 : ℝ)..1,t*γ.coeff t)=ε*(∫ t in (0 : ℝ)..1,t*(η.coeff t-γ.coeff t)) := by
+    rw [←intervalIntegral.integral_sub (γ.mix η ε hε).weighted_intervalIntegrable γ.weighted_intervalIntegrable,
+      ←intervalIntegral.integral_const_mul]
+    apply intervalIntegral.integral_congr
+    intro t ht
+    dsimp [OrderParameter.mix]
+    ring
+  dsimp only [parisi] at hh
+  linarith
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def gradientMoment (W : BrownianSpace) (γ : OrderParameter)
+    (X : ℝ → W.Ω → ℝ) (t : ℝ) : ℝ := ∫ ω,(gradient W γ t (X t ω))^2 ∂W.μ
+
+lemma gradientMoment_nonneg (W : BrownianSpace) (γ : OrderParameter)
+    (X : ℝ → W.Ω → ℝ) (t : ℝ) : 0≤gradientMoment W γ X t :=
+  integral_nonneg (fun _ ↦ sq_nonneg _)
+
+lemma gradientMoment_bound (W : BrownianSpace) (γ : OrderParameter)
+    (X : ℝ → W.Ω → ℝ) {t : ℝ} (ht : t∈Icc (0 : ℝ) 1) :
+    |gradientMoment W γ X t|≤1 := by
+  simpa only [gradientMoment,Real.norm_eq_abs,probReal_univ,mul_one] using
+    (norm_integral_le_of_norm_le_const (μ := W.μ) (C := (1 : ℝ))
+      (f := fun ω ↦ (gradient W γ t (X t ω))^2)
+      (Eventually.of_forall (fun ω ↦ by
+        rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+        exact (sq_le_one_iff_abs_le_one _).mpr (gradient_bound W γ ht _))))
+
+lemma IsDiffusion.gradientMoment_continuous {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) :
+    ContinuousOn (gradientMoment W γ X) (Ico (0 : ℝ) 1) := by
+  intro t ht
+  obtain ⟨T,htT,hT1⟩ := exists_between ht.2
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨ht.1.trans htT.le,hT1⟩
+  have hc := mean_derivative_square_continuous (by norm_num : (0 : ℝ)≤1) hLu
+    (fun s hs ↦ (hV.smooth s hs).continuous_deriv (by norm_num)) hV.bounded hu
+    (fun s hs ↦ (hX.measurable ⟨hs.1,hs.2.trans hT1.le⟩).aestronglyMeasurable)
+    (hX.2.mono (fun _ hω ↦ hω.1.mono (Icc_subset_Icc le_rfl hT1.le)))
+  apply (hc t ⟨ht.1,htT.le⟩).mono_of_mem_nhdsWithin
+  filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds htT)] with s hs hsT
+  exact ⟨hs.1,hsT.le⟩
+
+lemma IsDiffusion.weightedMoment_integrable {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter) :
+    IntervalIntegrable (fun t ↦ η.coeff t*gradientMoment W γ X t) volume 0 1 := by
+  apply (intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num)).mpr
+  have hm := hX.gradientMoment_continuous.aestronglyMeasurable (μ := volume) measurableSet_Ico
+  rw [restrict_Ico_eq_restrict_Ioc] at hm
+  exact η.integrable.mul_bdd hm (by
+    filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
+    simpa only [Real.norm_eq_abs] using gradientMoment_bound W γ X ⟨ht.1.le,ht.2⟩)
+
+lemma IsDiffusion.weightedMean_integrable {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter) :
+    IntervalIntegrable (fun t ↦ η.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ)) volume 0 1 := by
+  have hi := (hX.weightedMoment_integrable η).const_mul (1/2 : ℝ)
+  simpa only [integral_const_mul,mul_left_comm,gradientMoment] using hi
+
+lemma IsDiffusion.mix_moment_integral {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter)
+    {ε T : ℝ} (hε : ε∈Icc (0 : ℝ) 1) (hT : T∈Icc (0 : ℝ) 1) :
+    (∫ t in (0 : ℝ)..T,(γ.mix η ε hε).coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))-
+    (∫ t in (0 : ℝ)..T,γ.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))=
+    (ε/2)*(∫ t in (0 : ℝ)..T,(η.coeff t-γ.coeff t)*gradientMoment W γ X t) := by
+  have ht : uIcc (0 : ℝ) T⊆uIcc (0 : ℝ) 1 := by
+    simpa only [uIcc_of_le hT.1,uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using Icc_subset_Icc le_rfl hT.2
+  rw [←intervalIntegral.integral_sub ((hX.weightedMean_integrable _).mono_set ht)
+    ((hX.weightedMean_integrable _).mono_set ht),←intervalIntegral.integral_const_mul]
+  apply intervalIntegral.integral_congr
+  intro t ht
+  simp only [integral_const_mul]
+  dsimp only [OrderParameter.mix,gradientMoment]
+  ring
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SKValue/Variation/WeightedMoments.lean b/lean/OAI/Probability/SKValue/Variation/WeightedMoments.lean
index 26f44e98fe8d099beee0957f3654827dafc4302d..1daeb2e96fe3620a4c1867bcf082f88e545f6732 100644
--- a/lean/OAI/Probability/SKValue/Variation/WeightedMoments.lean
+++ b/lean/OAI/Probability/SKValue/Variation/WeightedMoments.lean
@@ -1,130 +1,130 @@
-import Mathlib
-import OAI.Probability.SKValue.Control.Regularity
-
-namespace OAI
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-noncomputable def weightedEulerMoment (W : BrownianSpace) (γ η : OrderParameter)
-    (T : ℝ) (N : ℕ) : ℝ :=
-  stepSize T N*∑ j : Fin N,η.coeff (meshTime T N j)*
-    (∫ z,(1/2 : ℝ)*(gradient W γ (meshTime T N j)
-      (euler T N γ.coeff (gradient W γ) z j))^2 ∂gaussianProduct (Fin (N+1)))
-
-lemma IsDiffusion.weightedEulerMoment_tendsto {W : BrownianSpace} {γ : OrderParameter}
-    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter)
-    {T : ℝ} (hT : 0<T) (hT1 : T<1) :
-    Tendsto (weightedEulerMoment W γ η T) atTop
-      (𝓝 (∫ t in (0 : ℝ)..T,η.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))) := by
-  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨hT.le,hT1⟩
-  let E (N : ℕ) := ∫ ω,(meshMaxError T X (coupledEuler T γ.coeff (gradient W γ) W.B) N ω)^2 ∂W.μ
-  let f (t : ℝ) := ∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ
-  have hXM (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : MemLp (X t) 2 W.μ := hX.memLp ⟨ht.1,ht.2.trans hT1.le⟩ (by norm_num)
-  have hpaths := hX.strip_paths hT.le hT1 hLu (fun t ht s hs x y ↦ hu s hs t ht y x)
-  have hEl : Tendsto E atTop (𝓝 0) := coupled_euler_L2_convergence W.brownian.toIsPreBrownianReal
-    hT hG hLu hu (fun t ht ↦ (hXM t ht).aestronglyMeasurable) hpaths
-  have hEs : Tendsto (fun N ↦ Real.sqrt (E N)) atTop (𝓝 0) := by
-    simpa only [Real.sqrt_zero,Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp hEl
-  have hf : ContinuousOn f (Icc (0 : ℝ) T) := by
-    have hc := (mean_derivative_square_continuous (by norm_num : (0 : ℝ)≤1) hLu
-      (fun t ht ↦ (hG.smooth t ht).continuous) hG.bounded hu
-      (fun t ht ↦ (hXM t ht).aestronglyMeasurable) (hpaths.mono (fun _ hω ↦ hω.1))).const_mul (1/2 : ℝ)
-    simpa only [f,gradient,integral_const_mul] using hc
-  have hfB (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : |f t|≤Kv := by
-    simpa only [f,Real.norm_eq_abs,probReal_univ,mul_one] using
-      (norm_integral_le_of_norm_le_const (μ := W.μ) (C := Kv)
-        (f := fun ω ↦ (1/2 : ℝ)*(gradient W γ t (X t ω))^2)
-        (Eventually.of_forall (fun ω ↦ by
-          simpa only [Real.norm_eq_abs,gradient] using hV.product_bound t ht (X t ω))))
-  apply perturbed_monotone_weighted_grid_tendsto (r := fun N ↦ Lv*Real.sqrt (E N)) hT hV.K_nonneg
-    (η.monotone.mono (fun t ht ↦ ⟨ht.1,ht.2.trans_lt hT1⟩)) (η.nonneg 0 (by norm_num)) hf hfB
-    (by simpa only [mul_zero] using hEs.const_mul Lv)
-  filter_upwards [eventually_gt_atTop 0] with N hN j
-  have hh := coupled_lipschitz_expectation_bound W.brownian.toIsPreBrownianReal hT hG
-    (fun t ht ↦ hXM t ht) (hpaths.mono (fun _ hω ↦ hω.2.2)) hN j.isLt.le
-    (hV.product_lipschitzWith (mesh_time_mem hT.le hN j.isLt.le))
-  simpa only [Real.coe_toNNReal _ hV.L_nonneg,E,f,gradient] using hh
-
-end SKValue
-
-end
-
-section
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal BigOperators
-namespace SKValue
-
-lemma phi_difference_bound (W : BrownianSpace) (γ β : OrderParameter)
-    {t : ℝ} (ht : t∈Icc (0 : ℝ) 1) (x : ℝ) :
-    |phi W β t x-phi W γ t x|≤(3/2 : ℝ)*(∫ s in t..1,|β.cutoff s-γ.cutoff s|) := by
-  have hψ : LipschitzWith 1 (abs : ℝ → ℝ) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro a b
-    simpa only [Real.dist_eq,NNReal.coe_one,one_mul] using abs_abs_sub_abs_le_abs_sub a b
-  have hi (η : OrderParameter) : IntervalIntegrable (fun s ↦ η.cutoff (t+s)) volume 0 (1-t) := by
-    simpa only [sub_self] using (η.cutoff_integrable.intervalIntegrable (a := t) (b := 1)).comp_add_left t
-  have he := elapsedValue_difference_bound (f := shiftedFiltration W t)
-    (brownian_increment_memLp W t) hψ hψ (fun y ↦ by simp : ∀ y : ℝ,|abs y-abs y|≤(0 : ℝ))
-    (β.cutoff_measurable.comp (measurable_const.add measurable_id))
-    (γ.cutoff_measurable.comp (measurable_const.add measurable_id)) (hi β) (hi γ)
-    (sub_nonneg.mpr ht.2) (x := x)
-  simp only [Function.comp_def,Pi.add_apply, id_eq] at he
-  rw [←phi_eq_elapsedValue W β ht x,←phi_eq_elapsedValue W γ ht x] at he
-  have hshift := intervalIntegral.integral_comp_add_left (fun s ↦ |β.cutoff s-γ.cutoff s|)
-    (a := (0 : ℝ)) (b := 1-t) t
-  simp only [add_zero,add_sub_cancel] at hshift
-  rw [hshift] at he
-  simpa only [zero_add] using he
-
-lemma convex_derivative_uniform_comparison {f g : ℝ → ℝ} {L ε h : ℝ}
-    (hf : ConvexOn ℝ univ f) (hg : ConvexOn ℝ univ g)
-    (hfd : Differentiable ℝ f) (hgd : Differentiable ℝ g)
-    (hL : ∀ x y,|deriv f x-deriv f y|≤L*|x-y|)
-    (he : ∀ x,|g x-f x|≤ε) (hh : 0<h) (x : ℝ) :
-    |deriv g x-deriv f x|≤L*h+2*ε/h := by
-  have hp1 := hg.deriv_le_slope (mem_univ x) (mem_univ (x+h)) (by linarith) (hgd x)
-  have hp2 := hf.slope_le_deriv (mem_univ x) (mem_univ (x+h)) (by linarith) (hfd (x+h))
-  have hm1 := hg.slope_le_deriv (mem_univ (x-h)) (mem_univ x) (by linarith) (hgd x)
-  have hm2 := hf.deriv_le_slope (mem_univ (x-h)) (mem_univ x) (by linarith) (hfd (x-h))
-  rw [slope_def_field,show x+h-x=h by ring] at hp1 hp2
-  rw [slope_def_field,show x-(x-h)=h by ring] at hm1 hm2
-  have hp1' := (le_div_iff₀ hh).mp hp1
-  have hp2' := (div_le_iff₀ hh).mp hp2
-  have hm1' := (div_le_iff₀ hh).mp hm1
-  have hm2' := (le_div_iff₀ hh).mp hm2
-  have hpL := (le_abs_self _).trans (hL (x+h) x)
-  have hmL := (neg_le_abs _).trans (hL (x-h) x)
-  rw [show x+h-x=h by ring,abs_of_pos hh] at hpL
-  rw [show x-h-x= -h by ring,abs_neg,abs_of_pos hh] at hmL
-  have hpLm := mul_le_mul_of_nonneg_right hpL hh.le
-  have hmLm := mul_le_mul_of_nonneg_right hmL hh.le
-  have hx := (abs_le.mp (he x))
-  have hxp := (abs_le.mp (he (x+h)))
-  have hxm := (abs_le.mp (he (x-h)))
-  have hcancel : (2*ε/h)*h=2*ε := div_mul_cancel₀ _ hh.ne'
-  apply abs_le.mpr
-  constructor <;> apply (mul_le_mul_iff_left₀ hh).mp <;> nlinarith
-
-lemma phi_gradient_difference_bound (W : BrownianSpace) (γ β : OrderParameter)
-    {T L ε h : ℝ} (hT : T∈Ico (0 : ℝ) 1)
-    (hL : ∀ t∈Icc (0 : ℝ) T,∀ x y,|gradient W γ t x-gradient W γ t y|≤L*|x-y|)
-    (he : ∀ t∈Icc (0 : ℝ) T,∀ x,|phi W β t x-phi W γ t x|≤ε)
-    (hh : 0<h) {t : ℝ} (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
-    |gradient W β t x-gradient W γ t x|≤L*h+2*ε/h := by
-  have ht1 : t∈Ico (0 : ℝ) 1 := ⟨ht.1,ht.2.trans_lt hT.2⟩
-  exact convex_derivative_uniform_comparison (phi_convex W γ ⟨ht1.1,ht1.2.le⟩)
-    (phi_convex W β ⟨ht1.1,ht1.2.le⟩)
-    ((phi_smoothTerminal W γ ht1).smooth.differentiable (by simp))
-    ((phi_smoothTerminal W β ht1).smooth.differentiable (by simp))
-    (hL t ht) (he t ht) hh x
-
-end SKValue
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SKValue.Control.Regularity
+
+namespace OAI
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+noncomputable def weightedEulerMoment (W : BrownianSpace) (γ η : OrderParameter)
+    (T : ℝ) (N : ℕ) : ℝ :=
+  stepSize T N*∑ j : Fin N,η.coeff (meshTime T N j)*
+    (∫ z,(1/2 : ℝ)*(gradient W γ (meshTime T N j)
+      (euler T N γ.coeff (gradient W γ) z j))^2 ∂gaussianProduct (Fin (N+1)))
+
+lemma IsDiffusion.weightedEulerMoment_tendsto {W : BrownianSpace} {γ : OrderParameter}
+    {X : ℝ → W.Ω → ℝ} (hX : IsDiffusion W γ X) (η : OrderParameter)
+    {T : ℝ} (hT : 0<T) (hT1 : T<1) :
+    Tendsto (weightedEulerMoment W γ η T) atTop
+      (𝓝 (∫ t in (0 : ℝ)..T,η.coeff t*(∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ))) := by
+  obtain ⟨Kv,Lv,K,L,D,Lu,La,hV,hG,hD,hLu,hLa,hDb,hu,ha⟩ := sourceStripRegularity W γ T ⟨hT.le,hT1⟩
+  let E (N : ℕ) := ∫ ω,(meshMaxError T X (coupledEuler T γ.coeff (gradient W γ) W.B) N ω)^2 ∂W.μ
+  let f (t : ℝ) := ∫ ω,(1/2 : ℝ)*(gradient W γ t (X t ω))^2 ∂W.μ
+  have hXM (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : MemLp (X t) 2 W.μ := hX.memLp ⟨ht.1,ht.2.trans hT1.le⟩ (by norm_num)
+  have hpaths := hX.strip_paths hT.le hT1 hLu (fun t ht s hs x y ↦ hu s hs t ht y x)
+  have hEl : Tendsto E atTop (𝓝 0) := coupled_euler_L2_convergence W.brownian.toIsPreBrownianReal
+    hT hG hLu hu (fun t ht ↦ (hXM t ht).aestronglyMeasurable) hpaths
+  have hEs : Tendsto (fun N ↦ Real.sqrt (E N)) atTop (𝓝 0) := by
+    simpa only [Real.sqrt_zero,Function.comp_def] using (Real.continuous_sqrt.tendsto 0).comp hEl
+  have hf : ContinuousOn f (Icc (0 : ℝ) T) := by
+    have hc := (mean_derivative_square_continuous (by norm_num : (0 : ℝ)≤1) hLu
+      (fun t ht ↦ (hG.smooth t ht).continuous) hG.bounded hu
+      (fun t ht ↦ (hXM t ht).aestronglyMeasurable) (hpaths.mono (fun _ hω ↦ hω.1))).const_mul (1/2 : ℝ)
+    simpa only [f,gradient,integral_const_mul] using hc
+  have hfB (t : ℝ) (ht : t∈Icc (0 : ℝ) T) : |f t|≤Kv := by
+    simpa only [f,Real.norm_eq_abs,probReal_univ,mul_one] using
+      (norm_integral_le_of_norm_le_const (μ := W.μ) (C := Kv)
+        (f := fun ω ↦ (1/2 : ℝ)*(gradient W γ t (X t ω))^2)
+        (Eventually.of_forall (fun ω ↦ by
+          simpa only [Real.norm_eq_abs,gradient] using hV.product_bound t ht (X t ω))))
+  apply perturbed_monotone_weighted_grid_tendsto (r := fun N ↦ Lv*Real.sqrt (E N)) hT hV.K_nonneg
+    (η.monotone.mono (fun t ht ↦ ⟨ht.1,ht.2.trans_lt hT1⟩)) (η.nonneg 0 (by norm_num)) hf hfB
+    (by simpa only [mul_zero] using hEs.const_mul Lv)
+  filter_upwards [eventually_gt_atTop 0] with N hN j
+  have hh := coupled_lipschitz_expectation_bound W.brownian.toIsPreBrownianReal hT hG
+    (fun t ht ↦ hXM t ht) (hpaths.mono (fun _ hω ↦ hω.2.2)) hN j.isLt.le
+    (hV.product_lipschitzWith (mesh_time_mem hT.le hN j.isLt.le))
+  simpa only [Real.coe_toNNReal _ hV.L_nonneg,E,f,gradient] using hh
+
+end SKValue
+
+end
+
+section
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal BigOperators
+namespace SKValue
+
+lemma phi_difference_bound (W : BrownianSpace) (γ β : OrderParameter)
+    {t : ℝ} (ht : t∈Icc (0 : ℝ) 1) (x : ℝ) :
+    |phi W β t x-phi W γ t x|≤(3/2 : ℝ)*(∫ s in t..1,|β.cutoff s-γ.cutoff s|) := by
+  have hψ : LipschitzWith 1 (abs : ℝ → ℝ) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro a b
+    simpa only [Real.dist_eq,NNReal.coe_one,one_mul] using abs_abs_sub_abs_le_abs_sub a b
+  have hi (η : OrderParameter) : IntervalIntegrable (fun s ↦ η.cutoff (t+s)) volume 0 (1-t) := by
+    simpa only [sub_self] using (η.cutoff_integrable.intervalIntegrable (a := t) (b := 1)).comp_add_left t
+  have he := elapsedValue_difference_bound (f := shiftedFiltration W t)
+    (brownian_increment_memLp W t) hψ hψ (fun y ↦ by simp : ∀ y : ℝ,|abs y-abs y|≤(0 : ℝ))
+    (β.cutoff_measurable.comp (measurable_const.add measurable_id))
+    (γ.cutoff_measurable.comp (measurable_const.add measurable_id)) (hi β) (hi γ)
+    (sub_nonneg.mpr ht.2) (x := x)
+  simp only [Function.comp_def,Pi.add_apply, id_eq] at he
+  rw [←phi_eq_elapsedValue W β ht x,←phi_eq_elapsedValue W γ ht x] at he
+  have hshift := intervalIntegral.integral_comp_add_left (fun s ↦ |β.cutoff s-γ.cutoff s|)
+    (a := (0 : ℝ)) (b := 1-t) t
+  simp only [add_zero,add_sub_cancel] at hshift
+  rw [hshift] at he
+  simpa only [zero_add] using he
+
+lemma convex_derivative_uniform_comparison {f g : ℝ → ℝ} {L ε h : ℝ}
+    (hf : ConvexOn ℝ univ f) (hg : ConvexOn ℝ univ g)
+    (hfd : Differentiable ℝ f) (hgd : Differentiable ℝ g)
+    (hL : ∀ x y,|deriv f x-deriv f y|≤L*|x-y|)
+    (he : ∀ x,|g x-f x|≤ε) (hh : 0<h) (x : ℝ) :
+    |deriv g x-deriv f x|≤L*h+2*ε/h := by
+  have hp1 := hg.deriv_le_slope (mem_univ x) (mem_univ (x+h)) (by linarith) (hgd x)
+  have hp2 := hf.slope_le_deriv (mem_univ x) (mem_univ (x+h)) (by linarith) (hfd (x+h))
+  have hm1 := hg.slope_le_deriv (mem_univ (x-h)) (mem_univ x) (by linarith) (hgd x)
+  have hm2 := hf.deriv_le_slope (mem_univ (x-h)) (mem_univ x) (by linarith) (hfd (x-h))
+  rw [slope_def_field,show x+h-x=h by ring] at hp1 hp2
+  rw [slope_def_field,show x-(x-h)=h by ring] at hm1 hm2
+  have hp1' := (le_div_iff₀ hh).mp hp1
+  have hp2' := (div_le_iff₀ hh).mp hp2
+  have hm1' := (div_le_iff₀ hh).mp hm1
+  have hm2' := (le_div_iff₀ hh).mp hm2
+  have hpL := (le_abs_self _).trans (hL (x+h) x)
+  have hmL := (neg_le_abs _).trans (hL (x-h) x)
+  rw [show x+h-x=h by ring,abs_of_pos hh] at hpL
+  rw [show x-h-x= -h by ring,abs_neg,abs_of_pos hh] at hmL
+  have hpLm := mul_le_mul_of_nonneg_right hpL hh.le
+  have hmLm := mul_le_mul_of_nonneg_right hmL hh.le
+  have hx := (abs_le.mp (he x))
+  have hxp := (abs_le.mp (he (x+h)))
+  have hxm := (abs_le.mp (he (x-h)))
+  have hcancel : (2*ε/h)*h=2*ε := div_mul_cancel₀ _ hh.ne'
+  apply abs_le.mpr
+  constructor <;> apply (mul_le_mul_iff_left₀ hh).mp <;> nlinarith
+
+lemma phi_gradient_difference_bound (W : BrownianSpace) (γ β : OrderParameter)
+    {T L ε h : ℝ} (hT : T∈Ico (0 : ℝ) 1)
+    (hL : ∀ t∈Icc (0 : ℝ) T,∀ x y,|gradient W γ t x-gradient W γ t y|≤L*|x-y|)
+    (he : ∀ t∈Icc (0 : ℝ) T,∀ x,|phi W β t x-phi W γ t x|≤ε)
+    (hh : 0<h) {t : ℝ} (ht : t∈Icc (0 : ℝ) T) (x : ℝ) :
+    |gradient W β t x-gradient W γ t x|≤L*h+2*ε/h := by
+  have ht1 : t∈Ico (0 : ℝ) 1 := ⟨ht.1,ht.2.trans_lt hT.2⟩
+  exact convex_derivative_uniform_comparison (phi_convex W γ ⟨ht1.1,ht1.2.le⟩)
+    (phi_convex W β ⟨ht1.1,ht1.2.le⟩)
+    ((phi_smoothTerminal W γ ht1).smooth.differentiable (by simp))
+    ((phi_smoothTerminal W β ht1).smooth.differentiable (by simp))
+    (hL t ht) (he t ht) hh x
+
+end SKValue
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/ActualRetainedKernel.lean b/lean/OAI/Probability/SLE/ActualRetainedKernel.lean
index c8104d669868c2f5f6727b4871a028584f3beb4f..e6b4bf8feea845f04ae0e4aac95c85420dcef7a1 100644
--- a/lean/OAI/Probability/SLE/ActualRetainedKernel.lean
+++ b/lean/OAI/Probability/SLE/ActualRetainedKernel.lean
@@ -1,763 +1,763 @@
-import Mathlib
-import OAI.Probability.SLE.HausdorffLimits
-
-namespace OAI
-
-/-! Actual Retained Kernel for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-variable (a : RegularSLEPresentation P κ γ)
-def sourceFiniteGreenKernel (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) : Kernel a.samples ℂ :=
-  (Kernel.const a.samples (volume.restrict (closedBall z₀ 1))).withDensity
-    (fun ω z => ENNReal.ofReal (if spatialHitTime (γ ω) z ((2:ℝ)^(-(N:ℝ))) ≤ T then
-      spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z ((2:ℝ)^(-(N:ℝ))) else 0))
-lemma sourceFiniteGreenDensity_measurable (r : ℝ) (T : ℝ≥0) :
-    Measurable (fun p : a.samples × ℂ => ENNReal.ofReal (if spatialHitTime (γ p.1) p.2 r ≤ T then
-      spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t p.1) (γ p.1) (a.G p.1) p.2 r else 0)) := by
-  have hm (t : ℝ≥0) : Measurable (fun ω : a.samples => γ ω t) :=
-    (a.trace_adapted t).mono (a.filtration.le t) le_rfl
-  have ht := (spatialHitTime_measurable (fun ω : a.samples => γ ω) a.trace_continuous hm r).comp measurable_swap
-  have hW := (a.spatialGreenWeight_measurable r).comp measurable_swap
-  exact (Measurable.ite (measurableSet_le ht measurable_const) hW measurable_const).ennreal_ofReal
-lemma sourceFiniteGreenKernel_apply (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples)
-    (S : Set ℂ) (hS : MeasurableSet S) :
-    a.sourceFiniteGreenKernel z₀ T N ω S = finiteSpatialGreenMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1 ∩ S) := by
-  rw [sourceFiniteGreenKernel,Kernel.withDensity_apply' _ (a.sourceFiniteGreenDensity_measurable _ T)]
-  simp only [Kernel.const_apply,finiteSpatialGreenMass,Measure.restrict_restrict hS,inter_comm]
-  rfl
-lemma sourceFiniteGreenKernel_univ (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
-    a.sourceFiniteGreenKernel z₀ T N ω univ = finiteSpatialGreenMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1) := by
-  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω univ MeasurableSet.univ,inter_univ]
-lemma sourceFiniteGreenKernel_K (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
-    a.sourceFiniteGreenKernel z₀ T N ω (closedBall z₀ 1) = finiteSpatialGreenMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1) := by
-  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω _ measurableSet_closedBall,inter_self]
-lemma sourceFiniteGreenKernel_supported (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
-    a.sourceFiniteGreenKernel z₀ T N ω (closedBall z₀ 1)ᶜ = 0 := by
-  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω _ measurableSet_closedBall.compl,inter_compl_self]
-  simp [finiteSpatialGreenMass]
-lemma sourceFiniteGreenKernel_finite (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz : 2 < z₀.im) (T : ℝ≥0) (N : ℕ) :
-    IsFiniteKernel (a.sourceFiniteGreenKernel z₀ T N) := by
-  let r := (2:ℝ)^(-(N:ℝ))
-  have hr : 0 < r := source_dyadic_side_pos N
-  have hr1 : r ≤ 1 := by
-    dsimp [r]
-    rw [source_dyadic_side_eq]
-    exact pow_le_one₀ (by norm_num) (by norm_num)
-  refine ⟨⟨ENNReal.ofReal (r^(-codimension κ))*volume (closedBall z₀ 1),
-    ENNReal.mul_lt_top ENNReal.ofReal_lt_top (measure_closedBall_lt_top),?_⟩⟩
-  intro ω
-  rw [a.sourceFiniteGreenKernel_univ]
-  change (∫⁻ z in closedBall z₀ 1, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
-    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0)) ≤ _
-  calc
-    _ ≤ ∫⁻ z in closedBall z₀ 1, ENNReal.ofReal (r^(-codimension κ)) := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem measurableSet_closedBall] with z hz'
-      apply ENNReal.ofReal_le_ofReal
-      split_ifs
-      · exact (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hκ hκ8 hr z
-          (hr1.trans (source_unit_ball_height hz hz').le) |>.2
-      · exact Real.rpow_nonneg hr.le _
-    _ = _ := by simp
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-variable (a : RegularSLEPresentation P κ γ)
-lemma sourceFiniteGreenKernel_carrier (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
-    a.sourceFiniteGreenKernel z₀ T N ω (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))ᶜ = 0 := by
-  have hm := (isClosed_cthickening : IsClosed (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))).measurableSet
-  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω _ hm.compl,finiteSpatialGreenMass]
-  apply lintegral_eq_zero_of_ae_eq_zero
-  filter_upwards [ae_restrict_mem (measurableSet_closedBall.inter hm.compl)] with z hz
-  have hn : ¬ spatialHitTime (γ ω) z ((2:ℝ)^(-(N:ℝ))) ≤ T := by
-    intro ht
-    obtain ⟨t,ht,hd⟩ := (spatialHitTime_le_iff (a.trace_continuous ω) z _ T).mp ht
-    exact hz.2 (mem_cthickening_of_dist_le z (γ ω t) _ _ ⟨t,ht,rfl⟩ (by simpa [dist_comm] using hd.le))
-  simp only [ite_eq_right hn, ENNReal.ofReal_zero,Pi.zero_apply]
-end RegularSLEPresentation
-lemma source_dyadic_tendsto : Tendsto (fun N : ℕ => (2:ℝ)^(-(N:ℝ))) atTop (𝓝 0) := by
-  simp_rw [source_dyadic_side_eq]
-  exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
-lemma source_dyadic_eventually_le {ε : ℝ} (hε : 0 < ε) :
-    ∃ n : ℕ, ∀ N ≥ n, (2:ℝ)^(-(N:ℝ)) ≤ ε := by
-  exact (eventually_atTop.mp (source_dyadic_tendsto.eventually (eventually_le_nhds hε)))
-lemma retainedMeasure_zero_of_zero {Ω : Type*} [MeasurableSpace Ω] (κ D : ℝ)
-    (ν : ℕ → Kernel Ω ℂ) (k₀ N : ℕ) (ω : Ω) {S : Set ℂ} (hS : ν N ω S = 0) :
-    retainedMeasure κ D ν k₀ N ω S = 0 := by
-  exact le_antisymm ((Measure.restrict_apply_le _ _).trans hS.le) bot_le
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-variable (a : RegularSLEPresentation P κ γ)
-lemma source_actual_retained_experiment [IsProbabilityMeasure P] (hκ : 4 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz₀ : 2 < z₀.im) :
-    ∃ (T : ℝ≥0) (D b p : ℝ) (k₀ : ℕ), 0 < T ∧ 1 ≤ D ∧ 0 < b ∧ 0 < p ∧
-      ENNReal.ofReal p ≤ (P.comap Subtype.val)
-        {ω : a.samples | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-          retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)} := by
-  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
-  have hk : 0 < κ := by linarith
-  obtain ⟨T,e₀,m₀,m₂,hT,he,hm₀,hm₂,hm⟩ := sourceRetainedSeed_proved κ hκ hκ8
-    Ω inferInstance P inferInstance γ a z₀ hz₀
-  obtain ⟨A,c,k₁,hA,hc,ht⟩ := sourceActualSquareTail_proved κ hκ hκ8
-    Ω inferInstance P inferInstance γ a z₀ hz₀
-  obtain ⟨k₂,hk₂⟩ := source_dyadic_eventually_le he
-  let kmin := max k₁ k₂
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  have hν (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
-  have hfirst (N : ℕ) (hN : kmin ≤ N) :
-      ENNReal.ofReal m₀ ≤ ∫⁻ ω, ν N ω (closedBall z₀ 1) ∂(P.comap Subtype.val) := by
-    simp only [ν,a.sourceFiniteGreenKernel_K]
-    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).1
-  have hsecond (N : ℕ) (hN : kmin ≤ N) :
-      (∫⁻ ω, (ν N ω (closedBall z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
-    simp only [ν,a.sourceFiniteGreenKernel_K]
-    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).2
-  have htail (N k : ℕ) (hkmin : kmin ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) (u : ℝ) (hu : 1 ≤ u) :
-      (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
-        (ν N ω (retainedDyadicSquare k ij)).toReal} ≤
-        ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ))) := by
-    simp only [ν,a.sourceFiniteGreenKernel_apply z₀ T N _ _ (retainedSquare_measurable k ij)]
-    exact ht T N k ((le_max_left _ _).trans hkmin) hkN ij u hu
-  obtain ⟨D,b,p,B,k₀,hD,hb,hp,hB,hk₀,hall,hlim⟩ := sourceRetainedConstruction a.samples
-    inferInstance (P.comap Subtype.val) inferInstance κ hk hκ8 ν hν
-    (closedBall z₀ 1) (isCompact_closedBall _ _) (fun N ω => a.sourceFiniteGreenKernel_supported z₀ T N ω)
-    A c m₀ m₂ hA hc hm₀ hm₂ kmin hfirst hsecond htail
-  exact ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-variable (a : RegularSLEPresentation P κ γ)
-lemma source_actual_retained_pathwise (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz₀ : 2 < z₀.im) (T : ℝ≥0) (D b : ℝ) (hD : 0 < D) (hb : 0 < b) (k₀ : ℕ)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    (ω : a.samples) (hω : ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-      retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)) :
-    0 < hausdorffGauge h (γ ω '' Icc 0 T) := by
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hκ hκ8 z₀ hz₀ T N
-  let μ (N : ℕ) : FiniteMeasure ℂ := ⟨retainedMeasure κ D ν k₀ N ω, by
-    dsimp [retainedMeasure]; infer_instance⟩
-  have hmass : ∃ᶠ N in atTop, (⟨b,hb.le⟩ : ℝ≥0) ≤ (μ N).mass := by
-    apply hω.mono
-    intro N hN
-    have hm : ENNReal.ofReal b ≤ (μ N : Measure ℂ) univ :=
-      hN.trans (measure_mono (subset_univ _))
-    change b ≤ ((μ N).mass : ℝ)
-    have ht := ENNReal.toReal_mono (measure_ne_top (μ N : Measure ℂ) univ) hm
-    simpa only [ENNReal.toReal_ofReal hb.le, ← FiniteMeasure.ennreal_mass,
-      ENNReal.coe_toReal] using ht
-  have hK (N : ℕ) : μ N (closedBall z₀ 1)ᶜ = 0 := by
-    apply (FiniteMeasure.null_iff_toMeasure_null _ _).mpr
-    exact retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_supported z₀ T N ω)
-  have hcarrier (N : ℕ) : (μ N : Measure ℂ) (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))ᶜ = 0 :=
-    retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_carrier z₀ T N ω)
-  have htest (N k : ℕ) (hk₀ : k₀ ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) :
-      (μ N (retainedDyadicSquare k ij) : ℝ≥0∞) ≤ ENNReal.ofReal (retainedThreshold κ D k) := by
-    rw [FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure]
-    exact retainedMeasure_square_bound κ D ν k₀ N ω k hk₀ hkN ij
-  exact sourceRetainedDyadic_proved κ hκ hκ8 h hh hf (closedBall z₀ 1) (γ ω '' Icc 0 T)
-    (isCompact_closedBall _ _) (isCompact_Icc.image (a.trace_continuous ω)).isClosed
-    μ ⟨b,hb.le⟩ hb hmass hK (fun N => (2:ℝ)^(-(N:ℝ))) source_dyadic_tendsto hcarrier D hD k₀ htest
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
- 
-
-def SourceActualRetainedLowerTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
-    0 < (P.comap Subtype.val) E ∧
-    ∀ ω ∈ E, 0 < hausdorffGauge h (γ ω '' Set.Icc 0 T)
-theorem sourceActualRetainedLower_proved : SourceActualRetainedLowerTarget := by
-  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
-  let : IsProbabilityMeasure P := hP
-  have hk : 0 < κ := by linarith
-  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment hκ hκ8 z₀ hz₀
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
-  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
-  have hE : MeasurableSet E := by
-    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
-        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
-      ext ω
-      simp [E,frequently_atTop]
-    rw [heq]
-    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
-      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
-  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
-  intro ω hω
-  exact a.source_actual_retained_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
-    k₀ h hh hf ω hω
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_gauge_lipschitz_image_le (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    {K : ℝ≥0} {f : ℂ → ℂ} {S : Set ℂ} (hLip : LipschitzOnWith K f S) :
-    hausdorffGauge h (f '' S) ≤
-      ENNReal.ofReal ((max (K : ℝ) 1)^dimension κ)*hausdorffGauge h S := by
-  let L : ℝ≥0 := max K 1
-  have hL : 1 ≤ (L : ℝ) := le_max_right _ _
-  have hLp : 0 < (L : ℝ) := zero_lt_one.trans_le hL
-  have hLC : LipschitzOnWith L f S := hLip.weaken (le_max_left _ _)
-  have himg (t : Set ℂ) : ediam (f '' (t ∩ S)) ≤ (L:ℝ≥0∞)*ediam t := by
-    simpa only [NNReal.coe_one,ENNReal.rpow_one] using hLC.holderOnWith.ediam_image_inter_le t
-  obtain ⟨a,ha,hscale⟩ := source_gauge_bounded_scaling κ hκ hκ8 h hh hf (L:ℝ) hL
-  let C : ℝ≥0∞ := ENNReal.ofReal ((L:ℝ)^dimension κ)
-  have hC0 : C ≠ 0 := (ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos hLp _)).ne'
-  have hCt : C ≠ ⊤ := ENNReal.ofReal_ne_top
-  change hausdorffGauge h (f '' S) ≤ C*hausdorffGauge h S
-  simp only [hausdorffGauge,Measure.mkMetric_apply,ENNReal.mul_iSup,
-    ENNReal.mul_iInf_of_ne hC0 hCt,←ENNReal.tsum_mul_left]
-  refine iSup_le fun R => iSup_le fun hR => ?_
-  have ht : Tendsto (fun d : ℝ≥0∞ => (L:ℝ≥0∞)*d) (𝓝 0) (𝓝 0) := by
-    simpa using (ENNReal.continuous_const_mul (a := (L:ℝ≥0∞)) ENNReal.coe_ne_top).tendsto 0
-  obtain ⟨δ,hδ,Hδ⟩ := ENNReal.nhds_zero_basis_Iic.eventually_iff.mp
-    (ht.eventually (gt_mem_nhds hR))
-  let ε := min δ (ENNReal.ofReal a)
-  have hε : 0 < ε := lt_min hδ (ENNReal.ofReal_pos.mpr ha)
-  have hεa : ε ≤ ENNReal.ofReal a := min_le_right _ _
-  have hεt : ε ≠ ⊤ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hεa
-  refine le_iSup₂_of_le ε hε <| iInf₂_mono' fun t hSt =>
-    ⟨fun n => f '' (t n ∩ S), ?_, iInf_mono' fun htε => ⟨?_,?_⟩⟩
-  · grw [←image_iUnion,←iUnion_inter,←hSt,inter_self]
-  · intro n
-    exact (himg (t n)).trans (Hδ ((htε n).trans (min_le_left _ _))).le
-  · apply ENNReal.tsum_le_tsum
-    intro n
-    simp only [iSup_le_iff,image_nonempty]
-    intro hn
-    simp only [Nonempty.mono ((t n).inter_subset_left) hn,ciSup_pos]
-    have hd : ediam (t n) ≠ ⊤ := ne_top_of_le_ne_top hεt (htε n)
-    have hda : (ediam (t n)).toReal ≤ a := by
-      simpa only [ENNReal.toReal_ofReal ha.le] using
-        ENNReal.toReal_mono ENNReal.ofReal_ne_top ((htε n).trans hεa)
-    have hprod : (L:ℝ≥0∞)*ediam (t n) ≠ ⊤ := ENNReal.mul_ne_top ENNReal.coe_ne_top hd
-    have him := ENNReal.toReal_mono hprod (himg (t n))
-    rw [ENNReal.toReal_mul,ENNReal.coe_toReal] at him
-    change ENNReal.ofReal (h (ediam (f '' (t n ∩ S))).toReal) ≤
-      ENNReal.ofReal ((L:ℝ)^dimension κ)*ENNReal.ofReal (h (ediam (t n)).toReal)
-    rw [←ENNReal.ofReal_mul (Real.rpow_nonneg (le_of_lt hLp) _)]
-    apply ENNReal.ofReal_le_ofReal
-    exact (hh.monotoneOn ENNReal.toReal_nonneg (mul_nonneg hLp.le ENNReal.toReal_nonneg) him).trans
-      (hscale _ ENNReal.toReal_nonneg hda)
-
-lemma source_gauge_lipschitz_null (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    {K : ℝ≥0} {f : ℂ → ℂ} {S : Set ℂ} (hLip : LipschitzOnWith K f S)
-    (hS : hausdorffGauge h S = 0) : hausdorffGauge h (f '' S) = 0 := by
-  have hb : hausdorffGauge h (f '' S) ≤ 0 := by
-    simpa only [hS,mul_zero] using source_gauge_lipschitz_image_le κ hκ hκ8 h hh hf hLip
-  exact le_antisymm hb zero_le
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_gauge_holomorphic_compact_null (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    {f : ℂ → ℂ} {D K : Set ℂ} (hD : IsOpen D) (hK : IsCompact K)
-    (hKD : K ⊆ D) (hd : DifferentiableOn ℂ f D) (hz : hausdorffGauge h K = 0) :
-    hausdorffGauge h (f '' K) = 0 := by
-  classical
-  have hlocal (x : K) : ∃ L : ℝ≥0, ∃ V : Set ℂ, V ∈ 𝓝 (x:ℂ) ∧ LipschitzOnWith L f V :=
-    (hd.analyticAt (hD.mem_nhds (hKD x.property))).contDiffAt.exists_lipschitzOnWith
-  choose L V hV hLip using hlocal
-  obtain ⟨I,hI⟩ := hK.elim_finite_subcover (fun x : K => interior (V x))
-    (fun _ => isOpen_interior) (fun x hx => mem_iUnion.mpr
-      ⟨⟨x,hx⟩,mem_interior_iff_mem_nhds.mpr (hV ⟨x,hx⟩)⟩)
-  apply measure_mono_null (t := ⋃ i : I, f '' (K ∩ V i.val))
-  · rintro _ ⟨x,hx,rfl⟩
-    obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (hI hx)
-    exact mem_iUnion.mpr ⟨⟨i,hi⟩,⟨x,⟨hx,interior_subset hxi⟩,rfl⟩⟩
-  · apply measure_iUnion_null
-    intro i
-    exact source_gauge_lipschitz_null κ hκ hκ8 h hh hf
-      ((hLip i.val).mono inter_subset_right) (measure_mono_null inter_subset_left hz)
-
-lemma source_inverse_chart_gauge_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    (U : ℝ≥0 → ℝ) (η : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U η G F) (s : ℝ≥0)
-    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
-    (hpos : 0 < hausdorffGauge h K) :
-    0 < hausdorffGauge h (F s '' K) := by
-  have hGc : DifferentiableOn ℂ (G s) (survivingDomain η s) := hw.2.2.2.1 s
-  have hFc : ContinuousOn (F s) upperHalfPlane := (hw.2.2.2.2.1 s).continuousOn
-  have hFK : IsCompact (F s '' K) := hK.image_of_continuousOn (hFc.mono hKH)
-  have hFD : F s '' K ⊆ survivingDomain η s := by
-    rintro _ ⟨z,hz,rfl⟩
-    exact hw.2.2.2.2.2.2.1 s (hKH hz)
-  have hId : G s '' (F s '' K) = K := by
-    ext z
-    constructor
-    · rintro ⟨_,⟨w,hwK,rfl⟩,rfl⟩
-      simpa only [hw.2.2.2.2.2.2.2.2.1 s w (hKH hwK)] using hwK
-    · intro hz
-      exact ⟨F s z,⟨z,hz,rfl⟩,hw.2.2.2.2.2.2.2.2.1 s z (hKH hz)⟩
-  by_contra hn
-  have hnull : hausdorffGauge h (F s '' K) = 0 := le_antisymm (le_of_not_gt hn) zero_le
-  have hhnull := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
-    (isOpen_survivingDomain hw.1 s) hFK hFD hGc hnull
-  rw [hId] at hhnull
-  exact (ne_of_gt hpos) hhnull
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-theorem source_retained_compact_interior_lower_bound
-    {h : ℝ → ℝ} (hh : IsGauge h) {K E : Set ℂ}
-    (hK : IsCompact K) (hE : IsClosed E)
-    (μ : ℕ → FiniteMeasure ℂ) (a C : ℝ≥0) (ha : 0 < a) (hC : 0 < C)
-    (ρ : ℝ) (hρ : 0 < ρ)
-    (hmass : ∀ n, a ≤ (μ n).mass)
-    (hKsupport : ∀ n, μ n Kᶜ = 0)
-    (e : ℕ → ℝ) (he : Tendsto e atTop (𝓝 0))
-    (hcarrier : ∀ n, (μ n : Measure ℂ) (Metric.cthickening (e n) E)ᶜ = 0)
-    (hball : ∀ x r, 0 < r → r < ρ →
-      ∀ᶠ n in atTop, (μ n : Measure ℂ) (Metric.ball x r) ≤
-        (C : ℝ≥0∞) * ENNReal.ofReal (h r)) :
-    0 < hausdorffGauge h (K ∩ E) := by
-  have hmpos (n : ℕ) : 0 < (μ n).mass := ha.trans_le (hmass n)
-  have hmne (n : ℕ) : μ n ≠ 0 := (μ n).mass_nonzero_iff.mp (hmpos n).ne'
-  let νs : ℕ → ProbabilityMeasure ℂ := fun n => (μ n).normalize
-  have hνK (n : ℕ) : νs n ∈ {ν : ProbabilityMeasure ℂ | ν Kᶜ = 0} := by
-    change (μ n).normalize Kᶜ = 0
-    rw [(μ n).normalize_eq_of_nonzero (hmne n), hKsupport n, mul_zero]
-  obtain ⟨ν, hνK, φ, hφ, hlim⟩ :=
-    (normalized_supported_probability_compact hK).tendsto_subseq hνK
-  have hνcarrier (n : ℕ) :
-      ((νs ∘ φ) n : Measure ℂ) (Metric.cthickening ((e ∘ φ) n) E)ᶜ = 0 := by
-    change ((μ (φ n)).normalize : Measure ℂ) (Metric.cthickening (e (φ n)) E)ᶜ = 0
-    rw [(μ (φ n)).toMeasure_normalize_eq_of_nonzero (hmne (φ n)),
-      Measure.coe_nnreal_smul_apply, hcarrier (φ n), mul_zero]
-  have hνE : (ν : Measure ℂ) E = 1 :=
-    (prob_compl_eq_zero_iff hE.measurableSet).mp
-      (probability_limit_shrinking_carriers hlim hE (e ∘ φ)
-        (he.comp hφ.tendsto_atTop) hνcarrier)
-  have hνKE : (ν : Measure ℂ) (K ∩ E) = 1 := by
-    apply (prob_compl_eq_zero_iff (hK.isClosed.inter hE).measurableSet).mp
-    rw [compl_inter]
-    apply measure_union_null
-    · exact (ProbabilityMeasure.null_iff_toMeasure_null _ _).mp hνK
-    · exact (prob_compl_eq_zero_iff hE.measurableSet).mpr hνE
-  have hνball (x : ℂ) (r : ℝ) (hr : 0 < r) (hrρ : r < ρ) :
-      (ν : Measure ℂ) (Metric.ball x r) ≤ (↑(C / a) : ℝ≥0∞) * ENNReal.ofReal (h r) := by
-    apply probability_limit_ball_bound hlim
-    filter_upwards [hφ.tendsto_atTop.eventually (hball x r hr hrρ)] with n hn
-    change ((μ (φ n)).normalize : Measure ℂ) (Metric.ball x r) ≤ _
-    rw [(μ (φ n)).toMeasure_normalize_eq_of_nonzero (hmne (φ n)),
-      Measure.coe_nnreal_smul_apply]
-    have hfactor : (↑((μ (φ n)).mass⁻¹) : ℝ≥0∞) ≤ (↑(a⁻¹) : ℝ≥0∞) := by
-      rw [ENNReal.coe_inv (hmpos (φ n)).ne', ENNReal.coe_inv ha.ne']
-      exact ENNReal.inv_le_inv.mpr (ENNReal.coe_le_coe.mpr (hmass (φ n)))
-    calc
-      (↑((μ (φ n)).mass⁻¹) : ℝ≥0∞) * (μ (φ n) : Measure ℂ) (Metric.ball x r)
-          ≤ (↑(a⁻¹) : ℝ≥0∞) * ((C : ℝ≥0∞) * ENNReal.ofReal (h r)) :=
-        mul_le_mul' hfactor hn
-      _ = (↑(C / a) : ℝ≥0∞) * ENNReal.ofReal (h r) := by
-        rw [div_eq_mul_inv, ENNReal.coe_mul]
-        ac_rfl
-  have hdom := gauge_mass_distribution hh (ν : Measure ℂ) (C / a)
-    (div_pos hC ha) ρ hρ hνball
-  by_contra hn
-  have hz : hausdorffGauge h (K ∩ E) = 0 := le_antisymm (le_of_not_gt hn) bot_le
-  have hbound := Measure.le_iff'.mp hdom (K ∩ E)
-  simp only [hνKE, Measure.smul_apply, hz, smul_eq_mul, mul_zero, not_le_of_gt zero_lt_one]
-    at hbound
-
- 
-theorem source_retained_compact_interior_lower_bound_of_frequently
-    {h : ℝ → ℝ} (hh : IsGauge h) {K E : Set ℂ}
-    (hK : IsCompact K) (hE : IsClosed E)
-    (μ : ℕ → FiniteMeasure ℂ) (a C : ℝ≥0) (ha : 0 < a) (hC : 0 < C)
-    (ρ : ℝ) (hρ : 0 < ρ)
-    (hmass : ∃ᶠ n in atTop, a ≤ (μ n).mass)
-    (hKsupport : ∀ n, μ n Kᶜ = 0)
-    (e : ℕ → ℝ) (he : Tendsto e atTop (𝓝 0))
-    (hcarrier : ∀ n, (μ n : Measure ℂ) (Metric.cthickening (e n) E)ᶜ = 0)
-    (hball : ∀ x r, 0 < r → r < ρ →
-      ∀ᶠ n in atTop, (μ n : Measure ℂ) (Metric.ball x r) ≤
-        (C : ℝ≥0∞) * ENNReal.ofReal (h r)) :
-    0 < hausdorffGauge h (K ∩ E) := by
-
-  obtain ⟨φ, hφ, hm⟩ := Filter.subseq_forall_of_frequently (p := fun n : ℕ => a ≤ (μ n).mass)
-    (x := id) tendsto_id hmass
-  apply source_retained_compact_interior_lower_bound hh hK hE (μ ∘ φ) a C ha hC ρ hρ hm
-    (fun n => hKsupport (φ n)) (e ∘ φ) (he.comp hφ) (fun n => hcarrier (φ n))
-  intro x r hr hrρ
-  exact hφ.eventually (hball x r hr hrρ)
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-def SourceRetainedDyadicInteriorTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∀ K E : Set ℂ, IsCompact K → IsClosed E →
-  ∀ μ : ℕ → FiniteMeasure ℂ, ∀ a : ℝ≥0, 0 < a →
-  (∃ᶠ N in atTop, a ≤ (μ N).mass) →
-  (∀ N, μ N Kᶜ = 0) →
-  ∀ e : ℕ → ℝ, Tendsto e atTop (𝓝 0) →
-  (∀ N, (μ N : Measure ℂ) (Metric.cthickening (e N) E)ᶜ = 0) →
-  ∀ D : ℝ, 0 < D → ∀ k₀ : ℕ,
-  (∀ N k, k₀ ≤ k → k ≤ N → ∀ ij : ℤ × ℤ,
-    μ N (retainedDyadicSquare k ij) ≤ ENNReal.ofReal (retainedThreshold κ D k)) →
-  0 < hausdorffGauge h (K ∩ E)
-theorem sourceRetainedDyadicInterior_proved : SourceRetainedDyadicInteriorTarget := by
-  intro κ hκ hκ8 h hh hf K E hK hE μ a ha hmass hKsupport e he hcarrier D hD k₀ htest
-  obtain ⟨b, hb, hscale⟩ := source_gauge_bounded_scaling κ hκ hκ8 h hh hf 2 (by norm_num)
-  obtain ⟨k₁, hk₁⟩ := eventually_atTop.mp
-    (retained_threshold_gauge_eventually κ hκ8 h hf D hD.le)
-  let J := max k₀ k₁
-  let ρ : ℝ := min 1 (min b ((1/2:ℝ)^J/2))
-  have hρ : 0 < ρ := lt_min (by norm_num) (lt_min hb (by positivity))
-  let C : ℝ≥0 := ⟨9 * D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ, by positivity⟩
-  have hC : 0 < C := by change 0 < 9 * D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ; positivity
-  apply source_retained_compact_interior_lower_bound_of_frequently hh hK hE μ a C ha hC ρ hρ
-    hmass hKsupport e he hcarrier
-  intro x r hr hrρ
-  have hr1 : r ≤ 1 := hrρ.le.trans (min_le_left _ _)
-  obtain ⟨k, hklo, hkhi⟩ := exists_nat_pow_near_of_lt_one hr hr1
-    (by norm_num : (0:ℝ)<1/2) (by norm_num : (1/2:ℝ)<1)
-  have hside : (1/2:ℝ)^k < 2*r := by rw [pow_succ] at hklo; linarith
-  have hrJ : 2*r < (1/2:ℝ)^J := by
-    have hv : r < (1/2:ℝ)^J/2 := hrρ.trans_le ((min_le_right _ _).trans (min_le_right _ _))
-    linarith
-  have hJk : J ≤ k := (pow_lt_pow_iff_right_of_lt_one₀
-    (by norm_num : (0:ℝ)<1/2) (by norm_num : (1/2:ℝ)<1)).mp (hside.trans hrJ) |>.le
-  have hbound : retainedThreshold κ D k ≤
-      D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ * h r := by
-    have hmono : h ((1/2:ℝ)^k) ≤ h (2*r) :=
-      hh.monotoneOn (show 0 ≤ (1/2:ℝ)^k by positivity)
-        (show 0 ≤ 2*r by positivity) hside.le
-    have hrb : r ≤ b := hrρ.le.trans ((min_le_right _ _).trans (min_le_left _ _))
-    calc
-      retainedThreshold κ D k ≤ D * (2:ℝ)^(codimension κ/2) * h ((1/2:ℝ)^k) :=
-        hk₁ k ((le_max_right _ _).trans hJk)
-      _ ≤ D * (2:ℝ)^(codimension κ/2) * ((2:ℝ)^dimension κ * h r) :=
-        mul_le_mul_of_nonneg_left (hmono.trans (hscale r hr.le hrb)) (by positivity)
-      _ = _ := by ring
-  filter_upwards [eventually_ge_atTop k] with N hN
-  calc
-    (μ N : Measure ℂ) (Metric.ball x r) ≤ 9 * ENNReal.ofReal (retainedThreshold κ D k) :=
-      retained_dyadic_ball_bound (μ N) k _ (by
-        simpa only [FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure] using
-          htest N k ((le_max_left _ _).trans hJk) hN) x r hkhi
-    _ = ENNReal.ofReal (9 * retainedThreshold κ D k) := by
-      rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤9)]
-      norm_num
-    _ ≤ ENNReal.ofReal ((C:ℝ)*h r) := by
-      apply ENNReal.ofReal_le_ofReal
-      change 9 * retainedThreshold κ D k ≤
-        (9 * D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ) * h r
-      calc
-        9 * retainedThreshold κ D k ≤ 9 * (D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ * h r) :=
-          mul_le_mul_of_nonneg_left hbound (by norm_num)
-        _ = _ := by ring
-    _ = (C:ℝ≥0∞) * ENNReal.ofReal (h r) := by
-      rw [ENNReal.ofReal_mul C.coe_nonneg, ENNReal.ofReal_coe_nnreal]
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-variable (a : RegularSLEPresentation P κ γ)
-lemma source_actual_retained_interior_pathwise (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz₀ : 2 < z₀.im) (T : ℝ≥0) (D b : ℝ) (hD : 0 < D) (hb : 0 < b) (k₀ : ℕ)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    (ω : a.samples) (hω : ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-      retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)) :
-    0 < hausdorffGauge h (closedBall z₀ 1 ∩ (γ ω '' Icc 0 T)) := by
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hκ hκ8 z₀ hz₀ T N
-  let μ (N : ℕ) : FiniteMeasure ℂ := ⟨retainedMeasure κ D ν k₀ N ω, by
-    dsimp [retainedMeasure]; infer_instance⟩
-  have hmass : ∃ᶠ N in atTop, (⟨b,hb.le⟩ : ℝ≥0) ≤ (μ N).mass := by
-    apply hω.mono
-    intro N hN
-    have hm : ENNReal.ofReal b ≤ (μ N : Measure ℂ) univ :=
-      hN.trans (measure_mono (subset_univ _))
-    change b ≤ ((μ N).mass : ℝ)
-    have ht := ENNReal.toReal_mono (measure_ne_top (μ N : Measure ℂ) univ) hm
-    simpa only [ENNReal.toReal_ofReal hb.le, ← FiniteMeasure.ennreal_mass,
-      ENNReal.coe_toReal] using ht
-  have hK (N : ℕ) : μ N (closedBall z₀ 1)ᶜ = 0 := by
-    apply (FiniteMeasure.null_iff_toMeasure_null _ _).mpr
-    exact retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_supported z₀ T N ω)
-  have hcarrier (N : ℕ) : (μ N : Measure ℂ) (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))ᶜ = 0 :=
-    retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_carrier z₀ T N ω)
-  have htest (N k : ℕ) (hk₀ : k₀ ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) :
-      (μ N (retainedDyadicSquare k ij) : ℝ≥0∞) ≤ ENNReal.ofReal (retainedThreshold κ D k) := by
-    rw [FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure]
-    exact retainedMeasure_square_bound κ D ν k₀ N ω k hk₀ hkN ij
-  exact sourceRetainedDyadicInterior_proved κ hκ hκ8 h hh hf (closedBall z₀ 1) (γ ω '' Icc 0 T)
-    (isCompact_closedBall _ _) (isCompact_Icc.image (a.trace_continuous ω)).isClosed
-    μ ⟨b,hb.le⟩ hb hmass hK (fun N => (2:ℝ)^(-(N:ℝ))) source_dyadic_tendsto hcarrier D hD k₀ htest
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
- 
-
-def SourceInteriorRetainedTransferTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
-    0 < (P.comap Subtype.val) E ∧ ∀ ω ∈ E,
-      0 < hausdorffGauge h (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)) ∧
-      ∀ (U : ℝ≥0 → ℝ) (η : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
-        IsCapacityTwoWitness U η G F → ∀ s : ℝ≥0,
-          0 < hausdorffGauge h (F s '' (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)))
-
-theorem sourceInteriorRetainedTransfer_proved : SourceInteriorRetainedTransferTarget := by
-  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
-  let : IsProbabilityMeasure P := hP
-  have hk : 0 < κ := by linarith
-  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment hκ hκ8 z₀ hz₀
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
-  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
-  have hE : MeasurableSet E := by
-    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
-        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
-      ext ω
-      simp [E,frequently_atTop]
-    rw [heq]
-    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
-      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
-  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
-  intro ω hω
-  have hpos := a.source_actual_retained_interior_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
-    k₀ h hh hf ω hω
-  refine ⟨hpos,?_⟩
-  intro U η G F hw s
-  exact source_inverse_chart_gauge_positive κ hk hκ8 h hh hf U η G F hw s
-    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed)
-    (fun z hz => zero_lt_one.trans (source_unit_ball_height hz₀ hz.1)) hpos
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance actualRetainedKernelDriverMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance actualRetainedKernelTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance actualRetainedKernelDriverBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance actualRetainedKernelTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_brownian_continuous_law_unique
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (B B' : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P)
-    (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hc' : ∀ ω, Continuous (fun t => B' t ω)) :
-    P.map (sourceDriverPath B hc) = P.map (sourceDriverPath B' hc') := by
-  have hL : IsProjectiveLimit (P.map (fun ω t => B t ω)) BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm)]
-    exact (hB.hasLaw I).map_eq
-  have hR : IsProjectiveLimit (P.map (fun ω t => B' t ω)) BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm')]
-    exact (hB'.hasLaw I).map_eq
-  apply source_path_evaluation_embedding.map_injective
-  have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n))) (hL.unique hR)
-  have hem : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) :=
-    Measurable.of_eval (fun _ => measurable_pi_apply _)
-  rw [Measure.map_map source_path_evaluation_embedding.measurable (source_path_measurable _ hm),
-    Measure.map_map source_path_evaluation_embedding.measurable (source_path_measurable _ hm')]
-  rw [Measure.map_map hem (Measurable.of_eval hm),
-    Measure.map_map hem (Measurable.of_eval hm')] at he
-  exact he
- 
-
-def SourceTraceLawUniqueTarget : Prop :=
-  ∀ {Ω : Type*} [_m : MeasurableSpace Ω] {P : Measure Ω} (κ : ℝ), 0 < κ → κ < 8 →
-  ∀ (B B' : ℝ≥0 → Ω → ℝ), IsBrownianReal B P → IsBrownianReal B' P →
-  ∀ (_hm : ∀ t, Measurable (B t)) (_hm' : ∀ t, Measurable (B' t))
-    (_hc : ∀ ω, Continuous (fun t => B t ω)) (_hc' : ∀ ω, Continuous (fun t => B' t ω))
-    (γ η : Ω → ℝ≥0 → ℂ) (G F G' F' : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hw' : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B' t ω) (η ω) (G' ω) (F' ω)),
-    P.map (sourceTracePath γ (fun ω => (hw ω).1)) =
-      P.map (sourceTracePath η (fun ω => (hw' ω).1))
-theorem sourceTraceLawUnique_proved : SourceTraceLawUniqueTarget := by
-  intro Ω m P κ hκ hκ8 B B' hB hB' hm hm' hc hc' γ η G F G' F' hw hw'
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let bp := sourceDriverPath B hc
-  let rp := sourceDriverPath B' hc'
-  let gp := sourceTracePath γ (fun ω => (hw ω).1)
-  let ep := sourceTracePath η (fun ω => (hw' ω).1)
-  have hbm : Measurable bp := source_path_measurable bp hm
-  have hrm : Measurable rp := source_path_measurable rp hm'
-  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t ω => gp ω t) := by
-    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω)
-      ?_ (fun ω => continuous_const.mul (hc ω)) γ G F hw
-    intro t
-    exact measurable_const.mul
-      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
-  obtain ⟨Φ,hΦm,hΦ⟩ := source_trace_factor B hm hc gp hγa
-  have hΦe (ω : Ω) : Φ (bp ω) = gp ω := (congrFun hΦ ω).symm
-  have hlaw : P.map bp = P.map rp := source_brownian_continuous_law_unique B B' hB hB' hm hm' hc hc'
-  choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) =>
-    source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity)
-  let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1)
-  have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
-  have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) :=
-    tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, Eventually.of_forall ha⟩
-  have hQorig (ω : Ω) (t : ℝ≥0) (n : ℕ) :
-      Q t n t (bp ω) = F ω t ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) := by
-    apply (hw ω).reverse_flow_identifies t
-      ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
-      (fun s => Q t n s (bp ω)) (hQ0 t n (bp ω))
-    intro s hs
-    exact (hQd t n (bp ω) s hs.1).mono Icc_subset_Ici_self
-  have hQnew (ω : Ω) (t : ℝ≥0) (n : ℕ) :
-      Q t n t (rp ω) = F' ω t ((Real.sqrt κ * B' t ω : ℝ) + (a n : ℂ) * I) := by
-    apply (hw' ω).reverse_flow_identifies t
-      ((Real.sqrt κ * B' t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
-      (fun s => Q t n s (rp ω)) (hQ0 t n (rp ω))
-    intro s hs
-    exact (hQd t n (rp ω) s hs.1).mono Icc_subset_Ici_self
-  have hQlimit (ω : Ω) (t : ℝ≥0) :
-      Tendsto (fun n => Q t n t (bp ω)) atTop (𝓝 (Φ (bp ω) t)) := by
-    have ht := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy
-    simpa only [hQorig,hΦe,gp,sourceTracePath,ContinuousMap.coe_mk,Function.comp_def] using ht
-  have hset (t : ℝ≥0) : MeasurableSet
-      {p : C(ℝ≥0,ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} :=
-    measurableSet_tendsto (𝓝 (0:ℂ)) (fun n => (hQm t n t).sub
-      ((continuous_eval_const t).measurable.comp hΦm))
-  have hQR (t : ℝ≥0) : ∀ᵐ ω ∂P, Tendsto (fun n => Q t n t (rp ω)) atTop (𝓝 (Φ (rp ω) t)) := by
-    have ho : ∀ᵐ p ∂P.map bp, Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by
-      apply (ae_map_iff hbm.aemeasurable (hset t)).2
-      exact Eventually.of_forall (fun ω => tendsto_sub_nhds_zero_iff.mpr (hQlimit ω t))
-    rw [hlaw] at ho
-    filter_upwards [(ae_map_iff hrm.aemeasurable (hset t)).1 ho] with ω hω
-    exact tendsto_sub_nhds_zero_iff.mp hω
-  have hEQ : ∀ᵐ ω ∂P, Φ (rp ω) = ep ω := by
-    filter_upwards [ae_all_iff.mpr (fun j : ℕ => hQR (denseSeq ℝ≥0 j))] with ω hω
-    apply DFunLike.coe_injective
-    apply (denseRange_denseSeq ℝ≥0).equalizer (Φ (rp ω)).continuous (ep ω).continuous
-    funext j
-    have ht := ((hw' ω).2.2.2.2.2.2.2.2.2.2.2.2 (denseSeq ℝ≥0 j)).comp hy
-    have hn := hω j
-    simp only [hQnew] at hn
-    exact tendsto_nhds_unique hn ht
-  have he := congrArg (Measure.map Φ) hlaw
-  rw [Measure.map_map hΦm hbm,Measure.map_map hΦm hrm] at he
-  have hcomp : Φ ∘ bp = gp := hΦ.symm
-  rw [hcomp] at he
-  exact he.trans (Measure.map_congr hEQ)
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.HausdorffLimits
+
+namespace OAI
+
+/-! Actual Retained Kernel for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+variable (a : RegularSLEPresentation P κ γ)
+def sourceFiniteGreenKernel (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) : Kernel a.samples ℂ :=
+  (Kernel.const a.samples (volume.restrict (closedBall z₀ 1))).withDensity
+    (fun ω z => ENNReal.ofReal (if spatialHitTime (γ ω) z ((2:ℝ)^(-(N:ℝ))) ≤ T then
+      spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z ((2:ℝ)^(-(N:ℝ))) else 0))
+lemma sourceFiniteGreenDensity_measurable (r : ℝ) (T : ℝ≥0) :
+    Measurable (fun p : a.samples × ℂ => ENNReal.ofReal (if spatialHitTime (γ p.1) p.2 r ≤ T then
+      spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t p.1) (γ p.1) (a.G p.1) p.2 r else 0)) := by
+  have hm (t : ℝ≥0) : Measurable (fun ω : a.samples => γ ω t) :=
+    (a.trace_adapted t).mono (a.filtration.le t) le_rfl
+  have ht := (spatialHitTime_measurable (fun ω : a.samples => γ ω) a.trace_continuous hm r).comp measurable_swap
+  have hW := (a.spatialGreenWeight_measurable r).comp measurable_swap
+  exact (Measurable.ite (measurableSet_le ht measurable_const) hW measurable_const).ennreal_ofReal
+lemma sourceFiniteGreenKernel_apply (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples)
+    (S : Set ℂ) (hS : MeasurableSet S) :
+    a.sourceFiniteGreenKernel z₀ T N ω S = finiteSpatialGreenMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1 ∩ S) := by
+  rw [sourceFiniteGreenKernel,Kernel.withDensity_apply' _ (a.sourceFiniteGreenDensity_measurable _ T)]
+  simp only [Kernel.const_apply,finiteSpatialGreenMass,Measure.restrict_restrict hS,inter_comm]
+  rfl
+lemma sourceFiniteGreenKernel_univ (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
+    a.sourceFiniteGreenKernel z₀ T N ω univ = finiteSpatialGreenMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1) := by
+  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω univ MeasurableSet.univ,inter_univ]
+lemma sourceFiniteGreenKernel_K (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
+    a.sourceFiniteGreenKernel z₀ T N ω (closedBall z₀ 1) = finiteSpatialGreenMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1) := by
+  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω _ measurableSet_closedBall,inter_self]
+lemma sourceFiniteGreenKernel_supported (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
+    a.sourceFiniteGreenKernel z₀ T N ω (closedBall z₀ 1)ᶜ = 0 := by
+  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω _ measurableSet_closedBall.compl,inter_compl_self]
+  simp [finiteSpatialGreenMass]
+lemma sourceFiniteGreenKernel_finite (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz : 2 < z₀.im) (T : ℝ≥0) (N : ℕ) :
+    IsFiniteKernel (a.sourceFiniteGreenKernel z₀ T N) := by
+  let r := (2:ℝ)^(-(N:ℝ))
+  have hr : 0 < r := source_dyadic_side_pos N
+  have hr1 : r ≤ 1 := by
+    dsimp [r]
+    rw [source_dyadic_side_eq]
+    exact pow_le_one₀ (by norm_num) (by norm_num)
+  refine ⟨⟨ENNReal.ofReal (r^(-codimension κ))*volume (closedBall z₀ 1),
+    ENNReal.mul_lt_top ENNReal.ofReal_lt_top (measure_closedBall_lt_top),?_⟩⟩
+  intro ω
+  rw [a.sourceFiniteGreenKernel_univ]
+  change (∫⁻ z in closedBall z₀ 1, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
+    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0)) ≤ _
+  calc
+    _ ≤ ∫⁻ z in closedBall z₀ 1, ENNReal.ofReal (r^(-codimension κ)) := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem measurableSet_closedBall] with z hz'
+      apply ENNReal.ofReal_le_ofReal
+      split_ifs
+      · exact (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hκ hκ8 hr z
+          (hr1.trans (source_unit_ball_height hz hz').le) |>.2
+      · exact Real.rpow_nonneg hr.le _
+    _ = _ := by simp
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+variable (a : RegularSLEPresentation P κ γ)
+lemma sourceFiniteGreenKernel_carrier (z₀ : ℂ) (T : ℝ≥0) (N : ℕ) (ω : a.samples) :
+    a.sourceFiniteGreenKernel z₀ T N ω (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))ᶜ = 0 := by
+  have hm := (isClosed_cthickening : IsClosed (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))).measurableSet
+  rw [a.sourceFiniteGreenKernel_apply z₀ T N ω _ hm.compl,finiteSpatialGreenMass]
+  apply lintegral_eq_zero_of_ae_eq_zero
+  filter_upwards [ae_restrict_mem (measurableSet_closedBall.inter hm.compl)] with z hz
+  have hn : ¬ spatialHitTime (γ ω) z ((2:ℝ)^(-(N:ℝ))) ≤ T := by
+    intro ht
+    obtain ⟨t,ht,hd⟩ := (spatialHitTime_le_iff (a.trace_continuous ω) z _ T).mp ht
+    exact hz.2 (mem_cthickening_of_dist_le z (γ ω t) _ _ ⟨t,ht,rfl⟩ (by simpa [dist_comm] using hd.le))
+  simp only [ite_eq_right hn, ENNReal.ofReal_zero,Pi.zero_apply]
+end RegularSLEPresentation
+lemma source_dyadic_tendsto : Tendsto (fun N : ℕ => (2:ℝ)^(-(N:ℝ))) atTop (𝓝 0) := by
+  simp_rw [source_dyadic_side_eq]
+  exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+lemma source_dyadic_eventually_le {ε : ℝ} (hε : 0 < ε) :
+    ∃ n : ℕ, ∀ N ≥ n, (2:ℝ)^(-(N:ℝ)) ≤ ε := by
+  exact (eventually_atTop.mp (source_dyadic_tendsto.eventually (eventually_le_nhds hε)))
+lemma retainedMeasure_zero_of_zero {Ω : Type*} [MeasurableSpace Ω] (κ D : ℝ)
+    (ν : ℕ → Kernel Ω ℂ) (k₀ N : ℕ) (ω : Ω) {S : Set ℂ} (hS : ν N ω S = 0) :
+    retainedMeasure κ D ν k₀ N ω S = 0 := by
+  exact le_antisymm ((Measure.restrict_apply_le _ _).trans hS.le) bot_le
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+variable (a : RegularSLEPresentation P κ γ)
+lemma source_actual_retained_experiment [IsProbabilityMeasure P] (hκ : 4 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz₀ : 2 < z₀.im) :
+    ∃ (T : ℝ≥0) (D b p : ℝ) (k₀ : ℕ), 0 < T ∧ 1 ≤ D ∧ 0 < b ∧ 0 < p ∧
+      ENNReal.ofReal p ≤ (P.comap Subtype.val)
+        {ω : a.samples | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+          retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)} := by
+  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
+  have hk : 0 < κ := by linarith
+  obtain ⟨T,e₀,m₀,m₂,hT,he,hm₀,hm₂,hm⟩ := sourceRetainedSeed_proved κ hκ hκ8
+    Ω inferInstance P inferInstance γ a z₀ hz₀
+  obtain ⟨A,c,k₁,hA,hc,ht⟩ := sourceActualSquareTail_proved κ hκ hκ8
+    Ω inferInstance P inferInstance γ a z₀ hz₀
+  obtain ⟨k₂,hk₂⟩ := source_dyadic_eventually_le he
+  let kmin := max k₁ k₂
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  have hν (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
+  have hfirst (N : ℕ) (hN : kmin ≤ N) :
+      ENNReal.ofReal m₀ ≤ ∫⁻ ω, ν N ω (closedBall z₀ 1) ∂(P.comap Subtype.val) := by
+    simp only [ν,a.sourceFiniteGreenKernel_K]
+    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).1
+  have hsecond (N : ℕ) (hN : kmin ≤ N) :
+      (∫⁻ ω, (ν N ω (closedBall z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
+    simp only [ν,a.sourceFiniteGreenKernel_K]
+    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).2
+  have htail (N k : ℕ) (hkmin : kmin ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) (u : ℝ) (hu : 1 ≤ u) :
+      (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
+        (ν N ω (retainedDyadicSquare k ij)).toReal} ≤
+        ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ))) := by
+    simp only [ν,a.sourceFiniteGreenKernel_apply z₀ T N _ _ (retainedSquare_measurable k ij)]
+    exact ht T N k ((le_max_left _ _).trans hkmin) hkN ij u hu
+  obtain ⟨D,b,p,B,k₀,hD,hb,hp,hB,hk₀,hall,hlim⟩ := sourceRetainedConstruction a.samples
+    inferInstance (P.comap Subtype.val) inferInstance κ hk hκ8 ν hν
+    (closedBall z₀ 1) (isCompact_closedBall _ _) (fun N ω => a.sourceFiniteGreenKernel_supported z₀ T N ω)
+    A c m₀ m₂ hA hc hm₀ hm₂ kmin hfirst hsecond htail
+  exact ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+variable (a : RegularSLEPresentation P κ γ)
+lemma source_actual_retained_pathwise (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz₀ : 2 < z₀.im) (T : ℝ≥0) (D b : ℝ) (hD : 0 < D) (hb : 0 < b) (k₀ : ℕ)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    (ω : a.samples) (hω : ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+      retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)) :
+    0 < hausdorffGauge h (γ ω '' Icc 0 T) := by
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hκ hκ8 z₀ hz₀ T N
+  let μ (N : ℕ) : FiniteMeasure ℂ := ⟨retainedMeasure κ D ν k₀ N ω, by
+    dsimp [retainedMeasure]; infer_instance⟩
+  have hmass : ∃ᶠ N in atTop, (⟨b,hb.le⟩ : ℝ≥0) ≤ (μ N).mass := by
+    apply hω.mono
+    intro N hN
+    have hm : ENNReal.ofReal b ≤ (μ N : Measure ℂ) univ :=
+      hN.trans (measure_mono (subset_univ _))
+    change b ≤ ((μ N).mass : ℝ)
+    have ht := ENNReal.toReal_mono (measure_ne_top (μ N : Measure ℂ) univ) hm
+    simpa only [ENNReal.toReal_ofReal hb.le, ← FiniteMeasure.ennreal_mass,
+      ENNReal.coe_toReal] using ht
+  have hK (N : ℕ) : μ N (closedBall z₀ 1)ᶜ = 0 := by
+    apply (FiniteMeasure.null_iff_toMeasure_null _ _).mpr
+    exact retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_supported z₀ T N ω)
+  have hcarrier (N : ℕ) : (μ N : Measure ℂ) (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))ᶜ = 0 :=
+    retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_carrier z₀ T N ω)
+  have htest (N k : ℕ) (hk₀ : k₀ ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) :
+      (μ N (retainedDyadicSquare k ij) : ℝ≥0∞) ≤ ENNReal.ofReal (retainedThreshold κ D k) := by
+    rw [FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure]
+    exact retainedMeasure_square_bound κ D ν k₀ N ω k hk₀ hkN ij
+  exact sourceRetainedDyadic_proved κ hκ hκ8 h hh hf (closedBall z₀ 1) (γ ω '' Icc 0 T)
+    (isCompact_closedBall _ _) (isCompact_Icc.image (a.trace_continuous ω)).isClosed
+    μ ⟨b,hb.le⟩ hb hmass hK (fun N => (2:ℝ)^(-(N:ℝ))) source_dyadic_tendsto hcarrier D hD k₀ htest
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+ 
+
+def SourceActualRetainedLowerTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
+    0 < (P.comap Subtype.val) E ∧
+    ∀ ω ∈ E, 0 < hausdorffGauge h (γ ω '' Set.Icc 0 T)
+theorem sourceActualRetainedLower_proved : SourceActualRetainedLowerTarget := by
+  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
+  let : IsProbabilityMeasure P := hP
+  have hk : 0 < κ := by linarith
+  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment hκ hκ8 z₀ hz₀
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
+  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
+  have hE : MeasurableSet E := by
+    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
+        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
+      ext ω
+      simp [E,frequently_atTop]
+    rw [heq]
+    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
+      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
+  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
+  intro ω hω
+  exact a.source_actual_retained_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
+    k₀ h hh hf ω hω
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_gauge_lipschitz_image_le (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    {K : ℝ≥0} {f : ℂ → ℂ} {S : Set ℂ} (hLip : LipschitzOnWith K f S) :
+    hausdorffGauge h (f '' S) ≤
+      ENNReal.ofReal ((max (K : ℝ) 1)^dimension κ)*hausdorffGauge h S := by
+  let L : ℝ≥0 := max K 1
+  have hL : 1 ≤ (L : ℝ) := le_max_right _ _
+  have hLp : 0 < (L : ℝ) := zero_lt_one.trans_le hL
+  have hLC : LipschitzOnWith L f S := hLip.weaken (le_max_left _ _)
+  have himg (t : Set ℂ) : ediam (f '' (t ∩ S)) ≤ (L:ℝ≥0∞)*ediam t := by
+    simpa only [NNReal.coe_one,ENNReal.rpow_one] using hLC.holderOnWith.ediam_image_inter_le t
+  obtain ⟨a,ha,hscale⟩ := source_gauge_bounded_scaling κ hκ hκ8 h hh hf (L:ℝ) hL
+  let C : ℝ≥0∞ := ENNReal.ofReal ((L:ℝ)^dimension κ)
+  have hC0 : C ≠ 0 := (ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos hLp _)).ne'
+  have hCt : C ≠ ⊤ := ENNReal.ofReal_ne_top
+  change hausdorffGauge h (f '' S) ≤ C*hausdorffGauge h S
+  simp only [hausdorffGauge,Measure.mkMetric_apply,ENNReal.mul_iSup,
+    ENNReal.mul_iInf_of_ne hC0 hCt,←ENNReal.tsum_mul_left]
+  refine iSup_le fun R => iSup_le fun hR => ?_
+  have ht : Tendsto (fun d : ℝ≥0∞ => (L:ℝ≥0∞)*d) (𝓝 0) (𝓝 0) := by
+    simpa using (ENNReal.continuous_const_mul (a := (L:ℝ≥0∞)) ENNReal.coe_ne_top).tendsto 0
+  obtain ⟨δ,hδ,Hδ⟩ := ENNReal.nhds_zero_basis_Iic.eventually_iff.mp
+    (ht.eventually (gt_mem_nhds hR))
+  let ε := min δ (ENNReal.ofReal a)
+  have hε : 0 < ε := lt_min hδ (ENNReal.ofReal_pos.mpr ha)
+  have hεa : ε ≤ ENNReal.ofReal a := min_le_right _ _
+  have hεt : ε ≠ ⊤ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hεa
+  refine le_iSup₂_of_le ε hε <| iInf₂_mono' fun t hSt =>
+    ⟨fun n => f '' (t n ∩ S), ?_, iInf_mono' fun htε => ⟨?_,?_⟩⟩
+  · grw [←image_iUnion,←iUnion_inter,←hSt,inter_self]
+  · intro n
+    exact (himg (t n)).trans (Hδ ((htε n).trans (min_le_left _ _))).le
+  · apply ENNReal.tsum_le_tsum
+    intro n
+    simp only [iSup_le_iff,image_nonempty]
+    intro hn
+    simp only [Nonempty.mono ((t n).inter_subset_left) hn,ciSup_pos]
+    have hd : ediam (t n) ≠ ⊤ := ne_top_of_le_ne_top hεt (htε n)
+    have hda : (ediam (t n)).toReal ≤ a := by
+      simpa only [ENNReal.toReal_ofReal ha.le] using
+        ENNReal.toReal_mono ENNReal.ofReal_ne_top ((htε n).trans hεa)
+    have hprod : (L:ℝ≥0∞)*ediam (t n) ≠ ⊤ := ENNReal.mul_ne_top ENNReal.coe_ne_top hd
+    have him := ENNReal.toReal_mono hprod (himg (t n))
+    rw [ENNReal.toReal_mul,ENNReal.coe_toReal] at him
+    change ENNReal.ofReal (h (ediam (f '' (t n ∩ S))).toReal) ≤
+      ENNReal.ofReal ((L:ℝ)^dimension κ)*ENNReal.ofReal (h (ediam (t n)).toReal)
+    rw [←ENNReal.ofReal_mul (Real.rpow_nonneg (le_of_lt hLp) _)]
+    apply ENNReal.ofReal_le_ofReal
+    exact (hh.monotoneOn ENNReal.toReal_nonneg (mul_nonneg hLp.le ENNReal.toReal_nonneg) him).trans
+      (hscale _ ENNReal.toReal_nonneg hda)
+
+lemma source_gauge_lipschitz_null (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    {K : ℝ≥0} {f : ℂ → ℂ} {S : Set ℂ} (hLip : LipschitzOnWith K f S)
+    (hS : hausdorffGauge h S = 0) : hausdorffGauge h (f '' S) = 0 := by
+  have hb : hausdorffGauge h (f '' S) ≤ 0 := by
+    simpa only [hS,mul_zero] using source_gauge_lipschitz_image_le κ hκ hκ8 h hh hf hLip
+  exact le_antisymm hb zero_le
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_gauge_holomorphic_compact_null (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    {f : ℂ → ℂ} {D K : Set ℂ} (hD : IsOpen D) (hK : IsCompact K)
+    (hKD : K ⊆ D) (hd : DifferentiableOn ℂ f D) (hz : hausdorffGauge h K = 0) :
+    hausdorffGauge h (f '' K) = 0 := by
+  classical
+  have hlocal (x : K) : ∃ L : ℝ≥0, ∃ V : Set ℂ, V ∈ 𝓝 (x:ℂ) ∧ LipschitzOnWith L f V :=
+    (hd.analyticAt (hD.mem_nhds (hKD x.property))).contDiffAt.exists_lipschitzOnWith
+  choose L V hV hLip using hlocal
+  obtain ⟨I,hI⟩ := hK.elim_finite_subcover (fun x : K => interior (V x))
+    (fun _ => isOpen_interior) (fun x hx => mem_iUnion.mpr
+      ⟨⟨x,hx⟩,mem_interior_iff_mem_nhds.mpr (hV ⟨x,hx⟩)⟩)
+  apply measure_mono_null (t := ⋃ i : I, f '' (K ∩ V i.val))
+  · rintro _ ⟨x,hx,rfl⟩
+    obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (hI hx)
+    exact mem_iUnion.mpr ⟨⟨i,hi⟩,⟨x,⟨hx,interior_subset hxi⟩,rfl⟩⟩
+  · apply measure_iUnion_null
+    intro i
+    exact source_gauge_lipschitz_null κ hκ hκ8 h hh hf
+      ((hLip i.val).mono inter_subset_right) (measure_mono_null inter_subset_left hz)
+
+lemma source_inverse_chart_gauge_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    (U : ℝ≥0 → ℝ) (η : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U η G F) (s : ℝ≥0)
+    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
+    (hpos : 0 < hausdorffGauge h K) :
+    0 < hausdorffGauge h (F s '' K) := by
+  have hGc : DifferentiableOn ℂ (G s) (survivingDomain η s) := hw.2.2.2.1 s
+  have hFc : ContinuousOn (F s) upperHalfPlane := (hw.2.2.2.2.1 s).continuousOn
+  have hFK : IsCompact (F s '' K) := hK.image_of_continuousOn (hFc.mono hKH)
+  have hFD : F s '' K ⊆ survivingDomain η s := by
+    rintro _ ⟨z,hz,rfl⟩
+    exact hw.2.2.2.2.2.2.1 s (hKH hz)
+  have hId : G s '' (F s '' K) = K := by
+    ext z
+    constructor
+    · rintro ⟨_,⟨w,hwK,rfl⟩,rfl⟩
+      simpa only [hw.2.2.2.2.2.2.2.2.1 s w (hKH hwK)] using hwK
+    · intro hz
+      exact ⟨F s z,⟨z,hz,rfl⟩,hw.2.2.2.2.2.2.2.2.1 s z (hKH hz)⟩
+  by_contra hn
+  have hnull : hausdorffGauge h (F s '' K) = 0 := le_antisymm (le_of_not_gt hn) zero_le
+  have hhnull := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
+    (isOpen_survivingDomain hw.1 s) hFK hFD hGc hnull
+  rw [hId] at hhnull
+  exact (ne_of_gt hpos) hhnull
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+theorem source_retained_compact_interior_lower_bound
+    {h : ℝ → ℝ} (hh : IsGauge h) {K E : Set ℂ}
+    (hK : IsCompact K) (hE : IsClosed E)
+    (μ : ℕ → FiniteMeasure ℂ) (a C : ℝ≥0) (ha : 0 < a) (hC : 0 < C)
+    (ρ : ℝ) (hρ : 0 < ρ)
+    (hmass : ∀ n, a ≤ (μ n).mass)
+    (hKsupport : ∀ n, μ n Kᶜ = 0)
+    (e : ℕ → ℝ) (he : Tendsto e atTop (𝓝 0))
+    (hcarrier : ∀ n, (μ n : Measure ℂ) (Metric.cthickening (e n) E)ᶜ = 0)
+    (hball : ∀ x r, 0 < r → r < ρ →
+      ∀ᶠ n in atTop, (μ n : Measure ℂ) (Metric.ball x r) ≤
+        (C : ℝ≥0∞) * ENNReal.ofReal (h r)) :
+    0 < hausdorffGauge h (K ∩ E) := by
+  have hmpos (n : ℕ) : 0 < (μ n).mass := ha.trans_le (hmass n)
+  have hmne (n : ℕ) : μ n ≠ 0 := (μ n).mass_nonzero_iff.mp (hmpos n).ne'
+  let νs : ℕ → ProbabilityMeasure ℂ := fun n => (μ n).normalize
+  have hνK (n : ℕ) : νs n ∈ {ν : ProbabilityMeasure ℂ | ν Kᶜ = 0} := by
+    change (μ n).normalize Kᶜ = 0
+    rw [(μ n).normalize_eq_of_nonzero (hmne n), hKsupport n, mul_zero]
+  obtain ⟨ν, hνK, φ, hφ, hlim⟩ :=
+    (normalized_supported_probability_compact hK).tendsto_subseq hνK
+  have hνcarrier (n : ℕ) :
+      ((νs ∘ φ) n : Measure ℂ) (Metric.cthickening ((e ∘ φ) n) E)ᶜ = 0 := by
+    change ((μ (φ n)).normalize : Measure ℂ) (Metric.cthickening (e (φ n)) E)ᶜ = 0
+    rw [(μ (φ n)).toMeasure_normalize_eq_of_nonzero (hmne (φ n)),
+      Measure.coe_nnreal_smul_apply, hcarrier (φ n), mul_zero]
+  have hνE : (ν : Measure ℂ) E = 1 :=
+    (prob_compl_eq_zero_iff hE.measurableSet).mp
+      (probability_limit_shrinking_carriers hlim hE (e ∘ φ)
+        (he.comp hφ.tendsto_atTop) hνcarrier)
+  have hνKE : (ν : Measure ℂ) (K ∩ E) = 1 := by
+    apply (prob_compl_eq_zero_iff (hK.isClosed.inter hE).measurableSet).mp
+    rw [compl_inter]
+    apply measure_union_null
+    · exact (ProbabilityMeasure.null_iff_toMeasure_null _ _).mp hνK
+    · exact (prob_compl_eq_zero_iff hE.measurableSet).mpr hνE
+  have hνball (x : ℂ) (r : ℝ) (hr : 0 < r) (hrρ : r < ρ) :
+      (ν : Measure ℂ) (Metric.ball x r) ≤ (↑(C / a) : ℝ≥0∞) * ENNReal.ofReal (h r) := by
+    apply probability_limit_ball_bound hlim
+    filter_upwards [hφ.tendsto_atTop.eventually (hball x r hr hrρ)] with n hn
+    change ((μ (φ n)).normalize : Measure ℂ) (Metric.ball x r) ≤ _
+    rw [(μ (φ n)).toMeasure_normalize_eq_of_nonzero (hmne (φ n)),
+      Measure.coe_nnreal_smul_apply]
+    have hfactor : (↑((μ (φ n)).mass⁻¹) : ℝ≥0∞) ≤ (↑(a⁻¹) : ℝ≥0∞) := by
+      rw [ENNReal.coe_inv (hmpos (φ n)).ne', ENNReal.coe_inv ha.ne']
+      exact ENNReal.inv_le_inv.mpr (ENNReal.coe_le_coe.mpr (hmass (φ n)))
+    calc
+      (↑((μ (φ n)).mass⁻¹) : ℝ≥0∞) * (μ (φ n) : Measure ℂ) (Metric.ball x r)
+          ≤ (↑(a⁻¹) : ℝ≥0∞) * ((C : ℝ≥0∞) * ENNReal.ofReal (h r)) :=
+        mul_le_mul' hfactor hn
+      _ = (↑(C / a) : ℝ≥0∞) * ENNReal.ofReal (h r) := by
+        rw [div_eq_mul_inv, ENNReal.coe_mul]
+        ac_rfl
+  have hdom := gauge_mass_distribution hh (ν : Measure ℂ) (C / a)
+    (div_pos hC ha) ρ hρ hνball
+  by_contra hn
+  have hz : hausdorffGauge h (K ∩ E) = 0 := le_antisymm (le_of_not_gt hn) bot_le
+  have hbound := Measure.le_iff'.mp hdom (K ∩ E)
+  simp only [hνKE, Measure.smul_apply, hz, smul_eq_mul, mul_zero, not_le_of_gt zero_lt_one]
+    at hbound
+
+ 
+theorem source_retained_compact_interior_lower_bound_of_frequently
+    {h : ℝ → ℝ} (hh : IsGauge h) {K E : Set ℂ}
+    (hK : IsCompact K) (hE : IsClosed E)
+    (μ : ℕ → FiniteMeasure ℂ) (a C : ℝ≥0) (ha : 0 < a) (hC : 0 < C)
+    (ρ : ℝ) (hρ : 0 < ρ)
+    (hmass : ∃ᶠ n in atTop, a ≤ (μ n).mass)
+    (hKsupport : ∀ n, μ n Kᶜ = 0)
+    (e : ℕ → ℝ) (he : Tendsto e atTop (𝓝 0))
+    (hcarrier : ∀ n, (μ n : Measure ℂ) (Metric.cthickening (e n) E)ᶜ = 0)
+    (hball : ∀ x r, 0 < r → r < ρ →
+      ∀ᶠ n in atTop, (μ n : Measure ℂ) (Metric.ball x r) ≤
+        (C : ℝ≥0∞) * ENNReal.ofReal (h r)) :
+    0 < hausdorffGauge h (K ∩ E) := by
+
+  obtain ⟨φ, hφ, hm⟩ := Filter.subseq_forall_of_frequently (p := fun n : ℕ => a ≤ (μ n).mass)
+    (x := id) tendsto_id hmass
+  apply source_retained_compact_interior_lower_bound hh hK hE (μ ∘ φ) a C ha hC ρ hρ hm
+    (fun n => hKsupport (φ n)) (e ∘ φ) (he.comp hφ) (fun n => hcarrier (φ n))
+  intro x r hr hrρ
+  exact hφ.eventually (hball x r hr hrρ)
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+def SourceRetainedDyadicInteriorTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∀ K E : Set ℂ, IsCompact K → IsClosed E →
+  ∀ μ : ℕ → FiniteMeasure ℂ, ∀ a : ℝ≥0, 0 < a →
+  (∃ᶠ N in atTop, a ≤ (μ N).mass) →
+  (∀ N, μ N Kᶜ = 0) →
+  ∀ e : ℕ → ℝ, Tendsto e atTop (𝓝 0) →
+  (∀ N, (μ N : Measure ℂ) (Metric.cthickening (e N) E)ᶜ = 0) →
+  ∀ D : ℝ, 0 < D → ∀ k₀ : ℕ,
+  (∀ N k, k₀ ≤ k → k ≤ N → ∀ ij : ℤ × ℤ,
+    μ N (retainedDyadicSquare k ij) ≤ ENNReal.ofReal (retainedThreshold κ D k)) →
+  0 < hausdorffGauge h (K ∩ E)
+theorem sourceRetainedDyadicInterior_proved : SourceRetainedDyadicInteriorTarget := by
+  intro κ hκ hκ8 h hh hf K E hK hE μ a ha hmass hKsupport e he hcarrier D hD k₀ htest
+  obtain ⟨b, hb, hscale⟩ := source_gauge_bounded_scaling κ hκ hκ8 h hh hf 2 (by norm_num)
+  obtain ⟨k₁, hk₁⟩ := eventually_atTop.mp
+    (retained_threshold_gauge_eventually κ hκ8 h hf D hD.le)
+  let J := max k₀ k₁
+  let ρ : ℝ := min 1 (min b ((1/2:ℝ)^J/2))
+  have hρ : 0 < ρ := lt_min (by norm_num) (lt_min hb (by positivity))
+  let C : ℝ≥0 := ⟨9 * D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ, by positivity⟩
+  have hC : 0 < C := by change 0 < 9 * D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ; positivity
+  apply source_retained_compact_interior_lower_bound_of_frequently hh hK hE μ a C ha hC ρ hρ
+    hmass hKsupport e he hcarrier
+  intro x r hr hrρ
+  have hr1 : r ≤ 1 := hrρ.le.trans (min_le_left _ _)
+  obtain ⟨k, hklo, hkhi⟩ := exists_nat_pow_near_of_lt_one hr hr1
+    (by norm_num : (0:ℝ)<1/2) (by norm_num : (1/2:ℝ)<1)
+  have hside : (1/2:ℝ)^k < 2*r := by rw [pow_succ] at hklo; linarith
+  have hrJ : 2*r < (1/2:ℝ)^J := by
+    have hv : r < (1/2:ℝ)^J/2 := hrρ.trans_le ((min_le_right _ _).trans (min_le_right _ _))
+    linarith
+  have hJk : J ≤ k := (pow_lt_pow_iff_right_of_lt_one₀
+    (by norm_num : (0:ℝ)<1/2) (by norm_num : (1/2:ℝ)<1)).mp (hside.trans hrJ) |>.le
+  have hbound : retainedThreshold κ D k ≤
+      D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ * h r := by
+    have hmono : h ((1/2:ℝ)^k) ≤ h (2*r) :=
+      hh.monotoneOn (show 0 ≤ (1/2:ℝ)^k by positivity)
+        (show 0 ≤ 2*r by positivity) hside.le
+    have hrb : r ≤ b := hrρ.le.trans ((min_le_right _ _).trans (min_le_left _ _))
+    calc
+      retainedThreshold κ D k ≤ D * (2:ℝ)^(codimension κ/2) * h ((1/2:ℝ)^k) :=
+        hk₁ k ((le_max_right _ _).trans hJk)
+      _ ≤ D * (2:ℝ)^(codimension κ/2) * ((2:ℝ)^dimension κ * h r) :=
+        mul_le_mul_of_nonneg_left (hmono.trans (hscale r hr.le hrb)) (by positivity)
+      _ = _ := by ring
+  filter_upwards [eventually_ge_atTop k] with N hN
+  calc
+    (μ N : Measure ℂ) (Metric.ball x r) ≤ 9 * ENNReal.ofReal (retainedThreshold κ D k) :=
+      retained_dyadic_ball_bound (μ N) k _ (by
+        simpa only [FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure] using
+          htest N k ((le_max_left _ _).trans hJk) hN) x r hkhi
+    _ = ENNReal.ofReal (9 * retainedThreshold κ D k) := by
+      rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤9)]
+      norm_num
+    _ ≤ ENNReal.ofReal ((C:ℝ)*h r) := by
+      apply ENNReal.ofReal_le_ofReal
+      change 9 * retainedThreshold κ D k ≤
+        (9 * D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ) * h r
+      calc
+        9 * retainedThreshold κ D k ≤ 9 * (D * (2:ℝ)^(codimension κ/2) * (2:ℝ)^dimension κ * h r) :=
+          mul_le_mul_of_nonneg_left hbound (by norm_num)
+        _ = _ := by ring
+    _ = (C:ℝ≥0∞) * ENNReal.ofReal (h r) := by
+      rw [ENNReal.ofReal_mul C.coe_nonneg, ENNReal.ofReal_coe_nnreal]
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+variable (a : RegularSLEPresentation P κ γ)
+lemma source_actual_retained_interior_pathwise (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz₀ : 2 < z₀.im) (T : ℝ≥0) (D b : ℝ) (hD : 0 < D) (hb : 0 < b) (k₀ : ℕ)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    (ω : a.samples) (hω : ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+      retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)) :
+    0 < hausdorffGauge h (closedBall z₀ 1 ∩ (γ ω '' Icc 0 T)) := by
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hκ hκ8 z₀ hz₀ T N
+  let μ (N : ℕ) : FiniteMeasure ℂ := ⟨retainedMeasure κ D ν k₀ N ω, by
+    dsimp [retainedMeasure]; infer_instance⟩
+  have hmass : ∃ᶠ N in atTop, (⟨b,hb.le⟩ : ℝ≥0) ≤ (μ N).mass := by
+    apply hω.mono
+    intro N hN
+    have hm : ENNReal.ofReal b ≤ (μ N : Measure ℂ) univ :=
+      hN.trans (measure_mono (subset_univ _))
+    change b ≤ ((μ N).mass : ℝ)
+    have ht := ENNReal.toReal_mono (measure_ne_top (μ N : Measure ℂ) univ) hm
+    simpa only [ENNReal.toReal_ofReal hb.le, ← FiniteMeasure.ennreal_mass,
+      ENNReal.coe_toReal] using ht
+  have hK (N : ℕ) : μ N (closedBall z₀ 1)ᶜ = 0 := by
+    apply (FiniteMeasure.null_iff_toMeasure_null _ _).mpr
+    exact retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_supported z₀ T N ω)
+  have hcarrier (N : ℕ) : (μ N : Measure ℂ) (cthickening ((2:ℝ)^(-(N:ℝ))) (γ ω '' Icc 0 T))ᶜ = 0 :=
+    retainedMeasure_zero_of_zero κ D ν k₀ N ω (a.sourceFiniteGreenKernel_carrier z₀ T N ω)
+  have htest (N k : ℕ) (hk₀ : k₀ ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) :
+      (μ N (retainedDyadicSquare k ij) : ℝ≥0∞) ≤ ENNReal.ofReal (retainedThreshold κ D k) := by
+    rw [FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure]
+    exact retainedMeasure_square_bound κ D ν k₀ N ω k hk₀ hkN ij
+  exact sourceRetainedDyadicInterior_proved κ hκ hκ8 h hh hf (closedBall z₀ 1) (γ ω '' Icc 0 T)
+    (isCompact_closedBall _ _) (isCompact_Icc.image (a.trace_continuous ω)).isClosed
+    μ ⟨b,hb.le⟩ hb hmass hK (fun N => (2:ℝ)^(-(N:ℝ))) source_dyadic_tendsto hcarrier D hD k₀ htest
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+ 
+
+def SourceInteriorRetainedTransferTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
+    0 < (P.comap Subtype.val) E ∧ ∀ ω ∈ E,
+      0 < hausdorffGauge h (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)) ∧
+      ∀ (U : ℝ≥0 → ℝ) (η : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
+        IsCapacityTwoWitness U η G F → ∀ s : ℝ≥0,
+          0 < hausdorffGauge h (F s '' (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)))
+
+theorem sourceInteriorRetainedTransfer_proved : SourceInteriorRetainedTransferTarget := by
+  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
+  let : IsProbabilityMeasure P := hP
+  have hk : 0 < κ := by linarith
+  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment hκ hκ8 z₀ hz₀
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
+  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
+  have hE : MeasurableSet E := by
+    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
+        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
+      ext ω
+      simp [E,frequently_atTop]
+    rw [heq]
+    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
+      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
+  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
+  intro ω hω
+  have hpos := a.source_actual_retained_interior_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
+    k₀ h hh hf ω hω
+  refine ⟨hpos,?_⟩
+  intro U η G F hw s
+  exact source_inverse_chart_gauge_positive κ hk hκ8 h hh hf U η G F hw s
+    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed)
+    (fun z hz => zero_lt_one.trans (source_unit_ball_height hz₀ hz.1)) hpos
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance actualRetainedKernelDriverMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance actualRetainedKernelTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance actualRetainedKernelDriverBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance actualRetainedKernelTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_brownian_continuous_law_unique
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (B B' : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P)
+    (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hc' : ∀ ω, Continuous (fun t => B' t ω)) :
+    P.map (sourceDriverPath B hc) = P.map (sourceDriverPath B' hc') := by
+  have hL : IsProjectiveLimit (P.map (fun ω t => B t ω)) BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm)]
+    exact (hB.hasLaw I).map_eq
+  have hR : IsProjectiveLimit (P.map (fun ω t => B' t ω)) BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm')]
+    exact (hB'.hasLaw I).map_eq
+  apply source_path_evaluation_embedding.map_injective
+  have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n))) (hL.unique hR)
+  have hem : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) :=
+    Measurable.of_eval (fun _ => measurable_pi_apply _)
+  rw [Measure.map_map source_path_evaluation_embedding.measurable (source_path_measurable _ hm),
+    Measure.map_map source_path_evaluation_embedding.measurable (source_path_measurable _ hm')]
+  rw [Measure.map_map hem (Measurable.of_eval hm),
+    Measure.map_map hem (Measurable.of_eval hm')] at he
+  exact he
+ 
+
+def SourceTraceLawUniqueTarget : Prop :=
+  ∀ {Ω : Type*} [_m : MeasurableSpace Ω] {P : Measure Ω} (κ : ℝ), 0 < κ → κ < 8 →
+  ∀ (B B' : ℝ≥0 → Ω → ℝ), IsBrownianReal B P → IsBrownianReal B' P →
+  ∀ (_hm : ∀ t, Measurable (B t)) (_hm' : ∀ t, Measurable (B' t))
+    (_hc : ∀ ω, Continuous (fun t => B t ω)) (_hc' : ∀ ω, Continuous (fun t => B' t ω))
+    (γ η : Ω → ℝ≥0 → ℂ) (G F G' F' : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hw' : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B' t ω) (η ω) (G' ω) (F' ω)),
+    P.map (sourceTracePath γ (fun ω => (hw ω).1)) =
+      P.map (sourceTracePath η (fun ω => (hw' ω).1))
+theorem sourceTraceLawUnique_proved : SourceTraceLawUniqueTarget := by
+  intro Ω m P κ hκ hκ8 B B' hB hB' hm hm' hc hc' γ η G F G' F' hw hw'
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let bp := sourceDriverPath B hc
+  let rp := sourceDriverPath B' hc'
+  let gp := sourceTracePath γ (fun ω => (hw ω).1)
+  let ep := sourceTracePath η (fun ω => (hw' ω).1)
+  have hbm : Measurable bp := source_path_measurable bp hm
+  have hrm : Measurable rp := source_path_measurable rp hm'
+  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t ω => gp ω t) := by
+    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω)
+      ?_ (fun ω => continuous_const.mul (hc ω)) γ G F hw
+    intro t
+    exact measurable_const.mul
+      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
+  obtain ⟨Φ,hΦm,hΦ⟩ := source_trace_factor B hm hc gp hγa
+  have hΦe (ω : Ω) : Φ (bp ω) = gp ω := (congrFun hΦ ω).symm
+  have hlaw : P.map bp = P.map rp := source_brownian_continuous_law_unique B B' hB hB' hm hm' hc hc'
+  choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) =>
+    source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity)
+  let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1)
+  have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
+  have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) :=
+    tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, Eventually.of_forall ha⟩
+  have hQorig (ω : Ω) (t : ℝ≥0) (n : ℕ) :
+      Q t n t (bp ω) = F ω t ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) := by
+    apply (hw ω).reverse_flow_identifies t
+      ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
+      (fun s => Q t n s (bp ω)) (hQ0 t n (bp ω))
+    intro s hs
+    exact (hQd t n (bp ω) s hs.1).mono Icc_subset_Ici_self
+  have hQnew (ω : Ω) (t : ℝ≥0) (n : ℕ) :
+      Q t n t (rp ω) = F' ω t ((Real.sqrt κ * B' t ω : ℝ) + (a n : ℂ) * I) := by
+    apply (hw' ω).reverse_flow_identifies t
+      ((Real.sqrt κ * B' t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
+      (fun s => Q t n s (rp ω)) (hQ0 t n (rp ω))
+    intro s hs
+    exact (hQd t n (rp ω) s hs.1).mono Icc_subset_Ici_self
+  have hQlimit (ω : Ω) (t : ℝ≥0) :
+      Tendsto (fun n => Q t n t (bp ω)) atTop (𝓝 (Φ (bp ω) t)) := by
+    have ht := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy
+    simpa only [hQorig,hΦe,gp,sourceTracePath,ContinuousMap.coe_mk,Function.comp_def] using ht
+  have hset (t : ℝ≥0) : MeasurableSet
+      {p : C(ℝ≥0,ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} :=
+    measurableSet_tendsto (𝓝 (0:ℂ)) (fun n => (hQm t n t).sub
+      ((continuous_eval_const t).measurable.comp hΦm))
+  have hQR (t : ℝ≥0) : ∀ᵐ ω ∂P, Tendsto (fun n => Q t n t (rp ω)) atTop (𝓝 (Φ (rp ω) t)) := by
+    have ho : ∀ᵐ p ∂P.map bp, Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by
+      apply (ae_map_iff hbm.aemeasurable (hset t)).2
+      exact Eventually.of_forall (fun ω => tendsto_sub_nhds_zero_iff.mpr (hQlimit ω t))
+    rw [hlaw] at ho
+    filter_upwards [(ae_map_iff hrm.aemeasurable (hset t)).1 ho] with ω hω
+    exact tendsto_sub_nhds_zero_iff.mp hω
+  have hEQ : ∀ᵐ ω ∂P, Φ (rp ω) = ep ω := by
+    filter_upwards [ae_all_iff.mpr (fun j : ℕ => hQR (denseSeq ℝ≥0 j))] with ω hω
+    apply DFunLike.coe_injective
+    apply (denseRange_denseSeq ℝ≥0).equalizer (Φ (rp ω)).continuous (ep ω).continuous
+    funext j
+    have ht := ((hw' ω).2.2.2.2.2.2.2.2.2.2.2.2 (denseSeq ℝ≥0 j)).comp hy
+    have hn := hω j
+    simp only [hQnew] at hn
+    exact tendsto_nhds_unique hn ht
+  have he := congrArg (Measure.map Φ) hlaw
+  rw [Measure.map_map hΦm hbm,Measure.map_map hΦm hrm] at he
+  have hcomp : Φ ∘ bp = gp := hΦ.symm
+  rw [hcomp] at he
+  exact he.trans (Measure.map_congr hEQ)
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/AllPublishedMain.lean b/lean/OAI/Probability/SLE/AllPublishedMain.lean
index e11b9cb2f47a7c6f50c2d444782c9b9e793cf118..c4d825fd792734d33f13c574f716c3b4e27c859c 100644
--- a/lean/OAI/Probability/SLE/AllPublishedMain.lean
+++ b/lean/OAI/Probability/SLE/AllPublishedMain.lean
@@ -1,6 +1,6 @@
-import OAI.Probability.SLE.Geometry.CanonicalRestartDischarge
-import OAI.Probability.SLE.Geometry.ExplicitPublishedMain
-import OAI.Probability.SLE.Gauge.LargerGaugeMain
-
-/-! Exact Hausdorff gauges, physical multipoint estimates, and canonical-mass
-restart identities for ordinary chordal SLE. -/
+import OAI.Probability.SLE.Geometry.CanonicalRestartDischarge
+import OAI.Probability.SLE.Geometry.ExplicitPublishedMain
+import OAI.Probability.SLE.Gauge.LargerGaugeMain
+
+/-! Exact Hausdorff gauges, physical multipoint estimates, and canonical-mass
+restart identities for ordinary chordal SLE. -/
diff --git a/lean/OAI/Probability/SLE/Angle/ActualAngleSpectral.lean b/lean/OAI/Probability/SLE/Angle/ActualAngleSpectral.lean
index f373b8fa60c4893fdb2ff6fd3b8f3d0c10ae4c42..240992055f4f2084d1ab74ce886299ff03871188 100644
--- a/lean/OAI/Probability/SLE/Angle/ActualAngleSpectral.lean
+++ b/lean/OAI/Probability/SLE/Angle/ActualAngleSpectral.lean
@@ -1,69 +1,69 @@
-import OAI.Probability.SLE.Angle.AngleEigenMoment
-import OAI.Probability.SLE.Angle.AngleMomentUniqueness
-import OAI.Probability.SLE.Angle.AngleSpectralLaw
-
-namespace OAI
-
-/-! Identification of the actual finite centered angle law with the
-explicit positive spectral kernel at every sufficiently large radius time. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ}
-  (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B))
-  (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-  (hκ : 0<κ) (hκ8 : κ<8)
-  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-
-include hB hBm hc h0 hκ hκ8 γ G F hw
-
-lemma actual_angle_law_eq_spectral (z : ℂ) (hz : z∈upperHalfPlane)
-    (u : ℝ) (hu : angleSpectralMixingTime κ≤u) :
-    (finiteCenteredLaw P κ B γ G z u).map
-      (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z)=
-        angleSpectralLaw κ u z.arg := by
-  have hu0 : 0≤u := (angleSpectralMixingTime_pos hκ).le.trans hu
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let S : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
-  have hS := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu0
-  let a := extendedMarkedAngle κ B γ G S z
-  have ham : Measurable a :=
-    (extendedMarkedAngle_measurable B hm hc κ γ G F hw S hS z).mono hS.measurableSpace_le le_rfl
-  let Q := finiteCenteredLaw P κ B γ G z u
-  have : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F
-    (Eventually.of_forall hw) z hz u hu0
-  have : IsProbabilityMeasure (angleSpectralLaw κ u z.arg) := angleSpectralLaw_isProbability hκ hu z.arg
-  have hang : ∀ᵐ ω ∂Q, a ω∈Ioo (0:ℝ) Real.pi :=
-    finiteCenteredLaw_extended_angle_mem hB hc hκ hκ8 γ G F hw z hz u u hu0 le_rfl
-  have hfinite : ∀ᵐ ω ∂Q, radiusLevelTime (γ ω) (G ω) z u<∞ := by
-    have hh := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F
-      (Eventually.of_forall hw) z hz u u hu0 le_rfl S (Eventually.of_forall fun _ => le_rfl)
-    exact hh.mono fun ω hω => lt_top_iff_ne_top.mpr hω
-  apply angleMeasures_eq_of_eigenMoments κ
-  · exact (ae_map_iff ham.aemeasurable measurableSet_Icc).mpr (hang.mono fun _ h => ⟨h.1.le,h.2.le⟩)
-  · exact (angleSpectralLaw_ae_mem z.arg).mono fun _ h => ⟨h.1.le,h.2.le⟩
-  · intro n
-    rw [integral_map ham.aemeasurable
-      (anglePolynomialTest_integrable (Q.map a) (angleEigenPolynomial κ n)).aestronglyMeasurable,
-      angleSpectralLaw_eigen_moment hκ hu]
-    change (∫ ω, (angleEigenPolynomial κ n).eval (Real.cos (a ω)) ∂Q)=_
-    calc
-      _ = ∫ ω, (angleEigenPolynomial κ n).eval (Real.cos
-          ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
-            (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg)) ∂Q := by
-        apply integral_congr_ae
-        filter_upwards [hfinite] with ω hω
-        rw [show a ω= _ from extendedMarkedAngle_at_finite_radius κ B hc γ G F hw z u ω hω]
-      _ = _ := finiteCenteredLaw_eigen_moment hB hBm hc h0 hκ hκ8 γ G F
-        (Eventually.of_forall hw) z hz u hu0 n
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenMoment
+import OAI.Probability.SLE.Angle.AngleMomentUniqueness
+import OAI.Probability.SLE.Angle.AngleSpectralLaw
+
+namespace OAI
+
+/-! Identification of the actual finite centered angle law with the
+explicit positive spectral kernel at every sufficiently large radius time. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ}
+  (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B))
+  (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+  (hκ : 0<κ) (hκ8 : κ<8)
+  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+
+include hB hBm hc h0 hκ hκ8 γ G F hw
+
+lemma actual_angle_law_eq_spectral (z : ℂ) (hz : z∈upperHalfPlane)
+    (u : ℝ) (hu : angleSpectralMixingTime κ≤u) :
+    (finiteCenteredLaw P κ B γ G z u).map
+      (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z)=
+        angleSpectralLaw κ u z.arg := by
+  have hu0 : 0≤u := (angleSpectralMixingTime_pos hκ).le.trans hu
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let S : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
+  have hS := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu0
+  let a := extendedMarkedAngle κ B γ G S z
+  have ham : Measurable a :=
+    (extendedMarkedAngle_measurable B hm hc κ γ G F hw S hS z).mono hS.measurableSpace_le le_rfl
+  let Q := finiteCenteredLaw P κ B γ G z u
+  have : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F
+    (Eventually.of_forall hw) z hz u hu0
+  have : IsProbabilityMeasure (angleSpectralLaw κ u z.arg) := angleSpectralLaw_isProbability hκ hu z.arg
+  have hang : ∀ᵐ ω ∂Q, a ω∈Ioo (0:ℝ) Real.pi :=
+    finiteCenteredLaw_extended_angle_mem hB hc hκ hκ8 γ G F hw z hz u u hu0 le_rfl
+  have hfinite : ∀ᵐ ω ∂Q, radiusLevelTime (γ ω) (G ω) z u<∞ := by
+    have hh := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F
+      (Eventually.of_forall hw) z hz u u hu0 le_rfl S (Eventually.of_forall fun _ => le_rfl)
+    exact hh.mono fun ω hω => lt_top_iff_ne_top.mpr hω
+  apply angleMeasures_eq_of_eigenMoments κ
+  · exact (ae_map_iff ham.aemeasurable measurableSet_Icc).mpr (hang.mono fun _ h => ⟨h.1.le,h.2.le⟩)
+  · exact (angleSpectralLaw_ae_mem z.arg).mono fun _ h => ⟨h.1.le,h.2.le⟩
+  · intro n
+    rw [integral_map ham.aemeasurable
+      (anglePolynomialTest_integrable (Q.map a) (angleEigenPolynomial κ n)).aestronglyMeasurable,
+      angleSpectralLaw_eigen_moment hκ hu]
+    change (∫ ω, (angleEigenPolynomial κ n).eval (Real.cos (a ω)) ∂Q)=_
+    calc
+      _ = ∫ ω, (angleEigenPolynomial κ n).eval (Real.cos
+          ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
+            (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg)) ∂Q := by
+        apply integral_congr_ae
+        filter_upwards [hfinite] with ω hω
+        rw [show a ω= _ from extendedMarkedAngle_at_finite_radius κ B hc γ G F hw z u ω hω]
+      _ = _ := finiteCenteredLaw_eigen_moment hB hBm hc h0 hκ hκ8 γ G F
+        (Eventually.of_forall hw) z hz u hu0 n
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/ActualAngleWarmup.lean b/lean/OAI/Probability/SLE/Angle/ActualAngleWarmup.lean
index 41d42799324705d47c47614d3af5272a072298cf..cb64a7e71052ccf1424cdb4152b52a9fde100306 100644
--- a/lean/OAI/Probability/SLE/Angle/ActualAngleWarmup.lean
+++ b/lean/OAI/Probability/SLE/Angle/ActualAngleWarmup.lean
@@ -1,250 +1,250 @@
-import OAI.Probability.SLE.Loewner.CanonicalInverseAngle
-import OAI.Probability.SLE.Geometry.NormalizedCanonicalRestart
-import OAI.Probability.SLE.Geometry.StoppedMarkedStateMeasurability
-import OAI.Probability.SLE.Geometry.FiniteDensityAtRadiusState
-import OAI.Probability.SLE.Mass.RestartStoppedMass
-import OAI.Probability.SLE.Geometry.RestartWeightedKernel
-
-namespace OAI
-
-/-! A one-unit conformal-radius warmup has a uniformly positive chance of
-producing a good angle, conditionally on every actual stopped-past event.
-The estimate uses the canonical Loewner witness. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance warmupDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance warmupDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance warmupPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance warmupPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-lemma IsCapacityTwoWitness.normalized_restart_stopped_sine
-    {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a)
-    (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) β G' F')
-    (w : ℂ) (u v : ℝ)
-    (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) =
-      (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u))
-    (hfinite : radiusLevelTime β G' w v < ∞) :
-    radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + v) < ∞ ∧
-    Real.sin (Complex.arg (G (radiusLevelTime γ G
-        (F T ((a : ℂ) * w + (U T : ℂ))) (u + v)).toNNReal
-        (F T ((a : ℂ) * w + (U T : ℂ))) -
-      (U (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ)))
-        (u + v)).toNNReal : ℂ))) =
-      Real.sin (Complex.arg (G' (radiusLevelTime β G' w v).toNNReal w -
-        (normalizedRestartDriver U T a (radiusLevelTime β G' w v).toNNReal : ℂ))) := by
-  have hV : Continuous (normalizedRestartDriver U T a) := by
-    unfold normalizedRestartDriver
-    fun_prop
-  obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top β G' w v hfinite
-  have hs := radiusLevelTime_eq_of_live_level hV hb w v t ht hR
-  have hp := (hw.normalized_restart_level_of_live_source hU T a ha hb w u v hbase t ht hR).2
-  refine ⟨by rw [hp]; exact ENNReal.coe_lt_top, ?_⟩
-  rw [hp, hs, ENNReal.toNNReal_coe, ENNReal.toNNReal_coe]
-  have hbup := normalized_restart_witness T a ha hb
-  have haC : (a : ℂ) ≠ 0 := by exact_mod_cast ha.ne'
-  have htu : (a : ℂ) * w ∈ survivingDomain (sourceScaleTrace a β) (a ^ 2 * t) := by
-    rw [source_scale_domain_iff a _ ha β, source_scale_time_mul_sq a t ha,
-      mul_div_cancel_left₀ w haC]
-    exact ht
-  have hc := hw.restart_centered_value T hbup (a ^ 2 * t) ((a : ℂ) * w) htu
-  rw [sourceScaleChart, source_scale_time_mul_sq a t ha,
-    mul_div_cancel_left₀ w haC] at hc
-  have hscale : (a : ℂ) * G' t w - ((U (T + a ^ 2 * t) - U T : ℝ) : ℂ) =
-      (a : ℂ) * (G' t w - (normalizedRestartDriver U T a t : ℂ)) := by
-    rw [normalizedRestartDriver, Complex.ofReal_div]
-    field_simp
-  rw [hc, hscale, sin_arg_pos_real_mul (a : ℝ) ha]
-
-/-- The canonical measurable event is exactly the geometric stopped-angle
-condition for the supplied actual witness. -/
-lemma canonical_good_angle_source_eq
-    {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (u s : ℝ) (hu : 0 ≤ u) :
-    {ω | canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω) < ∞ ∧
-      s ≤ canonicalStoppedSine κ (u, z, sourceDriverPath B hc ω)} =
-    {ω | radiusLevelTime (γ ω) (G ω) z u < ∞ ∧
-      s ≤ Real.sin (Complex.arg (G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z -
-        (Real.sqrt κ * B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω : ℝ)))} := by
-  ext ω
-  simp only [mem_ofPred_eq, canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) u hu z]
-  by_cases hf : radiusLevelTime (γ ω) (G ω) z u < ∞
-  · rw [canonicalStoppedSine_eq κ (sourceDriverPath B hc ω) (hw ω) u hu z hf]
-    rfl
-  · simp only [hf, false_and]
-
-/-- The actual finite centered law is compared on every stopped-past event.
-The scale and canonical witness may be fixed before choosing that event. -/
-theorem actual_one_unit_angle_warmup
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ))
-    (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ)
-    (hW : ∀ᵐ p ∂P.map (sourceDriverPath B hc), IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) :
-    ∃ s : ℝ, 0 < s ∧ ∀ (τ : Ω → ℝ≥0)
-      (hτ : IsStoppingTime (Filtration.natural B
-        (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable))
-        (fun ω => (τ ω : WithTop ℝ≥0)))
-      (z : ℂ), z ∈ upperHalfPlane → ∀ (u : ℝ), 0 ≤ u →
-      ∀ (E : Set Ω), MeasurableSet[hτ.measurableSpace] E →
-      (∀ ω ∈ E, z ∈ survivingDomain (γ ω) (τ ω)) →
-      (∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u)) →
-      ∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
-        ENNReal.ofReal (1 / 2 : ℝ) * finiteCenteredLaw P κ B γ G z (u + 1) (A ∩ E) ≤
-          finiteCenteredLaw P κ B γ G z (u + 1)
-            (A ∩ E ∩ {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
-              s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}) := by
-  classical
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let bp := sourceDriverPath B hc
-  let ν := P.map bp
-  have hbpm : Measurable bp := source_path_measurable _ hm
-  have : IsProbabilityMeasure ν :=
-    (Measure.isProbabilityMeasure_map_iff hbpm.aemeasurable).mpr inferInstance
-  obtain ⟨s, hs, hsections⟩ := canonical_uniform_good_angle ν
-    (canonical_driver_isBrownian B hB hm hc) κ hκ hκ8 Φ Gc Fc hW
-  refine ⟨s, hs, ?_⟩
-  intro τ hτ z hz u hu E hE hlive hbase A hA
-  let a := stoppedMarkedScale κ B γ G τ z
-  let w := stoppedMarkedUnit κ B γ G τ z
-  let Y := normalizedRestartDriverPath B hc τ a
-  have ham : Measurable[hτ.measurableSpace] a :=
-    source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z
-  have ha : ∀ ω, 0 < a ω := stoppedMarkedScale_pos κ B γ G τ z
-  have hwm : Measurable[hτ.measurableSpace] w :=
-    source_stoppedMarkedUnit_measurable B hm hc κ γ G F hw τ hτ z
-  have hYm : Measurable Y := by
-    have hrm : Measurable (sourceRestartDriverPath B hc τ) :=
-      source_path_measurable _ (fun t => (measurable_pi_apply t).comp
-        (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
-    exact measurable_scaleDriverPath.comp
-      ((ham.inv.mono hτ.measurableSpace_le le_rfl).prodMk hrm)
-  have hYlaw (D : Set Ω) (hD : MeasurableSet[hτ.measurableSpace] D) :
-      (P.restrict D).map Y = P D • ν :=
-    source_restart_random_scale_driver_law B hB hm hc τ hτ
-      (fun ω => (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) D hD
-  have hYlaw₀ : P.map Y = ν := by
-    simpa only [Measure.restrict_univ, measure_univ, one_smul] using hYlaw univ MeasurableSet.univ
-  have hYw := canonical_witness_pullback hW Y hYm hYlaw₀
-  have hgeometry (ω : Ω) (hω : ω ∈ E) :=
-    source_stoppedMarkedState_geometry B hc κ γ G F hw τ z ω (hlive ω hω)
-  have hpoint (ω : Ω) (hω : ω ∈ E) :
-      F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ)) = z := by
-    simpa only [normalizedInverse, add_comm] using (hgeometry ω hω).2.2.1
-  obtain ⟨M, hM, hMfinite, hMactual, hMpast⟩ :=
-    exists_finiteDensity_at_radius_state hB hBm hc hB0 κ hκ hκ8 γ G F hw
-      τ hτ z hz u 1 hu (by norm_num) E hE hlive hbase
-  let C : Set (Ω × C(ℝ≥0, ℝ)) := {q | q.1 ∈ E ∧
-    canonicalRadiusLevel κ (1, w q.1, q.2) < ∞ ∧
-      s ≤ canonicalStoppedSine κ (1, w q.1, q.2)}
-  let K : Ω × C(ℝ≥0, ℝ) → ℝ≥0∞ :=
-    fun q => M q.1 * canonicalFiniteDensity κ (1, w q.1, q.2)
-  let S : Set Ω := E ∩ {ω | canonicalRadiusLevel κ (u + 1, z, bp ω) < ∞ ∧
-    s ≤ canonicalStoppedSine κ (u + 1, z, bp ω)}
-  let Z : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal
-    (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z (u + 1) / initialGreen κ z)
-  have hCm : @MeasurableSet (Ω × C(ℝ≥0, ℝ)) (hτ.measurableSpace.prod _) C :=
-    (hE.preimage measurable_fst).inter
-      (@measurableSet_canonical_good_angle_random_state (Ω × C(ℝ≥0, ℝ))
-        (hτ.measurableSpace.prod _) κ s _ measurable_snd _ measurable_const _
-        (hwm.comp measurable_fst))
-  have hKm : @Measurable (Ω × C(ℝ≥0, ℝ)) ℝ≥0∞ (hτ.measurableSpace.prod _) _ K :=
-    (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp
-      (measurable_const.prodMk ((hwm.comp measurable_fst).prodMk measurable_snd)))
-  have hSm : MeasurableSet S := (hτ.measurableSpace_le E hE).inter
-    (measurableSet_canonical_good_angle_random_state κ s hbpm measurable_const measurable_const)
-  have hsec : ∀ᵐ ω ∂P, ENNReal.ofReal (1 / 2 : ℝ) * E.indicator M ω ≤
-      ∫⁻ p, C.indicator K (ω, p) ∂ν := by
-    apply Eventually.of_forall
-    intro ω
-    by_cases hω : ω ∈ E
-    · let D : Set C(ℝ≥0, ℝ) := {p | canonicalRadiusLevel κ (1, w ω, p) < ∞ ∧
-        s ≤ canonicalStoppedSine κ (1, w ω, p)}
-      have hDm : MeasurableSet D := measurableSet_canonical_good_angle_random_state κ s
-        measurable_id measurable_const measurable_const
-      have he : (fun p => C.indicator K (ω, p)) =
-          fun p => M ω * D.indicator (fun p => canonicalFiniteDensity κ (1, w ω, p)) p := by
-        funext p
-        by_cases hp : p ∈ D <;> simp only [C, K, D, mem_ofPred_eq] at hp ⊢ <;>
-          simp [hω, hp]
-      rw [he, lintegral_const_mul' _ _ (hMfinite ω), indicator_of_mem hω,
-        lintegral_indicator hDm, ← withDensity_apply _ hDm]
-      have hb := hsections (w ω) (hgeometry ω hω).2.1 1 (by norm_num)
-      change ENNReal.ofReal (1 / 2 : ℝ) ≤
-        (ν.withDensity (fun p => canonicalFiniteDensity κ (1, w ω, p))) D at hb
-      calc
-        _ = M ω * ENNReal.ofReal (1 / 2 : ℝ) := mul_comm _ _
-        _ ≤ _ := mul_le_mul' le_rfl hb
-    · simp [C, hω]
-  have hactual : ∀ᵐ ω ∂P, C.indicator K (ω, Y ω) ≤ S.indicator Z ω := by
-    filter_upwards [hYw, hMactual] with ω hy hMω
-    by_cases hC : (ω, Y ω) ∈ C
-    · rcases hC with ⟨hωE, hf, hsine⟩
-      have hdriver := normalizedRestartDriverPath_driver κ B hc τ a ω
-      have hy' : IsCapacityTwoWitness
-          (normalizedRestartDriver (fun t => Real.sqrt κ * B t ω) (τ ω) (a ω))
-          (Φ (Y ω)) (Gc (Y ω)) (Fc (Y ω)) := hdriver ▸ hy
-      have hf' : radiusLevelTime (Φ (Y ω)) (Gc (Y ω)) (w ω) 1 < ∞ := by
-        rwa [canonicalRadiusLevel_eq κ (Y ω) hy 1 (by norm_num) (w ω)] at hf
-      have hbase' : fullRadius (G ω) (τ ω)
-          (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))) =
-          (2 * (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))).im) *
-            Real.exp (-u) := by simpa only [hpoint ω hωE] using hbase ω hωE
-      have hang := (hw ω).normalized_restart_stopped_sine ((hc ω).const_mul _)
-        (τ ω) (a ω) (ha ω) hy' (w ω) u 1 hbase' hf'
-      rw [hpoint ω hωE] at hang
-      have hphysical : ω ∈ S := by
-        refine ⟨hωE, ?_, ?_⟩
-        · rw [canonicalRadiusLevel_eq κ (bp ω) (hw ω) (u + 1) (by linarith) z]
-          exact hang.1
-        · rw [canonicalStoppedSine_eq κ (bp ω) (hw ω) (u + 1) (by linarith) z hang.1]
-          change s ≤ Real.sin (Complex.arg (G ω (radiusLevelTime (γ ω) (G ω) z
-            (u + 1)).toNNReal z - (Real.sqrt κ * B (radiusLevelTime (γ ω) (G ω) z
-              (u + 1)).toNNReal ω : ℝ)))
-          rw [hang.2]
-          rw [canonicalStoppedSine_eq κ (Y ω) hy 1 (by norm_num) (w ω) hf'] at hsine
-          rw [← hdriver]
-          exact hsine
-      rw [indicator_of_mem (show (ω, Y ω) ∈ C from ⟨hωE, hf, hsine⟩),
-        indicator_of_mem hphysical]
-      change M ω * canonicalFiniteDensity κ (1, w ω, Y ω) ≤ Z ω
-      rw [hMω hωE, canonicalFiniteDensity_eq κ (Y ω) hy 1 (by norm_num) (w ω), hdriver,
-        ← ENNReal.ofReal_mul
-          (div_nonneg (greenDensity_nonneg (hw ω) κ (τ ω) z) (initialGreen_pos κ z hz).le)]
-      apply ENNReal.ofReal_le_ofReal
-      have hbound := (hw ω).normalized_restart_stopped_green_lower ((hc ω).const_mul _)
-        (τ ω) (a ω) (ha ω) hy' κ (w ω) (hgeometry ω hωE).2.1 u 1 hbase'
-      rw [hpoint ω hωE] at hbound
-      calc
-        _ = (greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (τ ω) z *
-          (stoppedGreen κ (normalizedRestartDriver (fun t => Real.sqrt κ * B t ω)
-            (τ ω) (a ω)) (Φ (Y ω)) (Gc (Y ω)) (w ω) 1 / initialGreen κ (w ω))) /
-            initialGreen κ z := by ring
-        _ ≤ _ := div_le_div_of_nonneg_right hbound (initialGreen_pos κ z hz).le
-    · rw [indicator_of_notMem hC]
-      exact bot_le
-  have h := restart_weighted_event_lower_bound P ν hτ.measurableSpace hτ.measurableSpace_le
-    Y hYm hYlaw K hKm C hCm (E.indicator M) Z (ENNReal.ofReal (1 / 2 : ℝ))
-    ENNReal.ofReal_ne_top S hSm hsec hactual A hA
-  rw [setLIntegral_indicator (hτ.measurableSpace_le E hE), inter_comm E A, hMpast A hA] at h
-  simpa only [Z, S, bp, finiteCenteredLaw, ← inter_assoc] using h
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.CanonicalInverseAngle
+import OAI.Probability.SLE.Geometry.NormalizedCanonicalRestart
+import OAI.Probability.SLE.Geometry.StoppedMarkedStateMeasurability
+import OAI.Probability.SLE.Geometry.FiniteDensityAtRadiusState
+import OAI.Probability.SLE.Mass.RestartStoppedMass
+import OAI.Probability.SLE.Geometry.RestartWeightedKernel
+
+namespace OAI
+
+/-! A one-unit conformal-radius warmup has a uniformly positive chance of
+producing a good angle, conditionally on every actual stopped-past event.
+The estimate uses the canonical Loewner witness. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance warmupDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance warmupDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance warmupPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance warmupPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+lemma IsCapacityTwoWitness.normalized_restart_stopped_sine
+    {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a)
+    (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) β G' F')
+    (w : ℂ) (u v : ℝ)
+    (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) =
+      (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u))
+    (hfinite : radiusLevelTime β G' w v < ∞) :
+    radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + v) < ∞ ∧
+    Real.sin (Complex.arg (G (radiusLevelTime γ G
+        (F T ((a : ℂ) * w + (U T : ℂ))) (u + v)).toNNReal
+        (F T ((a : ℂ) * w + (U T : ℂ))) -
+      (U (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ)))
+        (u + v)).toNNReal : ℂ))) =
+      Real.sin (Complex.arg (G' (radiusLevelTime β G' w v).toNNReal w -
+        (normalizedRestartDriver U T a (radiusLevelTime β G' w v).toNNReal : ℂ))) := by
+  have hV : Continuous (normalizedRestartDriver U T a) := by
+    unfold normalizedRestartDriver
+    fun_prop
+  obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top β G' w v hfinite
+  have hs := radiusLevelTime_eq_of_live_level hV hb w v t ht hR
+  have hp := (hw.normalized_restart_level_of_live_source hU T a ha hb w u v hbase t ht hR).2
+  refine ⟨by rw [hp]; exact ENNReal.coe_lt_top, ?_⟩
+  rw [hp, hs, ENNReal.toNNReal_coe, ENNReal.toNNReal_coe]
+  have hbup := normalized_restart_witness T a ha hb
+  have haC : (a : ℂ) ≠ 0 := by exact_mod_cast ha.ne'
+  have htu : (a : ℂ) * w ∈ survivingDomain (sourceScaleTrace a β) (a ^ 2 * t) := by
+    rw [source_scale_domain_iff a _ ha β, source_scale_time_mul_sq a t ha,
+      mul_div_cancel_left₀ w haC]
+    exact ht
+  have hc := hw.restart_centered_value T hbup (a ^ 2 * t) ((a : ℂ) * w) htu
+  rw [sourceScaleChart, source_scale_time_mul_sq a t ha,
+    mul_div_cancel_left₀ w haC] at hc
+  have hscale : (a : ℂ) * G' t w - ((U (T + a ^ 2 * t) - U T : ℝ) : ℂ) =
+      (a : ℂ) * (G' t w - (normalizedRestartDriver U T a t : ℂ)) := by
+    rw [normalizedRestartDriver, Complex.ofReal_div]
+    field_simp
+  rw [hc, hscale, sin_arg_pos_real_mul (a : ℝ) ha]
+
+/-- The canonical measurable event is exactly the geometric stopped-angle
+condition for the supplied actual witness. -/
+lemma canonical_good_angle_source_eq
+    {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (u s : ℝ) (hu : 0 ≤ u) :
+    {ω | canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω) < ∞ ∧
+      s ≤ canonicalStoppedSine κ (u, z, sourceDriverPath B hc ω)} =
+    {ω | radiusLevelTime (γ ω) (G ω) z u < ∞ ∧
+      s ≤ Real.sin (Complex.arg (G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z -
+        (Real.sqrt κ * B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω : ℝ)))} := by
+  ext ω
+  simp only [mem_ofPred_eq, canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) u hu z]
+  by_cases hf : radiusLevelTime (γ ω) (G ω) z u < ∞
+  · rw [canonicalStoppedSine_eq κ (sourceDriverPath B hc ω) (hw ω) u hu z hf]
+    rfl
+  · simp only [hf, false_and]
+
+/-- The actual finite centered law is compared on every stopped-past event.
+The scale and canonical witness may be fixed before choosing that event. -/
+theorem actual_one_unit_angle_warmup
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ))
+    (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ)
+    (hW : ∀ᵐ p ∂P.map (sourceDriverPath B hc), IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) :
+    ∃ s : ℝ, 0 < s ∧ ∀ (τ : Ω → ℝ≥0)
+      (hτ : IsStoppingTime (Filtration.natural B
+        (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable))
+        (fun ω => (τ ω : WithTop ℝ≥0)))
+      (z : ℂ), z ∈ upperHalfPlane → ∀ (u : ℝ), 0 ≤ u →
+      ∀ (E : Set Ω), MeasurableSet[hτ.measurableSpace] E →
+      (∀ ω ∈ E, z ∈ survivingDomain (γ ω) (τ ω)) →
+      (∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u)) →
+      ∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
+        ENNReal.ofReal (1 / 2 : ℝ) * finiteCenteredLaw P κ B γ G z (u + 1) (A ∩ E) ≤
+          finiteCenteredLaw P κ B γ G z (u + 1)
+            (A ∩ E ∩ {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
+              s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}) := by
+  classical
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let bp := sourceDriverPath B hc
+  let ν := P.map bp
+  have hbpm : Measurable bp := source_path_measurable _ hm
+  have : IsProbabilityMeasure ν :=
+    (Measure.isProbabilityMeasure_map_iff hbpm.aemeasurable).mpr inferInstance
+  obtain ⟨s, hs, hsections⟩ := canonical_uniform_good_angle ν
+    (canonical_driver_isBrownian B hB hm hc) κ hκ hκ8 Φ Gc Fc hW
+  refine ⟨s, hs, ?_⟩
+  intro τ hτ z hz u hu E hE hlive hbase A hA
+  let a := stoppedMarkedScale κ B γ G τ z
+  let w := stoppedMarkedUnit κ B γ G τ z
+  let Y := normalizedRestartDriverPath B hc τ a
+  have ham : Measurable[hτ.measurableSpace] a :=
+    source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z
+  have ha : ∀ ω, 0 < a ω := stoppedMarkedScale_pos κ B γ G τ z
+  have hwm : Measurable[hτ.measurableSpace] w :=
+    source_stoppedMarkedUnit_measurable B hm hc κ γ G F hw τ hτ z
+  have hYm : Measurable Y := by
+    have hrm : Measurable (sourceRestartDriverPath B hc τ) :=
+      source_path_measurable _ (fun t => (measurable_pi_apply t).comp
+        (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
+    exact measurable_scaleDriverPath.comp
+      ((ham.inv.mono hτ.measurableSpace_le le_rfl).prodMk hrm)
+  have hYlaw (D : Set Ω) (hD : MeasurableSet[hτ.measurableSpace] D) :
+      (P.restrict D).map Y = P D • ν :=
+    source_restart_random_scale_driver_law B hB hm hc τ hτ
+      (fun ω => (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) D hD
+  have hYlaw₀ : P.map Y = ν := by
+    simpa only [Measure.restrict_univ, measure_univ, one_smul] using hYlaw univ MeasurableSet.univ
+  have hYw := canonical_witness_pullback hW Y hYm hYlaw₀
+  have hgeometry (ω : Ω) (hω : ω ∈ E) :=
+    source_stoppedMarkedState_geometry B hc κ γ G F hw τ z ω (hlive ω hω)
+  have hpoint (ω : Ω) (hω : ω ∈ E) :
+      F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ)) = z := by
+    simpa only [normalizedInverse, add_comm] using (hgeometry ω hω).2.2.1
+  obtain ⟨M, hM, hMfinite, hMactual, hMpast⟩ :=
+    exists_finiteDensity_at_radius_state hB hBm hc hB0 κ hκ hκ8 γ G F hw
+      τ hτ z hz u 1 hu (by norm_num) E hE hlive hbase
+  let C : Set (Ω × C(ℝ≥0, ℝ)) := {q | q.1 ∈ E ∧
+    canonicalRadiusLevel κ (1, w q.1, q.2) < ∞ ∧
+      s ≤ canonicalStoppedSine κ (1, w q.1, q.2)}
+  let K : Ω × C(ℝ≥0, ℝ) → ℝ≥0∞ :=
+    fun q => M q.1 * canonicalFiniteDensity κ (1, w q.1, q.2)
+  let S : Set Ω := E ∩ {ω | canonicalRadiusLevel κ (u + 1, z, bp ω) < ∞ ∧
+    s ≤ canonicalStoppedSine κ (u + 1, z, bp ω)}
+  let Z : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal
+    (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z (u + 1) / initialGreen κ z)
+  have hCm : @MeasurableSet (Ω × C(ℝ≥0, ℝ)) (hτ.measurableSpace.prod _) C :=
+    (hE.preimage measurable_fst).inter
+      (@measurableSet_canonical_good_angle_random_state (Ω × C(ℝ≥0, ℝ))
+        (hτ.measurableSpace.prod _) κ s _ measurable_snd _ measurable_const _
+        (hwm.comp measurable_fst))
+  have hKm : @Measurable (Ω × C(ℝ≥0, ℝ)) ℝ≥0∞ (hτ.measurableSpace.prod _) _ K :=
+    (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp
+      (measurable_const.prodMk ((hwm.comp measurable_fst).prodMk measurable_snd)))
+  have hSm : MeasurableSet S := (hτ.measurableSpace_le E hE).inter
+    (measurableSet_canonical_good_angle_random_state κ s hbpm measurable_const measurable_const)
+  have hsec : ∀ᵐ ω ∂P, ENNReal.ofReal (1 / 2 : ℝ) * E.indicator M ω ≤
+      ∫⁻ p, C.indicator K (ω, p) ∂ν := by
+    apply Eventually.of_forall
+    intro ω
+    by_cases hω : ω ∈ E
+    · let D : Set C(ℝ≥0, ℝ) := {p | canonicalRadiusLevel κ (1, w ω, p) < ∞ ∧
+        s ≤ canonicalStoppedSine κ (1, w ω, p)}
+      have hDm : MeasurableSet D := measurableSet_canonical_good_angle_random_state κ s
+        measurable_id measurable_const measurable_const
+      have he : (fun p => C.indicator K (ω, p)) =
+          fun p => M ω * D.indicator (fun p => canonicalFiniteDensity κ (1, w ω, p)) p := by
+        funext p
+        by_cases hp : p ∈ D <;> simp only [C, K, D, mem_ofPred_eq] at hp ⊢ <;>
+          simp [hω, hp]
+      rw [he, lintegral_const_mul' _ _ (hMfinite ω), indicator_of_mem hω,
+        lintegral_indicator hDm, ← withDensity_apply _ hDm]
+      have hb := hsections (w ω) (hgeometry ω hω).2.1 1 (by norm_num)
+      change ENNReal.ofReal (1 / 2 : ℝ) ≤
+        (ν.withDensity (fun p => canonicalFiniteDensity κ (1, w ω, p))) D at hb
+      calc
+        _ = M ω * ENNReal.ofReal (1 / 2 : ℝ) := mul_comm _ _
+        _ ≤ _ := mul_le_mul' le_rfl hb
+    · simp [C, hω]
+  have hactual : ∀ᵐ ω ∂P, C.indicator K (ω, Y ω) ≤ S.indicator Z ω := by
+    filter_upwards [hYw, hMactual] with ω hy hMω
+    by_cases hC : (ω, Y ω) ∈ C
+    · rcases hC with ⟨hωE, hf, hsine⟩
+      have hdriver := normalizedRestartDriverPath_driver κ B hc τ a ω
+      have hy' : IsCapacityTwoWitness
+          (normalizedRestartDriver (fun t => Real.sqrt κ * B t ω) (τ ω) (a ω))
+          (Φ (Y ω)) (Gc (Y ω)) (Fc (Y ω)) := hdriver ▸ hy
+      have hf' : radiusLevelTime (Φ (Y ω)) (Gc (Y ω)) (w ω) 1 < ∞ := by
+        rwa [canonicalRadiusLevel_eq κ (Y ω) hy 1 (by norm_num) (w ω)] at hf
+      have hbase' : fullRadius (G ω) (τ ω)
+          (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))) =
+          (2 * (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))).im) *
+            Real.exp (-u) := by simpa only [hpoint ω hωE] using hbase ω hωE
+      have hang := (hw ω).normalized_restart_stopped_sine ((hc ω).const_mul _)
+        (τ ω) (a ω) (ha ω) hy' (w ω) u 1 hbase' hf'
+      rw [hpoint ω hωE] at hang
+      have hphysical : ω ∈ S := by
+        refine ⟨hωE, ?_, ?_⟩
+        · rw [canonicalRadiusLevel_eq κ (bp ω) (hw ω) (u + 1) (by linarith) z]
+          exact hang.1
+        · rw [canonicalStoppedSine_eq κ (bp ω) (hw ω) (u + 1) (by linarith) z hang.1]
+          change s ≤ Real.sin (Complex.arg (G ω (radiusLevelTime (γ ω) (G ω) z
+            (u + 1)).toNNReal z - (Real.sqrt κ * B (radiusLevelTime (γ ω) (G ω) z
+              (u + 1)).toNNReal ω : ℝ)))
+          rw [hang.2]
+          rw [canonicalStoppedSine_eq κ (Y ω) hy 1 (by norm_num) (w ω) hf'] at hsine
+          rw [← hdriver]
+          exact hsine
+      rw [indicator_of_mem (show (ω, Y ω) ∈ C from ⟨hωE, hf, hsine⟩),
+        indicator_of_mem hphysical]
+      change M ω * canonicalFiniteDensity κ (1, w ω, Y ω) ≤ Z ω
+      rw [hMω hωE, canonicalFiniteDensity_eq κ (Y ω) hy 1 (by norm_num) (w ω), hdriver,
+        ← ENNReal.ofReal_mul
+          (div_nonneg (greenDensity_nonneg (hw ω) κ (τ ω) z) (initialGreen_pos κ z hz).le)]
+      apply ENNReal.ofReal_le_ofReal
+      have hbound := (hw ω).normalized_restart_stopped_green_lower ((hc ω).const_mul _)
+        (τ ω) (a ω) (ha ω) hy' κ (w ω) (hgeometry ω hωE).2.1 u 1 hbase'
+      rw [hpoint ω hωE] at hbound
+      calc
+        _ = (greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (τ ω) z *
+          (stoppedGreen κ (normalizedRestartDriver (fun t => Real.sqrt κ * B t ω)
+            (τ ω) (a ω)) (Φ (Y ω)) (Gc (Y ω)) (w ω) 1 / initialGreen κ (w ω))) /
+            initialGreen κ z := by ring
+        _ ≤ _ := div_le_div_of_nonneg_right hbound (initialGreen_pos κ z hz).le
+    · rw [indicator_of_notMem hC]
+      exact bot_le
+  have h := restart_weighted_event_lower_bound P ν hτ.measurableSpace hτ.measurableSpace_le
+    Y hYm hYlaw K hKm C hCm (E.indicator M) Z (ENNReal.ofReal (1 / 2 : ℝ))
+    ENNReal.ofReal_ne_top S hSm hsec hactual A hA
+  rw [setLIntegral_indicator (hτ.measurableSpace_le E hE), inter_comm E A, hMpast A hA] at h
+  simpa only [Z, S, bp, finiteCenteredLaw, ← inter_assoc] using h
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/ActualAngleWarmupExtended.lean b/lean/OAI/Probability/SLE/Angle/ActualAngleWarmupExtended.lean
index 5b704fcddc059cfb66803e1d01c291f1c9665e76..d371342362b362922f7321a117eadd73bd6cbb0c 100644
--- a/lean/OAI/Probability/SLE/Angle/ActualAngleWarmupExtended.lean
+++ b/lean/OAI/Probability/SLE/Angle/ActualAngleWarmupExtended.lean
@@ -1,112 +1,112 @@
-import OAI.Probability.SLE.Geometry.ExtendedMarkedAngle
-import OAI.Probability.SLE.Geometry.FiniteCenteredConsistency
-import OAI.Probability.SLE.Geometry.ConditionalStopExtension
-import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
-
-namespace OAI
-
-/-! The one-unit angular warmup at a literal, possibly infinite, radius
-stop. Exact consistency transfers the bound to every later finite centered
-law used by the cover experiment. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance extendedWarmupDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance extendedWarmupDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance extendedWarmupPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance extendedWarmupPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem actual_radius_angle_warmup
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ))
-    (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ)
-    (hW : ∀ᵐ p ∂P.map (sourceDriverPath B hc), IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) :
-    ∃ s : ℝ, 0 < s ∧ s ≤ 1 ∧ ∀ (z : ℂ) (hz : z ∈ upperHalfPlane)
-      (u : ℝ) (hu : 0 ≤ u) (V : ℝ), u + 1 ≤ V →
-      ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB
-        (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
-        κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A →
-        ENNReal.ofReal (1 / 2 : ℝ) * finiteCenteredLaw P κ B γ G z V A ≤
-          finiteCenteredLaw P κ B γ G z V
-            (A ∩ {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
-              s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}) := by
-  classical
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  obtain ⟨s₀, hs₀, hstep⟩ := actual_one_unit_angle_warmup P κ hκ hκ8 B hB hBm hc hB0
-    γ G F hw Φ Gc Fc hW
-  let s := min s₀ 1
-  have hs : 0 < s := lt_min hs₀ zero_lt_one
-  have hss₀ : s ≤ s₀ := min_le_left _ _
-  refine ⟨s, hs, min_le_right _ _, ?_⟩
-  intro z hz u hu V hV A hA
-  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
-  have hσ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u hu
-  let Q := finiteCenteredLaw P κ B γ G z (u + 1)
-  let S : Set Ω := {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
-    s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}
-  let E := A ∩ {ω | σ ω ≠ ⊤}
-  have hE : MeasurableSet[hσ.measurableSpace] E :=
-    hA.inter ((measurableSet_singleton (⊤ : WithTop ℝ≥0)).preimage hσ.measurable).compl
-  have hfinite : ∀ᵐ ω ∂Q, σ ω ≠ ⊤ :=
-    finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw)
-      z hz u (u + 1) hu (by linarith) σ (Eventually.of_forall fun _ => le_rfl)
-  have hEeq : E =ᵐ[Q] A := by
-    filter_upwards [hfinite] with ω hω
-    exact propext (and_iff_left hω)
-  let state : Ω → ℝ≥0 → Prop := fun ω t =>
-    z ∈ survivingDomain (γ ω) t ∧ fullRadius (G ω) t z = (2 * z.im) * Real.exp (-u)
-  have hlocal : ∀ (τ : Ω → ℝ≥0)
-      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (τ ω : WithTop ℝ≥0)))
-      (D : Set Ω), MeasurableSet[hτ.measurableSpace] D →
-      (∀ ω ∈ D, state ω (τ ω)) → ENNReal.ofReal (1 / 2 : ℝ) * Q D ≤ Q (D ∩ S) := by
-    intro τ hτ D hD hstate
-    have h := hstep τ hτ z hz u hu D hD (fun ω hω => (hstate ω hω).1)
-      (fun ω hω => (hstate ω hω).2) D hD
-    simp only [inter_self] at h
-    apply h.trans (measure_mono ?_)
-    intro ω hω
-    exact ⟨hω.1, hω.2.1, hss₀.trans hω.2.2⟩
-  have hstate : ∀ ω ∈ E, state ω ((σ ω).untopD 0) := by
-    intro ω hω
-    exact actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u hω.2
-  have hext := conditional_event_bound_at_extended_stop Q _ (ENNReal.ofReal (1 / 2 : ℝ))
-    S state hlocal σ hσ E hE (hfinite.mono fun _ h _ => h) hstate
-  have hbase : ENNReal.ofReal (1 / 2 : ℝ) * Q A ≤ Q (A ∩ S) := by
-    rw [measure_congr hEeq] at hext
-    have hES : E ∩ S =ᵐ[Q] A ∩ S := by
-      filter_upwards [hEeq] with ω hω
-      exact propext (and_congr_left fun _ => Iff.of_eq hω)
-    rwa [measure_congr hES] at hext
-  have hσ₁ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz
-    (u + 1) (by linarith)
-  have hA₁ : MeasurableSet[hσ₁.measurableSpace] A :=
-    actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u (u + 1)
-      hu (by linarith) A hA
-  have hS₁ : MeasurableSet[hσ₁.measurableSpace] S :=
-    canonical_good_angle_measurable_at_radius hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz
-      (u + 1) s (by linarith)
-  have hcons (D : Set Ω) (hD : MeasurableSet[hσ₁.measurableSpace] D) :
-      finiteCenteredLaw P κ B γ G z V D = Q D :=
-    finiteCenteredLaw_consistent_at_radius_stop hB hBm hc hB0 κ hκ hκ8 γ G F
-      (Eventually.of_forall hw) z hz (u + 1) V (by linarith) hV _ hσ₁
-      (Eventually.of_forall fun _ => rfl) D hD
-  rw [hcons A hA₁, hcons (A ∩ S) (hA₁.inter hS₁)]
-  exact hbase
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.ExtendedMarkedAngle
+import OAI.Probability.SLE.Geometry.FiniteCenteredConsistency
+import OAI.Probability.SLE.Geometry.ConditionalStopExtension
+import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
+
+namespace OAI
+
+/-! The one-unit angular warmup at a literal, possibly infinite, radius
+stop. Exact consistency transfers the bound to every later finite centered
+law used by the cover experiment. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance extendedWarmupDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance extendedWarmupDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance extendedWarmupPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance extendedWarmupPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem actual_radius_angle_warmup
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ))
+    (Gc Fc : C(ℝ≥0, ℝ) → ℝ≥0 → ℂ → ℂ)
+    (hW : ∀ᵐ p ∂P.map (sourceDriverPath B hc), IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * p t) (Φ p) (Gc p) (Fc p)) :
+    ∃ s : ℝ, 0 < s ∧ s ≤ 1 ∧ ∀ (z : ℂ) (hz : z ∈ upperHalfPlane)
+      (u : ℝ) (hu : 0 ≤ u) (V : ℝ), u + 1 ≤ V →
+      ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB
+        (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
+        κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A →
+        ENNReal.ofReal (1 / 2 : ℝ) * finiteCenteredLaw P κ B γ G z V A ≤
+          finiteCenteredLaw P κ B γ G z V
+            (A ∩ {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
+              s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}) := by
+  classical
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  obtain ⟨s₀, hs₀, hstep⟩ := actual_one_unit_angle_warmup P κ hκ hκ8 B hB hBm hc hB0
+    γ G F hw Φ Gc Fc hW
+  let s := min s₀ 1
+  have hs : 0 < s := lt_min hs₀ zero_lt_one
+  have hss₀ : s ≤ s₀ := min_le_left _ _
+  refine ⟨s, hs, min_le_right _ _, ?_⟩
+  intro z hz u hu V hV A hA
+  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
+  have hσ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u hu
+  let Q := finiteCenteredLaw P κ B γ G z (u + 1)
+  let S : Set Ω := {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
+    s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}
+  let E := A ∩ {ω | σ ω ≠ ⊤}
+  have hE : MeasurableSet[hσ.measurableSpace] E :=
+    hA.inter ((measurableSet_singleton (⊤ : WithTop ℝ≥0)).preimage hσ.measurable).compl
+  have hfinite : ∀ᵐ ω ∂Q, σ ω ≠ ⊤ :=
+    finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw)
+      z hz u (u + 1) hu (by linarith) σ (Eventually.of_forall fun _ => le_rfl)
+  have hEeq : E =ᵐ[Q] A := by
+    filter_upwards [hfinite] with ω hω
+    exact propext (and_iff_left hω)
+  let state : Ω → ℝ≥0 → Prop := fun ω t =>
+    z ∈ survivingDomain (γ ω) t ∧ fullRadius (G ω) t z = (2 * z.im) * Real.exp (-u)
+  have hlocal : ∀ (τ : Ω → ℝ≥0)
+      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (τ ω : WithTop ℝ≥0)))
+      (D : Set Ω), MeasurableSet[hτ.measurableSpace] D →
+      (∀ ω ∈ D, state ω (τ ω)) → ENNReal.ofReal (1 / 2 : ℝ) * Q D ≤ Q (D ∩ S) := by
+    intro τ hτ D hD hstate
+    have h := hstep τ hτ z hz u hu D hD (fun ω hω => (hstate ω hω).1)
+      (fun ω hω => (hstate ω hω).2) D hD
+    simp only [inter_self] at h
+    apply h.trans (measure_mono ?_)
+    intro ω hω
+    exact ⟨hω.1, hω.2.1, hss₀.trans hω.2.2⟩
+  have hstate : ∀ ω ∈ E, state ω ((σ ω).untopD 0) := by
+    intro ω hω
+    exact actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u hω.2
+  have hext := conditional_event_bound_at_extended_stop Q _ (ENNReal.ofReal (1 / 2 : ℝ))
+    S state hlocal σ hσ E hE (hfinite.mono fun _ h _ => h) hstate
+  have hbase : ENNReal.ofReal (1 / 2 : ℝ) * Q A ≤ Q (A ∩ S) := by
+    rw [measure_congr hEeq] at hext
+    have hES : E ∩ S =ᵐ[Q] A ∩ S := by
+      filter_upwards [hEeq] with ω hω
+      exact propext (and_congr_left fun _ => Iff.of_eq hω)
+    rwa [measure_congr hES] at hext
+  have hσ₁ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz
+    (u + 1) (by linarith)
+  have hA₁ : MeasurableSet[hσ₁.measurableSpace] A :=
+    actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u (u + 1)
+      hu (by linarith) A hA
+  have hS₁ : MeasurableSet[hσ₁.measurableSpace] S :=
+    canonical_good_angle_measurable_at_radius hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz
+      (u + 1) s (by linarith)
+  have hcons (D : Set Ω) (hD : MeasurableSet[hσ₁.measurableSpace] D) :
+      finiteCenteredLaw P κ B γ G z V D = Q D :=
+    finiteCenteredLaw_consistent_at_radius_stop hB hBm hc hB0 κ hκ hκ8 γ G F
+      (Eventually.of_forall hw) z hz (u + 1) V (by linarith) hV _ hσ₁
+      (Eventually.of_forall fun _ => rfl) D hD
+  rw [hcons A hA₁, hcons (A ∩ S) (hA₁.inter hS₁)]
+  exact hbase
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenBasis.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenBasis.lean
index 3a8c3e455e3436df329fd5817f3a27d77ac60f6b..c004cb6f9d58f9b28516b9b613875a7f58758b3d 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenBasis.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenBasis.lean
@@ -1,43 +1,43 @@
-import OAI.Probability.SLE.Angle.AngleEigenPolynomials
-import Mathlib.Algebra.Polynomial.Sequence
-
-namespace OAI
-
-/-! The polynomial eigenfunctions form a basis, since each is monic of
-its prescribed degree. This will identify transition laws from moments. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def angleEigenSequence (κ : ℝ) : Polynomial.Sequence ℝ where
-  elems' := angleEigenPolynomial κ
-  degree_eq' n := by
-    rw [degree_eq_natDegree (angleEigenPolynomial_monic κ n).ne_zero,
-      angleEigenPolynomial_natDegree]
-
-def angleEigenBasis (κ : ℝ) : Module.Basis ℕ ℝ ℝ[X] :=
-  (angleEigenSequence κ).basis (fun n => by
-    change IsUnit (angleEigenPolynomial κ n).leadingCoeff
-    rw [show (angleEigenPolynomial κ n).leadingCoeff=1 from angleEigenPolynomial_monic κ n]
-    exact isUnit_one)
-
-lemma angleEigenBasis_apply (κ : ℝ) (n : ℕ) :
-    angleEigenBasis κ n=angleEigenPolynomial κ n := by
-  exact Polynomial.Sequence.basis_eq_self _ _ n
-
-lemma angleEigenBasis_eigen {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    anglePolynomialGenerator κ (angleEigenBasis κ n)=
-      -C (anglePolynomialEigenvalue κ n)*angleEigenBasis κ n := by
-  simp only [angleEigenBasis_apply]
-  exact angleEigenPolynomial_eigen hκ n
-
-lemma angleEigenPolynomial_zero (κ : ℝ) : angleEigenPolynomial κ 0=1 := by
-  exact eq_one_of_monic_natDegree_zero (angleEigenPolynomial_monic κ 0)
-    (angleEigenPolynomial_natDegree κ 0)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenPolynomials
+import Mathlib.Algebra.Polynomial.Sequence
+
+namespace OAI
+
+/-! The polynomial eigenfunctions form a basis, since each is monic of
+its prescribed degree. This will identify transition laws from moments. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def angleEigenSequence (κ : ℝ) : Polynomial.Sequence ℝ where
+  elems' := angleEigenPolynomial κ
+  degree_eq' n := by
+    rw [degree_eq_natDegree (angleEigenPolynomial_monic κ n).ne_zero,
+      angleEigenPolynomial_natDegree]
+
+def angleEigenBasis (κ : ℝ) : Module.Basis ℕ ℝ ℝ[X] :=
+  (angleEigenSequence κ).basis (fun n => by
+    change IsUnit (angleEigenPolynomial κ n).leadingCoeff
+    rw [show (angleEigenPolynomial κ n).leadingCoeff=1 from angleEigenPolynomial_monic κ n]
+    exact isUnit_one)
+
+lemma angleEigenBasis_apply (κ : ℝ) (n : ℕ) :
+    angleEigenBasis κ n=angleEigenPolynomial κ n := by
+  exact Polynomial.Sequence.basis_eq_self _ _ n
+
+lemma angleEigenBasis_eigen {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    anglePolynomialGenerator κ (angleEigenBasis κ n)=
+      -C (anglePolynomialEigenvalue κ n)*angleEigenBasis κ n := by
+  simp only [angleEigenBasis_apply]
+  exact angleEigenPolynomial_eigen hκ n
+
+lemma angleEigenPolynomial_zero (κ : ℝ) : angleEigenPolynomial κ 0=1 := by
+  exact eq_one_of_monic_natDegree_zero (angleEigenPolynomial_monic κ 0)
+    (angleEigenPolynomial_natDegree κ 0)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenBounds.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenBounds.lean
index 6da2876267ac7c35d8112be41f415d9f3d9d8b3d..7b78cc96d960c86ba99f832031ba5de05ab39d13 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenBounds.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenBounds.lean
@@ -1,92 +1,92 @@
-import OAI.Probability.SLE.Angle.AngleEigenBasis
-
-namespace OAI
-
-/-! Explicit coefficient and supremum bounds for the monic cosine
-eigenfunctions. Quadratic spectral decay will dominate this growth. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-lemma anglePolynomialEigenvalue_sub (κ : ℝ) (N k : ℕ) :
-    anglePolynomialEigenvalue κ N-anglePolynomialEigenvalue κ k =
-      ((N:ℝ)-(k:ℝ))*(1+(κ/8)*((N:ℝ)+(k:ℝ))) := by
-  unfold anglePolynomialEigenvalue
-  ring
-
-lemma angleEigenCoefficient_ratio_bound {κ : ℝ} (hκ : 0<κ) {N k : ℕ} (hk : k<N) :
-    |(κ/8)*((k:ℝ)+2)*((k:ℝ)+1)/
-      (anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N)| ≤ (N:ℝ)+1 := by
-  have hden := anglePolynomialEigenvalue_strictMono hκ hk
-  have hdenp : 0<anglePolynomialEigenvalue κ N-anglePolynomialEigenvalue κ k :=
-    sub_pos.mpr hden
-  have hkN : (k:ℝ)+1≤(N:ℝ) := by exact_mod_cast hk
-  have hk0 : (0:ℝ)≤k := Nat.cast_nonneg k
-  have hN0 : (0:ℝ)≤N := Nat.cast_nonneg N
-  have hnum : (κ/8)*((k:ℝ)+2)*((k:ℝ)+1) ≤ (κ/8)*((N:ℝ)+1)*(N:ℝ) := by
-    apply mul_le_mul _ hkN (by positivity) (by positivity)
-    exact mul_le_mul_of_nonneg_left (by linarith) (by positivity)
-  have hd : (κ/8)*(N:ℝ) ≤ anglePolynomialEigenvalue κ N-anglePolynomialEigenvalue κ k := by
-    rw [anglePolynomialEigenvalue_sub]
-    calc
-      _ ≤ 1+(κ/8)*((N:ℝ)+(k:ℝ)) := by nlinarith
-      _ ≤ ((N:ℝ)-(k:ℝ))*(1+(κ/8)*((N:ℝ)+(k:ℝ))) :=
-        le_mul_of_one_le_left (by positivity) (by linarith)
-  rw [abs_div,abs_of_nonneg (by positivity),abs_of_neg (sub_neg.mpr hden),neg_sub]
-  apply (div_le_iff₀ hdenp).mpr
-  calc
-    _ ≤ (κ/8)*((N:ℝ)+1)*(N:ℝ) := hnum
-    _ = ((N:ℝ)+1)*((κ/8)*(N:ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hd (by positivity)
-
-lemma angleEigenCoefficient_abs_le {κ : ℝ} (hκ : 0<κ) (N k : ℕ) :
-    |angleEigenCoefficient κ N k| ≤ ((N:ℝ)+1)^(N-k) := by
-  have hN1 : 1≤(N:ℝ)+1 := le_add_of_nonneg_left (Nat.cast_nonneg N)
-  by_cases he : k=N
-  · subst k
-    rw [angleEigenCoefficient_top]
-    simp
-  by_cases hk : k<N
-  · by_cases hk2 : k+2≤N
-    · have hi := angleEigenCoefficient_abs_le hκ N (k+2)
-      conv_lhs => rw [angleEigenCoefficient]
-      rw [ite_eq_right he,dite_eq_left hk,abs_mul]
-      calc
-        _ ≤ ((N:ℝ)+1)*((N:ℝ)+1)^(N-(k+2)) :=
-          mul_le_mul (angleEigenCoefficient_ratio_bound hκ hk) hi (abs_nonneg _) (by positivity)
-        _ = ((N:ℝ)+1)^(N-(k+2)+1) := by rw [pow_succ]; ring
-        _ ≤ ((N:ℝ)+1)^(N-k) :=
-          pow_le_pow_right₀ hN1 (by omega)
-    · have hz : angleEigenCoefficient κ N (k+2)=0 := angleEigenCoefficient_above κ (by omega)
-      conv_lhs => rw [angleEigenCoefficient]
-      rw [ite_eq_right he,dite_eq_left hk,hz,mul_zero,abs_zero]
-      positivity
-  · rw [angleEigenCoefficient_above κ (by omega),abs_zero]
-    positivity
-termination_by N-k
-  decreasing_by omega
-
-lemma angleEigenPolynomial_eval_abs_le {κ : ℝ} (hκ : 0<κ) (N : ℕ) {x : ℝ} (hx : |x|≤1) :
-    |(angleEigenPolynomial κ N).eval x| ≤ ((N:ℝ)+1)^(N+1) := by
-  have hN1 : 1≤(N:ℝ)+1 := le_add_of_nonneg_left (Nat.cast_nonneg N)
-  unfold angleEigenPolynomial
-  rw [eval_finsetSum]
-  calc
-    _ ≤ ∑ k ∈ Finset.range (N+1), |(monomial k (angleEigenCoefficient κ N k)).eval x| :=
-      Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ _k ∈ Finset.range (N+1), ((N:ℝ)+1)^N := by
-      apply Finset.sum_le_sum
-      intro k _hk
-      rw [eval_monomial,abs_mul,abs_pow]
-      have hc := (angleEigenCoefficient_abs_le hκ N k).trans
-        (pow_le_pow_right₀ hN1 (Nat.sub_le _ _))
-      have hpow : |x|^k≤1 := pow_le_one₀ (abs_nonneg _) hx
-      exact (mul_le_mul hc hpow (by positivity) (by positivity)).trans_eq (mul_one _)
-    _ = ((N:ℝ)+1)^(N+1) := by simp [pow_succ,mul_comm]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenBasis
+
+namespace OAI
+
+/-! Explicit coefficient and supremum bounds for the monic cosine
+eigenfunctions. Quadratic spectral decay will dominate this growth. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+lemma anglePolynomialEigenvalue_sub (κ : ℝ) (N k : ℕ) :
+    anglePolynomialEigenvalue κ N-anglePolynomialEigenvalue κ k =
+      ((N:ℝ)-(k:ℝ))*(1+(κ/8)*((N:ℝ)+(k:ℝ))) := by
+  unfold anglePolynomialEigenvalue
+  ring
+
+lemma angleEigenCoefficient_ratio_bound {κ : ℝ} (hκ : 0<κ) {N k : ℕ} (hk : k<N) :
+    |(κ/8)*((k:ℝ)+2)*((k:ℝ)+1)/
+      (anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N)| ≤ (N:ℝ)+1 := by
+  have hden := anglePolynomialEigenvalue_strictMono hκ hk
+  have hdenp : 0<anglePolynomialEigenvalue κ N-anglePolynomialEigenvalue κ k :=
+    sub_pos.mpr hden
+  have hkN : (k:ℝ)+1≤(N:ℝ) := by exact_mod_cast hk
+  have hk0 : (0:ℝ)≤k := Nat.cast_nonneg k
+  have hN0 : (0:ℝ)≤N := Nat.cast_nonneg N
+  have hnum : (κ/8)*((k:ℝ)+2)*((k:ℝ)+1) ≤ (κ/8)*((N:ℝ)+1)*(N:ℝ) := by
+    apply mul_le_mul _ hkN (by positivity) (by positivity)
+    exact mul_le_mul_of_nonneg_left (by linarith) (by positivity)
+  have hd : (κ/8)*(N:ℝ) ≤ anglePolynomialEigenvalue κ N-anglePolynomialEigenvalue κ k := by
+    rw [anglePolynomialEigenvalue_sub]
+    calc
+      _ ≤ 1+(κ/8)*((N:ℝ)+(k:ℝ)) := by nlinarith
+      _ ≤ ((N:ℝ)-(k:ℝ))*(1+(κ/8)*((N:ℝ)+(k:ℝ))) :=
+        le_mul_of_one_le_left (by positivity) (by linarith)
+  rw [abs_div,abs_of_nonneg (by positivity),abs_of_neg (sub_neg.mpr hden),neg_sub]
+  apply (div_le_iff₀ hdenp).mpr
+  calc
+    _ ≤ (κ/8)*((N:ℝ)+1)*(N:ℝ) := hnum
+    _ = ((N:ℝ)+1)*((κ/8)*(N:ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hd (by positivity)
+
+lemma angleEigenCoefficient_abs_le {κ : ℝ} (hκ : 0<κ) (N k : ℕ) :
+    |angleEigenCoefficient κ N k| ≤ ((N:ℝ)+1)^(N-k) := by
+  have hN1 : 1≤(N:ℝ)+1 := le_add_of_nonneg_left (Nat.cast_nonneg N)
+  by_cases he : k=N
+  · subst k
+    rw [angleEigenCoefficient_top]
+    simp
+  by_cases hk : k<N
+  · by_cases hk2 : k+2≤N
+    · have hi := angleEigenCoefficient_abs_le hκ N (k+2)
+      conv_lhs => rw [angleEigenCoefficient]
+      rw [ite_eq_right he,dite_eq_left hk,abs_mul]
+      calc
+        _ ≤ ((N:ℝ)+1)*((N:ℝ)+1)^(N-(k+2)) :=
+          mul_le_mul (angleEigenCoefficient_ratio_bound hκ hk) hi (abs_nonneg _) (by positivity)
+        _ = ((N:ℝ)+1)^(N-(k+2)+1) := by rw [pow_succ]; ring
+        _ ≤ ((N:ℝ)+1)^(N-k) :=
+          pow_le_pow_right₀ hN1 (by omega)
+    · have hz : angleEigenCoefficient κ N (k+2)=0 := angleEigenCoefficient_above κ (by omega)
+      conv_lhs => rw [angleEigenCoefficient]
+      rw [ite_eq_right he,dite_eq_left hk,hz,mul_zero,abs_zero]
+      positivity
+  · rw [angleEigenCoefficient_above κ (by omega),abs_zero]
+    positivity
+termination_by N-k
+  decreasing_by omega
+
+lemma angleEigenPolynomial_eval_abs_le {κ : ℝ} (hκ : 0<κ) (N : ℕ) {x : ℝ} (hx : |x|≤1) :
+    |(angleEigenPolynomial κ N).eval x| ≤ ((N:ℝ)+1)^(N+1) := by
+  have hN1 : 1≤(N:ℝ)+1 := le_add_of_nonneg_left (Nat.cast_nonneg N)
+  unfold angleEigenPolynomial
+  rw [eval_finsetSum]
+  calc
+    _ ≤ ∑ k ∈ Finset.range (N+1), |(monomial k (angleEigenCoefficient κ N k)).eval x| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ _k ∈ Finset.range (N+1), ((N:ℝ)+1)^N := by
+      apply Finset.sum_le_sum
+      intro k _hk
+      rw [eval_monomial,abs_mul,abs_pow]
+      have hc := (angleEigenCoefficient_abs_le hκ N k).trans
+        (pow_le_pow_right₀ hN1 (Nat.sub_le _ _))
+      have hpow : |x|^k≤1 := pow_le_one₀ (abs_nonneg _) hx
+      exact (mul_le_mul hc hpow (by positivity) (by positivity)).trans_eq (mul_one _)
+    _ = ((N:ℝ)+1)^(N+1) := by simp [pow_succ,mul_comm]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenDifferential.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenDifferential.lean
index 1f397d60c0cb12986d2397e25c97d28ea587e3f1..2cd326b1bee81930dc34912aa2366ab17eee8c5d 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenDifferential.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenDifferential.lean
@@ -1,63 +1,63 @@
-import OAI.Probability.SLE.Angle.AngleEigenBounds
-
-namespace OAI
-
-/-! The polynomial eigenfunctions satisfy the actual centered angular
-generator equation, with time measured by logarithmic conformal radius. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def anglePolynomialTest (p : ℝ[X]) (θ : ℝ) : ℝ := p.eval (Real.cos θ)
-
-lemma anglePolynomialTest_hasDerivAt (p : ℝ[X]) (θ : ℝ) :
-    HasDerivAt (anglePolynomialTest p)
-      (-Real.sin θ*p.derivative.eval (Real.cos θ)) θ := by
-  change HasDerivAt (fun x => p.eval (Real.cos x)) _ θ
-  apply ((p.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)).congr_deriv
-  ring
-
-lemma anglePolynomialTest_deriv (p : ℝ[X]) :
-    deriv (anglePolynomialTest p)=fun θ => -Real.sin θ*p.derivative.eval (Real.cos θ) := by
-  funext θ
-  exact (anglePolynomialTest_hasDerivAt p θ).deriv
-
-lemma anglePolynomialTest_second (p : ℝ[X]) (θ : ℝ) :
-    deriv (deriv (anglePolynomialTest p)) θ =
-      Real.sin θ^2*p.derivative.derivative.eval (Real.cos θ)-
-        Real.cos θ*p.derivative.eval (Real.cos θ) := by
-  rw [anglePolynomialTest_deriv]
-  have hh := (Real.hasDerivAt_sin θ).neg.mul
-    ((p.derivative.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ))
-  change HasDerivAt (fun x => -Real.sin x*p.derivative.eval (Real.cos x))
-    (-Real.cos θ*p.derivative.eval (Real.cos θ) +
-      (-Real.sin θ)*(p.derivative.derivative.eval (Real.cos θ)*(-Real.sin θ))) θ at hh
-  rw [hh.deriv]
-  ring
-
-lemma anglePolynomialTest_generator (κ : ℝ) (p : ℝ[X]) {θ : ℝ}
-    (hθ : Real.sin θ≠0) :
-    (κ/8)*deriv (deriv (anglePolynomialTest p)) θ +
-      (Real.cos θ/Real.sin θ)*deriv (anglePolynomialTest p) θ =
-        (anglePolynomialGenerator κ p).eval (Real.cos θ) := by
-  rw [anglePolynomialTest_second,anglePolynomialTest_deriv]
-  simp only [anglePolynomialGenerator,eval_sub,eval_mul,eval_C,eval_one,eval_pow,eval_X]
-  have hsin : Real.sin θ^2=1-Real.cos θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
-  rw [hsin]
-  field_simp
-  ring
-
-lemma angleEigenPolynomial_differential {κ : ℝ} (hκ : 0<κ) (N : ℕ) {θ : ℝ}
-    (hθ : Real.sin θ≠0) :
-    (κ/8)*deriv (deriv (anglePolynomialTest (angleEigenPolynomial κ N))) θ +
-      (Real.cos θ/Real.sin θ)*deriv (anglePolynomialTest (angleEigenPolynomial κ N)) θ =
-      -anglePolynomialEigenvalue κ N*anglePolynomialTest (angleEigenPolynomial κ N) θ := by
-  rw [anglePolynomialTest_generator κ _ hθ,angleEigenPolynomial_eigen hκ]
-  simp [anglePolynomialTest]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenBounds
+
+namespace OAI
+
+/-! The polynomial eigenfunctions satisfy the actual centered angular
+generator equation, with time measured by logarithmic conformal radius. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def anglePolynomialTest (p : ℝ[X]) (θ : ℝ) : ℝ := p.eval (Real.cos θ)
+
+lemma anglePolynomialTest_hasDerivAt (p : ℝ[X]) (θ : ℝ) :
+    HasDerivAt (anglePolynomialTest p)
+      (-Real.sin θ*p.derivative.eval (Real.cos θ)) θ := by
+  change HasDerivAt (fun x => p.eval (Real.cos x)) _ θ
+  apply ((p.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)).congr_deriv
+  ring
+
+lemma anglePolynomialTest_deriv (p : ℝ[X]) :
+    deriv (anglePolynomialTest p)=fun θ => -Real.sin θ*p.derivative.eval (Real.cos θ) := by
+  funext θ
+  exact (anglePolynomialTest_hasDerivAt p θ).deriv
+
+lemma anglePolynomialTest_second (p : ℝ[X]) (θ : ℝ) :
+    deriv (deriv (anglePolynomialTest p)) θ =
+      Real.sin θ^2*p.derivative.derivative.eval (Real.cos θ)-
+        Real.cos θ*p.derivative.eval (Real.cos θ) := by
+  rw [anglePolynomialTest_deriv]
+  have hh := (Real.hasDerivAt_sin θ).neg.mul
+    ((p.derivative.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ))
+  change HasDerivAt (fun x => -Real.sin x*p.derivative.eval (Real.cos x))
+    (-Real.cos θ*p.derivative.eval (Real.cos θ) +
+      (-Real.sin θ)*(p.derivative.derivative.eval (Real.cos θ)*(-Real.sin θ))) θ at hh
+  rw [hh.deriv]
+  ring
+
+lemma anglePolynomialTest_generator (κ : ℝ) (p : ℝ[X]) {θ : ℝ}
+    (hθ : Real.sin θ≠0) :
+    (κ/8)*deriv (deriv (anglePolynomialTest p)) θ +
+      (Real.cos θ/Real.sin θ)*deriv (anglePolynomialTest p) θ =
+        (anglePolynomialGenerator κ p).eval (Real.cos θ) := by
+  rw [anglePolynomialTest_second,anglePolynomialTest_deriv]
+  simp only [anglePolynomialGenerator,eval_sub,eval_mul,eval_C,eval_one,eval_pow,eval_X]
+  have hsin : Real.sin θ^2=1-Real.cos θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
+  rw [hsin]
+  field_simp
+  ring
+
+lemma angleEigenPolynomial_differential {κ : ℝ} (hκ : 0<κ) (N : ℕ) {θ : ℝ}
+    (hθ : Real.sin θ≠0) :
+    (κ/8)*deriv (deriv (anglePolynomialTest (angleEigenPolynomial κ N))) θ +
+      (Real.cos θ/Real.sin θ)*deriv (anglePolynomialTest (angleEigenPolynomial κ N)) θ =
+      -anglePolynomialEigenvalue κ N*anglePolynomialTest (angleEigenPolynomial κ N) θ := by
+  rw [anglePolynomialTest_generator κ _ hθ,angleEigenPolynomial_eigen hκ]
+  simp [anglePolynomialTest]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenMoment.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenMoment.lean
index 2085518efaa1b093570bde3132be129d5e9fd179..17c815116221784985d2268071c38bb1ac25f173 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenMoment.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenMoment.lean
@@ -1,116 +1,116 @@
-import OAI.Probability.SLE.Angle.AngleEigenSource
-import OAI.Probability.SLE.FiniteStoppedLaw
-
-namespace OAI
-
-/-! Exact exponential eigenfunction moments under the finite centered law,
-using angular localization and dominated convergence. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {B : ℝ≥0 → Ω → ℝ}
-  (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-  (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω=0)
-  {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-
-include hB hBm hBc hB0 hκ hκ8 γ G F hw
-
-lemma stoppedGreen_eigen_moment (z : ℂ) (hz : z∈upperHalfPlane)
-    (u : ℝ) (hu : 0≤u) (n : ℕ) :
-    (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u*
-      (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
-        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg)) ∂P)=
-    initialGreen κ z*Real.exp (-u*anglePolynomialEigenvalue κ n)*
-      (angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let A := logAngleSq κ (initialLogCoordinates z) 0
-  have hA : 0<A := logAngleSq_pos _ _ _
-  let ε (j : ℕ) := A/((j:ℝ)+1)
-  have hε (j : ℕ) : 0<ε j := by dsimp [ε]; positivity
-  have hεA (j : ℕ) : ε j≤A :=
-    div_le_self hA.le (by have hj := Nat.cast_nonneg (α := ℝ) j; linarith)
-  have hεlim : Tendsto ε atTop (𝓝 0) := by
-    simpa only [mul_one_div,mul_zero] using
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A
-  let a (j : ℕ) : AngularLocalization B hm κ z u (ε j) :=
-    (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε j) hu (hε j) (hεA j)).some
-  let ev := anglePolynomialEigenvalue κ n
-  let f (j : ℕ) (ω : Ω) := logPolynomialDensity κ (codimension κ+ev)
-    (greenExponent κ) (angleEigenPolynomial κ n) ((a j).solution.Q ((a j).τ ω) ω) (B ((a j).τ ω) ω)
-  let w (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u*
-    (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
-      (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg))
-  let K := Real.exp (ev*(u-Real.log 2-Real.log z.im))
-  let C := Real.exp ((codimension κ+ev)*(u-Real.log 2-Real.log z.im))*((n:ℝ)+1)^(n+1)
-  have hf (j : ℕ) : Integrable (f j) P ∧
-      (∫ ω, f j ω ∂P)=Real.exp (-ev*(Real.log 2+Real.log z.im))*initialGreen κ z*
-        (angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
-    exact ⟨((a j).mean_eigen hB hBm hBc hB0 hκ hκ8 n).1,
-      ((a j).mean_eigen hB hBm hBc hB0 hκ hκ8 n).2.trans
-        (logPolynomialDensity_initial κ ev _ z hz)⟩
-  have hb (j : ℕ) : ∀ᵐ ω ∂P, ‖f j ω‖≤C := Eventually.of_forall fun ω => by
-    simpa only [Real.norm_eq_abs] using
-      logEigenDensity_bound hκ hκ8 n (Real.log z.im) u _ _ ((a j).inside ω ((a j).τ ω) le_rfl).1
-  have hl : ∀ᵐ ω ∂P, Tendsto (fun j => f j ω) atTop (𝓝 (K*w ω)) := by
-    filter_upwards [hw] with ω hω
-    simpa only [f,w,K,ev,mul_assoc] using
-      localization_eigen_tendsto a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω n
-  have hi := tendsto_integral_of_dominated_convergence (fun _ : Ω => C)
-    (fun j => (hf j).1.aestronglyMeasurable) (integrable_const C) hb hl
-  have he : (∫ ω, K*w ω ∂P)=Real.exp (-ev*(Real.log 2+Real.log z.im))*initialGreen κ z*
-      (angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
-    apply tendsto_nhds_unique hi
-    exact tendsto_const_nhds.congr' (Eventually.of_forall fun j => (hf j).2.symm)
-  rw [integral_const_mul] at he
-  change (∫ ω, w ω ∂P)=_
-  apply (mul_left_cancel₀ (Real.exp_ne_zero (ev*(u-Real.log 2-Real.log z.im))))
-  change K*(∫ ω, w ω ∂P)=_
-  rw [he]
-  have hexp : K*Real.exp (-u*ev)=Real.exp (-ev*(Real.log 2+Real.log z.im)) := by
-    dsimp only [K]
-    rw [← Real.exp_add]
-    congr 1
-    ring
-  rw [← hexp]
-  ring
-
-lemma finiteCenteredLaw_eigen_moment (z : ℂ) (hz : z∈upperHalfPlane)
-    (u : ℝ) (hu : 0≤u) (n : ℕ) :
-    (∫ ω, (angleEigenPolynomial κ n).eval (Real.cos
-      ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
-        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg))
-      ∂finiteCenteredLaw P κ B γ G z u)=
-    Real.exp (-u*anglePolynomialEigenvalue κ n)*(angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
-  obtain ⟨hgi,_⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
-  have hnon : ∀ᵐ ω ∂P, 0≤ stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u/
-      initialGreen κ z := by
-    filter_upwards [hw] with ω hω
-    exact div_nonneg (stoppedGreen_nonneg hω κ z u) (initialGreen_pos κ z hz).le
-  rw [finiteCenteredLaw,integral_withDensity_eq_integral_toReal_smul₀
-    (hgi.div_const _).aemeasurable.ennreal_ofReal
-    (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
-  have he : (fun ω => (ENNReal.ofReal
-      (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u/initialGreen κ z)).toReal •
-      (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
-        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg))) =ᵐ[P]
-    fun ω => (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u*
-      (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
-        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg)))/initialGreen κ z := by
-    filter_upwards [hnon] with ω hω
-    rw [ENNReal.toReal_ofReal hω,smul_eq_mul]
-    ring
-  rw [integral_congr_ae he,integral_div,
-    stoppedGreen_eigen_moment hB hBm hBc hB0 hκ hκ8 γ G F hw z hz u hu n]
-  field_simp [(initialGreen_pos κ z hz).ne']
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenSource
+import OAI.Probability.SLE.FiniteStoppedLaw
+
+namespace OAI
+
+/-! Exact exponential eigenfunction moments under the finite centered law,
+using angular localization and dominated convergence. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {B : ℝ≥0 → Ω → ℝ}
+  (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+  (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω=0)
+  {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+
+include hB hBm hBc hB0 hκ hκ8 γ G F hw
+
+lemma stoppedGreen_eigen_moment (z : ℂ) (hz : z∈upperHalfPlane)
+    (u : ℝ) (hu : 0≤u) (n : ℕ) :
+    (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u*
+      (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
+        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg)) ∂P)=
+    initialGreen κ z*Real.exp (-u*anglePolynomialEigenvalue κ n)*
+      (angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let A := logAngleSq κ (initialLogCoordinates z) 0
+  have hA : 0<A := logAngleSq_pos _ _ _
+  let ε (j : ℕ) := A/((j:ℝ)+1)
+  have hε (j : ℕ) : 0<ε j := by dsimp [ε]; positivity
+  have hεA (j : ℕ) : ε j≤A :=
+    div_le_self hA.le (by have hj := Nat.cast_nonneg (α := ℝ) j; linarith)
+  have hεlim : Tendsto ε atTop (𝓝 0) := by
+    simpa only [mul_one_div,mul_zero] using
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A
+  let a (j : ℕ) : AngularLocalization B hm κ z u (ε j) :=
+    (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε j) hu (hε j) (hεA j)).some
+  let ev := anglePolynomialEigenvalue κ n
+  let f (j : ℕ) (ω : Ω) := logPolynomialDensity κ (codimension κ+ev)
+    (greenExponent κ) (angleEigenPolynomial κ n) ((a j).solution.Q ((a j).τ ω) ω) (B ((a j).τ ω) ω)
+  let w (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u*
+    (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
+      (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg))
+  let K := Real.exp (ev*(u-Real.log 2-Real.log z.im))
+  let C := Real.exp ((codimension κ+ev)*(u-Real.log 2-Real.log z.im))*((n:ℝ)+1)^(n+1)
+  have hf (j : ℕ) : Integrable (f j) P ∧
+      (∫ ω, f j ω ∂P)=Real.exp (-ev*(Real.log 2+Real.log z.im))*initialGreen κ z*
+        (angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
+    exact ⟨((a j).mean_eigen hB hBm hBc hB0 hκ hκ8 n).1,
+      ((a j).mean_eigen hB hBm hBc hB0 hκ hκ8 n).2.trans
+        (logPolynomialDensity_initial κ ev _ z hz)⟩
+  have hb (j : ℕ) : ∀ᵐ ω ∂P, ‖f j ω‖≤C := Eventually.of_forall fun ω => by
+    simpa only [Real.norm_eq_abs] using
+      logEigenDensity_bound hκ hκ8 n (Real.log z.im) u _ _ ((a j).inside ω ((a j).τ ω) le_rfl).1
+  have hl : ∀ᵐ ω ∂P, Tendsto (fun j => f j ω) atTop (𝓝 (K*w ω)) := by
+    filter_upwards [hw] with ω hω
+    simpa only [f,w,K,ev,mul_assoc] using
+      localization_eigen_tendsto a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω n
+  have hi := tendsto_integral_of_dominated_convergence (fun _ : Ω => C)
+    (fun j => (hf j).1.aestronglyMeasurable) (integrable_const C) hb hl
+  have he : (∫ ω, K*w ω ∂P)=Real.exp (-ev*(Real.log 2+Real.log z.im))*initialGreen κ z*
+      (angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
+    apply tendsto_nhds_unique hi
+    exact tendsto_const_nhds.congr' (Eventually.of_forall fun j => (hf j).2.symm)
+  rw [integral_const_mul] at he
+  change (∫ ω, w ω ∂P)=_
+  apply (mul_left_cancel₀ (Real.exp_ne_zero (ev*(u-Real.log 2-Real.log z.im))))
+  change K*(∫ ω, w ω ∂P)=_
+  rw [he]
+  have hexp : K*Real.exp (-u*ev)=Real.exp (-ev*(Real.log 2+Real.log z.im)) := by
+    dsimp only [K]
+    rw [← Real.exp_add]
+    congr 1
+    ring
+  rw [← hexp]
+  ring
+
+lemma finiteCenteredLaw_eigen_moment (z : ℂ) (hz : z∈upperHalfPlane)
+    (u : ℝ) (hu : 0≤u) (n : ℕ) :
+    (∫ ω, (angleEigenPolynomial κ n).eval (Real.cos
+      ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
+        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg))
+      ∂finiteCenteredLaw P κ B γ G z u)=
+    Real.exp (-u*anglePolynomialEigenvalue κ n)*(angleEigenPolynomial κ n).eval (Real.cos z.arg) := by
+  obtain ⟨hgi,_⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
+  have hnon : ∀ᵐ ω ∂P, 0≤ stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u/
+      initialGreen κ z := by
+    filter_upwards [hw] with ω hω
+    exact div_nonneg (stoppedGreen_nonneg hω κ z u) (initialGreen_pos κ z hz).le
+  rw [finiteCenteredLaw,integral_withDensity_eq_integral_toReal_smul₀
+    (hgi.div_const _).aemeasurable.ennreal_ofReal
+    (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
+  have he : (fun ω => (ENNReal.ofReal
+      (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u/initialGreen κ z)).toReal •
+      (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
+        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg))) =ᵐ[P]
+    fun ω => (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u*
+      (angleEigenPolynomial κ n).eval (Real.cos ((G ω (radiusLevelTime (γ ω) (G ω) z u).toNNReal z-
+        (Real.sqrt κ*B (radiusLevelTime (γ ω) (G ω) z u).toNNReal ω:ℝ)).arg)))/initialGreen κ z := by
+    filter_upwards [hnon] with ω hω
+    rw [ENNReal.toReal_ofReal hω,smul_eq_mul]
+    ring
+  rw [integral_congr_ae he,integral_div,
+    stoppedGreen_eigen_moment hB hBm hBc hB0 hκ hκ8 γ G F hw z hz u hu n]
+  field_simp [(initialGreen_pos κ z hz).ne']
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenNorm.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenNorm.lean
index 2381ada85b5fcaddb405e85739800c92074d851b..9a83111c32e9b8dbc0e954bc1f5febb51d91111c 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenNorm.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenNorm.lean
@@ -1,69 +1,69 @@
-import OAI.Probability.SLE.Angle.AnglePolynomialInner
-import OAI.Probability.SLE.Angle.AngleRecurrenceRemainders
-
-namespace OAI
-
-/-! Exact norm recurrence and a uniform exponential lower bound. Together
-with the coefficient bounds this controls the normalized spectral series. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def angleEigenSquareNorm (κ : ℝ) (n : ℕ) : ℝ :=
-  anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ n)
-
-lemma angleEigenSquareNorm_zero_pos {κ : ℝ} (hκ : 0<κ) :
-    0<angleEigenSquareNorm κ 0 := by
-  simpa only [angleEigenSquareNorm,angleEigenPolynomial_zero] using anglePolynomialInner_one_pos hκ
-
-lemma angleEigenSquareNorm_shift {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    anglePolynomialInner κ (angleEigenPolynomial κ n) (X*angleEigenPolynomial κ (n+1))=
-      angleEigenSquareNorm κ (n+1) := by
-  have he := anglePolynomialInner_lower_degree hκ (n+1)
-    (X*angleEigenPolynomial κ n-angleEigenPolynomial κ (n+1))
-    (angleEigenPolynomial_leading_remainder κ n)
-  rw [anglePolynomialInner_sub_right hκ] at he
-  rw [← anglePolynomialInner_X,anglePolynomialInner_comm κ
-    (X*angleEigenPolynomial κ n)]
-  exact sub_eq_zero.mp he
-
-lemma angleEigenSquareNorm_recurrence {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    angleEigenSquareNorm κ (n+1)=angleRecurrenceCoefficient κ n*angleEigenSquareNorm κ n := by
-  have hd : (X*angleEigenPolynomial κ (n+1)-angleEigenPolynomial κ (n+2)-
-      angleRecurrenceCoefficient κ n • angleEigenPolynomial κ n).degree < (n:WithBot ℕ) := by
-    cases n with
-    | zero => rw [angleEigenPolynomial_recurrence_zero hκ]; simp
-    | succ n => exact angleEigenPolynomial_recurrence_remainder hκ n
-  have he := anglePolynomialInner_lower_degree hκ n _ hd
-  rw [anglePolynomialInner_sub_right hκ,anglePolynomialInner_sub_right hκ,
-    anglePolynomialInner_smul_right,angleEigenSquareNorm_shift hκ] at he
-  have hz := angleEigenPolynomial_orthogonal hκ n (n+2) (by omega)
-  change anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ (n+2))=0 at hz
-  rw [hz,sub_zero] at he
-  exact sub_eq_zero.mp he
-
-lemma angleEigenSquareNorm_lower {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    angleEigenSquareNorm κ 0*(κ/(4*(κ+4)))^n≤angleEigenSquareNorm κ n := by
-  have hc : 0<κ/(4*(κ+4)) := by positivity
-  have h0 := angleEigenSquareNorm_zero_pos hκ
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [angleEigenSquareNorm_recurrence hκ,pow_succ]
-    calc
-      _ = κ/(4*(κ+4))*(angleEigenSquareNorm κ 0*(κ/(4*(κ+4)))^n) := by ring
-      _ ≤ angleRecurrenceCoefficient κ n*angleEigenSquareNorm κ n :=
-        mul_le_mul (angleRecurrenceCoefficient_lower hκ n) ih
-          (by positivity) (angleRecurrenceCoefficient_pos hκ n).le
-
-lemma angleEigenSquareNorm_pos {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    0<angleEigenSquareNorm κ n :=
-  lt_of_lt_of_le (by have := angleEigenSquareNorm_zero_pos hκ; positivity)
-    (angleEigenSquareNorm_lower hκ n)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AnglePolynomialInner
+import OAI.Probability.SLE.Angle.AngleRecurrenceRemainders
+
+namespace OAI
+
+/-! Exact norm recurrence and a uniform exponential lower bound. Together
+with the coefficient bounds this controls the normalized spectral series. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def angleEigenSquareNorm (κ : ℝ) (n : ℕ) : ℝ :=
+  anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ n)
+
+lemma angleEigenSquareNorm_zero_pos {κ : ℝ} (hκ : 0<κ) :
+    0<angleEigenSquareNorm κ 0 := by
+  simpa only [angleEigenSquareNorm,angleEigenPolynomial_zero] using anglePolynomialInner_one_pos hκ
+
+lemma angleEigenSquareNorm_shift {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    anglePolynomialInner κ (angleEigenPolynomial κ n) (X*angleEigenPolynomial κ (n+1))=
+      angleEigenSquareNorm κ (n+1) := by
+  have he := anglePolynomialInner_lower_degree hκ (n+1)
+    (X*angleEigenPolynomial κ n-angleEigenPolynomial κ (n+1))
+    (angleEigenPolynomial_leading_remainder κ n)
+  rw [anglePolynomialInner_sub_right hκ] at he
+  rw [← anglePolynomialInner_X,anglePolynomialInner_comm κ
+    (X*angleEigenPolynomial κ n)]
+  exact sub_eq_zero.mp he
+
+lemma angleEigenSquareNorm_recurrence {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    angleEigenSquareNorm κ (n+1)=angleRecurrenceCoefficient κ n*angleEigenSquareNorm κ n := by
+  have hd : (X*angleEigenPolynomial κ (n+1)-angleEigenPolynomial κ (n+2)-
+      angleRecurrenceCoefficient κ n • angleEigenPolynomial κ n).degree < (n:WithBot ℕ) := by
+    cases n with
+    | zero => rw [angleEigenPolynomial_recurrence_zero hκ]; simp
+    | succ n => exact angleEigenPolynomial_recurrence_remainder hκ n
+  have he := anglePolynomialInner_lower_degree hκ n _ hd
+  rw [anglePolynomialInner_sub_right hκ,anglePolynomialInner_sub_right hκ,
+    anglePolynomialInner_smul_right,angleEigenSquareNorm_shift hκ] at he
+  have hz := angleEigenPolynomial_orthogonal hκ n (n+2) (by omega)
+  change anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ (n+2))=0 at hz
+  rw [hz,sub_zero] at he
+  exact sub_eq_zero.mp he
+
+lemma angleEigenSquareNorm_lower {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    angleEigenSquareNorm κ 0*(κ/(4*(κ+4)))^n≤angleEigenSquareNorm κ n := by
+  have hc : 0<κ/(4*(κ+4)) := by positivity
+  have h0 := angleEigenSquareNorm_zero_pos hκ
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [angleEigenSquareNorm_recurrence hκ,pow_succ]
+    calc
+      _ = κ/(4*(κ+4))*(angleEigenSquareNorm κ 0*(κ/(4*(κ+4)))^n) := by ring
+      _ ≤ angleRecurrenceCoefficient κ n*angleEigenSquareNorm κ n :=
+        mul_le_mul (angleRecurrenceCoefficient_lower hκ n) ih
+          (by positivity) (angleRecurrenceCoefficient_pos hκ n).le
+
+lemma angleEigenSquareNorm_pos {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    0<angleEigenSquareNorm κ n :=
+  lt_of_lt_of_le (by have := angleEigenSquareNorm_zero_pos hκ; positivity)
+    (angleEigenSquareNorm_lower hκ n)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenOrthogonality.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenOrthogonality.lean
index 035503826736cadb50b9ed086c51f544a95b0af8..2a58daed2aaf6dc643767d13b3a016bd6087f524 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenOrthogonality.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenOrthogonality.lean
@@ -1,90 +1,90 @@
-import OAI.Probability.SLE.Angle.AngleStationaryFlux
-
-namespace OAI
-
-/-! Symmetry under the paper's stationary angular weight, and orthogonality
-of the actual centered-generator polynomial eigenfunctions. -/
-noncomputable section
-open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma anglePolynomialWronskian_hasDerivAt {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) {θ : ℝ}
-    (hθ : θ∈Icc (0:ℝ) Real.pi) :
-    HasDerivAt (fun θ => anglePolynomialTest q θ*anglePolynomialFlux κ p θ -
-      anglePolynomialTest p θ*anglePolynomialFlux κ q θ)
-      (Real.sin θ^(4*Target.a κ)*
-        (anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ) -
-          anglePolynomialTest p θ*(anglePolynomialGenerator κ q).eval (Real.cos θ))) θ := by
-  have hp := (anglePolynomialTest_hasDerivAt q θ).mul (anglePolynomialFlux_hasDerivAt hκ p hθ)
-  have hq := (anglePolynomialTest_hasDerivAt p θ).mul (anglePolynomialFlux_hasDerivAt hκ q hθ)
-  apply (hp.sub hq).congr_deriv
-  simp only [anglePolynomialTest,anglePolynomialFlux]
-  ring
-
-lemma anglePolynomialGenerator_symmetric {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) :
-    (∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
-      anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ)) =
-    ∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
-      anglePolynomialTest p θ*(anglePolynomialGenerator κ q).eval (Real.cos θ) := by
-  have hr : 0<4*Target.a κ := by unfold Target.a; positivity
-  have hc (p q : ℝ[X]) : Continuous (fun θ => Real.sin θ^(4*Target.a κ)*
-      anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ)) :=
-    (((Real.continuous_rpow_const hr.le).comp Real.continuous_sin).mul
-      (q.continuous.comp Real.continuous_cos)).mul
-      ((anglePolynomialGenerator κ p).continuous.comp Real.continuous_cos)
-  have hi : IntervalIntegrable (fun θ => Real.sin θ^(4*Target.a κ)*
-      (anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ) -
-        anglePolynomialTest p θ*(anglePolynomialGenerator κ q).eval (Real.cos θ)))
-      volume 0 Real.pi := by
-    convert (((hc p q).sub (hc q p)).intervalIntegrable (μ := volume) 0 Real.pi) using 1
-    ext θ
-    simp only [Pi.sub_apply]
-    ring
-  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt
-    (fun θ hθ => anglePolynomialWronskian_hasDerivAt hκ p q
-      (by simpa only [uIcc_of_le Real.pi_pos.le] using hθ))
-    hi
-  simp only [mul_sub,← mul_assoc] at he
-  rw [intervalIntegral.integral_sub ((hc p q).intervalIntegrable _ _)
-    ((hc q p).intervalIntegrable _ _)] at he
-  have hzero : ∀ p : ℝ[X], anglePolynomialFlux κ p Real.pi=0 ∧ anglePolynomialFlux κ p 0=0 := by
-    intro p
-    simp [anglePolynomialFlux,Real.zero_rpow (by linarith : 4*Target.a κ+1≠0)]
-  rw [(hzero p).1,(hzero q).1,(hzero p).2,(hzero q).2] at he
-  simpa only [mul_zero,sub_self,sub_eq_zero] using he
-
-lemma angleEigenPolynomial_orthogonal {κ : ℝ} (hκ : 0<κ) (n m : ℕ) (hnm : n≠m) :
-    (∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
-      anglePolynomialTest (angleEigenPolynomial κ n) θ*
-        anglePolynomialTest (angleEigenPolynomial κ m) θ)=0 := by
-  have he := anglePolynomialGenerator_symmetric hκ
-    (angleEigenPolynomial κ n) (angleEigenPolynomial κ m)
-  simp_rw [angleEigenPolynomial_eigen hκ,eval_mul,eval_neg,eval_C] at he
-  have hn (θ : ℝ) : Real.sin θ^(4*Target.a κ)*anglePolynomialTest (angleEigenPolynomial κ m) θ*
-      (-anglePolynomialEigenvalue κ n*(angleEigenPolynomial κ n).eval (Real.cos θ)) =
-      -anglePolynomialEigenvalue κ n*(Real.sin θ^(4*Target.a κ)*
-        anglePolynomialTest (angleEigenPolynomial κ n) θ*anglePolynomialTest (angleEigenPolynomial κ m) θ) := by
-    simp only [anglePolynomialTest]
-    ring
-  have hm (θ : ℝ) : Real.sin θ^(4*Target.a κ)*anglePolynomialTest (angleEigenPolynomial κ n) θ*
-      (-anglePolynomialEigenvalue κ m*(angleEigenPolynomial κ m).eval (Real.cos θ)) =
-      -anglePolynomialEigenvalue κ m*(Real.sin θ^(4*Target.a κ)*
-        anglePolynomialTest (angleEigenPolynomial κ n) θ*anglePolynomialTest (angleEigenPolynomial κ m) θ) := by
-    simp only [anglePolynomialTest]
-    ring
-  simp_rw [hn,hm,intervalIntegral.integral_const_mul] at he
-  have hd : anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ m≠0 :=
-    sub_ne_zero.mpr ((anglePolynomialEigenvalue_strictMono hκ).injective.ne hnm)
-  have hz : (anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ m)*
-      (∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
-        anglePolynomialTest (angleEigenPolynomial κ n) θ*anglePolynomialTest (angleEigenPolynomial κ m) θ)=0 := by
-    linarith
-  exact (mul_eq_zero.mp hz).resolve_left hd
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleStationaryFlux
+
+namespace OAI
+
+/-! Symmetry under the paper's stationary angular weight, and orthogonality
+of the actual centered-generator polynomial eigenfunctions. -/
+noncomputable section
+open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma anglePolynomialWronskian_hasDerivAt {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) {θ : ℝ}
+    (hθ : θ∈Icc (0:ℝ) Real.pi) :
+    HasDerivAt (fun θ => anglePolynomialTest q θ*anglePolynomialFlux κ p θ -
+      anglePolynomialTest p θ*anglePolynomialFlux κ q θ)
+      (Real.sin θ^(4*Target.a κ)*
+        (anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ) -
+          anglePolynomialTest p θ*(anglePolynomialGenerator κ q).eval (Real.cos θ))) θ := by
+  have hp := (anglePolynomialTest_hasDerivAt q θ).mul (anglePolynomialFlux_hasDerivAt hκ p hθ)
+  have hq := (anglePolynomialTest_hasDerivAt p θ).mul (anglePolynomialFlux_hasDerivAt hκ q hθ)
+  apply (hp.sub hq).congr_deriv
+  simp only [anglePolynomialTest,anglePolynomialFlux]
+  ring
+
+lemma anglePolynomialGenerator_symmetric {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) :
+    (∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
+      anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ)) =
+    ∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
+      anglePolynomialTest p θ*(anglePolynomialGenerator κ q).eval (Real.cos θ) := by
+  have hr : 0<4*Target.a κ := by unfold Target.a; positivity
+  have hc (p q : ℝ[X]) : Continuous (fun θ => Real.sin θ^(4*Target.a κ)*
+      anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ)) :=
+    (((Real.continuous_rpow_const hr.le).comp Real.continuous_sin).mul
+      (q.continuous.comp Real.continuous_cos)).mul
+      ((anglePolynomialGenerator κ p).continuous.comp Real.continuous_cos)
+  have hi : IntervalIntegrable (fun θ => Real.sin θ^(4*Target.a κ)*
+      (anglePolynomialTest q θ*(anglePolynomialGenerator κ p).eval (Real.cos θ) -
+        anglePolynomialTest p θ*(anglePolynomialGenerator κ q).eval (Real.cos θ)))
+      volume 0 Real.pi := by
+    convert (((hc p q).sub (hc q p)).intervalIntegrable (μ := volume) 0 Real.pi) using 1
+    ext θ
+    simp only [Pi.sub_apply]
+    ring
+  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (fun θ hθ => anglePolynomialWronskian_hasDerivAt hκ p q
+      (by simpa only [uIcc_of_le Real.pi_pos.le] using hθ))
+    hi
+  simp only [mul_sub,← mul_assoc] at he
+  rw [intervalIntegral.integral_sub ((hc p q).intervalIntegrable _ _)
+    ((hc q p).intervalIntegrable _ _)] at he
+  have hzero : ∀ p : ℝ[X], anglePolynomialFlux κ p Real.pi=0 ∧ anglePolynomialFlux κ p 0=0 := by
+    intro p
+    simp [anglePolynomialFlux,Real.zero_rpow (by linarith : 4*Target.a κ+1≠0)]
+  rw [(hzero p).1,(hzero q).1,(hzero p).2,(hzero q).2] at he
+  simpa only [mul_zero,sub_self,sub_eq_zero] using he
+
+lemma angleEigenPolynomial_orthogonal {κ : ℝ} (hκ : 0<κ) (n m : ℕ) (hnm : n≠m) :
+    (∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
+      anglePolynomialTest (angleEigenPolynomial κ n) θ*
+        anglePolynomialTest (angleEigenPolynomial κ m) θ)=0 := by
+  have he := anglePolynomialGenerator_symmetric hκ
+    (angleEigenPolynomial κ n) (angleEigenPolynomial κ m)
+  simp_rw [angleEigenPolynomial_eigen hκ,eval_mul,eval_neg,eval_C] at he
+  have hn (θ : ℝ) : Real.sin θ^(4*Target.a κ)*anglePolynomialTest (angleEigenPolynomial κ m) θ*
+      (-anglePolynomialEigenvalue κ n*(angleEigenPolynomial κ n).eval (Real.cos θ)) =
+      -anglePolynomialEigenvalue κ n*(Real.sin θ^(4*Target.a κ)*
+        anglePolynomialTest (angleEigenPolynomial κ n) θ*anglePolynomialTest (angleEigenPolynomial κ m) θ) := by
+    simp only [anglePolynomialTest]
+    ring
+  have hm (θ : ℝ) : Real.sin θ^(4*Target.a κ)*anglePolynomialTest (angleEigenPolynomial κ n) θ*
+      (-anglePolynomialEigenvalue κ m*(angleEigenPolynomial κ m).eval (Real.cos θ)) =
+      -anglePolynomialEigenvalue κ m*(Real.sin θ^(4*Target.a κ)*
+        anglePolynomialTest (angleEigenPolynomial κ n) θ*anglePolynomialTest (angleEigenPolynomial κ m) θ) := by
+    simp only [anglePolynomialTest]
+    ring
+  simp_rw [hn,hm,intervalIntegral.integral_const_mul] at he
+  have hd : anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ m≠0 :=
+    sub_ne_zero.mpr ((anglePolynomialEigenvalue_strictMono hκ).injective.ne hnm)
+  have hz : (anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ m)*
+      (∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
+        anglePolynomialTest (angleEigenPolynomial κ n) θ*anglePolynomialTest (angleEigenPolynomial κ m) θ)=0 := by
+    linarith
+  exact (mul_eq_zero.mp hz).resolve_left hd
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenPolynomials.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenPolynomials.lean
index ad206e7c6b5145618a05cfb083bcac6be4e79427..5390de9d9afaeb11f53b2109efb4c66f3d6e5a41 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenPolynomials.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenPolynomials.lean
@@ -1,81 +1,81 @@
-import OAI.Probability.SLE.Angle.AnglePolynomialCoefficients
-
-namespace OAI
-
-/-! Monic polynomial eigenfunctions for the centered cosine generator.
-The coefficients are solved downwards in degree, with no choice of an
-auxiliary stochastic process or additional published input. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def angleEigenCoefficient (κ : ℝ) (N k : ℕ) : ℝ :=
-  if k=N then 1 else if _h : k<N then
-    ((κ/8)*((k:ℝ)+2)*((k:ℝ)+1)/(anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N))*
-      angleEigenCoefficient κ N (k+2)
-  else 0
-termination_by N-k
-  decreasing_by omega
-
-lemma angleEigenCoefficient_top (κ : ℝ) (N : ℕ) : angleEigenCoefficient κ N N=1 := by
-  rw [angleEigenCoefficient,ite_eq_left rfl]
-
-lemma angleEigenCoefficient_above (κ : ℝ) {N k : ℕ} (h : N<k) :
-    angleEigenCoefficient κ N k=0 := by
-  rw [angleEigenCoefficient,ite_eq_right (ne_of_gt h),dite_eq_right (by omega)]
-
-lemma angleEigenCoefficient_recurrence {κ : ℝ} (hκ : 0<κ) (N k : ℕ) :
-    (κ/8)*((k:ℝ)+2)*((k:ℝ)+1)*angleEigenCoefficient κ N (k+2) =
-      (anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N)*angleEigenCoefficient κ N k := by
-  rcases lt_trichotomy k N with hk | heq | hk
-  · have he : anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N ≠ 0 :=
-      sub_ne_zero.mpr (ne_of_lt (anglePolynomialEigenvalue_strictMono hκ hk))
-    conv_rhs => rw [angleEigenCoefficient]
-    rw [ite_eq_right (ne_of_lt hk),dite_eq_left hk]
-    field_simp
-  · subst k
-    rw [angleEigenCoefficient_above κ (by omega : N<N+2),angleEigenCoefficient_top]
-    ring
-  · rw [angleEigenCoefficient_above κ (by omega : N<k+2),angleEigenCoefficient_above κ hk]
-    ring
-
-def angleEigenPolynomial (κ : ℝ) (N : ℕ) : ℝ[X] :=
-  ∑ k ∈ Finset.range (N+1), monomial k (angleEigenCoefficient κ N k)
-
-lemma angleEigenPolynomial_coeff (κ : ℝ) (N k : ℕ) :
-    (angleEigenPolynomial κ N).coeff k=angleEigenCoefficient κ N k := by
-  by_cases hk : k<N+1
-  · simp [angleEigenPolynomial,coeff_monomial,Finset.sum_ite_eq',hk]
-  · simp [angleEigenPolynomial,coeff_monomial,Finset.sum_ite_eq',hk,
-      angleEigenCoefficient_above κ (by omega : N<k)]
-
-lemma angleEigenPolynomial_natDegree_le (κ : ℝ) (N : ℕ) :
-    (angleEigenPolynomial κ N).natDegree ≤ N := by
-  apply natDegree_le_iff_coeff_eq_zero.mpr
-  intro k hk
-  rw [angleEigenPolynomial_coeff,angleEigenCoefficient_above κ hk]
-
-lemma angleEigenPolynomial_monic (κ : ℝ) (N : ℕ) : (angleEigenPolynomial κ N).Monic := by
-  apply monic_of_natDegree_le_of_coeff_eq_one N (angleEigenPolynomial_natDegree_le κ N)
-  rw [angleEigenPolynomial_coeff,angleEigenCoefficient_top]
-
-lemma angleEigenPolynomial_natDegree (κ : ℝ) (N : ℕ) : (angleEigenPolynomial κ N).natDegree=N := by
-  apply le_antisymm (angleEigenPolynomial_natDegree_le κ N)
-  apply le_natDegree_of_ne_zero
-  rw [angleEigenPolynomial_coeff,angleEigenCoefficient_top]
-  norm_num
-
-lemma angleEigenPolynomial_eigen {κ : ℝ} (hκ : 0<κ) (N : ℕ) :
-    anglePolynomialGenerator κ (angleEigenPolynomial κ N) =
-      -C (anglePolynomialEigenvalue κ N)*angleEigenPolynomial κ N := by
-  apply (anglePolynomial_eigen_iff_coeff κ _ N).mpr
-  intro k
-  simp only [angleEigenPolynomial_coeff]
-  exact angleEigenCoefficient_recurrence hκ N k
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AnglePolynomialCoefficients
+
+namespace OAI
+
+/-! Monic polynomial eigenfunctions for the centered cosine generator.
+The coefficients are solved downwards in degree, with no choice of an
+auxiliary stochastic process or additional published input. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def angleEigenCoefficient (κ : ℝ) (N k : ℕ) : ℝ :=
+  if k=N then 1 else if _h : k<N then
+    ((κ/8)*((k:ℝ)+2)*((k:ℝ)+1)/(anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N))*
+      angleEigenCoefficient κ N (k+2)
+  else 0
+termination_by N-k
+  decreasing_by omega
+
+lemma angleEigenCoefficient_top (κ : ℝ) (N : ℕ) : angleEigenCoefficient κ N N=1 := by
+  rw [angleEigenCoefficient,ite_eq_left rfl]
+
+lemma angleEigenCoefficient_above (κ : ℝ) {N k : ℕ} (h : N<k) :
+    angleEigenCoefficient κ N k=0 := by
+  rw [angleEigenCoefficient,ite_eq_right (ne_of_gt h),dite_eq_right (by omega)]
+
+lemma angleEigenCoefficient_recurrence {κ : ℝ} (hκ : 0<κ) (N k : ℕ) :
+    (κ/8)*((k:ℝ)+2)*((k:ℝ)+1)*angleEigenCoefficient κ N (k+2) =
+      (anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N)*angleEigenCoefficient κ N k := by
+  rcases lt_trichotomy k N with hk | heq | hk
+  · have he : anglePolynomialEigenvalue κ k-anglePolynomialEigenvalue κ N ≠ 0 :=
+      sub_ne_zero.mpr (ne_of_lt (anglePolynomialEigenvalue_strictMono hκ hk))
+    conv_rhs => rw [angleEigenCoefficient]
+    rw [ite_eq_right (ne_of_lt hk),dite_eq_left hk]
+    field_simp
+  · subst k
+    rw [angleEigenCoefficient_above κ (by omega : N<N+2),angleEigenCoefficient_top]
+    ring
+  · rw [angleEigenCoefficient_above κ (by omega : N<k+2),angleEigenCoefficient_above κ hk]
+    ring
+
+def angleEigenPolynomial (κ : ℝ) (N : ℕ) : ℝ[X] :=
+  ∑ k ∈ Finset.range (N+1), monomial k (angleEigenCoefficient κ N k)
+
+lemma angleEigenPolynomial_coeff (κ : ℝ) (N k : ℕ) :
+    (angleEigenPolynomial κ N).coeff k=angleEigenCoefficient κ N k := by
+  by_cases hk : k<N+1
+  · simp [angleEigenPolynomial,coeff_monomial,Finset.sum_ite_eq',hk]
+  · simp [angleEigenPolynomial,coeff_monomial,Finset.sum_ite_eq',hk,
+      angleEigenCoefficient_above κ (by omega : N<k)]
+
+lemma angleEigenPolynomial_natDegree_le (κ : ℝ) (N : ℕ) :
+    (angleEigenPolynomial κ N).natDegree ≤ N := by
+  apply natDegree_le_iff_coeff_eq_zero.mpr
+  intro k hk
+  rw [angleEigenPolynomial_coeff,angleEigenCoefficient_above κ hk]
+
+lemma angleEigenPolynomial_monic (κ : ℝ) (N : ℕ) : (angleEigenPolynomial κ N).Monic := by
+  apply monic_of_natDegree_le_of_coeff_eq_one N (angleEigenPolynomial_natDegree_le κ N)
+  rw [angleEigenPolynomial_coeff,angleEigenCoefficient_top]
+
+lemma angleEigenPolynomial_natDegree (κ : ℝ) (N : ℕ) : (angleEigenPolynomial κ N).natDegree=N := by
+  apply le_antisymm (angleEigenPolynomial_natDegree_le κ N)
+  apply le_natDegree_of_ne_zero
+  rw [angleEigenPolynomial_coeff,angleEigenCoefficient_top]
+  norm_num
+
+lemma angleEigenPolynomial_eigen {κ : ℝ} (hκ : 0<κ) (N : ℕ) :
+    anglePolynomialGenerator κ (angleEigenPolynomial κ N) =
+      -C (anglePolynomialEigenvalue κ N)*angleEigenPolynomial κ N := by
+  apply (anglePolynomial_eigen_iff_coeff κ _ N).mpr
+  intro k
+  simp only [angleEigenPolynomial_coeff]
+  exact angleEigenCoefficient_recurrence hκ N k
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenSource.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenSource.lean
index 5ca4a3db420b892ae956499e678c108abd4bd939..0f94bdd9d388ccbd4ced271eafc55604f2f2c06b 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenSource.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenSource.lean
@@ -1,140 +1,140 @@
-import OAI.Probability.SLE.Angle.AngleEigenStopped
-import OAI.Probability.SLE.KernelExpectation
-
-namespace OAI
-
-/-! Identification of the localized harmonic eigenfunctions with the
-actual stopped Green weight and angle, including paths missing the level. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma logPolynomialDensity_green_factor (κ c η₀ : ℝ) (P : ℝ[X])
-    (q : ℂ × ℝ) (x : ℝ) :
-    logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P q x =
-      Real.exp (c*(logRadiusClock η₀ q-Real.log 2-η₀))*greenLogDensity κ q x*
-        P.eval (logCosine κ q x) := by
-  unfold logPolynomialDensity powerLogDensity greenLogDensity
-  rw [← Real.exp_add]
-  congr 2
-  unfold logRadiusClock
-  ring
-
-lemma logCosine_logCoordinates (κ b : ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ)
-    (t : ℝ≥0) (hy : 0<(G t z).im) :
-    logCosine κ (logLoewnerCoordinates G z t) b=
-      Real.cos ((G t z-(Real.sqrt κ*b:ℝ)).arg) := by
-  rw [logCosine_eq_angle,Multipoint.logAngle]
-  congr 2
-  apply Complex.ext
-  · simp only [Multipoint.logPoint_re,logCoordinates_re,Complex.sub_re,Complex.ofReal_re]
-  · simp only [Multipoint.logPoint_im,logCoordinates_im,Real.exp_log hy,
-      Complex.sub_im,Complex.ofReal_im,sub_zero]
-
-lemma logCosine_initial (κ : ℝ) (z : ℂ) (hz : z∈upperHalfPlane) :
-    logCosine κ (initialLogCoordinates z) 0=Real.cos z.arg := by
-  rw [logCosine_eq_angle,Multipoint.logAngle]
-  congr 2
-  apply Complex.ext
-  · simp [Multipoint.logPoint_re,initialLogCoordinates]
-  · simpa [Multipoint.logPoint_im,initialLogCoordinates] using Real.exp_log hz
-
-lemma logPolynomialDensity_initial (κ c : ℝ) (P : ℝ[X])
-    (z : ℂ) (hz : z∈upperHalfPlane) :
-    logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P (initialLogCoordinates z) 0=
-      Real.exp (-c*(Real.log 2+Real.log z.im))*initialGreen κ z*P.eval (Real.cos z.arg) := by
-  rw [logPolynomialDensity_green_factor κ c (Real.log z.im),greenLogDensity_initial κ z hz,
-    logCosine_initial κ z hz]
-  simp only [logRadiusClock,initialLogCoordinates,Complex.add_im,Complex.ofReal_im,
-    Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero,
-    add_zero,zero_add]
-  congr 2
-  congr 1
-  ring
-
-namespace AngularLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ}
-
-lemma eigen_eq_source_of_hit (a : AngularLocalization B hm κ z u ε)
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z∈upperHalfPlane)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F)
-    (hh : ω∈a.hit) (n : ℕ) :
-    logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)=
-    Real.exp (anglePolynomialEigenvalue κ n*(u-Real.log 2-Real.log z.im))*
-      stoppedGreen κ (fun t => Real.sqrt κ*B t ω) γ G z u*
-      (angleEigenPolynomial κ n).eval (Real.cos ((G (radiusLevelTime γ G z u).toNNReal z-
-        (Real.sqrt κ*B (radiusLevelTime γ G z u).toNNReal ω:ℝ)).arg)) := by
-  have hU := (hBc ω).const_mul (Real.sqrt κ)
-  obtain ⟨hl, hR⟩ := a.radius_at_hit hBc hz ω γ G F hw hh
-  have ht := radiusLevelTime_eq_of_live_level hU hw z u (a.τ ω) hl hR
-  have hq := (a.chart_identification hBc hz ω γ G F hw).2 (a.τ ω) le_rfl
-  rw [logPolynomialDensity_green_factor κ _ (Real.log z.im),
-    (show logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω)=u from hh)]
-  change Real.exp _*a.weight ω*_=_
-  rw [a.weight_eq_stoppedGreen_of_hit hBc hz ω γ G F hw hh,hq,
-    logCosine_logCoordinates κ _ G z _ (hw.imag_positive _ z hl),ht,ENNReal.toNNReal_coe]
-
-lemma eigen_small_of_not_hit (a : AngularLocalization B hm κ z u ε)
-    (hκ : 0<κ) (hκ8 : κ<8) (n : ℕ) (ω : Ω) (hh : ω∉a.hit) :
-    |logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)| ≤
-    Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-Real.log z.im))*
-      ε^(greenExponent κ/2)*((n:ℝ)+1)^(n+1) := by
-  have hp : 0≤codimension κ+anglePolynomialEigenvalue κ n := by
-    apply add_nonneg (codimension_pos κ hκ8).le
-    unfold anglePolynomialEigenvalue
-    positivity
-  have ha := (a.boundary ω).resolve_left hh
-  have hε : 0<ε := ha ▸ logAngleSq_pos κ (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)
-  rw [logPolynomialDensity,abs_mul,abs_of_pos (powerLogDensity_pos _ _ _ _ _)]
-  apply mul_le_mul _ (angleEigenPolynomial_eval_abs_le hκ n (logCosine_abs_le_one _ _ _))
-    (abs_nonneg _) (by positivity)
-  rw [powerLogDensity_radius_angle κ _ _ (Real.log z.im),ha]
-  apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg hε.le _)
-  apply Real.exp_le_exp.mpr
-  exact mul_le_mul_of_nonneg_left (by linarith [(a.inside ω (a.τ ω) le_rfl).1]) hp
-
-end AngularLocalization
-
-lemma localization_eigen_tendsto
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-    (a : ∀ j, AngularLocalization B hm κ z u (ε j))
-    (hε : Tendsto ε atTop (𝓝 0)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0<κ) (hκ8 : κ<8) (hz : z∈upperHalfPlane)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F) (n : ℕ) :
-    Tendsto (fun j => logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) ((a j).solution.Q ((a j).τ ω) ω) (B ((a j).τ ω) ω)) atTop
-      (𝓝 (Real.exp (anglePolynomialEigenvalue κ n*(u-Real.log 2-Real.log z.im))*
-        stoppedGreen κ (fun t => Real.sqrt κ*B t ω) γ G z u*
-        (angleEigenPolynomial κ n).eval (Real.cos ((G (radiusLevelTime γ G z u).toNNReal z-
-          (Real.sqrt κ*B (radiusLevelTime γ G z u).toNNReal ω:ℝ)).arg)))) := by
-  by_cases hh : ∃ T : ℝ≥0, z∈survivingDomain γ T ∧ fullRadius G T z=(2*z.im)*Real.exp (-u)
-  · obtain ⟨T,hT,hR⟩ := hh
-    apply tendsto_const_nhds.congr'
-    filter_upwards [eventually_hit_of_live_level a hε hBc hz ω γ G F hw T hT hR] with j hj
-    exact ((a j).eigen_eq_source_of_hit hBc hz ω γ G F hw hj.1 n).symm
-  · rw [stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hh,mul_zero,zero_mul]
-    have hp : 0<greenExponent κ/2 := by positivity [greenExponent_pos κ hκ hκ8]
-    have ht := (((Real.continuousAt_rpow_const 0 (greenExponent κ/2) (Or.inr hp.le)).tendsto.comp hε).const_mul
-      (Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-Real.log z.im)))).mul_const
-      (((n:ℝ)+1)^(n+1))
-    simp only [Real.zero_rpow hp.ne',mul_zero,zero_mul] at ht
-    apply squeeze_zero_norm _ ht
-    intro j
-    simpa only [Real.norm_eq_abs,Function.comp_apply] using (a j).eigen_small_of_not_hit hκ hκ8 n ω
-      (fun hj => hh ⟨(a j).τ ω,(a j).radius_at_hit hBc hz ω γ G F hw hj⟩)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenStopped
+import OAI.Probability.SLE.KernelExpectation
+
+namespace OAI
+
+/-! Identification of the localized harmonic eigenfunctions with the
+actual stopped Green weight and angle, including paths missing the level. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma logPolynomialDensity_green_factor (κ c η₀ : ℝ) (P : ℝ[X])
+    (q : ℂ × ℝ) (x : ℝ) :
+    logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P q x =
+      Real.exp (c*(logRadiusClock η₀ q-Real.log 2-η₀))*greenLogDensity κ q x*
+        P.eval (logCosine κ q x) := by
+  unfold logPolynomialDensity powerLogDensity greenLogDensity
+  rw [← Real.exp_add]
+  congr 2
+  unfold logRadiusClock
+  ring
+
+lemma logCosine_logCoordinates (κ b : ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ)
+    (t : ℝ≥0) (hy : 0<(G t z).im) :
+    logCosine κ (logLoewnerCoordinates G z t) b=
+      Real.cos ((G t z-(Real.sqrt κ*b:ℝ)).arg) := by
+  rw [logCosine_eq_angle,Multipoint.logAngle]
+  congr 2
+  apply Complex.ext
+  · simp only [Multipoint.logPoint_re,logCoordinates_re,Complex.sub_re,Complex.ofReal_re]
+  · simp only [Multipoint.logPoint_im,logCoordinates_im,Real.exp_log hy,
+      Complex.sub_im,Complex.ofReal_im,sub_zero]
+
+lemma logCosine_initial (κ : ℝ) (z : ℂ) (hz : z∈upperHalfPlane) :
+    logCosine κ (initialLogCoordinates z) 0=Real.cos z.arg := by
+  rw [logCosine_eq_angle,Multipoint.logAngle]
+  congr 2
+  apply Complex.ext
+  · simp [Multipoint.logPoint_re,initialLogCoordinates]
+  · simpa [Multipoint.logPoint_im,initialLogCoordinates] using Real.exp_log hz
+
+lemma logPolynomialDensity_initial (κ c : ℝ) (P : ℝ[X])
+    (z : ℂ) (hz : z∈upperHalfPlane) :
+    logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P (initialLogCoordinates z) 0=
+      Real.exp (-c*(Real.log 2+Real.log z.im))*initialGreen κ z*P.eval (Real.cos z.arg) := by
+  rw [logPolynomialDensity_green_factor κ c (Real.log z.im),greenLogDensity_initial κ z hz,
+    logCosine_initial κ z hz]
+  simp only [logRadiusClock,initialLogCoordinates,Complex.add_im,Complex.ofReal_im,
+    Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero,
+    add_zero,zero_add]
+  congr 2
+  congr 1
+  ring
+
+namespace AngularLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ}
+
+lemma eigen_eq_source_of_hit (a : AngularLocalization B hm κ z u ε)
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z∈upperHalfPlane)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F)
+    (hh : ω∈a.hit) (n : ℕ) :
+    logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)=
+    Real.exp (anglePolynomialEigenvalue κ n*(u-Real.log 2-Real.log z.im))*
+      stoppedGreen κ (fun t => Real.sqrt κ*B t ω) γ G z u*
+      (angleEigenPolynomial κ n).eval (Real.cos ((G (radiusLevelTime γ G z u).toNNReal z-
+        (Real.sqrt κ*B (radiusLevelTime γ G z u).toNNReal ω:ℝ)).arg)) := by
+  have hU := (hBc ω).const_mul (Real.sqrt κ)
+  obtain ⟨hl, hR⟩ := a.radius_at_hit hBc hz ω γ G F hw hh
+  have ht := radiusLevelTime_eq_of_live_level hU hw z u (a.τ ω) hl hR
+  have hq := (a.chart_identification hBc hz ω γ G F hw).2 (a.τ ω) le_rfl
+  rw [logPolynomialDensity_green_factor κ _ (Real.log z.im),
+    (show logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω)=u from hh)]
+  change Real.exp _*a.weight ω*_=_
+  rw [a.weight_eq_stoppedGreen_of_hit hBc hz ω γ G F hw hh,hq,
+    logCosine_logCoordinates κ _ G z _ (hw.imag_positive _ z hl),ht,ENNReal.toNNReal_coe]
+
+lemma eigen_small_of_not_hit (a : AngularLocalization B hm κ z u ε)
+    (hκ : 0<κ) (hκ8 : κ<8) (n : ℕ) (ω : Ω) (hh : ω∉a.hit) :
+    |logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)| ≤
+    Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-Real.log z.im))*
+      ε^(greenExponent κ/2)*((n:ℝ)+1)^(n+1) := by
+  have hp : 0≤codimension κ+anglePolynomialEigenvalue κ n := by
+    apply add_nonneg (codimension_pos κ hκ8).le
+    unfold anglePolynomialEigenvalue
+    positivity
+  have ha := (a.boundary ω).resolve_left hh
+  have hε : 0<ε := ha ▸ logAngleSq_pos κ (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)
+  rw [logPolynomialDensity,abs_mul,abs_of_pos (powerLogDensity_pos _ _ _ _ _)]
+  apply mul_le_mul _ (angleEigenPolynomial_eval_abs_le hκ n (logCosine_abs_le_one _ _ _))
+    (abs_nonneg _) (by positivity)
+  rw [powerLogDensity_radius_angle κ _ _ (Real.log z.im),ha]
+  apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg hε.le _)
+  apply Real.exp_le_exp.mpr
+  exact mul_le_mul_of_nonneg_left (by linarith [(a.inside ω (a.τ ω) le_rfl).1]) hp
+
+end AngularLocalization
+
+lemma localization_eigen_tendsto
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+    (a : ∀ j, AngularLocalization B hm κ z u (ε j))
+    (hε : Tendsto ε atTop (𝓝 0)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0<κ) (hκ8 : κ<8) (hz : z∈upperHalfPlane)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F) (n : ℕ) :
+    Tendsto (fun j => logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) ((a j).solution.Q ((a j).τ ω) ω) (B ((a j).τ ω) ω)) atTop
+      (𝓝 (Real.exp (anglePolynomialEigenvalue κ n*(u-Real.log 2-Real.log z.im))*
+        stoppedGreen κ (fun t => Real.sqrt κ*B t ω) γ G z u*
+        (angleEigenPolynomial κ n).eval (Real.cos ((G (radiusLevelTime γ G z u).toNNReal z-
+          (Real.sqrt κ*B (radiusLevelTime γ G z u).toNNReal ω:ℝ)).arg)))) := by
+  by_cases hh : ∃ T : ℝ≥0, z∈survivingDomain γ T ∧ fullRadius G T z=(2*z.im)*Real.exp (-u)
+  · obtain ⟨T,hT,hR⟩ := hh
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_hit_of_live_level a hε hBc hz ω γ G F hw T hT hR] with j hj
+    exact ((a j).eigen_eq_source_of_hit hBc hz ω γ G F hw hj.1 n).symm
+  · rw [stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hh,mul_zero,zero_mul]
+    have hp : 0<greenExponent κ/2 := by positivity [greenExponent_pos κ hκ hκ8]
+    have ht := (((Real.continuousAt_rpow_const 0 (greenExponent κ/2) (Or.inr hp.le)).tendsto.comp hε).const_mul
+      (Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-Real.log z.im)))).mul_const
+      (((n:ℝ)+1)^(n+1))
+    simp only [Real.zero_rpow hp.ne',mul_zero,zero_mul] at ht
+    apply squeeze_zero_norm _ ht
+    intro j
+    simpa only [Real.norm_eq_abs,Function.comp_apply] using (a j).eigen_small_of_not_hit hκ hκ8 n ω
+      (fun hj => hh ⟨(a j).τ ω,(a j).radius_at_hit hBc hz ω γ G F hw hj⟩)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleEigenStopped.lean b/lean/OAI/Probability/SLE/Angle/AngleEigenStopped.lean
index 3d9fb89ca2df92be0c9ff59f2902ed868bf711ae..7816f7ba1f1b6dcd1dd770550fa5433b07e23947 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleEigenStopped.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleEigenStopped.lean
@@ -1,104 +1,104 @@
-import OAI.Probability.SLE.Angle.AngleHarmonicStopped
-import OAI.Probability.SLE.Angle.AngleEigenBounds
-import OAI.Probability.SLE.AngularPowerDynkin
-
-namespace OAI
-
-/-! Exact stopped mean identities for every angular eigenpolynomial. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma ClippedLogSolution.stopped_harmonic_bounded
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω=0)
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
-    (hgen : ∀ q x, fderiv ℝ (fun r => Φ (r,x)) q
-      (physicalLogLoewnerField (Real.sqrt κ*x) q) +
-      (1/2:ℝ)*iteratedDeriv 2 (fun y => Φ (q,y)) x=0)
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hin : ∀ ω t, t ≤ τ ω → (c.Q t ω).1.im ∈ Icc lo hi)
-    (K : ℝ) (hval : ∀ ω t, t ≤ τ ω → |Φ (c.Q t ω,B t ω)| ≤ K) :
-    Integrable (fun ω => Φ (c.Q (τ ω) ω,B (τ ω) ω)) P ∧
-      (∫ ω, Φ (c.Q (τ ω) ω,B (τ ω) ω) ∂P)=Φ (q₀,0) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨σn, hsn, hsle, hsB, hstable⟩ := exists_brownian_compact_stops B hm hBc hB0 τ hτ T hb
-  have hmean (n : ℕ) : Integrable (fun ω => Φ (c.Q (σn n ω) ω,B (σn n ω) ω)) P ∧
-      (∫ ω, Φ (c.Q (σn n ω) ω,B (σn n ω) ω) ∂P)=Φ (q₀,0) := by
-    let R := max (‖q₀‖+c.L*T) ((n:ℝ)+1)
-    apply c.stopped_harmonic_compact hB hBm hBc Φ hΦ hgen R
-      (le_trans (by positivity : (0:ℝ)≤(n:ℝ)+1) (le_max_right _ _))
-      (σn n) (hsn n) T (fun ω => (hsle n ω).trans (hb ω))
-    intro ω t ht
-    refine ⟨?_,hin ω t (ht.trans (hsle n ω))⟩
-    rw [Prod.norm_def, max_le_iff,Real.norm_eq_abs]
-    exact ⟨(c.norm_le ω t T (ht.trans ((hsle n ω).trans (hb ω)))).trans (le_max_left _ _),
-      (hsB n ω t ht).trans (le_max_right _ _)⟩
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hQm := c.measurable.comp ((measurable_coe_nnreal_real.comp hτm).prodMk measurable_id)
-  have hbm := hBm.comp (hτm.prodMk measurable_id)
-  have hi : Integrable (fun ω => Φ (c.Q (τ ω) ω,B (τ ω) ω)) P :=
-    Integrable.of_bound (hΦ.continuous.measurable.comp (hQm.prodMk hbm)).aestronglyMeasurable K
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hval ω (τ ω) le_rfl)
-  refine ⟨hi,?_⟩
-  have hlim := integral_tendsto_stopped_eval (fun t ω => Φ (c.Q t ω,B t ω)) σn τ
-    (fun n => (hmean n).1.aestronglyMeasurable) K
-    (fun n ω => hval ω (σn n ω) (hsle n ω)) hstable
-  have he : (fun n => ∫ ω, Φ (c.Q (σn n ω) ω,B (σn n ω) ω) ∂P)=fun _ => Φ (q₀,0) :=
-    funext (fun n => (hmean n).2)
-  rw [he] at hlim
-  exact tendsto_nhds_unique hlim tendsto_const_nhds
-
-lemma logEigenDensity_bound {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    (n : ℕ) (η₀ u : ℝ) (q : ℂ × ℝ) (x : ℝ)
-    (hu : logRadiusClock η₀ q ≤ u) :
-    |logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) q x| ≤
-      Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-η₀))*
-        ((n:ℝ)+1)^(n+1) := by
-  have hn : 0≤anglePolynomialEigenvalue κ n := by
-    unfold anglePolynomialEigenvalue
-    positivity
-  rw [logPolynomialDensity,abs_mul,abs_of_pos (powerLogDensity_pos _ _ _ _ _)]
-  exact mul_le_mul
-    (powerLogDensity_le_radius κ _ _ η₀ u (add_nonneg (codimension_pos κ hκ8).le hn)
-      (greenExponent_pos κ hκ hκ8).le q x hu)
-    (angleEigenPolynomial_eval_abs_le hκ n (logCosine_abs_le_one κ q x))
-    (abs_nonneg _) (Real.exp_pos _).le
-
-lemma AngularLocalization.mean_eigen
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω=0)
-    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) {z : ℂ} {u ε : ℝ}
-    (a : AngularLocalization B hm κ z u ε) (n : ℕ) :
-    Integrable (fun ω => logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
-    (∫ ω, logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P)=
-    logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) (initialLogCoordinates z) 0 := by
-  apply a.solution.stopped_harmonic_bounded hB hBm hBc hB0
-    (fun p => logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
-      (greenExponent κ) (angleEigenPolynomial κ n) p.1 p.2)
-    (logPolynomialDensity_contDiff _ _ _ _) (logEigenDensity_generator_zero hκ n)
-    a.τ a.stopping a.T a.bounded (fun ω t ht => (a.inside ω t ht).2.2)
-    (Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-Real.log z.im))*
-      ((n:ℝ)+1)^(n+1))
-  exact fun ω t ht => logEigenDensity_bound hκ hκ8 n _ _ _ _ (a.inside ω t ht).1
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleHarmonicStopped
+import OAI.Probability.SLE.Angle.AngleEigenBounds
+import OAI.Probability.SLE.AngularPowerDynkin
+
+namespace OAI
+
+/-! Exact stopped mean identities for every angular eigenpolynomial. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Polynomial
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma ClippedLogSolution.stopped_harmonic_bounded
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω=0)
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
+    (hgen : ∀ q x, fderiv ℝ (fun r => Φ (r,x)) q
+      (physicalLogLoewnerField (Real.sqrt κ*x) q) +
+      (1/2:ℝ)*iteratedDeriv 2 (fun y => Φ (q,y)) x=0)
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hin : ∀ ω t, t ≤ τ ω → (c.Q t ω).1.im ∈ Icc lo hi)
+    (K : ℝ) (hval : ∀ ω t, t ≤ τ ω → |Φ (c.Q t ω,B t ω)| ≤ K) :
+    Integrable (fun ω => Φ (c.Q (τ ω) ω,B (τ ω) ω)) P ∧
+      (∫ ω, Φ (c.Q (τ ω) ω,B (τ ω) ω) ∂P)=Φ (q₀,0) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨σn, hsn, hsle, hsB, hstable⟩ := exists_brownian_compact_stops B hm hBc hB0 τ hτ T hb
+  have hmean (n : ℕ) : Integrable (fun ω => Φ (c.Q (σn n ω) ω,B (σn n ω) ω)) P ∧
+      (∫ ω, Φ (c.Q (σn n ω) ω,B (σn n ω) ω) ∂P)=Φ (q₀,0) := by
+    let R := max (‖q₀‖+c.L*T) ((n:ℝ)+1)
+    apply c.stopped_harmonic_compact hB hBm hBc Φ hΦ hgen R
+      (le_trans (by positivity : (0:ℝ)≤(n:ℝ)+1) (le_max_right _ _))
+      (σn n) (hsn n) T (fun ω => (hsle n ω).trans (hb ω))
+    intro ω t ht
+    refine ⟨?_,hin ω t (ht.trans (hsle n ω))⟩
+    rw [Prod.norm_def, max_le_iff,Real.norm_eq_abs]
+    exact ⟨(c.norm_le ω t T (ht.trans ((hsle n ω).trans (hb ω)))).trans (le_max_left _ _),
+      (hsB n ω t ht).trans (le_max_right _ _)⟩
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hQm := c.measurable.comp ((measurable_coe_nnreal_real.comp hτm).prodMk measurable_id)
+  have hbm := hBm.comp (hτm.prodMk measurable_id)
+  have hi : Integrable (fun ω => Φ (c.Q (τ ω) ω,B (τ ω) ω)) P :=
+    Integrable.of_bound (hΦ.continuous.measurable.comp (hQm.prodMk hbm)).aestronglyMeasurable K
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hval ω (τ ω) le_rfl)
+  refine ⟨hi,?_⟩
+  have hlim := integral_tendsto_stopped_eval (fun t ω => Φ (c.Q t ω,B t ω)) σn τ
+    (fun n => (hmean n).1.aestronglyMeasurable) K
+    (fun n ω => hval ω (σn n ω) (hsle n ω)) hstable
+  have he : (fun n => ∫ ω, Φ (c.Q (σn n ω) ω,B (σn n ω) ω) ∂P)=fun _ => Φ (q₀,0) :=
+    funext (fun n => (hmean n).2)
+  rw [he] at hlim
+  exact tendsto_nhds_unique hlim tendsto_const_nhds
+
+lemma logEigenDensity_bound {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    (n : ℕ) (η₀ u : ℝ) (q : ℂ × ℝ) (x : ℝ)
+    (hu : logRadiusClock η₀ q ≤ u) :
+    |logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) q x| ≤
+      Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-η₀))*
+        ((n:ℝ)+1)^(n+1) := by
+  have hn : 0≤anglePolynomialEigenvalue κ n := by
+    unfold anglePolynomialEigenvalue
+    positivity
+  rw [logPolynomialDensity,abs_mul,abs_of_pos (powerLogDensity_pos _ _ _ _ _)]
+  exact mul_le_mul
+    (powerLogDensity_le_radius κ _ _ η₀ u (add_nonneg (codimension_pos κ hκ8).le hn)
+      (greenExponent_pos κ hκ hκ8).le q x hu)
+    (angleEigenPolynomial_eval_abs_le hκ n (logCosine_abs_le_one κ q x))
+    (abs_nonneg _) (Real.exp_pos _).le
+
+lemma AngularLocalization.mean_eigen
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω=0)
+    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) {z : ℂ} {u ε : ℝ}
+    (a : AngularLocalization B hm κ z u ε) (n : ℕ) :
+    Integrable (fun ω => logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
+    (∫ ω, logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P)=
+    logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) (initialLogCoordinates z) 0 := by
+  apply a.solution.stopped_harmonic_bounded hB hBm hBc hB0
+    (fun p => logPolynomialDensity κ (codimension κ+anglePolynomialEigenvalue κ n)
+      (greenExponent κ) (angleEigenPolynomial κ n) p.1 p.2)
+    (logPolynomialDensity_contDiff _ _ _ _) (logEigenDensity_generator_zero hκ n)
+    a.τ a.stopping a.T a.bounded (fun ω t ht => (a.inside ω t ht).2.2)
+    (Real.exp ((codimension κ+anglePolynomialEigenvalue κ n)*(u-Real.log 2-Real.log z.im))*
+      ((n:ℝ)+1)^(n+1))
+  exact fun ω t ht => logEigenDensity_bound hκ hκ8 n _ _ _ _ (a.inside ω t ht).1
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleHarmonicStopped.lean b/lean/OAI/Probability/SLE/Angle/AngleHarmonicStopped.lean
index 280ef4b06b5666333208ced7a2aa9ed1c7df9c08..7b860eebfe83cb969c77f0ea81ecd2ddd86fc7e6 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleHarmonicStopped.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleHarmonicStopped.lean
@@ -1,143 +1,143 @@
-import OAI.Probability.SLE.Angle.AnglePolynomialLogGenerator
-import OAI.Probability.SLE.StoppedPower
-
-namespace OAI
-
-/-! Bounded stopping for smooth harmonic tests of the logarithmic Loewner
-state and Brownian driver. This supplies the angular eigenfunction moments. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-private lemma compact_harmonic_field
-    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
-    (R : ℝ) (hR : 0 ≤ R) :
-    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
-      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = Φ (q, x)) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → fderiv ℝ φ (q, x) = fderiv ℝ Φ (q, x)) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
-        iteratedDeriv 2 (fun y => Φ (q, y)) x) := by
-  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
-  let φ (z : (ℂ × ℝ) × ℝ) := b z * Φ z
-  have hnear (z : (ℂ × ℝ) × ℝ) (hz : ‖z‖ ≤ R) : φ =ᶠ[𝓝 z] Φ := by
-    have hb : z ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
-      simp only [Metric.mem_ball, dist_zero_right]
-      change ‖z‖ < R + 1
-      linarith
-    filter_upwards [b.eventuallyEq_one_of_mem_ball hb] with w hw
-    simp only [φ, hw, Pi.one_apply, one_mul]
-  refine ⟨φ, b.contDiff.mul hΦ, b.hasCompactSupport.mul_right,
-    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
-  · exact fun q x h => (hnear (q, x) h).fderiv_eq
-  · intro q x h
-    exact ((hnear (q, x) h).comp_tendsto
-      ((continuous_const.prodMk continuous_id).tendsto x)).iteratedDeriv_eq 2
-
-private lemma harmonic_pair_directional
-    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
-    (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ Φ (q, x) (v, 0) = fderiv ℝ (fun r => Φ (r, x)) q v := by
-  have hd := (hΦ.differentiable (by simp)) (q, x)
-  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk
-    (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
-  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) (hd.hasFDerivAt.comp q hp).fderiv
-  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
-    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
-
-lemma ClippedLogSolution.stopped_harmonic_compact
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
-    (hgen : ∀ q x, fderiv ℝ (fun r => Φ (r, x)) q
-      (physicalLogLoewnerField (Real.sqrt κ*x) q) +
-      (1/2:ℝ)*iteratedDeriv 2 (fun y => Φ (q,y)) x = 0)
-    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hinside : ∀ ω t, t ≤ τ ω →
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ (c.Q t ω).1.im ∈ Icc lo hi) :
-    Integrable (fun ω => Φ (c.Q (τ ω) ω, B (τ ω) ω)) P ∧
-      (∫ ω, Φ (c.Q (τ ω) ω, B (τ ω) ω) ∂P) = Φ (q₀, 0) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := compact_harmonic_field Φ hΦ R hR
-  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) :=
-    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
-  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
-    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
-      measurable_snd)).const_mul _
-  have hpair := hBr.prodMk c.measurable
-  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
-    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
-  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
-    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
-  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
-    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
-    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
-  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
-  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
-  have hzero (ω : Ω) (t : ℝ) (ht : t ∈ Icc 0 (τ ω : ℝ)) :
-      brownianFieldDrift B F a t ω = 0 := by
-    have ht' : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    obtain ⟨hball, hin⟩ := hinside ω t.toNNReal ht'
-    have he : (t.toNNReal : ℝ) = t := Real.coe_toNNReal t ht.1
-    change fderiv ℝ φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
-      (V (t.toNNReal : ℝ) ω, 0) + _ = 0
-    dsimp only [V]
-    rw [Real.toNNReal_coe, hφD _ _ hball]
-    change fderiv ℝ Φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
-      (clippedLogLoewnerField lo hi hle (Real.sqrt κ*B t.toNNReal ω)
-        (c.Q (t.toNNReal : ℝ) ω), 0) +
-      (1/2:ℝ)*iteratedDeriv 2 (fun y => φ (c.Q (t.toNNReal : ℝ) ω,y)) (B t.toNNReal ω)=0
-    rw [hφ2 _ _ hball, harmonic_pair_directional Φ hΦ,
-      clippedLogLoewnerField_eq_physical lo hi hle _ _ hin]
-    exact hgen _ _
-  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hBmτ := hBm.comp (hτm.prodMk measurable_id)
-  have hQmτ := c.measurable.comp
-    ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
-  have hfm := hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
-  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
-    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
-    exact Eventually.of_forall fun ω => by
-      change ‖φ (c.Q (τ ω) ω, B (τ ω) ω)‖ ≤ (K : ℝ)
-      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
-        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
-  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) = Φ (c.Q (τ ω) ω,B (τ ω) ω) :=
-    hφeq _ _ (hinside ω (τ ω) le_rfl).1
-  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
-  rw [← integral_congr_ae (Eventually.of_forall hfeq),hopt]
-  have hz : (fun ω => ∫ t in (0:ℝ)..(τ ω:ℝ), brownianFieldDrift B F a t ω) = 0 := by
-    funext ω
-    rw [intervalIntegral.integral_congr
-      (fun t ht => hzero ω t (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using ht))]
-    simp
-  rw [hz]
-  change (∫ ω, F 0 ω (B 0 ω) ∂P)+(∫ _ : Ω, (0:ℝ) ∂P)=Φ (q₀,0)
-  rw [integral_zero,add_zero]
-  calc
-    _ = ∫ _, Φ (q₀,0) ∂P := by
-      apply integral_congr_ae
-      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
-      change φ (c.Q 0 ω,B 0 ω)=_
-      have hh : ‖(c.Q 0 ω,B 0 ω)‖ ≤ R := by
-        simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
-      rw [hφeq _ _ hh,c.initial ω,hB0]
-    _ = _ := by simp
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AnglePolynomialLogGenerator
+import OAI.Probability.SLE.StoppedPower
+
+namespace OAI
+
+/-! Bounded stopping for smooth harmonic tests of the logarithmic Loewner
+state and Brownian driver. This supplies the angular eigenfunction moments. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+private lemma compact_harmonic_field
+    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
+    (R : ℝ) (hR : 0 ≤ R) :
+    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
+      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = Φ (q, x)) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → fderiv ℝ φ (q, x) = fderiv ℝ Φ (q, x)) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
+        iteratedDeriv 2 (fun y => Φ (q, y)) x) := by
+  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let φ (z : (ℂ × ℝ) × ℝ) := b z * Φ z
+  have hnear (z : (ℂ × ℝ) × ℝ) (hz : ‖z‖ ≤ R) : φ =ᶠ[𝓝 z] Φ := by
+    have hb : z ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
+      simp only [Metric.mem_ball, dist_zero_right]
+      change ‖z‖ < R + 1
+      linarith
+    filter_upwards [b.eventuallyEq_one_of_mem_ball hb] with w hw
+    simp only [φ, hw, Pi.one_apply, one_mul]
+  refine ⟨φ, b.contDiff.mul hΦ, b.hasCompactSupport.mul_right,
+    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
+  · exact fun q x h => (hnear (q, x) h).fderiv_eq
+  · intro q x h
+    exact ((hnear (q, x) h).comp_tendsto
+      ((continuous_const.prodMk continuous_id).tendsto x)).iteratedDeriv_eq 2
+
+private lemma harmonic_pair_directional
+    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
+    (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ Φ (q, x) (v, 0) = fderiv ℝ (fun r => Φ (r, x)) q v := by
+  have hd := (hΦ.differentiable (by simp)) (q, x)
+  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk
+    (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
+  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) (hd.hasFDerivAt.comp q hp).fderiv
+  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
+    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
+
+lemma ClippedLogSolution.stopped_harmonic_compact
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P) (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (Φ : (ℂ × ℝ) × ℝ → ℝ) (hΦ : ContDiff ℝ (⊤ : ℕ∞) Φ)
+    (hgen : ∀ q x, fderiv ℝ (fun r => Φ (r, x)) q
+      (physicalLogLoewnerField (Real.sqrt κ*x) q) +
+      (1/2:ℝ)*iteratedDeriv 2 (fun y => Φ (q,y)) x = 0)
+    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hinside : ∀ ω t, t ≤ τ ω →
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ (c.Q t ω).1.im ∈ Icc lo hi) :
+    Integrable (fun ω => Φ (c.Q (τ ω) ω, B (τ ω) ω)) P ∧
+      (∫ ω, Φ (c.Q (τ ω) ω, B (τ ω) ω) ∂P) = Φ (q₀, 0) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := compact_harmonic_field Φ hΦ R hR
+  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) :=
+    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
+  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
+    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
+      measurable_snd)).const_mul _
+  have hpair := hBr.prodMk c.measurable
+  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
+    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
+  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
+    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
+  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
+    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
+    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
+  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
+  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
+  have hzero (ω : Ω) (t : ℝ) (ht : t ∈ Icc 0 (τ ω : ℝ)) :
+      brownianFieldDrift B F a t ω = 0 := by
+    have ht' : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    obtain ⟨hball, hin⟩ := hinside ω t.toNNReal ht'
+    have he : (t.toNNReal : ℝ) = t := Real.coe_toNNReal t ht.1
+    change fderiv ℝ φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
+      (V (t.toNNReal : ℝ) ω, 0) + _ = 0
+    dsimp only [V]
+    rw [Real.toNNReal_coe, hφD _ _ hball]
+    change fderiv ℝ Φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
+      (clippedLogLoewnerField lo hi hle (Real.sqrt κ*B t.toNNReal ω)
+        (c.Q (t.toNNReal : ℝ) ω), 0) +
+      (1/2:ℝ)*iteratedDeriv 2 (fun y => φ (c.Q (t.toNNReal : ℝ) ω,y)) (B t.toNNReal ω)=0
+    rw [hφ2 _ _ hball, harmonic_pair_directional Φ hΦ,
+      clippedLogLoewnerField_eq_physical lo hi hle _ _ hin]
+    exact hgen _ _
+  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hBmτ := hBm.comp (hτm.prodMk measurable_id)
+  have hQmτ := c.measurable.comp
+    ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
+  have hfm := hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
+  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
+    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
+    exact Eventually.of_forall fun ω => by
+      change ‖φ (c.Q (τ ω) ω, B (τ ω) ω)‖ ≤ (K : ℝ)
+      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
+        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
+  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) = Φ (c.Q (τ ω) ω,B (τ ω) ω) :=
+    hφeq _ _ (hinside ω (τ ω) le_rfl).1
+  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
+  rw [← integral_congr_ae (Eventually.of_forall hfeq),hopt]
+  have hz : (fun ω => ∫ t in (0:ℝ)..(τ ω:ℝ), brownianFieldDrift B F a t ω) = 0 := by
+    funext ω
+    rw [intervalIntegral.integral_congr
+      (fun t ht => hzero ω t (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using ht))]
+    simp
+  rw [hz]
+  change (∫ ω, F 0 ω (B 0 ω) ∂P)+(∫ _ : Ω, (0:ℝ) ∂P)=Φ (q₀,0)
+  rw [integral_zero,add_zero]
+  calc
+    _ = ∫ _, Φ (q₀,0) ∂P := by
+      apply integral_congr_ae
+      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
+      change φ (c.Q 0 ω,B 0 ω)=_
+      have hh : ‖(c.Q 0 ω,B 0 ω)‖ ≤ R := by
+        simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
+      rw [hφeq _ _ hh,c.initial ω,hB0]
+    _ = _ := by simp
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleLogCosine.lean b/lean/OAI/Probability/SLE/Angle/AngleLogCosine.lean
index df0b37895b659c8ac725d7034ef482747271d164..66838846d95cfa4da6ff29687e3bd94540e0a1bb 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleLogCosine.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleLogCosine.lean
@@ -1,86 +1,86 @@
-import OAI.Probability.SLE.Multipoint.MultipointLogAngular
-import OAI.Probability.SLE.Angle.AnglePolynomialGenerator
-
-namespace OAI
-
-/-! Smooth cosine coordinates in the actual logarithmic Loewner state.
-These formulas are used to apply the Brownian generator to eigenpolynomials. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def logInverseNorm (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  Real.exp (-q.1.im)*powerLogDensity κ 0 1 q x
-
-def logCosine (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  (q.1.re-Real.sqrt κ*x)*logInverseNorm κ q x
-
-lemma logInverseNorm_eq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    logInverseNorm κ q x=Real.exp (-Real.log (greenLogNormSq κ q x)/2) := by
-  unfold logInverseNorm powerLogDensity
-  rw [← Real.exp_add]
-  congr 1
-  ring
-
-lemma logInverseNorm_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    0<logInverseNorm κ q x := by rw [logInverseNorm_eq]; positivity
-
-lemma logInverseNorm_sq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    logInverseNorm κ q x^2=(greenLogNormSq κ q x)⁻¹ := by
-  rw [logInverseNorm_eq,pow_two,← Real.exp_add,
-    show -Real.log (greenLogNormSq κ q x)/2 + -Real.log (greenLogNormSq κ q x)/2 =
-      -Real.log (greenLogNormSq κ q x) by ring,Real.exp_neg,
-    Real.exp_log (greenLogNormSq_pos κ q x)]
-
-lemma logCosine_eq_angle (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    logCosine κ q x=Real.cos (Multipoint.logAngle κ q x) := by
-  have hn := Multipoint.logPoint_norm_pos κ q x
-  have he : logInverseNorm κ q x=‖Multipoint.logPoint κ q x‖⁻¹ := by
-    rw [logInverseNorm_eq,← Multipoint.logPoint_norm_sq,Real.log_pow]
-    norm_num only [Nat.cast_ofNat]
-    rw [show -(2*Real.log ‖Multipoint.logPoint κ q x‖)/2 =
-      -Real.log ‖Multipoint.logPoint κ q x‖ by ring,Real.exp_neg,Real.exp_log hn]
-  rw [logCosine,he,Multipoint.logAngle,
-    Complex.cos_arg (norm_pos_iff.mp hn),Multipoint.logPoint_re,div_eq_mul_inv]
-
-lemma logCosine_abs_le_one (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    |logCosine κ q x|≤1 := by
-  rw [logCosine_eq_angle]
-  exact Real.abs_cos_le_one _
-
-lemma logInverseNorm_spatial (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    HasDerivAt (logInverseNorm κ q)
-      (logInverseNorm κ q x*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x) x := by
-  apply ((powerLogDensity_spatial_derivative κ 0 1 q x).const_mul (Real.exp (-q.1.im))).congr_deriv
-  unfold logInverseNorm
-  ring
-
-lemma logCosine_spatial (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    HasDerivAt (logCosine κ q)
-      (-Real.sqrt κ*Real.exp (2*q.1.im)*logInverseNorm κ q x/greenLogNormSq κ q x) x := by
-  apply (((hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))).mul
-    (logInverseNorm_spatial κ q x)).congr_deriv
-  simp only [Pi.sub_apply,id_eq]
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  field_simp
-  unfold greenLogNormSq
-  ring
-
-lemma logInverseNorm_contDiff (κ : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => logInverseNorm κ p.1 p.2) := by
-  unfold logInverseNorm
-  exact ((Complex.imCLM.contDiff.comp contDiff_fst.fst).neg.exp).mul
-    (powerLogDensity_contDiff κ 0 1)
-
-lemma logCosine_contDiff (κ : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => logCosine κ p.1 p.2) := by
-  unfold logCosine
-  exact ((Complex.reCLM.contDiff.comp contDiff_fst.fst).sub
-    (contDiff_const.mul contDiff_snd)).mul (logInverseNorm_contDiff κ)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLogAngular
+import OAI.Probability.SLE.Angle.AnglePolynomialGenerator
+
+namespace OAI
+
+/-! Smooth cosine coordinates in the actual logarithmic Loewner state.
+These formulas are used to apply the Brownian generator to eigenpolynomials. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def logInverseNorm (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  Real.exp (-q.1.im)*powerLogDensity κ 0 1 q x
+
+def logCosine (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  (q.1.re-Real.sqrt κ*x)*logInverseNorm κ q x
+
+lemma logInverseNorm_eq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    logInverseNorm κ q x=Real.exp (-Real.log (greenLogNormSq κ q x)/2) := by
+  unfold logInverseNorm powerLogDensity
+  rw [← Real.exp_add]
+  congr 1
+  ring
+
+lemma logInverseNorm_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    0<logInverseNorm κ q x := by rw [logInverseNorm_eq]; positivity
+
+lemma logInverseNorm_sq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    logInverseNorm κ q x^2=(greenLogNormSq κ q x)⁻¹ := by
+  rw [logInverseNorm_eq,pow_two,← Real.exp_add,
+    show -Real.log (greenLogNormSq κ q x)/2 + -Real.log (greenLogNormSq κ q x)/2 =
+      -Real.log (greenLogNormSq κ q x) by ring,Real.exp_neg,
+    Real.exp_log (greenLogNormSq_pos κ q x)]
+
+lemma logCosine_eq_angle (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    logCosine κ q x=Real.cos (Multipoint.logAngle κ q x) := by
+  have hn := Multipoint.logPoint_norm_pos κ q x
+  have he : logInverseNorm κ q x=‖Multipoint.logPoint κ q x‖⁻¹ := by
+    rw [logInverseNorm_eq,← Multipoint.logPoint_norm_sq,Real.log_pow]
+    norm_num only [Nat.cast_ofNat]
+    rw [show -(2*Real.log ‖Multipoint.logPoint κ q x‖)/2 =
+      -Real.log ‖Multipoint.logPoint κ q x‖ by ring,Real.exp_neg,Real.exp_log hn]
+  rw [logCosine,he,Multipoint.logAngle,
+    Complex.cos_arg (norm_pos_iff.mp hn),Multipoint.logPoint_re,div_eq_mul_inv]
+
+lemma logCosine_abs_le_one (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    |logCosine κ q x|≤1 := by
+  rw [logCosine_eq_angle]
+  exact Real.abs_cos_le_one _
+
+lemma logInverseNorm_spatial (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    HasDerivAt (logInverseNorm κ q)
+      (logInverseNorm κ q x*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x) x := by
+  apply ((powerLogDensity_spatial_derivative κ 0 1 q x).const_mul (Real.exp (-q.1.im))).congr_deriv
+  unfold logInverseNorm
+  ring
+
+lemma logCosine_spatial (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    HasDerivAt (logCosine κ q)
+      (-Real.sqrt κ*Real.exp (2*q.1.im)*logInverseNorm κ q x/greenLogNormSq κ q x) x := by
+  apply (((hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))).mul
+    (logInverseNorm_spatial κ q x)).congr_deriv
+  simp only [Pi.sub_apply,id_eq]
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  field_simp
+  unfold greenLogNormSq
+  ring
+
+lemma logInverseNorm_contDiff (κ : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => logInverseNorm κ p.1 p.2) := by
+  unfold logInverseNorm
+  exact ((Complex.imCLM.contDiff.comp contDiff_fst.fst).neg.exp).mul
+    (powerLogDensity_contDiff κ 0 1)
+
+lemma logCosine_contDiff (κ : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => logCosine κ p.1 p.2) := by
+  unfold logCosine
+  exact ((Complex.reCLM.contDiff.comp contDiff_fst.fst).sub
+    (contDiff_const.mul contDiff_snd)).mul (logInverseNorm_contDiff κ)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleLogCosineDerivatives.lean b/lean/OAI/Probability/SLE/Angle/AngleLogCosineDerivatives.lean
index 04f003ae024ccbf68d03a8636dc4a0262c4b4b48..132ed562a633edf37c9fd4d137bebfe8db32f814 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleLogCosineDerivatives.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleLogCosineDerivatives.lean
@@ -1,107 +1,107 @@
-import OAI.Probability.SLE.Angle.AngleLogCosine
-
-namespace OAI
-
-/-! Brownian second derivative and physical Loewner directional derivative
-of the actual cosine coordinate. -/
-noncomputable section
-
-namespace SLEExactGauge
-
-lemma logCosine_spatial_second {κ : ℝ} (hκ : 0≤κ) (q : ℂ × ℝ) (x : ℝ) :
-    iteratedDeriv 2 (logCosine κ q) x=
-      -3*κ*(q.1.re-Real.sqrt κ*x)*Real.exp (2*q.1.im)*logInverseNorm κ q x/
-        greenLogNormSq κ q x^2 := by
-  have hd : deriv (logCosine κ q)=fun y =>
-      -Real.sqrt κ*Real.exp (2*q.1.im)*logInverseNorm κ q y/greenLogNormSq κ q y :=
-    funext (fun y => (logCosine_spatial κ q y).deriv)
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
-    simp
-  have hcalc := ((logInverseNorm_spatial κ q x).const_mul
-    (-Real.sqrt κ*Real.exp (2*q.1.im))).div hN (greenLogNormSq_pos κ q x).ne'
-  conv_lhs => rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
-  rw [hd]
-  change HasDerivAt (fun y => -Real.sqrt κ*Real.exp (2*q.1.im)*
-    logInverseNorm κ q y/greenLogNormSq κ q y) _ x at hcalc
-  rw [hcalc.deriv]
-  have hsq := Real.sq_sqrt hκ
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  generalize Real.sqrt κ = r at *
-  field_simp
-  rw [← hsq]
-  ring
-
-lemma logInverseNorm_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => logInverseNorm κ r x) q v =
-      -logInverseNorm κ q x*((q.1.re-Real.sqrt κ*x)*v.1.re+
-        Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x := by
-  have hr : HasDerivAt (fun t : ℝ => (q+t • v).1.re) v.1.re 0 := by
-    simpa only [Prod.fst_add,Prod.smul_fst,Complex.add_re,Complex.smul_re,smul_eq_mul,
-      id_eq,one_mul] using ((hasDerivAt_id (0:ℝ)).mul_const v.1.re).const_add q.1.re
-  have hi : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
-    simpa only [Prod.fst_add,Prod.smul_fst,Complex.add_im,Complex.smul_im,smul_eq_mul,
-      id_eq,one_mul] using ((hasDerivAt_id (0:ℝ)).mul_const v.1.im).const_add q.1.im
-  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q+t • v) x)
-      (2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im) 0 := by
-    convert ((hr.sub_const (Real.sqrt κ*x)).pow 2).add ((hi.const_mul 2).exp) using 1 <;> try rfl
-    simp only [zero_smul,add_zero,Nat.cast_ofNat]
-    ring
-  have hcalc := ((hN.log (by simpa only [zero_smul,add_zero] using
-    (greenLogNormSq_pos κ q x).ne')).neg.div_const 2).exp
-  have hline : HasDerivAt (fun t : ℝ => logInverseNorm κ (q+t • v) x)
-      (-logInverseNorm κ q x*((q.1.re-Real.sqrt κ*x)*v.1.re+
-        Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x) 0 := by
-    simp only [logInverseNorm_eq]
-    apply hcalc.congr_deriv
-    simp only [Pi.neg_apply,zero_smul,add_zero]
-    ring
-  have hcd : ContDiff ℝ (⊤:ℕ∞) (fun r : ℂ × ℝ => logInverseNorm κ r x) :=
-    (logInverseNorm_contDiff κ).comp (contDiff_id.prodMk contDiff_const)
-  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
-    simpa only [id_eq,one_smul] using ((hasDerivAt_id (0:ℝ)).smul_const v).const_add q
-  have hfd : HasFDerivAt (fun r : ℂ × ℝ => logInverseNorm κ r x)
-      (fderiv ℝ (fun r : ℂ × ℝ => logInverseNorm κ r x) q) (q+(0:ℝ) • v) := by
-    simpa only [zero_smul,add_zero] using ((hcd.differentiable (by simp)) q).hasFDerivAt
-  have hchain := hfd.comp_hasDerivAt 0 haff
-  simpa only [Function.comp_def] using hchain.unique hline
-
-lemma logCosine_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q v =
-      v.1.re*logInverseNorm κ q x-(q.1.re-Real.sqrt κ*x)*logInverseNorm κ q x*
-        ((q.1.re-Real.sqrt κ*x)*v.1.re+Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x := by
-  let X : (ℂ × ℝ) →L[ℝ] ℝ := Complex.reCLM.comp (ContinuousLinearMap.fst ℝ ℂ ℝ)
-  have hx := (X.hasFDerivAt (x := q)).sub_const (Real.sqrt κ*x)
-  have hcd : ContDiff ℝ (⊤:ℕ∞) (fun r : ℂ × ℝ => logInverseNorm κ r x) :=
-    (logInverseNorm_contDiff κ).comp (contDiff_id.prodMk contDiff_const)
-  have he := congrArg (fun L : (ℂ × ℝ) →L[ℝ] ℝ => L v)
-    (hx.mul ((hcd.differentiable (by simp)) q).hasFDerivAt).fderiv
-  change fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q v = _ at he
-  rw [he]
-  simp only [add_apply,smul_apply,smul_eq_mul,
-    X,ContinuousLinearMap.comp_apply,Complex.reCLM_apply,
-    logInverseNorm_directional]
-  change (q.1.re-Real.sqrt κ*x)*(-logInverseNorm κ q x*((q.1.re-Real.sqrt κ*x)*v.1.re+
-    Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x)+logInverseNorm κ q x*v.1.re = _
-  ring
-
-lemma logCosine_physical_direction (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q
-      (physicalLogLoewnerField (Real.sqrt κ*x) q)=
-      4*(q.1.re-Real.sqrt κ*x)*Real.exp (2*q.1.im)*logInverseNorm κ q x/
-        greenLogNormSq κ q x^2 := by
-  rw [logCosine_directional]
-  obtain ⟨hr,hi,_⟩ := physicalLogLoewnerField_components κ q x
-  rw [hr,hi]
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  field_simp
-  unfold greenLogNormSq
-  ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleLogCosine
+
+namespace OAI
+
+/-! Brownian second derivative and physical Loewner directional derivative
+of the actual cosine coordinate. -/
+noncomputable section
+
+namespace SLEExactGauge
+
+lemma logCosine_spatial_second {κ : ℝ} (hκ : 0≤κ) (q : ℂ × ℝ) (x : ℝ) :
+    iteratedDeriv 2 (logCosine κ q) x=
+      -3*κ*(q.1.re-Real.sqrt κ*x)*Real.exp (2*q.1.im)*logInverseNorm κ q x/
+        greenLogNormSq κ q x^2 := by
+  have hd : deriv (logCosine κ q)=fun y =>
+      -Real.sqrt κ*Real.exp (2*q.1.im)*logInverseNorm κ q y/greenLogNormSq κ q y :=
+    funext (fun y => (logCosine_spatial κ q y).deriv)
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
+    simp
+  have hcalc := ((logInverseNorm_spatial κ q x).const_mul
+    (-Real.sqrt κ*Real.exp (2*q.1.im))).div hN (greenLogNormSq_pos κ q x).ne'
+  conv_lhs => rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
+  rw [hd]
+  change HasDerivAt (fun y => -Real.sqrt κ*Real.exp (2*q.1.im)*
+    logInverseNorm κ q y/greenLogNormSq κ q y) _ x at hcalc
+  rw [hcalc.deriv]
+  have hsq := Real.sq_sqrt hκ
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  generalize Real.sqrt κ = r at *
+  field_simp
+  rw [← hsq]
+  ring
+
+lemma logInverseNorm_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => logInverseNorm κ r x) q v =
+      -logInverseNorm κ q x*((q.1.re-Real.sqrt κ*x)*v.1.re+
+        Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x := by
+  have hr : HasDerivAt (fun t : ℝ => (q+t • v).1.re) v.1.re 0 := by
+    simpa only [Prod.fst_add,Prod.smul_fst,Complex.add_re,Complex.smul_re,smul_eq_mul,
+      id_eq,one_mul] using ((hasDerivAt_id (0:ℝ)).mul_const v.1.re).const_add q.1.re
+  have hi : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
+    simpa only [Prod.fst_add,Prod.smul_fst,Complex.add_im,Complex.smul_im,smul_eq_mul,
+      id_eq,one_mul] using ((hasDerivAt_id (0:ℝ)).mul_const v.1.im).const_add q.1.im
+  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q+t • v) x)
+      (2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im) 0 := by
+    convert ((hr.sub_const (Real.sqrt κ*x)).pow 2).add ((hi.const_mul 2).exp) using 1 <;> try rfl
+    simp only [zero_smul,add_zero,Nat.cast_ofNat]
+    ring
+  have hcalc := ((hN.log (by simpa only [zero_smul,add_zero] using
+    (greenLogNormSq_pos κ q x).ne')).neg.div_const 2).exp
+  have hline : HasDerivAt (fun t : ℝ => logInverseNorm κ (q+t • v) x)
+      (-logInverseNorm κ q x*((q.1.re-Real.sqrt κ*x)*v.1.re+
+        Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x) 0 := by
+    simp only [logInverseNorm_eq]
+    apply hcalc.congr_deriv
+    simp only [Pi.neg_apply,zero_smul,add_zero]
+    ring
+  have hcd : ContDiff ℝ (⊤:ℕ∞) (fun r : ℂ × ℝ => logInverseNorm κ r x) :=
+    (logInverseNorm_contDiff κ).comp (contDiff_id.prodMk contDiff_const)
+  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
+    simpa only [id_eq,one_smul] using ((hasDerivAt_id (0:ℝ)).smul_const v).const_add q
+  have hfd : HasFDerivAt (fun r : ℂ × ℝ => logInverseNorm κ r x)
+      (fderiv ℝ (fun r : ℂ × ℝ => logInverseNorm κ r x) q) (q+(0:ℝ) • v) := by
+    simpa only [zero_smul,add_zero] using ((hcd.differentiable (by simp)) q).hasFDerivAt
+  have hchain := hfd.comp_hasDerivAt 0 haff
+  simpa only [Function.comp_def] using hchain.unique hline
+
+lemma logCosine_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q v =
+      v.1.re*logInverseNorm κ q x-(q.1.re-Real.sqrt κ*x)*logInverseNorm κ q x*
+        ((q.1.re-Real.sqrt κ*x)*v.1.re+Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x := by
+  let X : (ℂ × ℝ) →L[ℝ] ℝ := Complex.reCLM.comp (ContinuousLinearMap.fst ℝ ℂ ℝ)
+  have hx := (X.hasFDerivAt (x := q)).sub_const (Real.sqrt κ*x)
+  have hcd : ContDiff ℝ (⊤:ℕ∞) (fun r : ℂ × ℝ => logInverseNorm κ r x) :=
+    (logInverseNorm_contDiff κ).comp (contDiff_id.prodMk contDiff_const)
+  have he := congrArg (fun L : (ℂ × ℝ) →L[ℝ] ℝ => L v)
+    (hx.mul ((hcd.differentiable (by simp)) q).hasFDerivAt).fderiv
+  change fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q v = _ at he
+  rw [he]
+  simp only [add_apply,smul_apply,smul_eq_mul,
+    X,ContinuousLinearMap.comp_apply,Complex.reCLM_apply,
+    logInverseNorm_directional]
+  change (q.1.re-Real.sqrt κ*x)*(-logInverseNorm κ q x*((q.1.re-Real.sqrt κ*x)*v.1.re+
+    Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x)+logInverseNorm κ q x*v.1.re = _
+  ring
+
+lemma logCosine_physical_direction (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q
+      (physicalLogLoewnerField (Real.sqrt κ*x) q)=
+      4*(q.1.re-Real.sqrt κ*x)*Real.exp (2*q.1.im)*logInverseNorm κ q x/
+        greenLogNormSq κ q x^2 := by
+  rw [logCosine_directional]
+  obtain ⟨hr,hi,_⟩ := physicalLogLoewnerField_components κ q x
+  rw [hr,hi]
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  field_simp
+  unfold greenLogNormSq
+  ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleLogGeneratorCoefficients.lean b/lean/OAI/Probability/SLE/Angle/AngleLogGeneratorCoefficients.lean
index a8dca00c1988076371efe35d4b1dbb3bb9dfc912..07740b5bdb32eecc20da88ce622263d044cfdab3 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleLogGeneratorCoefficients.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleLogGeneratorCoefficients.lean
@@ -1,52 +1,52 @@
-import OAI.Probability.SLE.Angle.AngleLogCosineDerivatives
-
-namespace OAI
-
-/-! The exact drift and variance coefficients after the Green change of
-law, in the paper's logarithmic conformal-radius clock. -/
-noncomputable section
-
-namespace SLEExactGauge
-
-lemma shifted_green_generator {κ : ℝ} (hκ : 0<κ) (c : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ (codimension κ+c) (greenExponent κ) r x) q)
-        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
-      (1/2:ℝ)*iteratedDeriv 2 (powerLogDensity κ (codimension κ+c) (greenExponent κ) q) x =
-    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*logClockRate κ q x*c := by
-  rw [power_physical_generator κ _ _ hκ.le]
-  unfold logClockRate greenExponent codimension dimension
-  have h0 := hκ.ne'
-  field_simp
-  ring
-
-lemma logCosine_tilted_drift {κ : ℝ} (hκ : 0<κ) (q : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q
-        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
-      (1/2:ℝ)*iteratedDeriv 2 (logCosine κ q) x+
-      (greenExponent κ*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x)*
-        deriv (logCosine κ q) x =
-    logClockRate κ q x*(-(1+κ/8)*logCosine κ q x) := by
-  rw [logCosine_physical_direction,logCosine_spatial_second hκ.le,(logCosine_spatial κ q x).deriv]
-  unfold logClockRate logCosine greenExponent
-  have hN := (greenLogNormSq_pos κ q x).ne'
-  have hk := hκ.ne'
-  field_simp
-  rw [Real.sq_sqrt hκ.le]
-  ring
-
-lemma logCosine_tilted_variance {κ : ℝ} (hκ : 0≤κ) (q : ℂ × ℝ) (x : ℝ) :
-    (1/2:ℝ)*(deriv (logCosine κ q) x)^2=
-      logClockRate κ q x*(κ/8)*(1-logCosine κ q x^2) := by
-  rw [(logCosine_spatial κ q x).deriv]
-  unfold logClockRate logCosine
-  rw [div_pow,mul_pow,mul_pow,mul_pow,neg_sq,Real.sq_sqrt hκ,logInverseNorm_sq]
-  have hN := (greenLogNormSq_pos κ q x).ne'
-  field_simp
-  unfold greenLogNormSq
-  ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleLogCosineDerivatives
+
+namespace OAI
+
+/-! The exact drift and variance coefficients after the Green change of
+law, in the paper's logarithmic conformal-radius clock. -/
+noncomputable section
+
+namespace SLEExactGauge
+
+lemma shifted_green_generator {κ : ℝ} (hκ : 0<κ) (c : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ (codimension κ+c) (greenExponent κ) r x) q)
+        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
+      (1/2:ℝ)*iteratedDeriv 2 (powerLogDensity κ (codimension κ+c) (greenExponent κ) q) x =
+    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*logClockRate κ q x*c := by
+  rw [power_physical_generator κ _ _ hκ.le]
+  unfold logClockRate greenExponent codimension dimension
+  have h0 := hκ.ne'
+  field_simp
+  ring
+
+lemma logCosine_tilted_drift {κ : ℝ} (hκ : 0<κ) (q : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q
+        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
+      (1/2:ℝ)*iteratedDeriv 2 (logCosine κ q) x+
+      (greenExponent κ*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x)*
+        deriv (logCosine κ q) x =
+    logClockRate κ q x*(-(1+κ/8)*logCosine κ q x) := by
+  rw [logCosine_physical_direction,logCosine_spatial_second hκ.le,(logCosine_spatial κ q x).deriv]
+  unfold logClockRate logCosine greenExponent
+  have hN := (greenLogNormSq_pos κ q x).ne'
+  have hk := hκ.ne'
+  field_simp
+  rw [Real.sq_sqrt hκ.le]
+  ring
+
+lemma logCosine_tilted_variance {κ : ℝ} (hκ : 0≤κ) (q : ℂ × ℝ) (x : ℝ) :
+    (1/2:ℝ)*(deriv (logCosine κ q) x)^2=
+      logClockRate κ q x*(κ/8)*(1-logCosine κ q x^2) := by
+  rw [(logCosine_spatial κ q x).deriv]
+  unfold logClockRate logCosine
+  rw [div_pow,mul_pow,mul_pow,mul_pow,neg_sq,Real.sq_sqrt hκ,logInverseNorm_sq]
+  have hN := (greenLogNormSq_pos κ q x).ne'
+  field_simp
+  unfold greenLogNormSq
+  ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleMomentUniqueness.lean b/lean/OAI/Probability/SLE/Angle/AngleMomentUniqueness.lean
index 2441f944116557dcefef97e94df623ccdaf9e3ed..35a58326f9095fc4f932670106521b1aaa8fa380 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleMomentUniqueness.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleMomentUniqueness.lean
@@ -1,97 +1,97 @@
-import OAI.Probability.SLE.Angle.AngleSpectralPairing
-import Mathlib.Topology.ContinuousMap.Weierstrass
-import Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real
-
-namespace OAI
-
-/-! The cosine eigenpolynomials determine finite angle laws on [0, pi]. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Polynomial
-open scoped Topology
-
-namespace SLEExactGauge
-
-private def angleIntegralLinear (μ : Measure ℝ) [IsFiniteMeasure μ] : ℝ[X] →ₗ[ℝ] ℝ where
-  toFun p := ∫ θ, anglePolynomialTest p θ ∂μ
-  map_add' p q := by
-    simp only [anglePolynomialTest,eval_add]
-    exact integral_add (anglePolynomialTest_integrable μ p) (anglePolynomialTest_integrable μ q)
-  map_smul' c p := by
-    simp only [anglePolynomialTest,eval_smul,smul_eq_mul,integral_const_mul,RingHom.id_apply]
-
-lemma angle_polynomial_moments_of_eigen_moments (κ : ℝ)
-    (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
-    (hm : ∀ n, (∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂μ)=
-      ∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂ν) (p : ℝ[X]) :
-    (∫ θ, anglePolynomialTest p θ ∂μ)=∫ θ, anglePolynomialTest p θ ∂ν := by
-  have he : angleIntegralLinear μ=angleIntegralLinear ν := by
-    apply (angleEigenBasis κ).ext
-    intro n
-    change (∫ θ, anglePolynomialTest (angleEigenBasis κ n) θ ∂μ)=
-      ∫ θ, anglePolynomialTest (angleEigenBasis κ n) θ ∂ν
-    simpa only [angleEigenBasis_apply] using hm n
-  exact congrArg (fun L : ℝ[X] →ₗ[ℝ] ℝ => L p) he
-
-lemma angle_continuous_moments_of_polynomial_moments
-    (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
-    (hm : ∀ p : ℝ[X], (∫ θ, anglePolynomialTest p θ ∂μ)=∫ θ, anglePolynomialTest p θ ∂ν)
-    (g : ℝ → ℝ) (hg : Continuous g) :
-    (∫ θ, g (Real.cos θ) ∂μ)=∫ θ, g (Real.cos θ) ∂ν := by
-  let δ (j : ℕ) : ℝ := 1/((j:ℝ)+1)
-  have hδ (j : ℕ) : 0<δ j := by dsimp [δ]; positivity
-  have hδ1 (j : ℕ) : δ j≤1 := by
-    dsimp [δ]
-    apply (div_le_one (by positivity)).mpr
-    have hj := Nat.cast_nonneg (α := ℝ) j
-    linarith
-  have hδlim : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
-  choose p hp using fun j => exists_polynomial_near_of_continuousOn (-1) 1 g hg.continuousOn (δ j) (hδ j)
-  obtain ⟨M,hM⟩ := (isCompact_Icc : IsCompact (Icc (-1:ℝ) 1)).exists_bound_of_continuousOn hg.continuousOn
-  have hc (θ : ℝ) : Real.cos θ∈Icc (-1:ℝ) 1 := ⟨Real.neg_one_le_cos θ,Real.cos_le_one θ⟩
-  have hb (j : ℕ) (θ : ℝ) : ‖anglePolynomialTest (p j) θ‖≤M+1 := by
-    change |(p j).eval (Real.cos θ)|≤M+1
-    calc
-      _ = |((p j).eval (Real.cos θ)-g (Real.cos θ))+g (Real.cos θ)| := by ring_nf
-      _ ≤ |(p j).eval (Real.cos θ)-g (Real.cos θ)|+|g (Real.cos θ)| := abs_add_le _ _
-      _ ≤ δ j+M := add_le_add (hp j _ (hc θ)).le (hM _ (hc θ))
-      _ ≤ M+1 := by linarith [hδ1 j]
-  have hl (θ : ℝ) : Tendsto (fun j => anglePolynomialTest (p j) θ) atTop (𝓝 (g (Real.cos θ))) := by
-    have hh := squeeze_zero_norm (f := fun j => (p j).eval (Real.cos θ)-g (Real.cos θ))
-      (fun j => by simpa only [Real.norm_eq_abs] using (hp j _ (hc θ)).le) hδlim
-    simpa only [sub_add_cancel,zero_add,anglePolynomialTest] using hh.add_const (g (Real.cos θ))
-  have hi (ρ : Measure ℝ) [IsFiniteMeasure ρ] :
-      Tendsto (fun j => ∫ θ, anglePolynomialTest (p j) θ ∂ρ) atTop (𝓝 (∫ θ, g (Real.cos θ) ∂ρ)) :=
-    tendsto_integral_of_dominated_convergence (fun _ => M+1)
-      (fun j => (anglePolynomialTest_integrable ρ (p j)).aestronglyMeasurable)
-      (integrable_const _) (fun j => Eventually.of_forall (hb j)) (Eventually.of_forall hl)
-  apply tendsto_nhds_unique (hi μ)
-  exact (hi ν).congr' (Eventually.of_forall fun j => (hm (p j)).symm)
-
-lemma angleMeasures_eq_of_eigenMoments (κ : ℝ)
-    (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
-    (hμ : ∀ᵐ θ ∂μ, θ∈Icc (0:ℝ) Real.pi) (hν : ∀ᵐ θ ∂ν, θ∈Icc (0:ℝ) Real.pi)
-    (hm : ∀ n, (∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂μ)=
-      ∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂ν) : μ=ν := by
-  have hpoly := angle_polynomial_moments_of_eigen_moments κ μ ν hm
-  have hcos : μ.map Real.cos=ν.map Real.cos := by
-    apply Measure.ext_of_integral_eq_on_compactlySupported
-    intro f
-    rw [integral_map Real.continuous_cos.measurable.aemeasurable
-      (show AEStronglyMeasurable (fun x => f x) (μ.map Real.cos) from f.continuous.aestronglyMeasurable),
-      integral_map Real.continuous_cos.measurable.aemeasurable
-      (show AEStronglyMeasurable (fun x => f x) (ν.map Real.cos) from f.continuous.aestronglyMeasurable)]
-    exact angle_continuous_moments_of_polynomial_moments μ ν hpoly f f.continuous
-  have hrecover (ρ : Measure ℝ) (hρ : ∀ᵐ θ ∂ρ, θ∈Icc (0:ℝ) Real.pi) :
-      (ρ.map Real.cos).map Real.arccos=ρ := by
-    rw [Measure.map_map Real.continuous_arccos.measurable Real.continuous_cos.measurable]
-    have he : Real.arccos ∘ Real.cos =ᵐ[ρ] id := by
-      filter_upwards [hρ] with θ hθ
-      exact Real.arccos_cos hθ.1 hθ.2
-    rw [Measure.map_congr he,Measure.map_id]
-  rw [← hrecover μ hμ,hcos,hrecover ν hν]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleSpectralPairing
+import Mathlib.Topology.ContinuousMap.Weierstrass
+import Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real
+
+namespace OAI
+
+/-! The cosine eigenpolynomials determine finite angle laws on [0, pi]. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Polynomial
+open scoped Topology
+
+namespace SLEExactGauge
+
+private def angleIntegralLinear (μ : Measure ℝ) [IsFiniteMeasure μ] : ℝ[X] →ₗ[ℝ] ℝ where
+  toFun p := ∫ θ, anglePolynomialTest p θ ∂μ
+  map_add' p q := by
+    simp only [anglePolynomialTest,eval_add]
+    exact integral_add (anglePolynomialTest_integrable μ p) (anglePolynomialTest_integrable μ q)
+  map_smul' c p := by
+    simp only [anglePolynomialTest,eval_smul,smul_eq_mul,integral_const_mul,RingHom.id_apply]
+
+lemma angle_polynomial_moments_of_eigen_moments (κ : ℝ)
+    (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hm : ∀ n, (∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂μ)=
+      ∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂ν) (p : ℝ[X]) :
+    (∫ θ, anglePolynomialTest p θ ∂μ)=∫ θ, anglePolynomialTest p θ ∂ν := by
+  have he : angleIntegralLinear μ=angleIntegralLinear ν := by
+    apply (angleEigenBasis κ).ext
+    intro n
+    change (∫ θ, anglePolynomialTest (angleEigenBasis κ n) θ ∂μ)=
+      ∫ θ, anglePolynomialTest (angleEigenBasis κ n) θ ∂ν
+    simpa only [angleEigenBasis_apply] using hm n
+  exact congrArg (fun L : ℝ[X] →ₗ[ℝ] ℝ => L p) he
+
+lemma angle_continuous_moments_of_polynomial_moments
+    (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hm : ∀ p : ℝ[X], (∫ θ, anglePolynomialTest p θ ∂μ)=∫ θ, anglePolynomialTest p θ ∂ν)
+    (g : ℝ → ℝ) (hg : Continuous g) :
+    (∫ θ, g (Real.cos θ) ∂μ)=∫ θ, g (Real.cos θ) ∂ν := by
+  let δ (j : ℕ) : ℝ := 1/((j:ℝ)+1)
+  have hδ (j : ℕ) : 0<δ j := by dsimp [δ]; positivity
+  have hδ1 (j : ℕ) : δ j≤1 := by
+    dsimp [δ]
+    apply (div_le_one (by positivity)).mpr
+    have hj := Nat.cast_nonneg (α := ℝ) j
+    linarith
+  have hδlim : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  choose p hp using fun j => exists_polynomial_near_of_continuousOn (-1) 1 g hg.continuousOn (δ j) (hδ j)
+  obtain ⟨M,hM⟩ := (isCompact_Icc : IsCompact (Icc (-1:ℝ) 1)).exists_bound_of_continuousOn hg.continuousOn
+  have hc (θ : ℝ) : Real.cos θ∈Icc (-1:ℝ) 1 := ⟨Real.neg_one_le_cos θ,Real.cos_le_one θ⟩
+  have hb (j : ℕ) (θ : ℝ) : ‖anglePolynomialTest (p j) θ‖≤M+1 := by
+    change |(p j).eval (Real.cos θ)|≤M+1
+    calc
+      _ = |((p j).eval (Real.cos θ)-g (Real.cos θ))+g (Real.cos θ)| := by ring_nf
+      _ ≤ |(p j).eval (Real.cos θ)-g (Real.cos θ)|+|g (Real.cos θ)| := abs_add_le _ _
+      _ ≤ δ j+M := add_le_add (hp j _ (hc θ)).le (hM _ (hc θ))
+      _ ≤ M+1 := by linarith [hδ1 j]
+  have hl (θ : ℝ) : Tendsto (fun j => anglePolynomialTest (p j) θ) atTop (𝓝 (g (Real.cos θ))) := by
+    have hh := squeeze_zero_norm (f := fun j => (p j).eval (Real.cos θ)-g (Real.cos θ))
+      (fun j => by simpa only [Real.norm_eq_abs] using (hp j _ (hc θ)).le) hδlim
+    simpa only [sub_add_cancel,zero_add,anglePolynomialTest] using hh.add_const (g (Real.cos θ))
+  have hi (ρ : Measure ℝ) [IsFiniteMeasure ρ] :
+      Tendsto (fun j => ∫ θ, anglePolynomialTest (p j) θ ∂ρ) atTop (𝓝 (∫ θ, g (Real.cos θ) ∂ρ)) :=
+    tendsto_integral_of_dominated_convergence (fun _ => M+1)
+      (fun j => (anglePolynomialTest_integrable ρ (p j)).aestronglyMeasurable)
+      (integrable_const _) (fun j => Eventually.of_forall (hb j)) (Eventually.of_forall hl)
+  apply tendsto_nhds_unique (hi μ)
+  exact (hi ν).congr' (Eventually.of_forall fun j => (hm (p j)).symm)
+
+lemma angleMeasures_eq_of_eigenMoments (κ : ℝ)
+    (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hμ : ∀ᵐ θ ∂μ, θ∈Icc (0:ℝ) Real.pi) (hν : ∀ᵐ θ ∂ν, θ∈Icc (0:ℝ) Real.pi)
+    (hm : ∀ n, (∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂μ)=
+      ∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ ∂ν) : μ=ν := by
+  have hpoly := angle_polynomial_moments_of_eigen_moments κ μ ν hm
+  have hcos : μ.map Real.cos=ν.map Real.cos := by
+    apply Measure.ext_of_integral_eq_on_compactlySupported
+    intro f
+    rw [integral_map Real.continuous_cos.measurable.aemeasurable
+      (show AEStronglyMeasurable (fun x => f x) (μ.map Real.cos) from f.continuous.aestronglyMeasurable),
+      integral_map Real.continuous_cos.measurable.aemeasurable
+      (show AEStronglyMeasurable (fun x => f x) (ν.map Real.cos) from f.continuous.aestronglyMeasurable)]
+    exact angle_continuous_moments_of_polynomial_moments μ ν hpoly f f.continuous
+  have hrecover (ρ : Measure ℝ) (hρ : ∀ᵐ θ ∂ρ, θ∈Icc (0:ℝ) Real.pi) :
+      (ρ.map Real.cos).map Real.arccos=ρ := by
+    rw [Measure.map_map Real.continuous_arccos.measurable Real.continuous_cos.measurable]
+    have he : Real.arccos ∘ Real.cos =ᵐ[ρ] id := by
+      filter_upwards [hρ] with θ hθ
+      exact Real.arccos_cos hθ.1 hθ.2
+    rw [Measure.map_congr he,Measure.map_id]
+  rw [← hrecover μ hμ,hcos,hrecover ν hν]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleMomentWeight.lean b/lean/OAI/Probability/SLE/Angle/AngleMomentWeight.lean
index 8b41ad13cf96e7aa091993282b2e29fba505cc8c..2ae7ca5255ffe86e571bef7b2f5fd3eb38c9bd02 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleMomentWeight.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleMomentWeight.lean
@@ -1,51 +1,51 @@
-import OAI.Probability.SLE.Angle.TargetAngleLaw
-import OAI.Probability.SLE.BoundaryDensity
-
-namespace OAI
-
-/-! The angular singularity in the centered moment estimate is
-integrable against the exact invariant angle law. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma angle_inverse_sine_lintegral_lt_top (κ : ℝ) :
-    (∫⁻ θ, ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ) ∂angleLaw κ)<∞ := by
-  let μ := volume.restrict (Ioo (0:ℝ) Real.pi)
-  let w := fun θ => ENNReal.ofReal ((Real.sin θ)^(4*a κ))
-  have hm : Measurable w := (measurable_id.sin.pow_const (4*a κ)).ennreal_ofReal
-  have hf : Measurable (fun θ : ℝ => ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ)) :=
-    (measurable_const.div (measurable_id.sin.pow_const (greenExponent κ))).ennreal_ofReal
-  change (∫⁻ θ, ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ)
-    ∂((μ.withDensity w) univ)⁻¹ • μ.withDensity w)<∞
-  rw [lintegral_smul_measure,lintegral_withDensity_eq_lintegral_mul μ hm hf]
-  apply ENNReal.mul_lt_top
-  · exact ENNReal.inv_lt_top.mpr (angle_normalizer_pos κ)
-  · apply lt_of_le_of_lt (lintegral_mono_ae ?_) (show (∫⁻ _ : ℝ, (1:ℝ≥0∞) ∂μ)<∞ from by
-      simp only [μ,lintegral_const,one_mul,Measure.restrict_apply_univ,Real.volume_Ioo,sub_zero]
-      exact ENNReal.ofReal_lt_top)
-    filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
-    have hs : 0<Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-    change ENNReal.ofReal ((Real.sin θ)^(4*a κ))*
-      ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ)≤1
-    rw [← ENNReal.ofReal_mul (Real.rpow_nonneg hs.le _),mul_one_div,
-      ← Real.rpow_sub hs]
-    have he : 4*a κ-greenExponent κ=1 := by unfold a greenExponent; ring
-    rw [he,Real.rpow_one]
-    exact (ENNReal.ofReal_le_ofReal (Real.sin_le_one θ)).trans_eq (by norm_num)
-
-lemma angle_inverse_sine_integrable (κ : ℝ) :
-    Integrable (fun θ => 1/(Real.sin θ)^greenExponent κ) (angleLaw κ) := by
-  refine ⟨(measurable_const.div (measurable_id.sin.pow_const (greenExponent κ))).aestronglyMeasurable,?_⟩
-  apply (hasFiniteIntegral_iff_ofReal ?_).mpr (angle_inverse_sine_lintegral_lt_top κ)
-  filter_upwards [angleLaw_ae_mem κ] with θ hθ
-  exact one_div_nonneg.mpr (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetAngleLaw
+import OAI.Probability.SLE.BoundaryDensity
+
+namespace OAI
+
+/-! The angular singularity in the centered moment estimate is
+integrable against the exact invariant angle law. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma angle_inverse_sine_lintegral_lt_top (κ : ℝ) :
+    (∫⁻ θ, ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ) ∂angleLaw κ)<∞ := by
+  let μ := volume.restrict (Ioo (0:ℝ) Real.pi)
+  let w := fun θ => ENNReal.ofReal ((Real.sin θ)^(4*a κ))
+  have hm : Measurable w := (measurable_id.sin.pow_const (4*a κ)).ennreal_ofReal
+  have hf : Measurable (fun θ : ℝ => ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ)) :=
+    (measurable_const.div (measurable_id.sin.pow_const (greenExponent κ))).ennreal_ofReal
+  change (∫⁻ θ, ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ)
+    ∂((μ.withDensity w) univ)⁻¹ • μ.withDensity w)<∞
+  rw [lintegral_smul_measure,lintegral_withDensity_eq_lintegral_mul μ hm hf]
+  apply ENNReal.mul_lt_top
+  · exact ENNReal.inv_lt_top.mpr (angle_normalizer_pos κ)
+  · apply lt_of_le_of_lt (lintegral_mono_ae ?_) (show (∫⁻ _ : ℝ, (1:ℝ≥0∞) ∂μ)<∞ from by
+      simp only [μ,lintegral_const,one_mul,Measure.restrict_apply_univ,Real.volume_Ioo,sub_zero]
+      exact ENNReal.ofReal_lt_top)
+    filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
+    have hs : 0<Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+    change ENNReal.ofReal ((Real.sin θ)^(4*a κ))*
+      ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ)≤1
+    rw [← ENNReal.ofReal_mul (Real.rpow_nonneg hs.le _),mul_one_div,
+      ← Real.rpow_sub hs]
+    have he : 4*a κ-greenExponent κ=1 := by unfold a greenExponent; ring
+    rw [he,Real.rpow_one]
+    exact (ENNReal.ofReal_le_ofReal (Real.sin_le_one θ)).trans_eq (by norm_num)
+
+lemma angle_inverse_sine_integrable (κ : ℝ) :
+    Integrable (fun θ => 1/(Real.sin θ)^greenExponent κ) (angleLaw κ) := by
+  refine ⟨(measurable_const.div (measurable_id.sin.pow_const (greenExponent κ))).aestronglyMeasurable,?_⟩
+  apply (hasFiniteIntegral_iff_ofReal ?_).mpr (angle_inverse_sine_lintegral_lt_top κ)
+  filter_upwards [angleLaw_ae_mem κ] with θ hθ
+  exact one_div_nonneg.mpr (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AnglePolynomialCoefficients.lean b/lean/OAI/Probability/SLE/Angle/AnglePolynomialCoefficients.lean
index ab02f756a7b7a1964a70aefaac1e1a41c834a8be..d4042f270d89aca591173a9f45a74493137230c2 100644
--- a/lean/OAI/Probability/SLE/Angle/AnglePolynomialCoefficients.lean
+++ b/lean/OAI/Probability/SLE/Angle/AnglePolynomialCoefficients.lean
@@ -1,52 +1,52 @@
-import OAI.Probability.SLE.Angle.AnglePolynomialGenerator
-
-namespace OAI
-
-/-! The exact triangular coefficient relation for the cosine generator.
-This is the recurrence used to construct its polynomial eigenfunctions. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-lemma anglePolynomialGenerator_coeff (κ : ℝ) (p : ℝ[X]) (n : ℕ) :
-    (anglePolynomialGenerator κ p).coeff n =
-      (κ/8)*((n:ℝ)+2)*((n:ℝ)+1)*p.coeff (n+2) -
-        anglePolynomialEigenvalue κ n*p.coeff n := by
-  have he : anglePolynomialGenerator κ p =
-      C (κ/8)*p.derivative.derivative - C (κ/8)*(p.derivative.derivative*X^2) -
-        C (1+κ/8)*(p.derivative*X) := by
-    unfold anglePolynomialGenerator
-    ring
-  rw [he]
-  simp only [coeff_sub,coeff_C_mul]
-  rcases n with _ | (_ | n)
-  · simp [coeff_derivative,anglePolynomialEigenvalue]
-    ring
-  · simp [coeff_derivative,coeff_mul_X_pow',anglePolynomialEigenvalue]
-    ring
-  · simp [coeff_derivative,coeff_mul_X_pow',anglePolynomialEigenvalue,
-      show n+1+1-2=n by omega]
-    ring
-
-lemma anglePolynomial_eigen_iff_coeff (κ : ℝ) (p : ℝ[X]) (N : ℕ) :
-    anglePolynomialGenerator κ p = -C (anglePolynomialEigenvalue κ N)*p ↔
-      ∀ n : ℕ, (κ/8)*((n:ℝ)+2)*((n:ℝ)+1)*p.coeff (n+2) =
-        (anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ N)*p.coeff n := by
-  constructor
-  · intro h n
-    have he := congrArg (fun q : ℝ[X] => q.coeff n) h
-    rw [anglePolynomialGenerator_coeff] at he
-    simp only [neg_mul,coeff_neg,coeff_C_mul] at he
-    linarith
-  · intro h
-    ext n
-    rw [anglePolynomialGenerator_coeff]
-    simp only [neg_mul,coeff_neg,coeff_C_mul]
-    linarith [h n]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AnglePolynomialGenerator
+
+namespace OAI
+
+/-! The exact triangular coefficient relation for the cosine generator.
+This is the recurrence used to construct its polynomial eigenfunctions. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+lemma anglePolynomialGenerator_coeff (κ : ℝ) (p : ℝ[X]) (n : ℕ) :
+    (anglePolynomialGenerator κ p).coeff n =
+      (κ/8)*((n:ℝ)+2)*((n:ℝ)+1)*p.coeff (n+2) -
+        anglePolynomialEigenvalue κ n*p.coeff n := by
+  have he : anglePolynomialGenerator κ p =
+      C (κ/8)*p.derivative.derivative - C (κ/8)*(p.derivative.derivative*X^2) -
+        C (1+κ/8)*(p.derivative*X) := by
+    unfold anglePolynomialGenerator
+    ring
+  rw [he]
+  simp only [coeff_sub,coeff_C_mul]
+  rcases n with _ | (_ | n)
+  · simp [coeff_derivative,anglePolynomialEigenvalue]
+    ring
+  · simp [coeff_derivative,coeff_mul_X_pow',anglePolynomialEigenvalue]
+    ring
+  · simp [coeff_derivative,coeff_mul_X_pow',anglePolynomialEigenvalue,
+      show n+1+1-2=n by omega]
+    ring
+
+lemma anglePolynomial_eigen_iff_coeff (κ : ℝ) (p : ℝ[X]) (N : ℕ) :
+    anglePolynomialGenerator κ p = -C (anglePolynomialEigenvalue κ N)*p ↔
+      ∀ n : ℕ, (κ/8)*((n:ℝ)+2)*((n:ℝ)+1)*p.coeff (n+2) =
+        (anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ N)*p.coeff n := by
+  constructor
+  · intro h n
+    have he := congrArg (fun q : ℝ[X] => q.coeff n) h
+    rw [anglePolynomialGenerator_coeff] at he
+    simp only [neg_mul,coeff_neg,coeff_C_mul] at he
+    linarith
+  · intro h
+    ext n
+    rw [anglePolynomialGenerator_coeff]
+    simp only [neg_mul,coeff_neg,coeff_C_mul]
+    linarith [h n]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AnglePolynomialGenerator.lean b/lean/OAI/Probability/SLE/Angle/AnglePolynomialGenerator.lean
index c16aba944d535d30b73f8986943c6f721f85fc8b..eb570c0c3c88d8e7d595a8db18c1434a5215b5da 100644
--- a/lean/OAI/Probability/SLE/Angle/AnglePolynomialGenerator.lean
+++ b/lean/OAI/Probability/SLE/Angle/AnglePolynomialGenerator.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Angle.CanonicalAngleSemigroup
-
-namespace OAI
-
-/-! The cosine-coordinate generator in the actual logarithmic radius
-clock. Its polynomial eigenvalues retain the paper's time normalization. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def anglePolynomialGenerator (κ : ℝ) (p : ℝ[X]) : ℝ[X] :=
-  C (κ/8)*(1-X^2)*p.derivative.derivative - C (1+κ/8)*X*p.derivative
-
-def anglePolynomialEigenvalue (κ : ℝ) (n : ℕ) : ℝ :=
-  (n:ℝ)+(κ/8)*(n:ℝ)^2
-
-lemma anglePolynomialGenerator_add (κ : ℝ) (p q : ℝ[X]) :
-    anglePolynomialGenerator κ (p+q)=
-      anglePolynomialGenerator κ p+anglePolynomialGenerator κ q := by
-  simp only [anglePolynomialGenerator,map_add]
-  ring
-
-lemma anglePolynomialGenerator_C_mul (κ c : ℝ) (p : ℝ[X]) :
-    anglePolynomialGenerator κ (C c*p)=C c*anglePolynomialGenerator κ p := by
-  simp only [anglePolynomialGenerator,derivative_C_mul]
-  ring
-
-lemma anglePolynomialGenerator_one (κ : ℝ) : anglePolynomialGenerator κ 1=0 := by
-  simp [anglePolynomialGenerator]
-
-lemma anglePolynomialGenerator_X (κ : ℝ) :
-    anglePolynomialGenerator κ X = -C (anglePolynomialEigenvalue κ 1)*X := by
-  simp only [anglePolynomialGenerator,anglePolynomialEigenvalue,derivative_X,
-    derivative_one,mul_zero,Nat.cast_one,one_pow,add_comm (1:ℝ),
-    map_add,map_one]
-  ring_nf
-
-lemma anglePolynomialGenerator_X_pow_add_two (κ : ℝ) (n : ℕ) :
-    anglePolynomialGenerator κ (X^(n+2)) =
-      C ((κ/8)*((n:ℝ)+2)*((n:ℝ)+1))*X^n -
-        C (anglePolynomialEigenvalue κ (n+2))*X^(n+2) := by
-  simp only [anglePolynomialGenerator,derivative_X_pow,derivative_C_mul,
-    show n+2-1=n+1 by omega,show n+1-1=n by omega]
-  simp only [anglePolynomialEigenvalue,Nat.cast_add,Nat.cast_ofNat,Nat.cast_one]
-  simp only [pow_add,pow_one]
-  simp only [map_add,map_mul,map_pow,map_ofNat,map_one]
-  ring
-
-lemma anglePolynomialEigenvalue_strictMono {κ : ℝ} (hκ : 0<κ) :
-    StrictMono (anglePolynomialEigenvalue κ) := by
-  intro n m hnm
-  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
-  have hnm' : (n:ℝ) < m := Nat.cast_lt.mpr hnm
-  have hsq : (n:ℝ)^2<(m:ℝ)^2 := by nlinarith
-  have hmul := mul_lt_mul_of_pos_left hsq (by positivity : 0<κ/8)
-  unfold anglePolynomialEigenvalue
-  linarith
-
-lemma anglePolynomialEigenvalue_clock (κ : ℝ) (hκ : κ≠0) (n : ℕ) :
-    anglePolynomialEigenvalue κ n=(κ/4)*((n:ℝ)*((n:ℝ)+4*Target.a κ)/2) := by
-  unfold anglePolynomialEigenvalue Target.a
-  field_simp
-  ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.CanonicalAngleSemigroup
+
+namespace OAI
+
+/-! The cosine-coordinate generator in the actual logarithmic radius
+clock. Its polynomial eigenvalues retain the paper's time normalization. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def anglePolynomialGenerator (κ : ℝ) (p : ℝ[X]) : ℝ[X] :=
+  C (κ/8)*(1-X^2)*p.derivative.derivative - C (1+κ/8)*X*p.derivative
+
+def anglePolynomialEigenvalue (κ : ℝ) (n : ℕ) : ℝ :=
+  (n:ℝ)+(κ/8)*(n:ℝ)^2
+
+lemma anglePolynomialGenerator_add (κ : ℝ) (p q : ℝ[X]) :
+    anglePolynomialGenerator κ (p+q)=
+      anglePolynomialGenerator κ p+anglePolynomialGenerator κ q := by
+  simp only [anglePolynomialGenerator,map_add]
+  ring
+
+lemma anglePolynomialGenerator_C_mul (κ c : ℝ) (p : ℝ[X]) :
+    anglePolynomialGenerator κ (C c*p)=C c*anglePolynomialGenerator κ p := by
+  simp only [anglePolynomialGenerator,derivative_C_mul]
+  ring
+
+lemma anglePolynomialGenerator_one (κ : ℝ) : anglePolynomialGenerator κ 1=0 := by
+  simp [anglePolynomialGenerator]
+
+lemma anglePolynomialGenerator_X (κ : ℝ) :
+    anglePolynomialGenerator κ X = -C (anglePolynomialEigenvalue κ 1)*X := by
+  simp only [anglePolynomialGenerator,anglePolynomialEigenvalue,derivative_X,
+    derivative_one,mul_zero,Nat.cast_one,one_pow,add_comm (1:ℝ),
+    map_add,map_one]
+  ring_nf
+
+lemma anglePolynomialGenerator_X_pow_add_two (κ : ℝ) (n : ℕ) :
+    anglePolynomialGenerator κ (X^(n+2)) =
+      C ((κ/8)*((n:ℝ)+2)*((n:ℝ)+1))*X^n -
+        C (anglePolynomialEigenvalue κ (n+2))*X^(n+2) := by
+  simp only [anglePolynomialGenerator,derivative_X_pow,derivative_C_mul,
+    show n+2-1=n+1 by omega,show n+1-1=n by omega]
+  simp only [anglePolynomialEigenvalue,Nat.cast_add,Nat.cast_ofNat,Nat.cast_one]
+  simp only [pow_add,pow_one]
+  simp only [map_add,map_mul,map_pow,map_ofNat,map_one]
+  ring
+
+lemma anglePolynomialEigenvalue_strictMono {κ : ℝ} (hκ : 0<κ) :
+    StrictMono (anglePolynomialEigenvalue κ) := by
+  intro n m hnm
+  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
+  have hnm' : (n:ℝ) < m := Nat.cast_lt.mpr hnm
+  have hsq : (n:ℝ)^2<(m:ℝ)^2 := by nlinarith
+  have hmul := mul_lt_mul_of_pos_left hsq (by positivity : 0<κ/8)
+  unfold anglePolynomialEigenvalue
+  linarith
+
+lemma anglePolynomialEigenvalue_clock (κ : ℝ) (hκ : κ≠0) (n : ℕ) :
+    anglePolynomialEigenvalue κ n=(κ/4)*((n:ℝ)*((n:ℝ)+4*Target.a κ)/2) := by
+  unfold anglePolynomialEigenvalue Target.a
+  field_simp
+  ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AnglePolynomialInner.lean b/lean/OAI/Probability/SLE/Angle/AnglePolynomialInner.lean
index f90d35d2a70f2eed36a4a523a446566ef0c83b06..aaea06deaed8c8c6d1c0ecd6b57b0652ffa7005f 100644
--- a/lean/OAI/Probability/SLE/Angle/AnglePolynomialInner.lean
+++ b/lean/OAI/Probability/SLE/Angle/AnglePolynomialInner.lean
@@ -1,98 +1,98 @@
-import OAI.Probability.SLE.Angle.AngleEigenOrthogonality
-import OAI.Probability.SLE.Angle.AngleEigenBasis
-
-namespace OAI
-
-/-! The stationary weighted polynomial pairing and its finite-degree
-orthogonality. This uses the actual centered generator eigenpolynomials. -/
-noncomputable section
-open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-def anglePolynomialInner (κ : ℝ) (p q : ℝ[X]) : ℝ :=
-  ∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
-    anglePolynomialTest p θ*anglePolynomialTest q θ
-
-lemma anglePolynomialInner_integrable {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) :
-    IntervalIntegrable (fun θ => Real.sin θ^(4*Target.a κ)*
-      anglePolynomialTest p θ*anglePolynomialTest q θ) volume 0 Real.pi := by
-  have hr : 0<4*Target.a κ := by unfold Target.a; positivity
-  exact ((((Real.continuous_rpow_const hr.le).comp Real.continuous_sin).mul
-    (p.continuous.comp Real.continuous_cos)).mul
-    (q.continuous.comp Real.continuous_cos)).intervalIntegrable _ _
-
-lemma anglePolynomialInner_comm (κ : ℝ) (p q : ℝ[X]) :
-    anglePolynomialInner κ p q=anglePolynomialInner κ q p := by
-  unfold anglePolynomialInner
-  congr 1
-  funext θ
-  ring
-
-lemma anglePolynomialInner_add_right {κ : ℝ} (hκ : 0<κ) (p q r : ℝ[X]) :
-    anglePolynomialInner κ p (q+r)=anglePolynomialInner κ p q+anglePolynomialInner κ p r := by
-  unfold anglePolynomialInner
-  simp only [anglePolynomialTest,eval_add,mul_add]
-  exact intervalIntegral.integral_add (anglePolynomialInner_integrable hκ p q)
-    (anglePolynomialInner_integrable hκ p r)
-
-lemma anglePolynomialInner_smul_right (κ : ℝ) (p q : ℝ[X]) (r : ℝ) :
-    anglePolynomialInner κ p (r • q)=r*anglePolynomialInner κ p q := by
-  unfold anglePolynomialInner
-  simp only [anglePolynomialTest,eval_smul,smul_eq_mul]
-  simp_rw [show ∀ θ : ℝ, Real.sin θ^(4*Target.a κ)*p.eval (Real.cos θ)*
-      (r*q.eval (Real.cos θ))=r*(Real.sin θ^(4*Target.a κ)*p.eval (Real.cos θ)*
-        q.eval (Real.cos θ)) by intro θ; ring]
-  exact intervalIntegral.integral_const_mul _ _
-
-def anglePolynomialInnerRight {κ : ℝ} (hκ : 0<κ) (p : ℝ[X]) : ℝ[X] →ₗ[ℝ] ℝ where
-  toFun := anglePolynomialInner κ p
-  map_add' := anglePolynomialInner_add_right hκ p
-  map_smul' r q := anglePolynomialInner_smul_right κ p q r
-
-lemma anglePolynomialInner_lower_degree {κ : ℝ} (hκ : 0<κ) (n : ℕ) (q : ℝ[X])
-    (hq : q.degree<n) : anglePolynomialInner κ (angleEigenPolynomial κ n) q=0 := by
-  have hspan : Submodule.span ℝ ((angleEigenSequence κ) '' Iio n) ≤
-      LinearMap.ker (anglePolynomialInnerRight hκ (angleEigenPolynomial κ n)) := by
-    apply Submodule.span_le.mpr
-    rintro p ⟨i,hi,rfl⟩
-    exact angleEigenPolynomial_orthogonal hκ n i (ne_of_gt hi)
-  apply hspan
-  rw [(angleEigenSequence κ).span_degreeLT]
-  · exact Polynomial.mem_degreeLT.mpr hq
-  · intro i _hi
-    change IsUnit (angleEigenPolynomial κ i).leadingCoeff
-    rw [show (angleEigenPolynomial κ i).leadingCoeff=1 from angleEigenPolynomial_monic κ i]
-    exact isUnit_one
-
-lemma anglePolynomialInner_X (κ : ℝ) (p q : ℝ[X]) :
-    anglePolynomialInner κ (X*p) q=anglePolynomialInner κ p (X*q) := by
-  unfold anglePolynomialInner
-  congr 1
-  funext θ
-  simp only [anglePolynomialTest,eval_mul,eval_X]
-  ring
-
-lemma anglePolynomialInner_sub_right {κ : ℝ} (hκ : 0<κ) (p q r : ℝ[X]) :
-    anglePolynomialInner κ p (q-r)=anglePolynomialInner κ p q-anglePolynomialInner κ p r :=
-  map_sub (anglePolynomialInnerRight hκ p) q r
-
-lemma anglePolynomialInner_one_pos {κ : ℝ} (hκ : 0<κ) :
-    0<anglePolynomialInner κ 1 1 := by
-  have hr : 0<4*Target.a κ := by unfold Target.a; positivity
-  simp only [anglePolynomialInner,anglePolynomialTest,eval_one,mul_one]
-  apply intervalIntegral.integral_pos Real.pi_pos
-    (((Real.continuous_rpow_const hr.le).comp Real.continuous_sin).continuousOn)
-  · intro θ hθ
-    exact Real.rpow_nonneg (Real.sin_nonneg_of_nonneg_of_le_pi hθ.1.le hθ.2) _
-  · refine ⟨Real.pi/2,⟨by positivity,by linarith [Real.pi_pos]⟩,?_⟩
-    change 0 < Real.sin (Real.pi/2)^(4*Target.a κ)
-    rw [Real.sin_pi_div_two,Real.one_rpow]
-    exact zero_lt_one
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenOrthogonality
+import OAI.Probability.SLE.Angle.AngleEigenBasis
+
+namespace OAI
+
+/-! The stationary weighted polynomial pairing and its finite-degree
+orthogonality. This uses the actual centered generator eigenpolynomials. -/
+noncomputable section
+open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+def anglePolynomialInner (κ : ℝ) (p q : ℝ[X]) : ℝ :=
+  ∫ θ in (0:ℝ)..Real.pi, Real.sin θ^(4*Target.a κ)*
+    anglePolynomialTest p θ*anglePolynomialTest q θ
+
+lemma anglePolynomialInner_integrable {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) :
+    IntervalIntegrable (fun θ => Real.sin θ^(4*Target.a κ)*
+      anglePolynomialTest p θ*anglePolynomialTest q θ) volume 0 Real.pi := by
+  have hr : 0<4*Target.a κ := by unfold Target.a; positivity
+  exact ((((Real.continuous_rpow_const hr.le).comp Real.continuous_sin).mul
+    (p.continuous.comp Real.continuous_cos)).mul
+    (q.continuous.comp Real.continuous_cos)).intervalIntegrable _ _
+
+lemma anglePolynomialInner_comm (κ : ℝ) (p q : ℝ[X]) :
+    anglePolynomialInner κ p q=anglePolynomialInner κ q p := by
+  unfold anglePolynomialInner
+  congr 1
+  funext θ
+  ring
+
+lemma anglePolynomialInner_add_right {κ : ℝ} (hκ : 0<κ) (p q r : ℝ[X]) :
+    anglePolynomialInner κ p (q+r)=anglePolynomialInner κ p q+anglePolynomialInner κ p r := by
+  unfold anglePolynomialInner
+  simp only [anglePolynomialTest,eval_add,mul_add]
+  exact intervalIntegral.integral_add (anglePolynomialInner_integrable hκ p q)
+    (anglePolynomialInner_integrable hκ p r)
+
+lemma anglePolynomialInner_smul_right (κ : ℝ) (p q : ℝ[X]) (r : ℝ) :
+    anglePolynomialInner κ p (r • q)=r*anglePolynomialInner κ p q := by
+  unfold anglePolynomialInner
+  simp only [anglePolynomialTest,eval_smul,smul_eq_mul]
+  simp_rw [show ∀ θ : ℝ, Real.sin θ^(4*Target.a κ)*p.eval (Real.cos θ)*
+      (r*q.eval (Real.cos θ))=r*(Real.sin θ^(4*Target.a κ)*p.eval (Real.cos θ)*
+        q.eval (Real.cos θ)) by intro θ; ring]
+  exact intervalIntegral.integral_const_mul _ _
+
+def anglePolynomialInnerRight {κ : ℝ} (hκ : 0<κ) (p : ℝ[X]) : ℝ[X] →ₗ[ℝ] ℝ where
+  toFun := anglePolynomialInner κ p
+  map_add' := anglePolynomialInner_add_right hκ p
+  map_smul' r q := anglePolynomialInner_smul_right κ p q r
+
+lemma anglePolynomialInner_lower_degree {κ : ℝ} (hκ : 0<κ) (n : ℕ) (q : ℝ[X])
+    (hq : q.degree<n) : anglePolynomialInner κ (angleEigenPolynomial κ n) q=0 := by
+  have hspan : Submodule.span ℝ ((angleEigenSequence κ) '' Iio n) ≤
+      LinearMap.ker (anglePolynomialInnerRight hκ (angleEigenPolynomial κ n)) := by
+    apply Submodule.span_le.mpr
+    rintro p ⟨i,hi,rfl⟩
+    exact angleEigenPolynomial_orthogonal hκ n i (ne_of_gt hi)
+  apply hspan
+  rw [(angleEigenSequence κ).span_degreeLT]
+  · exact Polynomial.mem_degreeLT.mpr hq
+  · intro i _hi
+    change IsUnit (angleEigenPolynomial κ i).leadingCoeff
+    rw [show (angleEigenPolynomial κ i).leadingCoeff=1 from angleEigenPolynomial_monic κ i]
+    exact isUnit_one
+
+lemma anglePolynomialInner_X (κ : ℝ) (p q : ℝ[X]) :
+    anglePolynomialInner κ (X*p) q=anglePolynomialInner κ p (X*q) := by
+  unfold anglePolynomialInner
+  congr 1
+  funext θ
+  simp only [anglePolynomialTest,eval_mul,eval_X]
+  ring
+
+lemma anglePolynomialInner_sub_right {κ : ℝ} (hκ : 0<κ) (p q r : ℝ[X]) :
+    anglePolynomialInner κ p (q-r)=anglePolynomialInner κ p q-anglePolynomialInner κ p r :=
+  map_sub (anglePolynomialInnerRight hκ p) q r
+
+lemma anglePolynomialInner_one_pos {κ : ℝ} (hκ : 0<κ) :
+    0<anglePolynomialInner κ 1 1 := by
+  have hr : 0<4*Target.a κ := by unfold Target.a; positivity
+  simp only [anglePolynomialInner,anglePolynomialTest,eval_one,mul_one]
+  apply intervalIntegral.integral_pos Real.pi_pos
+    (((Real.continuous_rpow_const hr.le).comp Real.continuous_sin).continuousOn)
+  · intro θ hθ
+    exact Real.rpow_nonneg (Real.sin_nonneg_of_nonneg_of_le_pi hθ.1.le hθ.2) _
+  · refine ⟨Real.pi/2,⟨by positivity,by linarith [Real.pi_pos]⟩,?_⟩
+    change 0 < Real.sin (Real.pi/2)^(4*Target.a κ)
+    rw [Real.sin_pi_div_two,Real.one_rpow]
+    exact zero_lt_one
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogField.lean b/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogField.lean
index 5aabbf67fe5f292967853b31fd1ce54c1915c2a4..db63fc9cbf051871ccba93fa2b6bbef867e7145a 100644
--- a/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogField.lean
+++ b/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogField.lean
@@ -1,96 +1,96 @@
-import OAI.Probability.SLE.Angle.AngleLogGeneratorCoefficients
-
-namespace OAI
-
-/-! Smooth Green-weighted polynomial tests in the actual Loewner state. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def logPolynomialDensity (κ p a : ℝ) (P : ℝ[X]) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  powerLogDensity κ p a q x*P.eval (logCosine κ q x)
-
-private lemma polynomial_eval_contDiff (P : ℝ[X]) (n : WithTop ℕ∞) :
-    ContDiff ℝ n (fun x => P.eval x) := by
-  simpa using P.contDiff_aeval (𝕜 := ℝ) n
-
-lemma logPolynomialDensity_contDiff (κ p a : ℝ) (P : ℝ[X]) :
-    ContDiff ℝ (⊤:ℕ∞) (fun z : (ℂ × ℝ) × ℝ => logPolynomialDensity κ p a P z.1 z.2) :=
-  (powerLogDensity_contDiff κ p a).mul
-    ((polynomial_eval_contDiff P _).comp (logCosine_contDiff κ))
-
-private lemma real_second_mul {f g : ℝ → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) (x : ℝ) :
-    iteratedDeriv 2 (fun y => f y*g y) x=
-      iteratedDeriv 2 f x*g x+2*deriv f x*deriv g x+f x*iteratedDeriv 2 g x := by
-  have hf1 := hf.differentiable (by norm_num)
-  have hg1 := hg.differentiable (by norm_num)
-  have hd : deriv (fun y => f y*g y)=fun y => deriv f y*g y+f y*deriv g y :=
-    funext (fun y => deriv_mul (hf1 y) (hg1 y))
-  have hh := (((hf.differentiable_deriv_two x).hasDerivAt).mul (hg1 x).hasDerivAt).add
-    ((hf1 x).hasDerivAt.mul (hg.differentiable_deriv_two x).hasDerivAt)
-  change HasDerivAt (fun y => deriv f y*g y+f y*deriv g y) _ x at hh
-  conv_lhs => rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
-  rw [hd,hh.deriv]
-  have hff : iteratedDeriv 2 f x=deriv (deriv f) x := by
-    rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
-  have hgg : iteratedDeriv 2 g x=deriv (deriv g) x := by
-    rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
-  rw [hff,hgg]
-  ring
-
-private lemma polynomial_second (P : ℝ[X]) (x : ℝ) :
-    iteratedDeriv 2 (fun y => P.eval y) x=P.derivative.derivative.eval x := by
-  have hd : deriv (fun y => P.eval y)=fun y => P.derivative.eval y :=
-    funext (fun y => P.deriv)
-  rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one,hd]
-  exact P.derivative.deriv
-
-lemma logPolynomialDensity_second (κ p a : ℝ) (P : ℝ[X]) (q : ℂ × ℝ) (x : ℝ) :
-    iteratedDeriv 2 (logPolynomialDensity κ p a P q) x=
-      iteratedDeriv 2 (powerLogDensity κ p a q) x*P.eval (logCosine κ q x)+
-      2*deriv (powerLogDensity κ p a q) x*P.derivative.eval (logCosine κ q x)*
-        deriv (logCosine κ q) x+
-      powerLogDensity κ p a q x*(P.derivative.derivative.eval (logCosine κ q x)*
-        (deriv (logCosine κ q) x)^2+
-        P.derivative.eval (logCosine κ q x)*iteratedDeriv 2 (logCosine κ q) x) := by
-  have hH : ContDiff ℝ 2 (powerLogDensity κ p a q) :=
-    ((powerLogDensity_contDiff κ p a).comp (contDiff_const.prodMk contDiff_id)).of_le (by simp)
-  have hC : ContDiff ℝ 2 (logCosine κ q) :=
-    ((logCosine_contDiff κ).comp (contDiff_const.prodMk contDiff_id)).of_le (by simp)
-  have hP := (polynomial_eval_contDiff P 2).comp hC
-  change ContDiff ℝ 2 (fun y => P.eval (logCosine κ q y)) at hP
-  have hp1 := (P.hasDerivAt (logCosine κ q x)).comp x
-    (hC.differentiable (by norm_num) x).hasDerivAt
-  have hp2 := iteratedDeriv_comp_two (x := x)
-    (polynomial_eval_contDiff P 2).contDiffAt hC.contDiffAt
-  change iteratedDeriv 2 (fun y => P.eval (logCosine κ q y)) x = _ at hp2
-  have hp1d := hp1.deriv
-  change deriv (fun y => P.eval (logCosine κ q y)) x = _ at hp1d
-  change iteratedDeriv 2 (fun y => powerLogDensity κ p a q y*P.eval (logCosine κ q y)) x=_
-  rw [real_second_mul hH hP x,hp1d,hp2,polynomial_second,P.deriv]
-  ring
-
-lemma logPolynomialDensity_directional (κ p a : ℝ) (P : ℝ[X]) (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ p a P r x) q v=
-      fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q v*P.eval (logCosine κ q x)+
-      powerLogDensity κ p a q x*P.derivative.eval (logCosine κ q x)*
-        fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q v := by
-  have hH : Differentiable ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) :=
-    ((powerLogDensity_contDiff κ p a).comp
-    (contDiff_id.prodMk contDiff_const)).differentiable (by simp)
-  have hC : Differentiable ℝ (fun r : ℂ × ℝ => logCosine κ r x) :=
-    ((logCosine_contDiff κ).comp
-    (contDiff_id.prodMk contDiff_const)).differentiable (by simp)
-  have hp := (P.hasDerivAt (logCosine κ q x)).comp_hasFDerivAt q (hC q).hasFDerivAt
-  have he := congrArg (fun L : (ℂ × ℝ) →L[ℝ] ℝ => L v) ((hH q).hasFDerivAt.mul hp).fderiv
-  change fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ p a P r x) q v=_ at he
-  rw [he]
-  simp only [add_apply,smul_apply,smul_eq_mul,Function.comp_apply]
-  ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleLogGeneratorCoefficients
+
+namespace OAI
+
+/-! Smooth Green-weighted polynomial tests in the actual Loewner state. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def logPolynomialDensity (κ p a : ℝ) (P : ℝ[X]) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  powerLogDensity κ p a q x*P.eval (logCosine κ q x)
+
+private lemma polynomial_eval_contDiff (P : ℝ[X]) (n : WithTop ℕ∞) :
+    ContDiff ℝ n (fun x => P.eval x) := by
+  simpa using P.contDiff_aeval (𝕜 := ℝ) n
+
+lemma logPolynomialDensity_contDiff (κ p a : ℝ) (P : ℝ[X]) :
+    ContDiff ℝ (⊤:ℕ∞) (fun z : (ℂ × ℝ) × ℝ => logPolynomialDensity κ p a P z.1 z.2) :=
+  (powerLogDensity_contDiff κ p a).mul
+    ((polynomial_eval_contDiff P _).comp (logCosine_contDiff κ))
+
+private lemma real_second_mul {f g : ℝ → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) (x : ℝ) :
+    iteratedDeriv 2 (fun y => f y*g y) x=
+      iteratedDeriv 2 f x*g x+2*deriv f x*deriv g x+f x*iteratedDeriv 2 g x := by
+  have hf1 := hf.differentiable (by norm_num)
+  have hg1 := hg.differentiable (by norm_num)
+  have hd : deriv (fun y => f y*g y)=fun y => deriv f y*g y+f y*deriv g y :=
+    funext (fun y => deriv_mul (hf1 y) (hg1 y))
+  have hh := (((hf.differentiable_deriv_two x).hasDerivAt).mul (hg1 x).hasDerivAt).add
+    ((hf1 x).hasDerivAt.mul (hg.differentiable_deriv_two x).hasDerivAt)
+  change HasDerivAt (fun y => deriv f y*g y+f y*deriv g y) _ x at hh
+  conv_lhs => rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
+  rw [hd,hh.deriv]
+  have hff : iteratedDeriv 2 f x=deriv (deriv f) x := by
+    rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
+  have hgg : iteratedDeriv 2 g x=deriv (deriv g) x := by
+    rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one]
+  rw [hff,hgg]
+  ring
+
+private lemma polynomial_second (P : ℝ[X]) (x : ℝ) :
+    iteratedDeriv 2 (fun y => P.eval y) x=P.derivative.derivative.eval x := by
+  have hd : deriv (fun y => P.eval y)=fun y => P.derivative.eval y :=
+    funext (fun y => P.deriv)
+  rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one,hd]
+  exact P.derivative.deriv
+
+lemma logPolynomialDensity_second (κ p a : ℝ) (P : ℝ[X]) (q : ℂ × ℝ) (x : ℝ) :
+    iteratedDeriv 2 (logPolynomialDensity κ p a P q) x=
+      iteratedDeriv 2 (powerLogDensity κ p a q) x*P.eval (logCosine κ q x)+
+      2*deriv (powerLogDensity κ p a q) x*P.derivative.eval (logCosine κ q x)*
+        deriv (logCosine κ q) x+
+      powerLogDensity κ p a q x*(P.derivative.derivative.eval (logCosine κ q x)*
+        (deriv (logCosine κ q) x)^2+
+        P.derivative.eval (logCosine κ q x)*iteratedDeriv 2 (logCosine κ q) x) := by
+  have hH : ContDiff ℝ 2 (powerLogDensity κ p a q) :=
+    ((powerLogDensity_contDiff κ p a).comp (contDiff_const.prodMk contDiff_id)).of_le (by simp)
+  have hC : ContDiff ℝ 2 (logCosine κ q) :=
+    ((logCosine_contDiff κ).comp (contDiff_const.prodMk contDiff_id)).of_le (by simp)
+  have hP := (polynomial_eval_contDiff P 2).comp hC
+  change ContDiff ℝ 2 (fun y => P.eval (logCosine κ q y)) at hP
+  have hp1 := (P.hasDerivAt (logCosine κ q x)).comp x
+    (hC.differentiable (by norm_num) x).hasDerivAt
+  have hp2 := iteratedDeriv_comp_two (x := x)
+    (polynomial_eval_contDiff P 2).contDiffAt hC.contDiffAt
+  change iteratedDeriv 2 (fun y => P.eval (logCosine κ q y)) x = _ at hp2
+  have hp1d := hp1.deriv
+  change deriv (fun y => P.eval (logCosine κ q y)) x = _ at hp1d
+  change iteratedDeriv 2 (fun y => powerLogDensity κ p a q y*P.eval (logCosine κ q y)) x=_
+  rw [real_second_mul hH hP x,hp1d,hp2,polynomial_second,P.deriv]
+  ring
+
+lemma logPolynomialDensity_directional (κ p a : ℝ) (P : ℝ[X]) (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ p a P r x) q v=
+      fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q v*P.eval (logCosine κ q x)+
+      powerLogDensity κ p a q x*P.derivative.eval (logCosine κ q x)*
+        fderiv ℝ (fun r : ℂ × ℝ => logCosine κ r x) q v := by
+  have hH : Differentiable ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) :=
+    ((powerLogDensity_contDiff κ p a).comp
+    (contDiff_id.prodMk contDiff_const)).differentiable (by simp)
+  have hC : Differentiable ℝ (fun r : ℂ × ℝ => logCosine κ r x) :=
+    ((logCosine_contDiff κ).comp
+    (contDiff_id.prodMk contDiff_const)).differentiable (by simp)
+  have hp := (P.hasDerivAt (logCosine κ q x)).comp_hasFDerivAt q (hC q).hasFDerivAt
+  have he := congrArg (fun L : (ℂ × ℝ) →L[ℝ] ℝ => L v) ((hH q).hasFDerivAt.mul hp).fderiv
+  change fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ p a P r x) q v=_ at he
+  rw [he]
+  simp only [add_apply,smul_apply,smul_eq_mul,Function.comp_apply]
+  ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogGenerator.lean b/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogGenerator.lean
index d4bf4dc18812bc728f95d1039c581933740950ef..9f78000f0121d70f51fa3ca8d3c6b3d379b8bb9b 100644
--- a/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogGenerator.lean
+++ b/lean/OAI/Probability/SLE/Angle/AnglePolynomialLogGenerator.lean
@@ -1,46 +1,46 @@
-import OAI.Probability.SLE.Angle.AnglePolynomialLogField
-import OAI.Probability.SLE.Angle.AngleEigenPolynomials
-
-namespace OAI
-
-/-! Exact cancellation for the Green-weighted angular eigenfunctions in
-the original Brownian driver and actual capacity-two coordinates. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-lemma logPolynomialDensity_generator {κ : ℝ} (hκ : 0<κ) (c : ℝ) (P : ℝ[X])
-    (q : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P r x) q
-        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
-      (1/2:ℝ)*iteratedDeriv 2 (logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P q) x =
-    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*logClockRate κ q x*
-      ((anglePolynomialGenerator κ P).eval (logCosine κ q x)+c*P.eval (logCosine κ q x)) := by
-  have hH := shifted_green_generator hκ c q x
-  have hC := logCosine_tilted_drift hκ q x
-  have hV := logCosine_tilted_variance hκ.le q x
-  rw [logPolynomialDensity_directional,logPolynomialDensity_second,
-    (powerLogDensity_spatial_derivative κ (codimension κ+c) (greenExponent κ) q x).deriv]
-  simp only [anglePolynomialGenerator,eval_sub,eval_mul,eval_C,eval_pow,eval_X,eval_one]
-  linear_combination P.eval (logCosine κ q x)*hH +
-    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*
-      P.derivative.eval (logCosine κ q x)*hC +
-    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*
-      P.derivative.derivative.eval (logCosine κ q x)*hV
-
-lemma logEigenDensity_generator_zero {κ : ℝ} (hκ : 0<κ) (n : ℕ) (q : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ
-      (codimension κ+anglePolynomialEigenvalue κ n) (greenExponent κ) (angleEigenPolynomial κ n) r x) q
-        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
-      (1/2:ℝ)*iteratedDeriv 2 (logPolynomialDensity κ
-        (codimension κ+anglePolynomialEigenvalue κ n) (greenExponent κ) (angleEigenPolynomial κ n) q) x=0 := by
-  rw [logPolynomialDensity_generator hκ,angleEigenPolynomial_eigen hκ]
-  simp only [eval_mul,eval_neg,eval_C]
-  ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AnglePolynomialLogField
+import OAI.Probability.SLE.Angle.AngleEigenPolynomials
+
+namespace OAI
+
+/-! Exact cancellation for the Green-weighted angular eigenfunctions in
+the original Brownian driver and actual capacity-two coordinates. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+lemma logPolynomialDensity_generator {κ : ℝ} (hκ : 0<κ) (c : ℝ) (P : ℝ[X])
+    (q : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P r x) q
+        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
+      (1/2:ℝ)*iteratedDeriv 2 (logPolynomialDensity κ (codimension κ+c) (greenExponent κ) P q) x =
+    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*logClockRate κ q x*
+      ((anglePolynomialGenerator κ P).eval (logCosine κ q x)+c*P.eval (logCosine κ q x)) := by
+  have hH := shifted_green_generator hκ c q x
+  have hC := logCosine_tilted_drift hκ q x
+  have hV := logCosine_tilted_variance hκ.le q x
+  rw [logPolynomialDensity_directional,logPolynomialDensity_second,
+    (powerLogDensity_spatial_derivative κ (codimension κ+c) (greenExponent κ) q x).deriv]
+  simp only [anglePolynomialGenerator,eval_sub,eval_mul,eval_C,eval_pow,eval_X,eval_one]
+  linear_combination P.eval (logCosine κ q x)*hH +
+    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*
+      P.derivative.eval (logCosine κ q x)*hC +
+    powerLogDensity κ (codimension κ+c) (greenExponent κ) q x*
+      P.derivative.derivative.eval (logCosine κ q x)*hV
+
+lemma logEigenDensity_generator_zero {κ : ℝ} (hκ : 0<κ) (n : ℕ) (q : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => logPolynomialDensity κ
+      (codimension κ+anglePolynomialEigenvalue κ n) (greenExponent κ) (angleEigenPolynomial κ n) r x) q
+        (physicalLogLoewnerField (Real.sqrt κ*x) q)+
+      (1/2:ℝ)*iteratedDeriv 2 (logPolynomialDensity κ
+        (codimension κ+anglePolynomialEigenvalue κ n) (greenExponent κ) (angleEigenPolynomial κ n) q) x=0 := by
+  rw [logPolynomialDensity_generator hκ,angleEigenPolynomial_eigen hκ]
+  simp only [eval_mul,eval_neg,eval_C]
+  ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleRecurrenceCoefficients.lean b/lean/OAI/Probability/SLE/Angle/AngleRecurrenceCoefficients.lean
index 4752fa1ce582651644405f932a2362095b622fd0..5fabac96a14f01383fcb5ca9bfb4e472b5782992 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleRecurrenceCoefficients.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleRecurrenceCoefficients.lean
@@ -1,75 +1,75 @@
-import OAI.Probability.SLE.Angle.AngleEigenBounds
-
-namespace OAI
-
-/-! The coefficients needed for the monic three-term recurrence, including
-a uniform positive lower bound for its squared-norm ratio. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-lemma angleEigenPolynomial_penultimate (κ : ℝ) (n : ℕ) :
-    (angleEigenPolynomial κ (n+1)).coeff n=0 := by
-  rw [angleEigenPolynomial_coeff,angleEigenCoefficient,
-    ite_eq_right (by omega),dite_eq_left (by omega),
-    angleEigenCoefficient_above κ (by omega),mul_zero]
-
-lemma angleEigenPolynomial_second_coeff {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    (angleEigenPolynomial κ (n+2)).coeff n=
-      -κ*((n:ℝ)+2)*((n:ℝ)+1)/(4*(κ*((n:ℝ)+1)+4)) := by
-  rw [angleEigenPolynomial_coeff,angleEigenCoefficient,
-    ite_eq_right (by omega),dite_eq_left (by omega),angleEigenCoefficient_top,mul_one]
-  have hd : 4*(κ*((n:ℝ)+1)+4)≠0 := by positivity
-  have he : anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ (n+2)≠0 :=
-    sub_ne_zero.mpr (ne_of_lt (anglePolynomialEigenvalue_strictMono hκ (by omega)))
-  apply (div_eq_div_iff he hd).mpr
-  unfold anglePolynomialEigenvalue
-  push_cast
-  ring
-
-def angleRecurrenceCoefficient (κ : ℝ) (n : ℕ) : ℝ :=
-  κ*((n:ℝ)+1)*(κ*(n:ℝ)+8)/(4*(κ*((n:ℝ)+1)+4)*(κ*(n:ℝ)+4))
-
-lemma angleRecurrenceCoefficient_zero (κ : ℝ) :
-    angleRecurrenceCoefficient κ 0=κ/(2*(κ+4)) := by
-  unfold angleRecurrenceCoefficient
-  by_cases h : κ+4=0
-  · have hk : κ = -4 := by linarith
-    subst κ
-    norm_num
-  · norm_num
-    field_simp
-    ring
-
-lemma angleRecurrenceCoefficient_sub {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    (angleEigenPolynomial κ (n+2)).coeff n-
-      (angleEigenPolynomial κ (n+3)).coeff (n+1)=angleRecurrenceCoefficient κ (n+1) := by
-  rw [angleEigenPolynomial_second_coeff hκ,
-    show n+3=(n+1)+2 by omega,angleEigenPolynomial_second_coeff hκ]
-  unfold angleRecurrenceCoefficient
-  push_cast
-  have h1 : κ*((n:ℝ)+1)+4≠0 := by positivity
-  have h2 : κ*((n:ℝ)+1+1)+4≠0 := by positivity
-  field_simp
-  ring
-
-lemma angleRecurrenceCoefficient_lower {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    κ/(4*(κ+4))≤angleRecurrenceCoefficient κ n := by
-  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
-  have hd : 0<4*(κ*((n:ℝ)+1)+4)*(κ*(n:ℝ)+4) := by positivity
-  have hc : 0<4*(κ+4) := by positivity
-  unfold angleRecurrenceCoefficient
-  apply (div_le_div_iff₀ hc hd).mpr
-  have hh : 0≤4*κ*(n:ℝ)^2+(4*κ+32)*(n:ℝ)+4*κ+16 := by positivity
-  nlinarith [mul_nonneg hκ.le hh]
-
-lemma angleRecurrenceCoefficient_pos {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    0<angleRecurrenceCoefficient κ n :=
-  lt_of_lt_of_le (by positivity) (angleRecurrenceCoefficient_lower hκ n)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenBounds
+
+namespace OAI
+
+/-! The coefficients needed for the monic three-term recurrence, including
+a uniform positive lower bound for its squared-norm ratio. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+lemma angleEigenPolynomial_penultimate (κ : ℝ) (n : ℕ) :
+    (angleEigenPolynomial κ (n+1)).coeff n=0 := by
+  rw [angleEigenPolynomial_coeff,angleEigenCoefficient,
+    ite_eq_right (by omega),dite_eq_left (by omega),
+    angleEigenCoefficient_above κ (by omega),mul_zero]
+
+lemma angleEigenPolynomial_second_coeff {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    (angleEigenPolynomial κ (n+2)).coeff n=
+      -κ*((n:ℝ)+2)*((n:ℝ)+1)/(4*(κ*((n:ℝ)+1)+4)) := by
+  rw [angleEigenPolynomial_coeff,angleEigenCoefficient,
+    ite_eq_right (by omega),dite_eq_left (by omega),angleEigenCoefficient_top,mul_one]
+  have hd : 4*(κ*((n:ℝ)+1)+4)≠0 := by positivity
+  have he : anglePolynomialEigenvalue κ n-anglePolynomialEigenvalue κ (n+2)≠0 :=
+    sub_ne_zero.mpr (ne_of_lt (anglePolynomialEigenvalue_strictMono hκ (by omega)))
+  apply (div_eq_div_iff he hd).mpr
+  unfold anglePolynomialEigenvalue
+  push_cast
+  ring
+
+def angleRecurrenceCoefficient (κ : ℝ) (n : ℕ) : ℝ :=
+  κ*((n:ℝ)+1)*(κ*(n:ℝ)+8)/(4*(κ*((n:ℝ)+1)+4)*(κ*(n:ℝ)+4))
+
+lemma angleRecurrenceCoefficient_zero (κ : ℝ) :
+    angleRecurrenceCoefficient κ 0=κ/(2*(κ+4)) := by
+  unfold angleRecurrenceCoefficient
+  by_cases h : κ+4=0
+  · have hk : κ = -4 := by linarith
+    subst κ
+    norm_num
+  · norm_num
+    field_simp
+    ring
+
+lemma angleRecurrenceCoefficient_sub {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    (angleEigenPolynomial κ (n+2)).coeff n-
+      (angleEigenPolynomial κ (n+3)).coeff (n+1)=angleRecurrenceCoefficient κ (n+1) := by
+  rw [angleEigenPolynomial_second_coeff hκ,
+    show n+3=(n+1)+2 by omega,angleEigenPolynomial_second_coeff hκ]
+  unfold angleRecurrenceCoefficient
+  push_cast
+  have h1 : κ*((n:ℝ)+1)+4≠0 := by positivity
+  have h2 : κ*((n:ℝ)+1+1)+4≠0 := by positivity
+  field_simp
+  ring
+
+lemma angleRecurrenceCoefficient_lower {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    κ/(4*(κ+4))≤angleRecurrenceCoefficient κ n := by
+  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
+  have hd : 0<4*(κ*((n:ℝ)+1)+4)*(κ*(n:ℝ)+4) := by positivity
+  have hc : 0<4*(κ+4) := by positivity
+  unfold angleRecurrenceCoefficient
+  apply (div_le_div_iff₀ hc hd).mpr
+  have hh : 0≤4*κ*(n:ℝ)^2+(4*κ+32)*(n:ℝ)+4*κ+16 := by positivity
+  nlinarith [mul_nonneg hκ.le hh]
+
+lemma angleRecurrenceCoefficient_pos {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    0<angleRecurrenceCoefficient κ n :=
+  lt_of_lt_of_le (by positivity) (angleRecurrenceCoefficient_lower hκ n)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleRecurrenceRemainders.lean b/lean/OAI/Probability/SLE/Angle/AngleRecurrenceRemainders.lean
index eade402c5ba398c9793fa136dc126645f92d532f..e3dd0cea5172f4c0c282222afa5926eeecbba7d3 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleRecurrenceRemainders.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleRecurrenceRemainders.lean
@@ -1,93 +1,93 @@
-import OAI.Probability.SLE.Angle.AngleRecurrenceCoefficients
-
-namespace OAI
-
-/-! Finite-degree cancellations used to recover the squared norms of the
-monic angular eigenpolynomials from their leading coefficients. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-lemma angleEigenPolynomial_leading_remainder (κ : ℝ) (n : ℕ) :
-    (X*angleEigenPolynomial κ n-angleEigenPolynomial κ (n+1)).degree < ((n+1:ℕ):WithBot ℕ) := by
-  have hm := monic_X.mul (angleEigenPolynomial_monic κ n)
-  have hd : (X*angleEigenPolynomial κ n).degree=(angleEigenPolynomial κ (n+1)).degree := by
-    rw [mul_comm,degree_mul_X,degree_eq_natDegree (angleEigenPolynomial_monic κ n).ne_zero,
-      degree_eq_natDegree (angleEigenPolynomial_monic κ (n+1)).ne_zero,
-      angleEigenPolynomial_natDegree,angleEigenPolynomial_natDegree]
-    norm_cast
-  have hh := degree_sub_lt_left hd hm.ne_zero
-    (by rw [hm,angleEigenPolynomial_monic])
-  rw [hd,degree_eq_natDegree (angleEigenPolynomial_monic κ (n+1)).ne_zero,
-    angleEigenPolynomial_natDegree] at hh
-  exact hh
-
-lemma angleEigenPolynomial_recurrence_remainder {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    (X*angleEigenPolynomial κ (n+2)-angleEigenPolynomial κ (n+3)-
-      angleRecurrenceCoefficient κ (n+1) • angleEigenPolynomial κ (n+1)).degree < ((n+1:ℕ):WithBot ℕ) := by
-  apply (degree_lt_iff_coeff_zero _ _).mpr
-  intro m hm
-  rcases eq_or_lt_of_le hm with rfl | hm
-  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul]
-    rw [angleEigenPolynomial_coeff κ (n+1) (n+1),angleEigenCoefficient_top]
-    have he := angleRecurrenceCoefficient_sub hκ n
-    rw [mul_one]
-    exact sub_eq_zero.mpr he
-  rcases eq_or_lt_of_le (Nat.succ_le_iff.mpr hm) with hm | hm
-  · subst m
-    rw [show n+1+1=n+2 by omega]
-    simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul]
-    rw [angleEigenPolynomial_penultimate,
-      show n+3=(n+2)+1 by omega,angleEigenPolynomial_penultimate,
-      angleEigenPolynomial_coeff,angleEigenCoefficient_above κ (by omega)]
-    ring
-  rcases eq_or_lt_of_le (Nat.succ_le_iff.mpr hm) with hm | hm
-  · subst m
-    rw [show n+1+1+1=n+3 by omega,show n+3=(n+2)+1 by omega]
-    simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,angleEigenPolynomial_coeff]
-    rw [angleEigenCoefficient_top,angleEigenCoefficient_top,
-      angleEigenCoefficient_above κ (by omega)]
-    ring
-  · obtain ⟨k,rfl⟩ := Nat.exists_eq_add_of_le (by omega : 1 ≤ m)
-    rw [Nat.add_comm 1 k]
-    simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,angleEigenPolynomial_coeff]
-    rw [angleEigenCoefficient_above κ (by omega),
-      angleEigenCoefficient_above κ (by omega),angleEigenCoefficient_above κ (by omega)]
-    ring
-
-lemma angleEigenPolynomial_recurrence_zero {κ : ℝ} (hκ : 0<κ) :
-    X*angleEigenPolynomial κ 1-angleEigenPolynomial κ 2-
-      angleRecurrenceCoefficient κ 0 • angleEigenPolynomial κ 0=0 := by
-  ext m
-  rcases m with _|m
-  · simp only [coeff_sub,coeff_X_mul_zero,coeff_smul,smul_eq_mul,coeff_zero]
-    rw [angleEigenPolynomial_coeff κ 0 0,angleEigenCoefficient_top,
-      angleEigenPolynomial_second_coeff hκ 0,
-      angleRecurrenceCoefficient_zero]
-    norm_num
-    field_simp
-    ring
-  rcases m with _|m
-  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,coeff_zero]
-    rw [show 1=0+1 from rfl,angleEigenPolynomial_penultimate,
-      show 2=1+1 from rfl,angleEigenPolynomial_penultimate,
-      angleEigenPolynomial_coeff,angleEigenCoefficient_above κ (by omega)]
-    ring
-  rcases m with _|m
-  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,coeff_zero,
-      angleEigenPolynomial_coeff]
-    rw [angleEigenCoefficient_top,angleEigenCoefficient_top,
-      angleEigenCoefficient_above κ (by omega)]
-    ring
-  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,coeff_zero,
-      angleEigenPolynomial_coeff]
-    rw [angleEigenCoefficient_above κ (by omega),
-      angleEigenCoefficient_above κ (by omega),angleEigenCoefficient_above κ (by omega)]
-    ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleRecurrenceCoefficients
+
+namespace OAI
+
+/-! Finite-degree cancellations used to recover the squared norms of the
+monic angular eigenpolynomials from their leading coefficients. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+lemma angleEigenPolynomial_leading_remainder (κ : ℝ) (n : ℕ) :
+    (X*angleEigenPolynomial κ n-angleEigenPolynomial κ (n+1)).degree < ((n+1:ℕ):WithBot ℕ) := by
+  have hm := monic_X.mul (angleEigenPolynomial_monic κ n)
+  have hd : (X*angleEigenPolynomial κ n).degree=(angleEigenPolynomial κ (n+1)).degree := by
+    rw [mul_comm,degree_mul_X,degree_eq_natDegree (angleEigenPolynomial_monic κ n).ne_zero,
+      degree_eq_natDegree (angleEigenPolynomial_monic κ (n+1)).ne_zero,
+      angleEigenPolynomial_natDegree,angleEigenPolynomial_natDegree]
+    norm_cast
+  have hh := degree_sub_lt_left hd hm.ne_zero
+    (by rw [hm,angleEigenPolynomial_monic])
+  rw [hd,degree_eq_natDegree (angleEigenPolynomial_monic κ (n+1)).ne_zero,
+    angleEigenPolynomial_natDegree] at hh
+  exact hh
+
+lemma angleEigenPolynomial_recurrence_remainder {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    (X*angleEigenPolynomial κ (n+2)-angleEigenPolynomial κ (n+3)-
+      angleRecurrenceCoefficient κ (n+1) • angleEigenPolynomial κ (n+1)).degree < ((n+1:ℕ):WithBot ℕ) := by
+  apply (degree_lt_iff_coeff_zero _ _).mpr
+  intro m hm
+  rcases eq_or_lt_of_le hm with rfl | hm
+  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul]
+    rw [angleEigenPolynomial_coeff κ (n+1) (n+1),angleEigenCoefficient_top]
+    have he := angleRecurrenceCoefficient_sub hκ n
+    rw [mul_one]
+    exact sub_eq_zero.mpr he
+  rcases eq_or_lt_of_le (Nat.succ_le_iff.mpr hm) with hm | hm
+  · subst m
+    rw [show n+1+1=n+2 by omega]
+    simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul]
+    rw [angleEigenPolynomial_penultimate,
+      show n+3=(n+2)+1 by omega,angleEigenPolynomial_penultimate,
+      angleEigenPolynomial_coeff,angleEigenCoefficient_above κ (by omega)]
+    ring
+  rcases eq_or_lt_of_le (Nat.succ_le_iff.mpr hm) with hm | hm
+  · subst m
+    rw [show n+1+1+1=n+3 by omega,show n+3=(n+2)+1 by omega]
+    simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,angleEigenPolynomial_coeff]
+    rw [angleEigenCoefficient_top,angleEigenCoefficient_top,
+      angleEigenCoefficient_above κ (by omega)]
+    ring
+  · obtain ⟨k,rfl⟩ := Nat.exists_eq_add_of_le (by omega : 1 ≤ m)
+    rw [Nat.add_comm 1 k]
+    simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,angleEigenPolynomial_coeff]
+    rw [angleEigenCoefficient_above κ (by omega),
+      angleEigenCoefficient_above κ (by omega),angleEigenCoefficient_above κ (by omega)]
+    ring
+
+lemma angleEigenPolynomial_recurrence_zero {κ : ℝ} (hκ : 0<κ) :
+    X*angleEigenPolynomial κ 1-angleEigenPolynomial κ 2-
+      angleRecurrenceCoefficient κ 0 • angleEigenPolynomial κ 0=0 := by
+  ext m
+  rcases m with _|m
+  · simp only [coeff_sub,coeff_X_mul_zero,coeff_smul,smul_eq_mul,coeff_zero]
+    rw [angleEigenPolynomial_coeff κ 0 0,angleEigenCoefficient_top,
+      angleEigenPolynomial_second_coeff hκ 0,
+      angleRecurrenceCoefficient_zero]
+    norm_num
+    field_simp
+    ring
+  rcases m with _|m
+  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,coeff_zero]
+    rw [show 1=0+1 from rfl,angleEigenPolynomial_penultimate,
+      show 2=1+1 from rfl,angleEigenPolynomial_penultimate,
+      angleEigenPolynomial_coeff,angleEigenCoefficient_above κ (by omega)]
+    ring
+  rcases m with _|m
+  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,coeff_zero,
+      angleEigenPolynomial_coeff]
+    rw [angleEigenCoefficient_top,angleEigenCoefficient_top,
+      angleEigenCoefficient_above κ (by omega)]
+    ring
+  · simp only [coeff_sub,coeff_X_mul,coeff_smul,smul_eq_mul,coeff_zero,
+      angleEigenPolynomial_coeff]
+    rw [angleEigenCoefficient_above κ (by omega),
+      angleEigenCoefficient_above κ (by omega),angleEigenCoefficient_above κ (by omega)]
+    ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleSpectralBounds.lean b/lean/OAI/Probability/SLE/Angle/AngleSpectralBounds.lean
index 88fbb9348403466a70a8cfae89b111c08767ca46..ad81f94b0acc04cd61984bdfcf39b760931180b0 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleSpectralBounds.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleSpectralBounds.lean
@@ -1,74 +1,74 @@
-import OAI.Probability.SLE.Angle.AngleEigenNorm
-
-namespace OAI
-
-/-! Quantitative bounds on the normalized polynomial kernel terms. The
-coarse quadratic exponential envelope is enough at a fixed large time. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def angleSpectralFactor (κ : ℝ) (n : ℕ) (x y : ℝ) : ℝ :=
-  (angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n)*
-    (angleEigenPolynomial κ n).eval x*(angleEigenPolynomial κ n).eval y
-
-lemma angleSpectralFactor_zero {κ : ℝ} (hκ : 0<κ) (x y : ℝ) :
-    angleSpectralFactor κ 0 x y=1 := by
-  simp [angleSpectralFactor,angleEigenPolynomial_zero,(angleEigenSquareNorm_zero_pos hκ).ne']
-
-lemma angleSpectralFactor_bound {κ : ℝ} (hκ : 0<κ) (n : ℕ) {x y : ℝ}
-    (hx : |x|≤1) (hy : |y|≤1) :
-    |angleSpectralFactor κ n x y|≤(4*(κ+4)/κ)^n*((n:ℝ)+1)^(2*(n+1)) := by
-  have hc : 0<κ/(4*(κ+4)) := by positivity
-  have hn := angleEigenSquareNorm_pos hκ n
-  have h0 := angleEigenSquareNorm_zero_pos hκ
-  have hr : angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n≤
-      (4*(κ+4)/κ)^n := by
-    have he : (4*(κ+4)/κ)^n=(1/(κ/(4*(κ+4))))^n := by
-      congr 1
-      field_simp
-    rw [he,one_div_pow]
-    apply (div_le_div_iff₀ hn (pow_pos hc n)).mpr
-    simpa only [one_mul] using angleEigenSquareNorm_lower hκ n
-  unfold angleSpectralFactor
-  rw [abs_mul,abs_mul,abs_of_pos (div_pos h0 hn)]
-  calc
-    _ ≤ (4*(κ+4)/κ)^n*((n:ℝ)+1)^(n+1)*((n:ℝ)+1)^(n+1) :=
-      mul_le_mul (mul_le_mul hr (angleEigenPolynomial_eval_abs_le hκ n hx)
-        (abs_nonneg _) (by positivity)) (angleEigenPolynomial_eval_abs_le hκ n hy)
-          (abs_nonneg _) (by positivity)
-    _ = _ := by rw [mul_assoc,← pow_add]; congr 2; omega
-
-lemma angleSpectralFactor_exp_bound {κ : ℝ} (hκ : 0<κ) {n : ℕ} (hn : 1≤n)
-    {x y : ℝ} (hx : |x|≤1) (hy : |y|≤1) :
-    |angleSpectralFactor κ n x y|≤
-      Real.exp ((4+Real.log (4*(κ+4)/κ))*(n:ℝ)^2) := by
-  have hn0 : (0:ℝ)≤n := Nat.cast_nonneg n
-  have hn1 : (1:ℝ)≤n := by exact_mod_cast hn
-  have hb : 1≤4*(κ+4)/κ := (le_div_iff₀ hκ).mpr (by linarith)
-  have hb0 : 0<4*(κ+4)/κ := by positivity
-  have hl := Real.log_nonneg hb
-  have hnpos : 0<(n:ℝ)+1 := by positivity
-  have hlog : Real.log ((n:ℝ)+1)≤(n:ℝ) := by
-    simpa only [add_sub_cancel_right] using Real.log_le_sub_one_of_pos hnpos
-  calc
-    _ ≤ (4*(κ+4)/κ)^n*((n:ℝ)+1)^(2*(n+1)) := angleSpectralFactor_bound hκ n hx hy
-    _ = Real.exp ((n:ℝ)*Real.log (4*(κ+4)/κ)+
-        (2*((n:ℝ)+1))*Real.log ((n:ℝ)+1)) := by
-      rw [Real.exp_add,Real.exp_nat_mul,Real.exp_log hb0,
-        show 2*((n:ℝ)+1)=((2*(n+1):ℕ):ℝ) by push_cast; ring,
-        Real.exp_nat_mul,Real.exp_log hnpos]
-    _ ≤ _ := by
-      apply Real.exp_le_exp.mpr
-      have h1 : (2*((n:ℝ)+1))*Real.log ((n:ℝ)+1)≤(2*((n:ℝ)+1))*(n:ℝ) :=
-        mul_le_mul_of_nonneg_left hlog (by positivity)
-      have h2 : (n:ℝ)*Real.log (4*(κ+4)/κ)≤(n:ℝ)^2*Real.log (4*(κ+4)/κ) :=
-        mul_le_mul_of_nonneg_right (by nlinarith) hl
-      nlinarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenNorm
+
+namespace OAI
+
+/-! Quantitative bounds on the normalized polynomial kernel terms. The
+coarse quadratic exponential envelope is enough at a fixed large time. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def angleSpectralFactor (κ : ℝ) (n : ℕ) (x y : ℝ) : ℝ :=
+  (angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n)*
+    (angleEigenPolynomial κ n).eval x*(angleEigenPolynomial κ n).eval y
+
+lemma angleSpectralFactor_zero {κ : ℝ} (hκ : 0<κ) (x y : ℝ) :
+    angleSpectralFactor κ 0 x y=1 := by
+  simp [angleSpectralFactor,angleEigenPolynomial_zero,(angleEigenSquareNorm_zero_pos hκ).ne']
+
+lemma angleSpectralFactor_bound {κ : ℝ} (hκ : 0<κ) (n : ℕ) {x y : ℝ}
+    (hx : |x|≤1) (hy : |y|≤1) :
+    |angleSpectralFactor κ n x y|≤(4*(κ+4)/κ)^n*((n:ℝ)+1)^(2*(n+1)) := by
+  have hc : 0<κ/(4*(κ+4)) := by positivity
+  have hn := angleEigenSquareNorm_pos hκ n
+  have h0 := angleEigenSquareNorm_zero_pos hκ
+  have hr : angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n≤
+      (4*(κ+4)/κ)^n := by
+    have he : (4*(κ+4)/κ)^n=(1/(κ/(4*(κ+4))))^n := by
+      congr 1
+      field_simp
+    rw [he,one_div_pow]
+    apply (div_le_div_iff₀ hn (pow_pos hc n)).mpr
+    simpa only [one_mul] using angleEigenSquareNorm_lower hκ n
+  unfold angleSpectralFactor
+  rw [abs_mul,abs_mul,abs_of_pos (div_pos h0 hn)]
+  calc
+    _ ≤ (4*(κ+4)/κ)^n*((n:ℝ)+1)^(n+1)*((n:ℝ)+1)^(n+1) :=
+      mul_le_mul (mul_le_mul hr (angleEigenPolynomial_eval_abs_le hκ n hx)
+        (abs_nonneg _) (by positivity)) (angleEigenPolynomial_eval_abs_le hκ n hy)
+          (abs_nonneg _) (by positivity)
+    _ = _ := by rw [mul_assoc,← pow_add]; congr 2; omega
+
+lemma angleSpectralFactor_exp_bound {κ : ℝ} (hκ : 0<κ) {n : ℕ} (hn : 1≤n)
+    {x y : ℝ} (hx : |x|≤1) (hy : |y|≤1) :
+    |angleSpectralFactor κ n x y|≤
+      Real.exp ((4+Real.log (4*(κ+4)/κ))*(n:ℝ)^2) := by
+  have hn0 : (0:ℝ)≤n := Nat.cast_nonneg n
+  have hn1 : (1:ℝ)≤n := by exact_mod_cast hn
+  have hb : 1≤4*(κ+4)/κ := (le_div_iff₀ hκ).mpr (by linarith)
+  have hb0 : 0<4*(κ+4)/κ := by positivity
+  have hl := Real.log_nonneg hb
+  have hnpos : 0<(n:ℝ)+1 := by positivity
+  have hlog : Real.log ((n:ℝ)+1)≤(n:ℝ) := by
+    simpa only [add_sub_cancel_right] using Real.log_le_sub_one_of_pos hnpos
+  calc
+    _ ≤ (4*(κ+4)/κ)^n*((n:ℝ)+1)^(2*(n+1)) := angleSpectralFactor_bound hκ n hx hy
+    _ = Real.exp ((n:ℝ)*Real.log (4*(κ+4)/κ)+
+        (2*((n:ℝ)+1))*Real.log ((n:ℝ)+1)) := by
+      rw [Real.exp_add,Real.exp_nat_mul,Real.exp_log hb0,
+        show 2*((n:ℝ)+1)=((2*(n+1):ℕ):ℝ) by push_cast; ring,
+        Real.exp_nat_mul,Real.exp_log hnpos]
+    _ ≤ _ := by
+      apply Real.exp_le_exp.mpr
+      have h1 : (2*((n:ℝ)+1))*Real.log ((n:ℝ)+1)≤(2*((n:ℝ)+1))*(n:ℝ) :=
+        mul_le_mul_of_nonneg_left hlog (by positivity)
+      have h2 : (n:ℝ)*Real.log (4*(κ+4)/κ)≤(n:ℝ)^2*Real.log (4*(κ+4)/κ) :=
+        mul_le_mul_of_nonneg_right (by nlinarith) hl
+      nlinarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleSpectralKernel.lean b/lean/OAI/Probability/SLE/Angle/AngleSpectralKernel.lean
index aec14b4a7a8aa5c845297937cbd1d18d6edcd8a3..44a80eae8b31c0f3cbd2604b6cb8cf3d8ebc0599 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleSpectralKernel.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleSpectralKernel.lean
@@ -1,98 +1,98 @@
-import OAI.Probability.SLE.Angle.AngleSpectralBounds
-
-namespace OAI
-
-/-! An explicit large-time uniformly positive polynomial spectral kernel.
-Identification with the actual stopped-angle transition is separate. -/
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge
-
-def angleSpectralMixingTime (κ : ℝ) : ℝ :=
-  (8/κ)*(4+Real.log (4*(κ+4)/κ))+2
-
-lemma angleSpectralMixingTime_pos {κ : ℝ} (hκ : 0<κ) :
-    0<angleSpectralMixingTime κ := by
-  have hb : 1≤4*(κ+4)/κ := (le_div_iff₀ hκ).mpr (by linarith)
-  have hl := Real.log_nonneg hb
-  unfold angleSpectralMixingTime
-  positivity
-
-lemma angleSpectralTerm_bound {κ u : ℝ} (hκ : 0<κ) (hu : angleSpectralMixingTime κ≤u)
-    {n : ℕ} (hn : 1≤n) {x y : ℝ} (hx : |x|≤1) (hy : |y|≤1) :
-    |Real.exp (-u*anglePolynomialEigenvalue κ n)*angleSpectralFactor κ n x y|≤
-      (1/3:ℝ)^n := by
-  let C := 4+Real.log (4*(κ+4)/κ)
-  have hb : 1≤4*(κ+4)/κ := (le_div_iff₀ hκ).mpr (by linarith)
-  have hC : 0≤C := by dsimp [C]; linarith [Real.log_nonneg hb]
-  have hn0 : (0:ℝ)≤n := Nat.cast_nonneg n
-  have heig : 0≤anglePolynomialEigenvalue κ n := by unfold anglePolynomialEigenvalue; positivity
-  have he : angleSpectralMixingTime κ*anglePolynomialEigenvalue κ n =
-      C*(n:ℝ)^2+2*(n:ℝ)+(8/κ)*C*(n:ℝ)+(κ/4)*(n:ℝ)^2 := by
-    unfold angleSpectralMixingTime anglePolynomialEigenvalue
-    dsimp [C]
-    field_simp
-    ring
-  have hexp : -u*anglePolynomialEigenvalue κ n+C*(n:ℝ)^2≤-2*(n:ℝ) := by
-    have hh := mul_le_mul_of_nonneg_right hu heig
-    rw [he] at hh
-    have hz : 0≤(8/κ)*C*(n:ℝ)+(κ/4)*(n:ℝ)^2 := by positivity
-    linarith
-  have he2 : Real.exp (-2)≤(1/3:ℝ) := by
-    rw [Real.exp_neg,inv_eq_one_div]
-    exact (div_le_iff₀ (Real.exp_pos 2)).mpr (by nlinarith [Real.add_one_le_exp (2:ℝ)])
-  calc
-    _ = Real.exp (-u*anglePolynomialEigenvalue κ n)*|angleSpectralFactor κ n x y| := by
-      rw [abs_mul,abs_of_pos (Real.exp_pos _)]
-    _ ≤ Real.exp (-u*anglePolynomialEigenvalue κ n)*Real.exp (C*(n:ℝ)^2) :=
-      mul_le_mul_of_nonneg_left (angleSpectralFactor_exp_bound hκ hn hx hy) (Real.exp_pos _).le
-    _ = Real.exp (-u*anglePolynomialEigenvalue κ n+C*(n:ℝ)^2) := (Real.exp_add _ _).symm
-    _ ≤ Real.exp (-2*(n:ℝ)) := Real.exp_le_exp.mpr hexp
-    _ = Real.exp (-2)^n := by rw [mul_comm,Real.exp_nat_mul]
-    _ ≤ _ := pow_le_pow_left₀ (Real.exp_pos _).le he2 n
-
-def angleSpectralKernel (κ u θ φ : ℝ) : ℝ :=
-  1+∑' n : ℕ, Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
-    angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)
-
-lemma angleSpectralKernel_summable {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ φ : ℝ) :
-    Summable (fun n : ℕ => Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
-      angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)) := by
-  have hs : Summable (fun n : ℕ => (1/3:ℝ)^(n+1)) := by
-    simpa only [pow_succ] using
-      (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3) (by norm_num : (1/3:ℝ)<1)).mul_right (1/3)
-  apply hs.of_norm_bounded
-  intro n
-  simpa only [Real.norm_eq_abs] using angleSpectralTerm_bound hκ hu (by omega : 1≤n+1)
-    (Real.abs_cos_le_one θ) (Real.abs_cos_le_one φ)
-
-lemma angleSpectralKernel_bounds {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ φ : ℝ) :
-    1/2≤angleSpectralKernel κ u θ φ ∧ angleSpectralKernel κ u θ φ≤2 := by
-  have hs : Summable (fun n : ℕ => (1/3:ℝ)^(n+1)) := by
-    simpa only [pow_succ] using
-      (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3) (by norm_num : (1/3:ℝ)<1)).mul_right (1/3)
-  have hsum : (∑' n : ℕ, (1/3:ℝ)^(n+1))=1/2 := by
-    simp_rw [pow_succ]
-    rw [tsum_mul_right,tsum_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
-      (by norm_num : (1/3:ℝ)<1)]
-    norm_num
-  have he : |∑' n : ℕ, Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
-      angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)|≤1/2 := by
-    rw [← hsum]
-    have hbnd (n : ℕ) : ‖Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
-        angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)‖≤(1/3:ℝ)^(n+1) := by
-      simpa only [Real.norm_eq_abs] using angleSpectralTerm_bound hκ hu (by omega : 1≤n+1)
-        (Real.abs_cos_le_one θ) (Real.abs_cos_le_one φ)
-    have hsn := hs.of_nonneg_of_le (fun n => norm_nonneg _) hbnd
-    exact (norm_tsum_le_tsum_norm hsn).trans (hsn.tsum_le_tsum hbnd hs)
-  unfold angleSpectralKernel
-  constructor <;> linarith [(abs_le.mp he).1,(abs_le.mp he).2]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleSpectralBounds
+
+namespace OAI
+
+/-! An explicit large-time uniformly positive polynomial spectral kernel.
+Identification with the actual stopped-angle transition is separate. -/
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge
+
+def angleSpectralMixingTime (κ : ℝ) : ℝ :=
+  (8/κ)*(4+Real.log (4*(κ+4)/κ))+2
+
+lemma angleSpectralMixingTime_pos {κ : ℝ} (hκ : 0<κ) :
+    0<angleSpectralMixingTime κ := by
+  have hb : 1≤4*(κ+4)/κ := (le_div_iff₀ hκ).mpr (by linarith)
+  have hl := Real.log_nonneg hb
+  unfold angleSpectralMixingTime
+  positivity
+
+lemma angleSpectralTerm_bound {κ u : ℝ} (hκ : 0<κ) (hu : angleSpectralMixingTime κ≤u)
+    {n : ℕ} (hn : 1≤n) {x y : ℝ} (hx : |x|≤1) (hy : |y|≤1) :
+    |Real.exp (-u*anglePolynomialEigenvalue κ n)*angleSpectralFactor κ n x y|≤
+      (1/3:ℝ)^n := by
+  let C := 4+Real.log (4*(κ+4)/κ)
+  have hb : 1≤4*(κ+4)/κ := (le_div_iff₀ hκ).mpr (by linarith)
+  have hC : 0≤C := by dsimp [C]; linarith [Real.log_nonneg hb]
+  have hn0 : (0:ℝ)≤n := Nat.cast_nonneg n
+  have heig : 0≤anglePolynomialEigenvalue κ n := by unfold anglePolynomialEigenvalue; positivity
+  have he : angleSpectralMixingTime κ*anglePolynomialEigenvalue κ n =
+      C*(n:ℝ)^2+2*(n:ℝ)+(8/κ)*C*(n:ℝ)+(κ/4)*(n:ℝ)^2 := by
+    unfold angleSpectralMixingTime anglePolynomialEigenvalue
+    dsimp [C]
+    field_simp
+    ring
+  have hexp : -u*anglePolynomialEigenvalue κ n+C*(n:ℝ)^2≤-2*(n:ℝ) := by
+    have hh := mul_le_mul_of_nonneg_right hu heig
+    rw [he] at hh
+    have hz : 0≤(8/κ)*C*(n:ℝ)+(κ/4)*(n:ℝ)^2 := by positivity
+    linarith
+  have he2 : Real.exp (-2)≤(1/3:ℝ) := by
+    rw [Real.exp_neg,inv_eq_one_div]
+    exact (div_le_iff₀ (Real.exp_pos 2)).mpr (by nlinarith [Real.add_one_le_exp (2:ℝ)])
+  calc
+    _ = Real.exp (-u*anglePolynomialEigenvalue κ n)*|angleSpectralFactor κ n x y| := by
+      rw [abs_mul,abs_of_pos (Real.exp_pos _)]
+    _ ≤ Real.exp (-u*anglePolynomialEigenvalue κ n)*Real.exp (C*(n:ℝ)^2) :=
+      mul_le_mul_of_nonneg_left (angleSpectralFactor_exp_bound hκ hn hx hy) (Real.exp_pos _).le
+    _ = Real.exp (-u*anglePolynomialEigenvalue κ n+C*(n:ℝ)^2) := (Real.exp_add _ _).symm
+    _ ≤ Real.exp (-2*(n:ℝ)) := Real.exp_le_exp.mpr hexp
+    _ = Real.exp (-2)^n := by rw [mul_comm,Real.exp_nat_mul]
+    _ ≤ _ := pow_le_pow_left₀ (Real.exp_pos _).le he2 n
+
+def angleSpectralKernel (κ u θ φ : ℝ) : ℝ :=
+  1+∑' n : ℕ, Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
+    angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)
+
+lemma angleSpectralKernel_summable {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ φ : ℝ) :
+    Summable (fun n : ℕ => Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
+      angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)) := by
+  have hs : Summable (fun n : ℕ => (1/3:ℝ)^(n+1)) := by
+    simpa only [pow_succ] using
+      (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3) (by norm_num : (1/3:ℝ)<1)).mul_right (1/3)
+  apply hs.of_norm_bounded
+  intro n
+  simpa only [Real.norm_eq_abs] using angleSpectralTerm_bound hκ hu (by omega : 1≤n+1)
+    (Real.abs_cos_le_one θ) (Real.abs_cos_le_one φ)
+
+lemma angleSpectralKernel_bounds {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ φ : ℝ) :
+    1/2≤angleSpectralKernel κ u θ φ ∧ angleSpectralKernel κ u θ φ≤2 := by
+  have hs : Summable (fun n : ℕ => (1/3:ℝ)^(n+1)) := by
+    simpa only [pow_succ] using
+      (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3) (by norm_num : (1/3:ℝ)<1)).mul_right (1/3)
+  have hsum : (∑' n : ℕ, (1/3:ℝ)^(n+1))=1/2 := by
+    simp_rw [pow_succ]
+    rw [tsum_mul_right,tsum_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
+      (by norm_num : (1/3:ℝ)<1)]
+    norm_num
+  have he : |∑' n : ℕ, Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
+      angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)|≤1/2 := by
+    rw [← hsum]
+    have hbnd (n : ℕ) : ‖Real.exp (-u*anglePolynomialEigenvalue κ (n+1))*
+        angleSpectralFactor κ (n+1) (Real.cos θ) (Real.cos φ)‖≤(1/3:ℝ)^(n+1) := by
+      simpa only [Real.norm_eq_abs] using angleSpectralTerm_bound hκ hu (by omega : 1≤n+1)
+        (Real.abs_cos_le_one θ) (Real.abs_cos_le_one φ)
+    have hsn := hs.of_nonneg_of_le (fun n => norm_nonneg _) hbnd
+    exact (norm_tsum_le_tsum_norm hsn).trans (hsn.tsum_le_tsum hbnd hs)
+  unfold angleSpectralKernel
+  constructor <;> linarith [(abs_le.mp he).1,(abs_le.mp he).2]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleSpectralLaw.lean b/lean/OAI/Probability/SLE/Angle/AngleSpectralLaw.lean
index a3bb1c7cdcec3c66f2552a3ec8caf0c13c85bb60..717599f6e516a87944f32f6b9cc4dc1339a114e1 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleSpectralLaw.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleSpectralLaw.lean
@@ -1,74 +1,74 @@
-import OAI.Probability.SLE.Angle.AngleSpectralMoments
-
-namespace OAI
-
-/-! The explicitly positive spectral density defines a probability law
-with the required polynomial moments. -/
-noncomputable section
-open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge
-
-def angleSpectralLaw (κ u θ : ℝ) : Measure ℝ :=
-  (Target.angleLaw κ).withDensity (fun φ => ENNReal.ofReal (angleSpectralKernel κ u θ φ))
-
-lemma angleSpectralKernel_integrable {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
-    Integrable (angleSpectralKernel κ u θ) (Target.angleLaw κ) := by
-  have : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
-  apply Integrable.of_bound (angleSpectralKernel_measurable hκ hu θ).aestronglyMeasurable 2
-  apply Filter.Eventually.of_forall
-  intro φ
-  have hb := angleSpectralKernel_bounds hκ hu θ φ
-  simpa only [Real.norm_eq_abs,abs_of_nonneg (by linarith : 0≤angleSpectralKernel κ u θ φ)] using hb.2
-
-lemma angleSpectralKernel_mean {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
-    (∫ φ, angleSpectralKernel κ u θ φ ∂Target.angleLaw κ)=1 := by
-  simpa only [angleEigenPolynomial_zero,anglePolynomialTest,eval_one,mul_one,
-    anglePolynomialEigenvalue,Nat.cast_zero,mul_zero,zero_pow (by decide : 2≠0),
-    add_zero,neg_mul,Real.exp_zero] using angleSpectralKernel_pairing hκ hu θ 0
-
-lemma angleSpectralLaw_isProbability {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
-    IsProbabilityMeasure (angleSpectralLaw κ u θ) := by
-  constructor
-  rw [angleSpectralLaw,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,
-    ← ofReal_integral_eq_lintegral_ofReal (angleSpectralKernel_integrable hκ hu θ)
-      (Filter.Eventually.of_forall (fun φ => by
-        change (0:ℝ)≤angleSpectralKernel κ u θ φ
-        have hb := (angleSpectralKernel_bounds hκ hu θ φ).1
-        linarith)),angleSpectralKernel_mean hκ hu θ]
-  exact ENNReal.ofReal_one
-
-lemma angleSpectralLaw_integral {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (f : ℝ → ℝ) :
-    (∫ φ, f φ ∂angleSpectralLaw κ u θ)=
-      ∫ φ, angleSpectralKernel κ u θ φ*f φ ∂Target.angleLaw κ := by
-  rw [angleSpectralLaw,integral_withDensity_eq_integral_toReal_smul
-    (angleSpectralKernel_measurable hκ hu θ).ennreal_ofReal
-    (Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
-  apply integral_congr_ae
-  apply Filter.Eventually.of_forall
-  intro φ
-  dsimp only
-  rw [ENNReal.toReal_ofReal (by have hb := (angleSpectralKernel_bounds hκ hu θ φ).1; linarith)]
-  rfl
-
-lemma angleSpectralLaw_eigen_moment {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (n : ℕ) :
-    (∫ φ, anglePolynomialTest (angleEigenPolynomial κ n) φ ∂angleSpectralLaw κ u θ)=
-      Real.exp (-u*anglePolynomialEigenvalue κ n)*anglePolynomialTest (angleEigenPolynomial κ n) θ := by
-  rw [angleSpectralLaw_integral hκ hu θ]
-  exact angleSpectralKernel_pairing hκ hu θ n
-
-lemma angleSpectralLaw_ae_mem {κ u : ℝ} (θ : ℝ) :
-    ∀ᵐ φ ∂angleSpectralLaw κ u θ, φ∈Ioo (0:ℝ) Real.pi :=
-  (Target.angleLaw_ae_mem κ).filter_mono (withDensity_absolutelyContinuous _ _).ae_le
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleSpectralMoments
+
+namespace OAI
+
+/-! The explicitly positive spectral density defines a probability law
+with the required polynomial moments. -/
+noncomputable section
+open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge
+
+def angleSpectralLaw (κ u θ : ℝ) : Measure ℝ :=
+  (Target.angleLaw κ).withDensity (fun φ => ENNReal.ofReal (angleSpectralKernel κ u θ φ))
+
+lemma angleSpectralKernel_integrable {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
+    Integrable (angleSpectralKernel κ u θ) (Target.angleLaw κ) := by
+  have : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
+  apply Integrable.of_bound (angleSpectralKernel_measurable hκ hu θ).aestronglyMeasurable 2
+  apply Filter.Eventually.of_forall
+  intro φ
+  have hb := angleSpectralKernel_bounds hκ hu θ φ
+  simpa only [Real.norm_eq_abs,abs_of_nonneg (by linarith : 0≤angleSpectralKernel κ u θ φ)] using hb.2
+
+lemma angleSpectralKernel_mean {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
+    (∫ φ, angleSpectralKernel κ u θ φ ∂Target.angleLaw κ)=1 := by
+  simpa only [angleEigenPolynomial_zero,anglePolynomialTest,eval_one,mul_one,
+    anglePolynomialEigenvalue,Nat.cast_zero,mul_zero,zero_pow (by decide : 2≠0),
+    add_zero,neg_mul,Real.exp_zero] using angleSpectralKernel_pairing hκ hu θ 0
+
+lemma angleSpectralLaw_isProbability {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
+    IsProbabilityMeasure (angleSpectralLaw κ u θ) := by
+  constructor
+  rw [angleSpectralLaw,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,
+    ← ofReal_integral_eq_lintegral_ofReal (angleSpectralKernel_integrable hκ hu θ)
+      (Filter.Eventually.of_forall (fun φ => by
+        change (0:ℝ)≤angleSpectralKernel κ u θ φ
+        have hb := (angleSpectralKernel_bounds hκ hu θ φ).1
+        linarith)),angleSpectralKernel_mean hκ hu θ]
+  exact ENNReal.ofReal_one
+
+lemma angleSpectralLaw_integral {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (f : ℝ → ℝ) :
+    (∫ φ, f φ ∂angleSpectralLaw κ u θ)=
+      ∫ φ, angleSpectralKernel κ u θ φ*f φ ∂Target.angleLaw κ := by
+  rw [angleSpectralLaw,integral_withDensity_eq_integral_toReal_smul
+    (angleSpectralKernel_measurable hκ hu θ).ennreal_ofReal
+    (Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
+  apply integral_congr_ae
+  apply Filter.Eventually.of_forall
+  intro φ
+  dsimp only
+  rw [ENNReal.toReal_ofReal (by have hb := (angleSpectralKernel_bounds hκ hu θ φ).1; linarith)]
+  rfl
+
+lemma angleSpectralLaw_eigen_moment {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (n : ℕ) :
+    (∫ φ, anglePolynomialTest (angleEigenPolynomial κ n) φ ∂angleSpectralLaw κ u θ)=
+      Real.exp (-u*anglePolynomialEigenvalue κ n)*anglePolynomialTest (angleEigenPolynomial κ n) θ := by
+  rw [angleSpectralLaw_integral hκ hu θ]
+  exact angleSpectralKernel_pairing hκ hu θ n
+
+lemma angleSpectralLaw_ae_mem {κ u : ℝ} (θ : ℝ) :
+    ∀ᵐ φ ∂angleSpectralLaw κ u θ, φ∈Ioo (0:ℝ) Real.pi :=
+  (Target.angleLaw_ae_mem κ).filter_mono (withDensity_absolutelyContinuous _ _).ae_le
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleSpectralMoments.lean b/lean/OAI/Probability/SLE/Angle/AngleSpectralMoments.lean
index 163b1e45922e5ad250c51f03de953f421ccc84a9..9c932432a2a4f32a17f763e9db834ea2040ddeef 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleSpectralMoments.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleSpectralMoments.lean
@@ -1,113 +1,113 @@
-import OAI.Probability.SLE.Angle.AngleSpectralPairing
-import OAI.Probability.SLE.Angle.AngleSpectralKernel
-
-namespace OAI
-
-/-! The positive spectral kernel has exactly the eigenfunction moments
-required of the centered angular transition. -/
-noncomputable section
-open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-
-namespace SLEExactGauge
-
-def angleSpectralTerm (κ u : ℝ) (n : ℕ) (θ φ : ℝ) : ℝ :=
-  Real.exp (-u*anglePolynomialEigenvalue κ n)*
-    angleSpectralFactor κ n (Real.cos θ) (Real.cos φ)
-
-lemma angleSpectralTerm_measurable (κ u : ℝ) (n : ℕ) (θ : ℝ) :
-    Measurable (angleSpectralTerm κ u n θ) := by
-  unfold angleSpectralTerm angleSpectralFactor
-  fun_prop
-
-lemma angleSpectralTerm_zero {κ : ℝ} (hκ : 0<κ) (u θ φ : ℝ) :
-    angleSpectralTerm κ u 0 θ φ=1 := by
-  simp [angleSpectralTerm,anglePolynomialEigenvalue,angleSpectralFactor_zero hκ]
-
-lemma angleSpectralTerm_bound_all {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (n : ℕ) (θ φ : ℝ) :
-    ‖angleSpectralTerm κ u n θ φ‖≤(1/3:ℝ)^n := by
-  cases n with
-  | zero => simp [angleSpectralTerm_zero hκ]
-  | succ n =>
-    exact angleSpectralTerm_bound hκ hu (by omega)
-      (Real.abs_cos_le_one θ) (Real.abs_cos_le_one φ)
-
-lemma angleSpectralKernel_eq_tsum {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ φ : ℝ) :
-    angleSpectralKernel κ u θ φ=∑' n : ℕ, angleSpectralTerm κ u n θ φ := by
-  have hs : Summable (fun n : ℕ => angleSpectralTerm κ u n θ φ) :=
-    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
-      (by norm_num : (1/3:ℝ)<1)).of_norm_bounded
-        (fun n => angleSpectralTerm_bound_all hκ hu n θ φ)
-  calc
-    _ = angleSpectralTerm κ u 0 θ φ+∑' n : ℕ, angleSpectralTerm κ u (n+1) θ φ := by
-      rw [angleSpectralTerm_zero hκ]
-      rfl
-    _ = _ := hs.tsum_eq_zero_add.symm
-
-lemma angleSpectralKernel_measurable {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
-    Measurable (angleSpectralKernel κ u θ) := by
-  have he : angleSpectralKernel κ u θ=(fun φ => ∑' n : ℕ, angleSpectralTerm κ u n θ φ) :=
-    funext (angleSpectralKernel_eq_tsum hκ hu θ)
-  rw [he]
-  exact Measurable.tsum (fun n => angleSpectralTerm_measurable κ u n θ)
-
-lemma angleSpectralTerm_pairing {κ : ℝ} (hκ : 0<κ) (u θ : ℝ) (n m : ℕ) :
-    (∫ φ, angleSpectralTerm κ u n θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ
-      ∂Target.angleLaw κ)=
-      if n=m then Real.exp (-u*anglePolynomialEigenvalue κ m)*
-        anglePolynomialTest (angleEigenPolynomial κ m) θ else 0 := by
-  unfold angleSpectralTerm
-  simp_rw [mul_assoc]
-  rw [integral_const_mul]
-  rw [angleSpectralFactor_pairing hκ]
-  split_ifs with h
-  · subst m
-    rfl
-  · exact mul_zero _
-
-lemma angleSpectralKernel_pairing {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (m : ℕ) :
-    (∫ φ, angleSpectralKernel κ u θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ
-      ∂Target.angleLaw κ)=Real.exp (-u*anglePolynomialEigenvalue κ m)*
-        anglePolynomialTest (angleEigenPolynomial κ m) θ := by
-  have : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
-  obtain ⟨M,hM⟩ := anglePolynomialTest_bound (angleEigenPolynomial κ m)
-  have hM0 : 0≤M := (norm_nonneg _).trans (hM 0)
-  let F := fun n φ => angleSpectralTerm κ u n θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ
-  have hFm (n : ℕ) : AEStronglyMeasurable (F n) (Target.angleLaw κ) :=
-    ((angleSpectralTerm_measurable κ u n θ).mul
-      ((angleEigenPolynomial κ m).continuous.comp Real.continuous_cos).measurable).aestronglyMeasurable
-  have hb (n : ℕ) (φ : ℝ) : ‖F n φ‖≤(1/3:ℝ)^n*M := by
-    dsimp [F]
-    rw [abs_mul]
-    exact mul_le_mul (angleSpectralTerm_bound_all hκ hu n θ φ) (hM φ)
-      (abs_nonneg _) (by positivity)
-  have hs : Summable (fun n : ℕ => (1/3:ℝ)^n*M) :=
-    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
-      (by norm_num : (1/3:ℝ)<1)).mul_right M
-  have hi : Integrable (fun _φ : ℝ => ∑' n : ℕ, (1/3:ℝ)^n*M) (Target.angleLaw κ) :=
-    integrable_const _
-  have hsum := hasSum_integral_of_dominated_convergence (fun (n : ℕ) (_φ : ℝ) => (1/3:ℝ)^n*M)
-    hFm (fun n => Filter.Eventually.of_forall (hb n))
-    (Filter.Eventually.of_forall (fun _φ => hs)) hi
-    (show ∀ᵐ φ ∂Target.angleLaw κ, HasSum (fun n => F n φ)
-      (angleSpectralKernel κ u θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ) from by
-      apply Filter.Eventually.of_forall
-      intro φ
-      have hs' : Summable (fun n => angleSpectralTerm κ u n θ φ) :=
-        (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
-          (by norm_num : (1/3:ℝ)<1)).of_norm_bounded
-            (fun n => angleSpectralTerm_bound_all hκ hu n θ φ)
-      simpa only [← angleSpectralKernel_eq_tsum hκ hu] using
-        hs'.hasSum.mul_right (anglePolynomialTest (angleEigenPolynomial κ m) φ))
-  rw [← hsum.tsum_eq]
-  simp only [F,angleSpectralTerm_pairing hκ]
-  simp
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleSpectralPairing
+import OAI.Probability.SLE.Angle.AngleSpectralKernel
+
+namespace OAI
+
+/-! The positive spectral kernel has exactly the eigenfunction moments
+required of the centered angular transition. -/
+noncomputable section
+open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+
+namespace SLEExactGauge
+
+def angleSpectralTerm (κ u : ℝ) (n : ℕ) (θ φ : ℝ) : ℝ :=
+  Real.exp (-u*anglePolynomialEigenvalue κ n)*
+    angleSpectralFactor κ n (Real.cos θ) (Real.cos φ)
+
+lemma angleSpectralTerm_measurable (κ u : ℝ) (n : ℕ) (θ : ℝ) :
+    Measurable (angleSpectralTerm κ u n θ) := by
+  unfold angleSpectralTerm angleSpectralFactor
+  fun_prop
+
+lemma angleSpectralTerm_zero {κ : ℝ} (hκ : 0<κ) (u θ φ : ℝ) :
+    angleSpectralTerm κ u 0 θ φ=1 := by
+  simp [angleSpectralTerm,anglePolynomialEigenvalue,angleSpectralFactor_zero hκ]
+
+lemma angleSpectralTerm_bound_all {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (n : ℕ) (θ φ : ℝ) :
+    ‖angleSpectralTerm κ u n θ φ‖≤(1/3:ℝ)^n := by
+  cases n with
+  | zero => simp [angleSpectralTerm_zero hκ]
+  | succ n =>
+    exact angleSpectralTerm_bound hκ hu (by omega)
+      (Real.abs_cos_le_one θ) (Real.abs_cos_le_one φ)
+
+lemma angleSpectralKernel_eq_tsum {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ φ : ℝ) :
+    angleSpectralKernel κ u θ φ=∑' n : ℕ, angleSpectralTerm κ u n θ φ := by
+  have hs : Summable (fun n : ℕ => angleSpectralTerm κ u n θ φ) :=
+    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
+      (by norm_num : (1/3:ℝ)<1)).of_norm_bounded
+        (fun n => angleSpectralTerm_bound_all hκ hu n θ φ)
+  calc
+    _ = angleSpectralTerm κ u 0 θ φ+∑' n : ℕ, angleSpectralTerm κ u (n+1) θ φ := by
+      rw [angleSpectralTerm_zero hκ]
+      rfl
+    _ = _ := hs.tsum_eq_zero_add.symm
+
+lemma angleSpectralKernel_measurable {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) :
+    Measurable (angleSpectralKernel κ u θ) := by
+  have he : angleSpectralKernel κ u θ=(fun φ => ∑' n : ℕ, angleSpectralTerm κ u n θ φ) :=
+    funext (angleSpectralKernel_eq_tsum hκ hu θ)
+  rw [he]
+  exact Measurable.tsum (fun n => angleSpectralTerm_measurable κ u n θ)
+
+lemma angleSpectralTerm_pairing {κ : ℝ} (hκ : 0<κ) (u θ : ℝ) (n m : ℕ) :
+    (∫ φ, angleSpectralTerm κ u n θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ
+      ∂Target.angleLaw κ)=
+      if n=m then Real.exp (-u*anglePolynomialEigenvalue κ m)*
+        anglePolynomialTest (angleEigenPolynomial κ m) θ else 0 := by
+  unfold angleSpectralTerm
+  simp_rw [mul_assoc]
+  rw [integral_const_mul]
+  rw [angleSpectralFactor_pairing hκ]
+  split_ifs with h
+  · subst m
+    rfl
+  · exact mul_zero _
+
+lemma angleSpectralKernel_pairing {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (m : ℕ) :
+    (∫ φ, angleSpectralKernel κ u θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ
+      ∂Target.angleLaw κ)=Real.exp (-u*anglePolynomialEigenvalue κ m)*
+        anglePolynomialTest (angleEigenPolynomial κ m) θ := by
+  have : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
+  obtain ⟨M,hM⟩ := anglePolynomialTest_bound (angleEigenPolynomial κ m)
+  have hM0 : 0≤M := (norm_nonneg _).trans (hM 0)
+  let F := fun n φ => angleSpectralTerm κ u n θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ
+  have hFm (n : ℕ) : AEStronglyMeasurable (F n) (Target.angleLaw κ) :=
+    ((angleSpectralTerm_measurable κ u n θ).mul
+      ((angleEigenPolynomial κ m).continuous.comp Real.continuous_cos).measurable).aestronglyMeasurable
+  have hb (n : ℕ) (φ : ℝ) : ‖F n φ‖≤(1/3:ℝ)^n*M := by
+    dsimp [F]
+    rw [abs_mul]
+    exact mul_le_mul (angleSpectralTerm_bound_all hκ hu n θ φ) (hM φ)
+      (abs_nonneg _) (by positivity)
+  have hs : Summable (fun n : ℕ => (1/3:ℝ)^n*M) :=
+    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
+      (by norm_num : (1/3:ℝ)<1)).mul_right M
+  have hi : Integrable (fun _φ : ℝ => ∑' n : ℕ, (1/3:ℝ)^n*M) (Target.angleLaw κ) :=
+    integrable_const _
+  have hsum := hasSum_integral_of_dominated_convergence (fun (n : ℕ) (_φ : ℝ) => (1/3:ℝ)^n*M)
+    hFm (fun n => Filter.Eventually.of_forall (hb n))
+    (Filter.Eventually.of_forall (fun _φ => hs)) hi
+    (show ∀ᵐ φ ∂Target.angleLaw κ, HasSum (fun n => F n φ)
+      (angleSpectralKernel κ u θ φ*anglePolynomialTest (angleEigenPolynomial κ m) φ) from by
+      apply Filter.Eventually.of_forall
+      intro φ
+      have hs' : Summable (fun n => angleSpectralTerm κ u n θ φ) :=
+        (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/3)
+          (by norm_num : (1/3:ℝ)<1)).of_norm_bounded
+            (fun n => angleSpectralTerm_bound_all hκ hu n θ φ)
+      simpa only [← angleSpectralKernel_eq_tsum hκ hu] using
+        hs'.hasSum.mul_right (anglePolynomialTest (angleEigenPolynomial κ m) φ))
+  rw [← hsum.tsum_eq]
+  simp only [F,angleSpectralTerm_pairing hκ]
+  simp
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleSpectralPairing.lean b/lean/OAI/Probability/SLE/Angle/AngleSpectralPairing.lean
index ad98a201051a932b98faf2ef3826d458e01fd30e..0b864498adfb504608b47eb169bc0143c5e42041 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleSpectralPairing.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleSpectralPairing.lean
@@ -1,80 +1,80 @@
-import OAI.Probability.SLE.Angle.AngleStationaryPairing
-import OAI.Probability.SLE.Angle.AngleSpectralBounds
-
-namespace OAI
-
-/-! Exact stationary moments of the normalized polynomial kernel terms. -/
-noncomputable section
-open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-
-namespace SLEExactGauge
-
-lemma anglePolynomialTest_bound (p : ℝ[X]) :
-    ∃ M : ℝ, ∀ θ : ℝ, ‖anglePolynomialTest p θ‖≤M := by
-  obtain ⟨M,hM⟩ := (isCompact_Icc : IsCompact (Icc (-1:ℝ) 1)).exists_bound_of_continuousOn
-    p.continuous.continuousOn
-  exact ⟨M,fun θ => hM (Real.cos θ) ⟨Real.neg_one_le_cos θ,Real.cos_le_one θ⟩⟩
-
-lemma anglePolynomialTest_integrable (μ : Measure ℝ) [IsFiniteMeasure μ] (p : ℝ[X]) :
-    Integrable (anglePolynomialTest p) μ := by
-  obtain ⟨M,hM⟩ := anglePolynomialTest_bound p
-  exact Integrable.of_bound (p.continuous.comp Real.continuous_cos).aestronglyMeasurable M
-    (Filter.Eventually.of_forall hM)
-
-lemma anglePolynomialTest_mul_integrable (μ : Measure ℝ) [IsFiniteMeasure μ] (p q : ℝ[X]) :
-    Integrable (fun θ => anglePolynomialTest p θ*anglePolynomialTest q θ) μ := by
-  have he := anglePolynomialTest_integrable μ (p*q)
-  change Integrable (fun θ => (p*q).eval (Real.cos θ)) μ at he
-  simpa only [anglePolynomialTest,eval_mul] using he
-
-lemma angleSpectralFactor_pairing {κ : ℝ} (hκ : 0<κ) (n m : ℕ) (x : ℝ) :
-    (∫ φ, angleSpectralFactor κ n x (Real.cos φ)*
-      anglePolynomialTest (angleEigenPolynomial κ m) φ ∂Target.angleLaw κ)=
-        if n=m then (angleEigenPolynomial κ m).eval x else 0 := by
-  have he : (fun φ => angleSpectralFactor κ n x (Real.cos φ)*
-      anglePolynomialTest (angleEigenPolynomial κ m) φ)=
-      fun φ => (angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n*
-        (angleEigenPolynomial κ n).eval x)*
-      (anglePolynomialTest (angleEigenPolynomial κ n) φ*
-        anglePolynomialTest (angleEigenPolynomial κ m) φ) := by
-    funext φ
-    unfold angleSpectralFactor anglePolynomialTest
-    ring
-  rw [he,integral_const_mul,angleLaw_polynomial_pairing hκ]
-  by_cases hnm : n=m
-  · subst m
-    rw [ite_eq_left rfl]
-    change (angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n*
-        (angleEigenPolynomial κ n).eval x)*
-      ((angleEigenSquareNorm κ 0)⁻¹*angleEigenSquareNorm κ n)=_
-    field_simp [(angleEigenSquareNorm_pos hκ n).ne',
-      (angleEigenSquareNorm_zero_pos hκ).ne']
-  · rw [ite_eq_right hnm]
-    have hz : anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ m)=0 :=
-      angleEigenPolynomial_orthogonal hκ n m hnm
-    rw [hz,mul_zero,mul_zero]
-
-lemma angleLaw_eigen_mean {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
-    (∫ φ, anglePolynomialTest (angleEigenPolynomial κ n) φ ∂Target.angleLaw κ)=
-      if n=0 then 1 else 0 := by
-  have he := angleLaw_polynomial_pairing hκ (angleEigenPolynomial κ n) (angleEigenPolynomial κ 0)
-  simp only [angleEigenPolynomial_zero,anglePolynomialTest,eval_one,mul_one] at he
-  change (∫ φ, (angleEigenPolynomial κ n).eval (Real.cos φ) ∂Target.angleLaw κ)=_
-  rw [he]
-  by_cases hn : n=0
-  · subst n
-    rw [ite_eq_left rfl]
-    rw [← angleEigenPolynomial_zero κ]
-    change (angleEigenSquareNorm κ 0)⁻¹*angleEigenSquareNorm κ 0=1
-    exact inv_mul_cancel₀ (angleEigenSquareNorm_zero_pos hκ).ne'
-  · rw [ite_eq_right hn]
-    have hz : anglePolynomialInner κ (angleEigenPolynomial κ n) 1=0 := by
-      rw [← angleEigenPolynomial_zero κ]
-      exact angleEigenPolynomial_orthogonal hκ n 0 hn
-    rw [hz,mul_zero]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleStationaryPairing
+import OAI.Probability.SLE.Angle.AngleSpectralBounds
+
+namespace OAI
+
+/-! Exact stationary moments of the normalized polynomial kernel terms. -/
+noncomputable section
+open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+
+namespace SLEExactGauge
+
+lemma anglePolynomialTest_bound (p : ℝ[X]) :
+    ∃ M : ℝ, ∀ θ : ℝ, ‖anglePolynomialTest p θ‖≤M := by
+  obtain ⟨M,hM⟩ := (isCompact_Icc : IsCompact (Icc (-1:ℝ) 1)).exists_bound_of_continuousOn
+    p.continuous.continuousOn
+  exact ⟨M,fun θ => hM (Real.cos θ) ⟨Real.neg_one_le_cos θ,Real.cos_le_one θ⟩⟩
+
+lemma anglePolynomialTest_integrable (μ : Measure ℝ) [IsFiniteMeasure μ] (p : ℝ[X]) :
+    Integrable (anglePolynomialTest p) μ := by
+  obtain ⟨M,hM⟩ := anglePolynomialTest_bound p
+  exact Integrable.of_bound (p.continuous.comp Real.continuous_cos).aestronglyMeasurable M
+    (Filter.Eventually.of_forall hM)
+
+lemma anglePolynomialTest_mul_integrable (μ : Measure ℝ) [IsFiniteMeasure μ] (p q : ℝ[X]) :
+    Integrable (fun θ => anglePolynomialTest p θ*anglePolynomialTest q θ) μ := by
+  have he := anglePolynomialTest_integrable μ (p*q)
+  change Integrable (fun θ => (p*q).eval (Real.cos θ)) μ at he
+  simpa only [anglePolynomialTest,eval_mul] using he
+
+lemma angleSpectralFactor_pairing {κ : ℝ} (hκ : 0<κ) (n m : ℕ) (x : ℝ) :
+    (∫ φ, angleSpectralFactor κ n x (Real.cos φ)*
+      anglePolynomialTest (angleEigenPolynomial κ m) φ ∂Target.angleLaw κ)=
+        if n=m then (angleEigenPolynomial κ m).eval x else 0 := by
+  have he : (fun φ => angleSpectralFactor κ n x (Real.cos φ)*
+      anglePolynomialTest (angleEigenPolynomial κ m) φ)=
+      fun φ => (angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n*
+        (angleEigenPolynomial κ n).eval x)*
+      (anglePolynomialTest (angleEigenPolynomial κ n) φ*
+        anglePolynomialTest (angleEigenPolynomial κ m) φ) := by
+    funext φ
+    unfold angleSpectralFactor anglePolynomialTest
+    ring
+  rw [he,integral_const_mul,angleLaw_polynomial_pairing hκ]
+  by_cases hnm : n=m
+  · subst m
+    rw [ite_eq_left rfl]
+    change (angleEigenSquareNorm κ 0/angleEigenSquareNorm κ n*
+        (angleEigenPolynomial κ n).eval x)*
+      ((angleEigenSquareNorm κ 0)⁻¹*angleEigenSquareNorm κ n)=_
+    field_simp [(angleEigenSquareNorm_pos hκ n).ne',
+      (angleEigenSquareNorm_zero_pos hκ).ne']
+  · rw [ite_eq_right hnm]
+    have hz : anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ m)=0 :=
+      angleEigenPolynomial_orthogonal hκ n m hnm
+    rw [hz,mul_zero,mul_zero]
+
+lemma angleLaw_eigen_mean {κ : ℝ} (hκ : 0<κ) (n : ℕ) :
+    (∫ φ, anglePolynomialTest (angleEigenPolynomial κ n) φ ∂Target.angleLaw κ)=
+      if n=0 then 1 else 0 := by
+  have he := angleLaw_polynomial_pairing hκ (angleEigenPolynomial κ n) (angleEigenPolynomial κ 0)
+  simp only [angleEigenPolynomial_zero,anglePolynomialTest,eval_one,mul_one] at he
+  change (∫ φ, (angleEigenPolynomial κ n).eval (Real.cos φ) ∂Target.angleLaw κ)=_
+  rw [he]
+  by_cases hn : n=0
+  · subst n
+    rw [ite_eq_left rfl]
+    rw [← angleEigenPolynomial_zero κ]
+    change (angleEigenSquareNorm κ 0)⁻¹*angleEigenSquareNorm κ 0=1
+    exact inv_mul_cancel₀ (angleEigenSquareNorm_zero_pos hκ).ne'
+  · rw [ite_eq_right hn]
+    have hz : anglePolynomialInner κ (angleEigenPolynomial κ n) 1=0 := by
+      rw [← angleEigenPolynomial_zero κ]
+      exact angleEigenPolynomial_orthogonal hκ n 0 hn
+    rw [hz,mul_zero]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleStationaryFlux.lean b/lean/OAI/Probability/SLE/Angle/AngleStationaryFlux.lean
index 7562116288cdc4bdf651f32a4d230850b0dcdbea..ee15658e55eb17c52cd16180766b55626283119c 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleStationaryFlux.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleStationaryFlux.lean
@@ -1,58 +1,58 @@
-import OAI.Probability.SLE.Angle.AngleEigenDifferential
-
-namespace OAI
-
-/-! The stationary angular weight puts the cosine generator in divergence
-form. Its flux vanishes at both endpoints of the actual angle interval. -/
-noncomputable section
-open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-def anglePolynomialFlux (κ : ℝ) (p : ℝ[X]) (θ : ℝ) : ℝ :=
-  -(κ/8)*Real.sin θ^(4*Target.a κ+1)*p.derivative.eval (Real.cos θ)
-
-lemma anglePolynomialFlux_hasDerivAt {κ : ℝ} (hκ : 0<κ) (p : ℝ[X]) {θ : ℝ}
-    (hθ : θ∈Icc (0:ℝ) Real.pi) :
-    HasDerivAt (anglePolynomialFlux κ p)
-      (Real.sin θ^(4*Target.a κ)*(anglePolynomialGenerator κ p).eval (Real.cos θ)) θ := by
-  have hp : 0<4*Target.a κ := by unfold Target.a; positivity
-  have hpow := (Real.hasDerivAt_sin θ).rpow_const
-    (Or.inr (by linarith : 1≤4*Target.a κ+1))
-  have hpol := (p.derivative.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)
-  apply ((hpow.const_mul (-(κ/8))).mul hpol).congr_deriv
-  dsimp only [Function.comp_apply]
-  have hs : 0≤Real.sin θ := Real.sin_nonneg_of_nonneg_of_le_pi hθ.1 hθ.2
-  rw [show 4*Target.a κ+1-1=4*Target.a κ by ring]
-  by_cases hz : Real.sin θ=0
-  · simp only [hz,Real.zero_rpow hp.ne',Real.zero_rpow (by linarith : 4*Target.a κ+1≠0),
-      mul_zero,zero_mul,add_zero]
-  · have hsp : 0<Real.sin θ := lt_of_le_of_ne hs (Ne.symm hz)
-    rw [Real.rpow_add hsp,Real.rpow_one]
-    simp only [anglePolynomialGenerator,eval_sub,eval_mul,eval_C,eval_one,eval_pow,eval_X]
-    have hsin : Real.sin θ^2=1-Real.cos θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
-    rw [← hsin]
-    unfold Target.a
-    field_simp
-    ring
-
-lemma anglePolynomialGenerator_stationary_integral {κ : ℝ} (hκ : 0<κ) (p : ℝ[X]) :
-    (∫ θ in (0:ℝ)..Real.pi,
-      Real.sin θ^(4*Target.a κ)*(anglePolynomialGenerator κ p).eval (Real.cos θ))=0 := by
-  have hp : 0<4*Target.a κ := by unfold Target.a; positivity
-  have hc : Continuous (fun θ : ℝ =>
-      Real.sin θ^(4*Target.a κ)*(anglePolynomialGenerator κ p).eval (Real.cos θ)) :=
-    ((Real.continuous_rpow_const hp.le).comp Real.continuous_sin).mul
-      ((anglePolynomialGenerator κ p).continuous.comp Real.continuous_cos)
-  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt
-    (fun θ hθ => anglePolynomialFlux_hasDerivAt hκ p
-      (by simpa only [uIcc_of_le Real.pi_pos.le] using hθ)) (hc.intervalIntegrable 0 Real.pi)
-  simpa only [anglePolynomialFlux,Real.sin_pi,Real.sin_zero,
-    Real.zero_rpow (by linarith : 4*Target.a κ+1≠0),mul_zero,zero_mul,sub_self] using he
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenDifferential
+
+namespace OAI
+
+/-! The stationary angular weight puts the cosine generator in divergence
+form. Its flux vanishes at both endpoints of the actual angle interval. -/
+noncomputable section
+open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+def anglePolynomialFlux (κ : ℝ) (p : ℝ[X]) (θ : ℝ) : ℝ :=
+  -(κ/8)*Real.sin θ^(4*Target.a κ+1)*p.derivative.eval (Real.cos θ)
+
+lemma anglePolynomialFlux_hasDerivAt {κ : ℝ} (hκ : 0<κ) (p : ℝ[X]) {θ : ℝ}
+    (hθ : θ∈Icc (0:ℝ) Real.pi) :
+    HasDerivAt (anglePolynomialFlux κ p)
+      (Real.sin θ^(4*Target.a κ)*(anglePolynomialGenerator κ p).eval (Real.cos θ)) θ := by
+  have hp : 0<4*Target.a κ := by unfold Target.a; positivity
+  have hpow := (Real.hasDerivAt_sin θ).rpow_const
+    (Or.inr (by linarith : 1≤4*Target.a κ+1))
+  have hpol := (p.derivative.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)
+  apply ((hpow.const_mul (-(κ/8))).mul hpol).congr_deriv
+  dsimp only [Function.comp_apply]
+  have hs : 0≤Real.sin θ := Real.sin_nonneg_of_nonneg_of_le_pi hθ.1 hθ.2
+  rw [show 4*Target.a κ+1-1=4*Target.a κ by ring]
+  by_cases hz : Real.sin θ=0
+  · simp only [hz,Real.zero_rpow hp.ne',Real.zero_rpow (by linarith : 4*Target.a κ+1≠0),
+      mul_zero,zero_mul,add_zero]
+  · have hsp : 0<Real.sin θ := lt_of_le_of_ne hs (Ne.symm hz)
+    rw [Real.rpow_add hsp,Real.rpow_one]
+    simp only [anglePolynomialGenerator,eval_sub,eval_mul,eval_C,eval_one,eval_pow,eval_X]
+    have hsin : Real.sin θ^2=1-Real.cos θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
+    rw [← hsin]
+    unfold Target.a
+    field_simp
+    ring
+
+lemma anglePolynomialGenerator_stationary_integral {κ : ℝ} (hκ : 0<κ) (p : ℝ[X]) :
+    (∫ θ in (0:ℝ)..Real.pi,
+      Real.sin θ^(4*Target.a κ)*(anglePolynomialGenerator κ p).eval (Real.cos θ))=0 := by
+  have hp : 0<4*Target.a κ := by unfold Target.a; positivity
+  have hc : Continuous (fun θ : ℝ =>
+      Real.sin θ^(4*Target.a κ)*(anglePolynomialGenerator κ p).eval (Real.cos θ)) :=
+    ((Real.continuous_rpow_const hp.le).comp Real.continuous_sin).mul
+      ((anglePolynomialGenerator κ p).continuous.comp Real.continuous_cos)
+  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (fun θ hθ => anglePolynomialFlux_hasDerivAt hκ p
+      (by simpa only [uIcc_of_le Real.pi_pos.le] using hθ)) (hc.intervalIntegrable 0 Real.pi)
+  simpa only [anglePolynomialFlux,Real.sin_pi,Real.sin_zero,
+    Real.zero_rpow (by linarith : 4*Target.a κ+1≠0),mul_zero,zero_mul,sub_self] using he
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/AngleStationaryPairing.lean b/lean/OAI/Probability/SLE/Angle/AngleStationaryPairing.lean
index 4a30f172b6821d092b002ccbe6862cc4c855a263..4d440ed00781b3b60eaae910656b1dbb79e1ddac 100644
--- a/lean/OAI/Probability/SLE/Angle/AngleStationaryPairing.lean
+++ b/lean/OAI/Probability/SLE/Angle/AngleStationaryPairing.lean
@@ -1,74 +1,74 @@
-import OAI.Probability.SLE.Angle.AngleEigenNorm
-import OAI.Probability.SLE.Angle.TargetAngleLaw
-
-namespace OAI
-
-/-! Match the polynomial pairing to the exact angle probability measure
-appearing in the prescribed coefficients. -/
-noncomputable section
-open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge
-
-lemma anglePolynomialInner_eq_setIntegral (κ : ℝ) (p q : ℝ[X]) :
-    anglePolynomialInner κ p q=∫ θ in Ioo (0:ℝ) Real.pi,
-      Real.sin θ^(4*Target.a κ)*anglePolynomialTest p θ*anglePolynomialTest q θ := by
-  rw [anglePolynomialInner,intervalIntegral.integral_of_le Real.pi_pos.le,
-    integral_Ioc_eq_integral_Ioo]
-
-lemma angle_normalizer_eq_norm {κ : ℝ} (hκ : 0<κ) :
-    ((volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
-      (fun θ => ENNReal.ofReal (Real.sin θ^(4*Target.a κ)))) univ=
-        ENNReal.ofReal (angleEigenSquareNorm κ 0) := by
-  have hi := anglePolynomialInner_integrable hκ (1:ℝ[X]) 1
-  simp only [anglePolynomialTest,eval_one,mul_one] at hi
-  have hi' : Integrable (fun θ : ℝ => Real.sin θ^(4*Target.a κ))
-      (volume.restrict (Ioo (0:ℝ) Real.pi)) := by
-    rw [intervalIntegrable_iff_integrableOn_Ioc_of_le Real.pi_pos.le] at hi
-    exact hi.mono_set Ioo_subset_Ioc_self
-  rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
-  rw [angleEigenSquareNorm,angleEigenPolynomial_zero,anglePolynomialInner_eq_setIntegral]
-  simp only [anglePolynomialTest,eval_one,mul_one]
-  exact (ofReal_integral_eq_lintegral_ofReal hi' (by
-    filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
-    exact Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)).symm
-
-lemma angleLaw_integral (κ : ℝ) (hκ : 0<κ) (f : ℝ → ℝ) :
-    (∫ θ, f θ ∂Target.angleLaw κ)=
-      (angleEigenSquareNorm κ 0)⁻¹*∫ θ in Ioo (0:ℝ) Real.pi,
-        Real.sin θ^(4*Target.a κ)*f θ := by
-  rw [Target.angleLaw,integral_smul_measure,ENNReal.toReal_inv,
-    angle_normalizer_eq_norm hκ,ENNReal.toReal_ofReal (angleEigenSquareNorm_zero_pos hκ).le]
-  congr 1
-  have hm : Measurable (fun θ : ℝ => ENNReal.ofReal (Real.sin θ^(4*Target.a κ))) := by
-    fun_prop
-  rw [integral_withDensity_eq_integral_toReal_smul
-    hm
-    (Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
-  apply integral_congr_ae
-  filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
-  rw [ENNReal.toReal_ofReal (Real.rpow_nonneg
-    (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)]
-  rfl
-
-lemma angleLaw_polynomial_pairing {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) :
-    (∫ θ, anglePolynomialTest p θ*anglePolynomialTest q θ ∂Target.angleLaw κ)=
-      (angleEigenSquareNorm κ 0)⁻¹*anglePolynomialInner κ p q := by
-  rw [angleLaw_integral κ hκ,anglePolynomialInner_eq_setIntegral]
-  congr 1
-  apply integral_congr_ae
-  exact Filter.Eventually.of_forall (fun _ => by ring)
-
-lemma angleLaw_eigen_orthogonal {κ : ℝ} (hκ : 0<κ) (n m : ℕ) (hnm : n≠m) :
-    (∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ*
-      anglePolynomialTest (angleEigenPolynomial κ m) θ ∂Target.angleLaw κ)=0 := by
-  rw [angleLaw_polynomial_pairing hκ]
-  rw [show anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ m)=0
-    from angleEigenPolynomial_orthogonal hκ n m hnm,mul_zero]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.AngleEigenNorm
+import OAI.Probability.SLE.Angle.TargetAngleLaw
+
+namespace OAI
+
+/-! Match the polynomial pairing to the exact angle probability measure
+appearing in the prescribed coefficients. -/
+noncomputable section
+open Polynomial Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge
+
+lemma anglePolynomialInner_eq_setIntegral (κ : ℝ) (p q : ℝ[X]) :
+    anglePolynomialInner κ p q=∫ θ in Ioo (0:ℝ) Real.pi,
+      Real.sin θ^(4*Target.a κ)*anglePolynomialTest p θ*anglePolynomialTest q θ := by
+  rw [anglePolynomialInner,intervalIntegral.integral_of_le Real.pi_pos.le,
+    integral_Ioc_eq_integral_Ioo]
+
+lemma angle_normalizer_eq_norm {κ : ℝ} (hκ : 0<κ) :
+    ((volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
+      (fun θ => ENNReal.ofReal (Real.sin θ^(4*Target.a κ)))) univ=
+        ENNReal.ofReal (angleEigenSquareNorm κ 0) := by
+  have hi := anglePolynomialInner_integrable hκ (1:ℝ[X]) 1
+  simp only [anglePolynomialTest,eval_one,mul_one] at hi
+  have hi' : Integrable (fun θ : ℝ => Real.sin θ^(4*Target.a κ))
+      (volume.restrict (Ioo (0:ℝ) Real.pi)) := by
+    rw [intervalIntegrable_iff_integrableOn_Ioc_of_le Real.pi_pos.le] at hi
+    exact hi.mono_set Ioo_subset_Ioc_self
+  rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
+  rw [angleEigenSquareNorm,angleEigenPolynomial_zero,anglePolynomialInner_eq_setIntegral]
+  simp only [anglePolynomialTest,eval_one,mul_one]
+  exact (ofReal_integral_eq_lintegral_ofReal hi' (by
+    filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
+    exact Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)).symm
+
+lemma angleLaw_integral (κ : ℝ) (hκ : 0<κ) (f : ℝ → ℝ) :
+    (∫ θ, f θ ∂Target.angleLaw κ)=
+      (angleEigenSquareNorm κ 0)⁻¹*∫ θ in Ioo (0:ℝ) Real.pi,
+        Real.sin θ^(4*Target.a κ)*f θ := by
+  rw [Target.angleLaw,integral_smul_measure,ENNReal.toReal_inv,
+    angle_normalizer_eq_norm hκ,ENNReal.toReal_ofReal (angleEigenSquareNorm_zero_pos hκ).le]
+  congr 1
+  have hm : Measurable (fun θ : ℝ => ENNReal.ofReal (Real.sin θ^(4*Target.a κ))) := by
+    fun_prop
+  rw [integral_withDensity_eq_integral_toReal_smul
+    hm
+    (Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
+  apply integral_congr_ae
+  filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
+  rw [ENNReal.toReal_ofReal (Real.rpow_nonneg
+    (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)]
+  rfl
+
+lemma angleLaw_polynomial_pairing {κ : ℝ} (hκ : 0<κ) (p q : ℝ[X]) :
+    (∫ θ, anglePolynomialTest p θ*anglePolynomialTest q θ ∂Target.angleLaw κ)=
+      (angleEigenSquareNorm κ 0)⁻¹*anglePolynomialInner κ p q := by
+  rw [angleLaw_integral κ hκ,anglePolynomialInner_eq_setIntegral]
+  congr 1
+  apply integral_congr_ae
+  exact Filter.Eventually.of_forall (fun _ => by ring)
+
+lemma angleLaw_eigen_orthogonal {κ : ℝ} (hκ : 0<κ) (n m : ℕ) (hnm : n≠m) :
+    (∫ θ, anglePolynomialTest (angleEigenPolynomial κ n) θ*
+      anglePolynomialTest (angleEigenPolynomial κ m) θ ∂Target.angleLaw κ)=0 := by
+  rw [angleLaw_polynomial_pairing hκ]
+  rw [show anglePolynomialInner κ (angleEigenPolynomial κ n) (angleEigenPolynomial κ m)=0
+    from angleEigenPolynomial_orthogonal hκ n m hnm,mul_zero]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/CanonicalAngleKernel.lean b/lean/OAI/Probability/SLE/Angle/CanonicalAngleKernel.lean
index b4292e6b608358e104b44b6a54d41e4830c6d6a4..148ca9184f55f7bd13b12477076dda35a27196ba 100644
--- a/lean/OAI/Probability/SLE/Angle/CanonicalAngleKernel.lean
+++ b/lean/OAI/Probability/SLE/Angle/CanonicalAngleKernel.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Geometry.NormalizedDiskAngle
-
-namespace OAI
-
-/-! The actual finite-centered transition of the full angle, represented
-by a measurable raw-driver kernel. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance angleKernelMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance angleKernelBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-def canonicalAngleAverage (ν : Measure C(ℝ≥0,ℝ)) (κ v : ℝ)
-    (f : ℝ → ℝ≥0∞) (θ : ℝ) : ℝ≥0∞ :=
-  ∫⁻ p, f (canonicalStoppedAngle κ (v,Target.diskMapInv θ 0,p))
-    ∂ν.withDensity (fun p => canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p))
-
-lemma canonicalAngleAverage_measurable (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν]
-    (κ v : ℝ) (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
-    Measurable (canonicalAngleAverage ν κ v f) := by
-  have hz : Measurable (fun θ => Target.diskMapInv θ 0) := by
-    unfold Target.diskMapInv
-    fun_prop
-  let q : ℝ × C(ℝ≥0,ℝ) → ℝ × (ℂ × C(ℝ≥0,ℝ)) :=
-    fun x => (v,Target.diskMapInv x.1 0,x.2)
-  have hq : Measurable q := measurable_const.prodMk
-    ((hz.comp measurable_fst).prodMk measurable_snd)
-  have h := (((canonicalFiniteDensity_measurable κ).comp hq).mul
-    (hf.comp ((canonicalStoppedAngle_measurable κ).comp hq))).lintegral_prod_right' (ν := ν)
-  have he (θ : ℝ) : (∫⁻ p, canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p)*
-      f (canonicalStoppedAngle κ (v,Target.diskMapInv θ 0,p)) ∂ν)=
-      canonicalAngleAverage ν κ v f θ := by
-    symm
-    exact lintegral_withDensity_eq_lintegral_mul _
-      ((canonicalFiniteDensity_measurable κ).comp
-        (measurable_const.prodMk (measurable_const.prodMk measurable_id)))
-      (hf.comp ((canonicalStoppedAngle_measurable κ).comp
-        (measurable_const.prodMk (measurable_const.prodMk measurable_id))))
-  simpa only [q,Pi.mul_apply,Function.comp_apply,he] using h
-
-lemma IsCapacityTwoWitness.normalized_disk_angle_test
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (κ : ℝ) (T a : ℝ≥0) (ha : 0 < a) (p : C(ℝ≥0,ℝ))
-    {β : ℝ≥0 → ℂ} {G' F' : ℝ≥0 → ℂ → ℂ}
-    (hb : IsCapacityTwoWitness (fun t => Real.sqrt κ*p t) β G' F')
-    (hdriver : (fun t => Real.sqrt κ*p t)=normalizedRestartDriver U T a)
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (z : ℂ)
-    (hz : z ∈ survivingDomain γ T) (hzH : z∈upperHalfPlane)
-    (hpolar : G T z=(U T:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I))
-    (u v : ℝ) (hu : 0 ≤ u) (hv : 0 < v)
-    (hbase : fullRadius G T z=(2*z.im)*Real.exp (-u)) (f : ℝ → ℝ≥0∞) :
-    ENNReal.ofReal (greenDensity κ U γ G T z/initialGreen κ z)*
-        canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p)*
-        f (canonicalStoppedAngle κ (v,Target.diskMapInv θ 0,p)) =
-      ENNReal.ofReal (stoppedGreen κ U γ G z (u+v)/initialGreen κ z)*
-        f ((G (radiusLevelTime γ G z (u+v)).toNNReal z-
-          (U (radiusLevelTime γ G z (u+v)).toNNReal:ℂ)).arg) := by
-  rw [← hw.normalized_disk_density hU κ T a ha p hb hdriver θ hθ z hz hzH hpolar u v hu hv hbase]
-  by_cases ht : radiusLevelTime γ G z (u+v)<∞
-  · rw [hw.normalized_disk_stopped_arg hU κ T a ha p hb hdriver θ hθ z hz hpolar u v hu hv hbase ht]
-  · simp only [stoppedGreen,ite_eq_right ht,zero_div,ENNReal.ofReal_zero,zero_mul]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.NormalizedDiskAngle
+
+namespace OAI
+
+/-! The actual finite-centered transition of the full angle, represented
+by a measurable raw-driver kernel. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance angleKernelMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance angleKernelBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+def canonicalAngleAverage (ν : Measure C(ℝ≥0,ℝ)) (κ v : ℝ)
+    (f : ℝ → ℝ≥0∞) (θ : ℝ) : ℝ≥0∞ :=
+  ∫⁻ p, f (canonicalStoppedAngle κ (v,Target.diskMapInv θ 0,p))
+    ∂ν.withDensity (fun p => canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p))
+
+lemma canonicalAngleAverage_measurable (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν]
+    (κ v : ℝ) (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
+    Measurable (canonicalAngleAverage ν κ v f) := by
+  have hz : Measurable (fun θ => Target.diskMapInv θ 0) := by
+    unfold Target.diskMapInv
+    fun_prop
+  let q : ℝ × C(ℝ≥0,ℝ) → ℝ × (ℂ × C(ℝ≥0,ℝ)) :=
+    fun x => (v,Target.diskMapInv x.1 0,x.2)
+  have hq : Measurable q := measurable_const.prodMk
+    ((hz.comp measurable_fst).prodMk measurable_snd)
+  have h := (((canonicalFiniteDensity_measurable κ).comp hq).mul
+    (hf.comp ((canonicalStoppedAngle_measurable κ).comp hq))).lintegral_prod_right' (ν := ν)
+  have he (θ : ℝ) : (∫⁻ p, canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p)*
+      f (canonicalStoppedAngle κ (v,Target.diskMapInv θ 0,p)) ∂ν)=
+      canonicalAngleAverage ν κ v f θ := by
+    symm
+    exact lintegral_withDensity_eq_lintegral_mul _
+      ((canonicalFiniteDensity_measurable κ).comp
+        (measurable_const.prodMk (measurable_const.prodMk measurable_id)))
+      (hf.comp ((canonicalStoppedAngle_measurable κ).comp
+        (measurable_const.prodMk (measurable_const.prodMk measurable_id))))
+  simpa only [q,Pi.mul_apply,Function.comp_apply,he] using h
+
+lemma IsCapacityTwoWitness.normalized_disk_angle_test
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (κ : ℝ) (T a : ℝ≥0) (ha : 0 < a) (p : C(ℝ≥0,ℝ))
+    {β : ℝ≥0 → ℂ} {G' F' : ℝ≥0 → ℂ → ℂ}
+    (hb : IsCapacityTwoWitness (fun t => Real.sqrt κ*p t) β G' F')
+    (hdriver : (fun t => Real.sqrt κ*p t)=normalizedRestartDriver U T a)
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (z : ℂ)
+    (hz : z ∈ survivingDomain γ T) (hzH : z∈upperHalfPlane)
+    (hpolar : G T z=(U T:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I))
+    (u v : ℝ) (hu : 0 ≤ u) (hv : 0 < v)
+    (hbase : fullRadius G T z=(2*z.im)*Real.exp (-u)) (f : ℝ → ℝ≥0∞) :
+    ENNReal.ofReal (greenDensity κ U γ G T z/initialGreen κ z)*
+        canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p)*
+        f (canonicalStoppedAngle κ (v,Target.diskMapInv θ 0,p)) =
+      ENNReal.ofReal (stoppedGreen κ U γ G z (u+v)/initialGreen κ z)*
+        f ((G (radiusLevelTime γ G z (u+v)).toNNReal z-
+          (U (radiusLevelTime γ G z (u+v)).toNNReal:ℂ)).arg) := by
+  rw [← hw.normalized_disk_density hU κ T a ha p hb hdriver θ hθ z hz hzH hpolar u v hu hv hbase]
+  by_cases ht : radiusLevelTime γ G z (u+v)<∞
+  · rw [hw.normalized_disk_stopped_arg hU κ T a ha p hb hdriver θ hθ z hz hpolar u v hu hv hbase ht]
+  · simp only [stoppedGreen,ite_eq_right ht,zero_div,ENNReal.ofReal_zero,zero_mul]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/CanonicalAngleMixing.lean b/lean/OAI/Probability/SLE/Angle/CanonicalAngleMixing.lean
index 3f592a3700e1cf76e33ecd6da0ee149c01a8bb36..fd9f7b579e7b6689e69785819ae377db54b0dbb3 100644
--- a/lean/OAI/Probability/SLE/Angle/CanonicalAngleMixing.lean
+++ b/lean/OAI/Probability/SLE/Angle/CanonicalAngleMixing.lean
@@ -1,82 +1,82 @@
-import OAI.Probability.SLE.Angle.ActualAngleSpectral
-
-namespace OAI
-
-/-! Uniform finite-centered mixing for the actual canonical angle kernel.
-The constants are derived from the positive spectral law, not assumed. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance angleMixingMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance angleMixingBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-lemma angleSpectralLaw_lintegral_bounds {κ u : ℝ} (hκ : 0<κ)
-    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ (∫⁻ φ, f φ ∂angleSpectralLaw κ u θ) ∧
-    (∫⁻ φ, f φ ∂angleSpectralLaw κ u θ) ≤ 2*(∫⁻ φ, f φ ∂Target.angleLaw κ) := by
-  rw [angleSpectralLaw,lintegral_withDensity_eq_lintegral_mul _
-    (angleSpectralKernel_measurable hκ hu θ).ennreal_ofReal hf]
-  have hb (φ : ℝ) : (1/2:ℝ≥0∞)≤ENNReal.ofReal (angleSpectralKernel κ u θ φ) ∧
-      ENNReal.ofReal (angleSpectralKernel κ u θ φ)≤2 := by
-    have h := angleSpectralKernel_bounds hκ hu θ φ
-    constructor
-    · simpa using ENNReal.ofReal_le_ofReal h.1
-    · simpa using ENNReal.ofReal_le_ofReal h.2
-  constructor
-  · calc
-      _ = (1/2:ℝ≥0∞)*(∫⁻ φ, f φ ∂Target.angleLaw κ) := by
-        rw [div_eq_mul_inv,one_div,mul_comm]
-      _ = ∫⁻ φ, (1/2:ℝ≥0∞)*f φ ∂Target.angleLaw κ := (lintegral_const_mul _ hf).symm
-      _ ≤ _ := lintegral_mono fun φ => mul_le_mul' (hb φ).1 le_rfl
-  · calc
-      _ ≤ ∫⁻ φ, (2:ℝ≥0∞)*f φ ∂Target.angleLaw κ :=
-        lintegral_mono fun φ => mul_le_mul' (hb φ).2 le_rfl
-      _ = _ := lintegral_const_mul _ hf
-
-lemma Target.diskMapInv_zero_arg (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) :
-    (Target.diskMapInv θ 0).arg=θ := by
-  simp only [Target.diskMapInv,zero_mul,sub_zero,div_one]
-  rw [Complex.arg_exp_mul_I]
-  apply (toIocMod_eq_self Real.two_pi_pos).mpr
-  constructor <;> linarith [Real.pi_pos,hθ.1,hθ.2]
-
-lemma canonicalAngleAverage_uniform_bounds
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (u : ℝ) (hu : angleSpectralMixingTime κ≤u) (f : ℝ → ℝ≥0∞) (hf : Measurable f)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) :
-    (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤
-      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ ∧
-    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ ≤
-      2*(∫⁻ φ, f φ ∂Target.angleLaw κ) := by
-  have hu0 : 0≤u := (angleSpectralMixingTime_pos hκ).le.trans hu
-  let z := Target.diskMapInv θ 0
-  have hz : z∈upperHalfPlane := Target.diskMapInv_maps hθ (by simp)
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let S : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
-  have hS := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu0
-  let a := extendedMarkedAngle κ B γ G S z
-  have ham : Measurable a :=
-    (extendedMarkedAngle_measurable B hm hc κ γ G F hw S hS z).mono hS.measurableSpace_le le_rfl
-  have he : canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ=
-      ∫⁻ φ, f φ ∂angleSpectralLaw κ u θ := by
-    rw [canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw u hu0 θ hθ f hf]
-    change (∫⁻ ω, f (a ω) ∂finiteCenteredLaw P κ B γ G z u)=_
-    rw [← lintegral_map' hf.aemeasurable ham.aemeasurable,
-      actual_angle_law_eq_spectral hB hBm hc h0 hκ hκ8 γ G F hw z hz u hu]
-    rw [Target.diskMapInv_zero_arg θ hθ]
-  rw [he]
-  exact angleSpectralLaw_lintegral_bounds hκ hu θ f hf
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.ActualAngleSpectral
+
+namespace OAI
+
+/-! Uniform finite-centered mixing for the actual canonical angle kernel.
+The constants are derived from the positive spectral law, not assumed. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance angleMixingMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance angleMixingBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+lemma angleSpectralLaw_lintegral_bounds {κ u : ℝ} (hκ : 0<κ)
+    (hu : angleSpectralMixingTime κ≤u) (θ : ℝ) (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤ (∫⁻ φ, f φ ∂angleSpectralLaw κ u θ) ∧
+    (∫⁻ φ, f φ ∂angleSpectralLaw κ u θ) ≤ 2*(∫⁻ φ, f φ ∂Target.angleLaw κ) := by
+  rw [angleSpectralLaw,lintegral_withDensity_eq_lintegral_mul _
+    (angleSpectralKernel_measurable hκ hu θ).ennreal_ofReal hf]
+  have hb (φ : ℝ) : (1/2:ℝ≥0∞)≤ENNReal.ofReal (angleSpectralKernel κ u θ φ) ∧
+      ENNReal.ofReal (angleSpectralKernel κ u θ φ)≤2 := by
+    have h := angleSpectralKernel_bounds hκ hu θ φ
+    constructor
+    · simpa using ENNReal.ofReal_le_ofReal h.1
+    · simpa using ENNReal.ofReal_le_ofReal h.2
+  constructor
+  · calc
+      _ = (1/2:ℝ≥0∞)*(∫⁻ φ, f φ ∂Target.angleLaw κ) := by
+        rw [div_eq_mul_inv,one_div,mul_comm]
+      _ = ∫⁻ φ, (1/2:ℝ≥0∞)*f φ ∂Target.angleLaw κ := (lintegral_const_mul _ hf).symm
+      _ ≤ _ := lintegral_mono fun φ => mul_le_mul' (hb φ).1 le_rfl
+  · calc
+      _ ≤ ∫⁻ φ, (2:ℝ≥0∞)*f φ ∂Target.angleLaw κ :=
+        lintegral_mono fun φ => mul_le_mul' (hb φ).2 le_rfl
+      _ = _ := lintegral_const_mul _ hf
+
+lemma Target.diskMapInv_zero_arg (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) :
+    (Target.diskMapInv θ 0).arg=θ := by
+  simp only [Target.diskMapInv,zero_mul,sub_zero,div_one]
+  rw [Complex.arg_exp_mul_I]
+  apply (toIocMod_eq_self Real.two_pi_pos).mpr
+  constructor <;> linarith [Real.pi_pos,hθ.1,hθ.2]
+
+lemma canonicalAngleAverage_uniform_bounds
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (u : ℝ) (hu : angleSpectralMixingTime κ≤u) (f : ℝ → ℝ≥0∞) (hf : Measurable f)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) :
+    (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤
+      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ ∧
+    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ ≤
+      2*(∫⁻ φ, f φ ∂Target.angleLaw κ) := by
+  have hu0 : 0≤u := (angleSpectralMixingTime_pos hκ).le.trans hu
+  let z := Target.diskMapInv θ 0
+  have hz : z∈upperHalfPlane := Target.diskMapInv_maps hθ (by simp)
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let S : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
+  have hS := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu0
+  let a := extendedMarkedAngle κ B γ G S z
+  have ham : Measurable a :=
+    (extendedMarkedAngle_measurable B hm hc κ γ G F hw S hS z).mono hS.measurableSpace_le le_rfl
+  have he : canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ=
+      ∫⁻ φ, f φ ∂angleSpectralLaw κ u θ := by
+    rw [canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw u hu0 θ hθ f hf]
+    change (∫⁻ ω, f (a ω) ∂finiteCenteredLaw P κ B γ G z u)=_
+    rw [← lintegral_map' hf.aemeasurable ham.aemeasurable,
+      actual_angle_law_eq_spectral hB hBm hc h0 hκ hκ8 γ G F hw z hz u hu]
+    rw [Target.diskMapInv_zero_arg θ hθ]
+  rw [he]
+  exact angleSpectralLaw_lintegral_bounds hκ hu θ f hf
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/CanonicalAngleSemigroup.lean b/lean/OAI/Probability/SLE/Angle/CanonicalAngleSemigroup.lean
index eef5d44dbe0f52dd5cc04b085e50408a05b19f99..aca430a08dab2c8177c452599faddf44d0069513 100644
--- a/lean/OAI/Probability/SLE/Angle/CanonicalAngleSemigroup.lean
+++ b/lean/OAI/Probability/SLE/Angle/CanonicalAngleSemigroup.lean
@@ -1,105 +1,105 @@
-import OAI.Probability.SLE.Angle.RadiusAngleBounds
-
-namespace OAI
-
-/-! Probability and composition of the actual finite centered angle
-transition, before its quantitative convergence is estimated. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance angleSemigroupMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance angleSemigroupBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ}
-  (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B))
-  (hc : ∀ ω, Continuous (fun t => B t ω))
-  (hκ : 0<κ) (hκ8 : κ<8)
-  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-
-include hB hBm hc hκ hκ8 γ G F hw
-
-lemma canonicalAngleAverage_eq_finiteCentered
-    (u : ℝ) (hu : 0≤u) (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi)
-    (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
-    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ =
-      ∫⁻ ω, f (extendedMarkedAngle κ B γ G
-        (fun ω => radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u)
-        (Target.diskMapInv θ 0) ω) ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u := by
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  have hz := Target.diskMapInv_maps hθ (show (0:ℂ)∈ball 0 1 by simp)
-  have hfm : Measurable (fun p : C(ℝ≥0,ℝ) =>
-      f (canonicalStoppedAngle κ (u,Target.diskMapInv θ 0,p))) :=
-    hf.comp ((canonicalStoppedAngle_measurable κ).comp
-      (measurable_const.prodMk (measurable_const.prodMk measurable_id)))
-  unfold canonicalAngleAverage
-  rw [← canonical_finiteCenteredLaw_map P κ B hm hc γ G F (Eventually.of_forall hw) u hu,
-    lintegral_map' hfm.aemeasurable (source_path_measurable _ hm).aemeasurable]
-  have ht := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F
-    (Eventually.of_forall hw) (Target.diskMapInv θ 0) hz u u hu le_rfl
-    (fun ω => radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u)
-    (Eventually.of_forall fun _ => le_rfl)
-  apply lintegral_congr_ae
-  filter_upwards [ht] with ω hω
-  have hfin : radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u<∞ :=
-    lt_top_iff_ne_top.mpr hω
-  rw [canonicalStoppedAngle_eq κ (sourceDriverPath B hc ω) (hw ω) u hu _ hfin,
-    extendedMarkedAngle_at_finite_radius κ B hc γ G F hw _ u ω hfin]
-  rfl
-
-lemma canonicalAngleAverage_one
-    (u : ℝ) (hu : 0≤u) (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) :
-    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u (fun _ => 1) θ=1 := by
-  rw [canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw u hu θ hθ
-    (fun _ => 1) measurable_const]
-  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw)
-    (Target.diskMapInv θ 0) (Target.diskMapInv_maps hθ (by simp)) u hu
-  simp
-
-lemma canonicalAngleAverage_comp
-    (h0 : ∀ ω, B 0 ω=0)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (u v : ℝ) (hu : 0≤u) (hv : 0<v)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi)
-    (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
-    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ (u+v) f θ =
-      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u
-        (canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f) θ := by
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let ν := P.map (sourceDriverPath B hc)
-  let : IsProbabilityMeasure ν :=
-    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
-  let z := Target.diskMapInv θ 0
-  have hz : z∈upperHalfPlane := Target.diskMapInv_maps hθ (by simp)
-  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let S : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
-  have hS : IsStoppingTime ℱ S :=
-    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu
-  let g : Ω → ℝ≥0∞ := fun ω => canonicalAngleAverage ν κ v f
-    (extendedMarkedAngle κ B γ G S z ω)
-  have hfm : Measurable (canonicalAngleAverage ν κ v f) :=
-    canonicalAngleAverage_measurable ν κ v f hf
-  have hgm : Measurable[hS.measurableSpace] g := hfm.comp
-    (extendedMarkedAngle_measurable B hm hc κ γ G F hw S hS z)
-  rw [canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw
-    (u+v) (by linarith) θ hθ f hf,
-    canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw u hu θ hθ _ hfm]
-  have hk := radius_angle_kernel hB hBm hc h0 hκ hκ8 γ G F hw hpublished
-    z hz u v (u+v) hu hv le_rfl f hf univ MeasurableSet.univ
-  simp only [setLIntegral_univ] at hk
-  rw [← hk]
-  exact @lintegral_eq_of_stopped_events Ω mΩ
-    (finiteCenteredLaw P κ B γ G z (u+v)) (finiteCenteredLaw P κ B γ G z u)
-    hS.measurableSpace hS.measurableSpace_le
-    (fun E hE => finiteCenteredLaw_consistent_at_extended_stop hB hBm hc h0 κ hκ hκ8
-      γ G F (Eventually.of_forall hw) z hz u (u+v) hu (by linarith) S hS
-      (Eventually.of_forall fun _ => le_rfl) E hE) g hgm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.RadiusAngleBounds
+
+namespace OAI
+
+/-! Probability and composition of the actual finite centered angle
+transition, before its quantitative convergence is estimated. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance angleSemigroupMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance angleSemigroupBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ}
+  (hB : IsBrownianReal B P) (hBm : Measurable (Function.uncurry B))
+  (hc : ∀ ω, Continuous (fun t => B t ω))
+  (hκ : 0<κ) (hκ8 : κ<8)
+  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+
+include hB hBm hc hκ hκ8 γ G F hw
+
+lemma canonicalAngleAverage_eq_finiteCentered
+    (u : ℝ) (hu : 0≤u) (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi)
+    (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
+    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u f θ =
+      ∫⁻ ω, f (extendedMarkedAngle κ B γ G
+        (fun ω => radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u)
+        (Target.diskMapInv θ 0) ω) ∂finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u := by
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  have hz := Target.diskMapInv_maps hθ (show (0:ℂ)∈ball 0 1 by simp)
+  have hfm : Measurable (fun p : C(ℝ≥0,ℝ) =>
+      f (canonicalStoppedAngle κ (u,Target.diskMapInv θ 0,p))) :=
+    hf.comp ((canonicalStoppedAngle_measurable κ).comp
+      (measurable_const.prodMk (measurable_const.prodMk measurable_id)))
+  unfold canonicalAngleAverage
+  rw [← canonical_finiteCenteredLaw_map P κ B hm hc γ G F (Eventually.of_forall hw) u hu,
+    lintegral_map' hfm.aemeasurable (source_path_measurable _ hm).aemeasurable]
+  have ht := finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F
+    (Eventually.of_forall hw) (Target.diskMapInv θ 0) hz u u hu le_rfl
+    (fun ω => radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u)
+    (Eventually.of_forall fun _ => le_rfl)
+  apply lintegral_congr_ae
+  filter_upwards [ht] with ω hω
+  have hfin : radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) u<∞ :=
+    lt_top_iff_ne_top.mpr hω
+  rw [canonicalStoppedAngle_eq κ (sourceDriverPath B hc ω) (hw ω) u hu _ hfin,
+    extendedMarkedAngle_at_finite_radius κ B hc γ G F hw _ u ω hfin]
+  rfl
+
+lemma canonicalAngleAverage_one
+    (u : ℝ) (hu : 0≤u) (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) :
+    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u (fun _ => 1) θ=1 := by
+  rw [canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw u hu θ hθ
+    (fun _ => 1) measurable_const]
+  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw)
+    (Target.diskMapInv θ 0) (Target.diskMapInv_maps hθ (by simp)) u hu
+  simp
+
+lemma canonicalAngleAverage_comp
+    (h0 : ∀ ω, B 0 ω=0)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (u v : ℝ) (hu : 0≤u) (hv : 0<v)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi)
+    (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
+    canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ (u+v) f θ =
+      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ u
+        (canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f) θ := by
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let ν := P.map (sourceDriverPath B hc)
+  let : IsProbabilityMeasure ν :=
+    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
+  let z := Target.diskMapInv θ 0
+  have hz : z∈upperHalfPlane := Target.diskMapInv_maps hθ (by simp)
+  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let S : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
+  have hS : IsStoppingTime ℱ S :=
+    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu
+  let g : Ω → ℝ≥0∞ := fun ω => canonicalAngleAverage ν κ v f
+    (extendedMarkedAngle κ B γ G S z ω)
+  have hfm : Measurable (canonicalAngleAverage ν κ v f) :=
+    canonicalAngleAverage_measurable ν κ v f hf
+  have hgm : Measurable[hS.measurableSpace] g := hfm.comp
+    (extendedMarkedAngle_measurable B hm hc κ γ G F hw S hS z)
+  rw [canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw
+    (u+v) (by linarith) θ hθ f hf,
+    canonicalAngleAverage_eq_finiteCentered hB hBm hc hκ hκ8 γ G F hw u hu θ hθ _ hfm]
+  have hk := radius_angle_kernel hB hBm hc h0 hκ hκ8 γ G F hw hpublished
+    z hz u v (u+v) hu hv le_rfl f hf univ MeasurableSet.univ
+  simp only [setLIntegral_univ] at hk
+  rw [← hk]
+  exact @lintegral_eq_of_stopped_events Ω mΩ
+    (finiteCenteredLaw P κ B γ G z (u+v)) (finiteCenteredLaw P κ B γ G z u)
+    hS.measurableSpace hS.measurableSpace_le
+    (fun E hE => finiteCenteredLaw_consistent_at_extended_stop hB hBm hc h0 κ hκ hκ8
+      γ G F (Eventually.of_forall hw) z hz u (u+v) hu (by linarith) S hS
+      (Eventually.of_forall fun _ => le_rfl) E hE) g hgm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/RadiusAngleBounds.lean b/lean/OAI/Probability/SLE/Angle/RadiusAngleBounds.lean
index 846f2097b857e0442060e4f0053696b98e62ac71..a58c278d9a853731d1027f2f1ec901686ea7b13e 100644
--- a/lean/OAI/Probability/SLE/Angle/RadiusAngleBounds.lean
+++ b/lean/OAI/Probability/SLE/Angle/RadiusAngleBounds.lean
@@ -1,83 +1,83 @@
-import OAI.Probability.SLE.Angle.RadiusAngleKernel
-
-namespace OAI
-
-/-! Bounds on the canonical angle transition transfer to every event in
-the actual radius-stopped past. No angular mixing bound is assumed globally. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance angleBoundsMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance angleBoundsBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-lemma finiteCenteredLaw_extended_angle_mem
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z∈upperHalfPlane) (u V : ℝ) (hu : 0≤u) (huV : u≤V) :
-    ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z V,
-      extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω∈
-        Ioo (0:ℝ) Real.pi := by
-  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
-  have hfinite : ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z V, σ ω≠⊤ :=
-    finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw)
-      z hz u V hu huV σ (Eventually.of_forall fun _ => le_rfl)
-  filter_upwards [hfinite] with ω hω
-  have hstate := actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u hω
-  have hp := (hw ω).live_marked_polar ((σ ω).untopD 0) z hstate.1
-  change extendedMarkedAngle κ B γ G σ z ω∈Ioo (0:ℝ) Real.pi
-  dsimp only [extendedMarkedAngle]
-  rw [ite_eq_right hω,stoppedMarkedCenter_eq κ B γ G _ z ω hstate.1]
-  exact hp.2.1
-
-theorem radius_angle_average_bounds
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (Function.uncurry B))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (z : ℂ) (hz : z∈upperHalfPlane) (u v V : ℝ) (hu : 0≤u) (hv : 0<v) (hV : u+v≤V)
-    (f : ℝ → ℝ≥0∞) (hf : Measurable f) (c C : ℝ≥0∞)
-    (hbound : ∀ θ∈Ioo (0:ℝ) Real.pi,
-      c≤canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ∧
-      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ≤C)
-    (A : Set Ω) (hA : MeasurableSet[(actual_radiusLevelTime_stopping hB
-      (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
-      κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A) :
-    c*finiteCenteredLaw P κ B γ G z V A ≤
-      (∫⁻ ω in A, f (extendedMarkedAngle κ B γ G
-        (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω)
-        ∂finiteCenteredLaw P κ B γ G z V) ∧
-    (∫⁻ ω in A, f (extendedMarkedAngle κ B γ G
-        (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω)
-        ∂finiteCenteredLaw P κ B γ G z V) ≤ C*finiteCenteredLaw P κ B γ G z V A := by
-  rw [← radius_angle_kernel hB hBm hc h0 hκ hκ8 γ G F hw hpublished z hz u v V hu hv hV f hf A hA]
-  have hang := finiteCenteredLaw_extended_angle_mem hB hc hκ hκ8 γ G F hw z hz u V hu
-    (by linarith)
-  constructor
-  · calc
-      _ = ∫⁻ _ω in A, c ∂finiteCenteredLaw P κ B γ G z V := by simp
-      _ ≤ _ := by
-        apply lintegral_mono_ae
-        filter_upwards [ae_restrict_of_ae hang] with ω hω
-        exact (hbound _ hω).1
-  · calc
-      _ ≤ ∫⁻ _ω in A, C ∂finiteCenteredLaw P κ B γ G z V := by
-        apply lintegral_mono_ae
-        filter_upwards [ae_restrict_of_ae hang] with ω hω
-        exact (hbound _ hω).2
-      _ = _ := by simp
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.RadiusAngleKernel
+
+namespace OAI
+
+/-! Bounds on the canonical angle transition transfer to every event in
+the actual radius-stopped past. No angular mixing bound is assumed globally. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance angleBoundsMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance angleBoundsBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+lemma finiteCenteredLaw_extended_angle_mem
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z∈upperHalfPlane) (u V : ℝ) (hu : 0≤u) (huV : u≤V) :
+    ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z V,
+      extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω∈
+        Ioo (0:ℝ) Real.pi := by
+  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
+  have hfinite : ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z V, σ ω≠⊤ :=
+    finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw)
+      z hz u V hu huV σ (Eventually.of_forall fun _ => le_rfl)
+  filter_upwards [hfinite] with ω hω
+  have hstate := actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u hω
+  have hp := (hw ω).live_marked_polar ((σ ω).untopD 0) z hstate.1
+  change extendedMarkedAngle κ B γ G σ z ω∈Ioo (0:ℝ) Real.pi
+  dsimp only [extendedMarkedAngle]
+  rw [ite_eq_right hω,stoppedMarkedCenter_eq κ B γ G _ z ω hstate.1]
+  exact hp.2.1
+
+theorem radius_angle_average_bounds
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (Function.uncurry B))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (z : ℂ) (hz : z∈upperHalfPlane) (u v V : ℝ) (hu : 0≤u) (hv : 0<v) (hV : u+v≤V)
+    (f : ℝ → ℝ≥0∞) (hf : Measurable f) (c C : ℝ≥0∞)
+    (hbound : ∀ θ∈Ioo (0:ℝ) Real.pi,
+      c≤canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ∧
+      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ≤C)
+    (A : Set Ω) (hA : MeasurableSet[(actual_radiusLevelTime_stopping hB
+      (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
+      κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A) :
+    c*finiteCenteredLaw P κ B γ G z V A ≤
+      (∫⁻ ω in A, f (extendedMarkedAngle κ B γ G
+        (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω)
+        ∂finiteCenteredLaw P κ B γ G z V) ∧
+    (∫⁻ ω in A, f (extendedMarkedAngle κ B γ G
+        (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω)
+        ∂finiteCenteredLaw P κ B γ G z V) ≤ C*finiteCenteredLaw P κ B γ G z V A := by
+  rw [← radius_angle_kernel hB hBm hc h0 hκ hκ8 γ G F hw hpublished z hz u v V hu hv hV f hf A hA]
+  have hang := finiteCenteredLaw_extended_angle_mem hB hc hκ hκ8 γ G F hw z hz u V hu
+    (by linarith)
+  constructor
+  · calc
+      _ = ∫⁻ _ω in A, c ∂finiteCenteredLaw P κ B γ G z V := by simp
+      _ ≤ _ := by
+        apply lintegral_mono_ae
+        filter_upwards [ae_restrict_of_ae hang] with ω hω
+        exact (hbound _ hω).1
+  · calc
+      _ ≤ ∫⁻ _ω in A, C ∂finiteCenteredLaw P κ B γ G z V := by
+        apply lintegral_mono_ae
+        filter_upwards [ae_restrict_of_ae hang] with ω hω
+        exact (hbound _ hω).2
+      _ = _ := by simp
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/RadiusAngleKernel.lean b/lean/OAI/Probability/SLE/Angle/RadiusAngleKernel.lean
index 24f6c5fccaf2f33995eaae01fc00ace4c7356105..45b1ba4fa105cfa0ff6eea5fd320afee838d37da 100644
--- a/lean/OAI/Probability/SLE/Angle/RadiusAngleKernel.lean
+++ b/lean/OAI/Probability/SLE/Angle/RadiusAngleKernel.lean
@@ -1,122 +1,122 @@
-import OAI.Probability.SLE.Geometry.FiniteCenteredAngleKernel
-import OAI.Probability.SLE.Gauge.PrescribedRadiusFinalLaw
-
-namespace OAI
-
-/-! The finite-centered angle transition on the full past of a literal
-radius time, and under every later finite centered law. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance radiusAngleMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance radiusAngleBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-theorem radius_angle_kernel
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (Function.uncurry B))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (z : ℂ) (hz : z∈upperHalfPlane) (u v V : ℝ) (hu : 0 ≤ u) (hv : 0 < v)
-    (hV : u+v ≤ V) (f : ℝ → ℝ≥0∞) (hf : Measurable f)
-    (A : Set Ω) (hA : MeasurableSet[(actual_radiusLevelTime_stopping hB
-      (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
-      κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A) :
-    (∫⁻ ω in A, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f
-      (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω)
-      ∂finiteCenteredLaw P κ B γ G z V) =
-    ∫⁻ ω in A, f (extendedMarkedAngle κ B γ G
-      (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω)
-      ∂finiteCenteredLaw P κ B γ G z V := by
-  classical
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
-  let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)
-  have hσ : IsStoppingTime ℱ σ :=
-    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu
-  have hT : IsStoppingTime ℱ T :=
-    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz (u+v) (by linarith)
-  let ν := P.map (sourceDriverPath B hc)
-  let : IsProbabilityMeasure ν :=
-    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
-  let Q := finiteCenteredLaw P κ B γ G z (u+v)
-  let QV := finiteCenteredLaw P κ B γ G z V
-  let L := fun ω => canonicalAngleAverage ν κ v f (extendedMarkedAngle κ B γ G σ z ω)
-  let R := fun ω => f (extendedMarkedAngle κ B γ G T z ω)
-  let E := fun n : ℕ => A∩{ω | σ ω≤((n:ℝ≥0):WithTop ℝ≥0)}
-  have hEm (n : ℕ) : MeasurableSet (E n) :=
-    hσ.measurableSpace_le _ (hA.inter (hσ.measurableSet_le' n))
-  have hmono : Monotone E := by
-    intro n k hnk ω hω
-    change ω∈A ∧ σ ω≤((n:ℝ≥0):WithTop ℝ≥0) at hω
-    change ω∈A ∧ σ ω≤((k:ℝ≥0):WithTop ℝ≥0)
-    exact ⟨hω.1,hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))⟩
-  have hfinite : ∀ᵐ ω ∂Q, σ ω≠⊤ :=
-    finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw)
-      z hz u (u+v) hu (by linarith) σ (Eventually.of_forall fun _ => le_rfl)
-  have hcover := stopped_past_cover_of_ae_finite Q σ hfinite A
-  have hlocal (n : ℕ) : (∫⁻ ω in E n, L ω ∂Q)=(∫⁻ ω in E n, R ω ∂Q) := by
-    let τ := cutoffStop σ n
-    have hτ := cutoffStop_stopping ℱ σ hσ (n:ℝ≥0)
-    have hEn := cutoffStop_success_measurable ℱ σ hσ n A hA
-    have hstate (ω : Ω) (hω : ω∈E n) :
-        z∈survivingDomain (γ ω) (τ ω) ∧
-        fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u) := by
-      rw [show τ ω=(σ ω).untopD 0 from cutoffStop_eq_on_success σ n ω hω.2]
-      exact actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u
-        (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2)
-    have hh := finite_centered_angle_kernel hB hBm hc h0 hκ hκ8 γ G F hw hpublished
-      τ hτ z hz u v hu hv (E n) hEn (fun ω hω => (hstate ω hω).1)
-      (fun ω hω => (hstate ω hω).2) f hf univ MeasurableSet.univ
-    simp only [univ_inter] at hh
-    have hid (ω : Ω) (hω : ω∈E n) : L ω=
-        canonicalAngleAverage ν κ v f (stoppedMarkedCenter κ B γ G τ z ω).arg := by
-      dsimp only [L,extendedMarkedAngle]
-      rw [ite_eq_right (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2)]
-      congr 2
-      exact (stoppedMarkedCenter_time_congr κ B γ G τ
-        (fun ω => (σ ω).untopD 0) z ω (cutoffStop_eq_on_success σ n ω hω.2)).symm
-    rw [setLIntegral_congr_fun (hEm n) hid]
-    exact hh
-  have hexhaust (g : Ω → ℝ≥0∞) : (∫⁻ ω in A, g ω ∂Q)=⨆ n, ∫⁻ ω in E n, g ω ∂Q := by
-    have habs : (Q.withDensity g).AbsolutelyContinuous Q := withDensity_absolutelyContinuous Q g
-    calc
-      _ = Q.withDensity g A := (withDensity_apply _ (hσ.measurableSpace_le _ hA)).symm
-      _ = Q.withDensity g (⋃ n, E n) := measure_congr (habs.ae_le hcover)
-      _ = ⨆ n, Q.withDensity g (E n) := hmono.measure_iUnion
-      _ = _ := by congr 1; funext n; exact withDensity_apply _ (hEm n)
-  have hbase : (∫⁻ ω in A, L ω ∂Q)=(∫⁻ ω in A, R ω ∂Q) := by
-    rw [hexhaust L,hexhaust R]
-    exact iSup_congr hlocal
-  have hσT (ω : Ω) : σ ω≤T ω :=
-    radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z hu (by linarith)
-  have hLT : Measurable[hT.measurableSpace] L :=
-    ((canonicalAngleAverage_measurable ν κ v f hf).comp
-      (extendedMarkedAngle_measurable B hm hc κ γ G F hw σ hσ z)).mono
-        (hσ.measurableSpace_mono hT hσT) le_rfl
-  have hRT : Measurable[hT.measurableSpace] R :=
-    hf.comp (extendedMarkedAngle_measurable B hm hc κ γ G F hw T hT z)
-  have hAT : MeasurableSet[hT.measurableSpace] A := hσ.measurableSpace_mono hT hσT A hA
-  have hcons (g : Ω → ℝ≥0∞) (hg : Measurable[hT.measurableSpace] g) :
-      (∫⁻ ω in A, g ω ∂QV)=(∫⁻ ω in A, g ω ∂Q) := by
-    have hh := @lintegral_eq_of_stopped_events Ω mΩ QV Q hT.measurableSpace
-      hT.measurableSpace_le (fun D hD => finiteCenteredLaw_consistent_at_radius_stop
-        hB hBm hc h0 κ hκ hκ8 γ G F (Eventually.of_forall hw) z hz (u+v) V
-        (by linarith) hV T hT (Eventually.of_forall fun _ => rfl) D hD)
-      (A.indicator g) (hg.indicator hAT)
-    simpa only [lintegral_indicator (hT.measurableSpace_le _ hAT)] using hh
-  change (∫⁻ ω in A, L ω ∂QV)=(∫⁻ ω in A, R ω ∂QV)
-  rw [hcons L hLT,hcons R hRT]
-  exact hbase
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.FiniteCenteredAngleKernel
+import OAI.Probability.SLE.Gauge.PrescribedRadiusFinalLaw
+
+namespace OAI
+
+/-! The finite-centered angle transition on the full past of a literal
+radius time, and under every later finite centered law. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance radiusAngleMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance radiusAngleBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+theorem radius_angle_kernel
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (Function.uncurry B))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (z : ℂ) (hz : z∈upperHalfPlane) (u v V : ℝ) (hu : 0 ≤ u) (hv : 0 < v)
+    (hV : u+v ≤ V) (f : ℝ → ℝ≥0∞) (hf : Measurable f)
+    (A : Set Ω) (hA : MeasurableSet[(actual_radiusLevelTime_stopping hB
+      (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
+      κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A) :
+    (∫⁻ ω in A, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f
+      (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z u) z ω)
+      ∂finiteCenteredLaw P κ B γ G z V) =
+    ∫⁻ ω in A, f (extendedMarkedAngle κ B γ G
+      (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)) z ω)
+      ∂finiteCenteredLaw P κ B γ G z V := by
+  classical
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z u
+  let T : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)
+  have hσ : IsStoppingTime ℱ σ :=
+    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu
+  have hT : IsStoppingTime ℱ T :=
+    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz (u+v) (by linarith)
+  let ν := P.map (sourceDriverPath B hc)
+  let : IsProbabilityMeasure ν :=
+    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
+  let Q := finiteCenteredLaw P κ B γ G z (u+v)
+  let QV := finiteCenteredLaw P κ B γ G z V
+  let L := fun ω => canonicalAngleAverage ν κ v f (extendedMarkedAngle κ B γ G σ z ω)
+  let R := fun ω => f (extendedMarkedAngle κ B γ G T z ω)
+  let E := fun n : ℕ => A∩{ω | σ ω≤((n:ℝ≥0):WithTop ℝ≥0)}
+  have hEm (n : ℕ) : MeasurableSet (E n) :=
+    hσ.measurableSpace_le _ (hA.inter (hσ.measurableSet_le' n))
+  have hmono : Monotone E := by
+    intro n k hnk ω hω
+    change ω∈A ∧ σ ω≤((n:ℝ≥0):WithTop ℝ≥0) at hω
+    change ω∈A ∧ σ ω≤((k:ℝ≥0):WithTop ℝ≥0)
+    exact ⟨hω.1,hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))⟩
+  have hfinite : ∀ᵐ ω ∂Q, σ ω≠⊤ :=
+    finiteCenteredLaw_earlier_stop_ae_finite hB hc κ hκ hκ8 γ G F (Eventually.of_forall hw)
+      z hz u (u+v) hu (by linarith) σ (Eventually.of_forall fun _ => le_rfl)
+  have hcover := stopped_past_cover_of_ae_finite Q σ hfinite A
+  have hlocal (n : ℕ) : (∫⁻ ω in E n, L ω ∂Q)=(∫⁻ ω in E n, R ω ∂Q) := by
+    let τ := cutoffStop σ n
+    have hτ := cutoffStop_stopping ℱ σ hσ (n:ℝ≥0)
+    have hEn := cutoffStop_success_measurable ℱ σ hσ n A hA
+    have hstate (ω : Ω) (hω : ω∈E n) :
+        z∈survivingDomain (γ ω) (τ ω) ∧
+        fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u) := by
+      rw [show τ ω=(σ ω).untopD 0 from cutoffStop_eq_on_success σ n ω hω.2]
+      exact actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z u
+        (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2)
+    have hh := finite_centered_angle_kernel hB hBm hc h0 hκ hκ8 γ G F hw hpublished
+      τ hτ z hz u v hu hv (E n) hEn (fun ω hω => (hstate ω hω).1)
+      (fun ω hω => (hstate ω hω).2) f hf univ MeasurableSet.univ
+    simp only [univ_inter] at hh
+    have hid (ω : Ω) (hω : ω∈E n) : L ω=
+        canonicalAngleAverage ν κ v f (stoppedMarkedCenter κ B γ G τ z ω).arg := by
+      dsimp only [L,extendedMarkedAngle]
+      rw [ite_eq_right (ne_top_of_le_ne_top WithTop.coe_ne_top hω.2)]
+      congr 2
+      exact (stoppedMarkedCenter_time_congr κ B γ G τ
+        (fun ω => (σ ω).untopD 0) z ω (cutoffStop_eq_on_success σ n ω hω.2)).symm
+    rw [setLIntegral_congr_fun (hEm n) hid]
+    exact hh
+  have hexhaust (g : Ω → ℝ≥0∞) : (∫⁻ ω in A, g ω ∂Q)=⨆ n, ∫⁻ ω in E n, g ω ∂Q := by
+    have habs : (Q.withDensity g).AbsolutelyContinuous Q := withDensity_absolutelyContinuous Q g
+    calc
+      _ = Q.withDensity g A := (withDensity_apply _ (hσ.measurableSpace_le _ hA)).symm
+      _ = Q.withDensity g (⋃ n, E n) := measure_congr (habs.ae_le hcover)
+      _ = ⨆ n, Q.withDensity g (E n) := hmono.measure_iUnion
+      _ = _ := by congr 1; funext n; exact withDensity_apply _ (hEm n)
+  have hbase : (∫⁻ ω in A, L ω ∂Q)=(∫⁻ ω in A, R ω ∂Q) := by
+    rw [hexhaust L,hexhaust R]
+    exact iSup_congr hlocal
+  have hσT (ω : Ω) : σ ω≤T ω :=
+    radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z hu (by linarith)
+  have hLT : Measurable[hT.measurableSpace] L :=
+    ((canonicalAngleAverage_measurable ν κ v f hf).comp
+      (extendedMarkedAngle_measurable B hm hc κ γ G F hw σ hσ z)).mono
+        (hσ.measurableSpace_mono hT hσT) le_rfl
+  have hRT : Measurable[hT.measurableSpace] R :=
+    hf.comp (extendedMarkedAngle_measurable B hm hc κ γ G F hw T hT z)
+  have hAT : MeasurableSet[hT.measurableSpace] A := hσ.measurableSpace_mono hT hσT A hA
+  have hcons (g : Ω → ℝ≥0∞) (hg : Measurable[hT.measurableSpace] g) :
+      (∫⁻ ω in A, g ω ∂QV)=(∫⁻ ω in A, g ω ∂Q) := by
+    have hh := @lintegral_eq_of_stopped_events Ω mΩ QV Q hT.measurableSpace
+      hT.measurableSpace_le (fun D hD => finiteCenteredLaw_consistent_at_radius_stop
+        hB hBm hc h0 κ hκ hκ8 γ G F (Eventually.of_forall hw) z hz (u+v) V
+        (by linarith) hV T hT (Eventually.of_forall fun _ => rfl) D hD)
+      (A.indicator g) (hg.indicator hAT)
+    simpa only [lintegral_indicator (hT.measurableSpace_le _ hAT)] using hh
+  change (∫⁻ ω in A, L ω ∂QV)=(∫⁻ ω in A, R ω ∂QV)
+  rw [hcons L hLT,hcons R hRT]
+  exact hbase
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetAngleLaw.lean b/lean/OAI/Probability/SLE/Angle/TargetAngleLaw.lean
index 5550c045d286aca0bf0d4817ff997535694f171f..25e665fac70d42a80ff0b425bb5edc5c5bc6b5be 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetAngleLaw.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetAngleLaw.lean
@@ -1,63 +1,63 @@
-import OAI.Probability.SLE.MomentGauge
-
-namespace OAI
-
-/-! The angle law in the prescribed coefficient formula is a probability
-measure supported on the nondegenerate disk charts. -/
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma angleLaw_ae_mem (κ : ℝ) : ∀ᵐ θ ∂angleLaw κ, θ ∈ Ioo (0:ℝ) Real.pi := by
-  have h : ∀ᵐ θ ∂(volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
-      (fun θ => ENNReal.ofReal (Real.sin θ ^ (4*a κ))), θ ∈ Ioo (0:ℝ) Real.pi :=
-    (ae_restrict_mem measurableSet_Ioo).filter_mono
-      (withDensity_absolutelyContinuous _ _).ae_le
-  exact Measure.ae_smul_measure h _
-
-lemma angle_normalizer_pos (κ : ℝ) :
-    0 < ((volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
-      (fun θ => ENNReal.ofReal (Real.sin θ ^ (4*a κ)))) univ := by
-  rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
-  have hm : Measurable (fun θ : ℝ => ENNReal.ofReal (Real.sin θ ^ (4*a κ))) :=
-    (measurable_id.sin.pow_const (4*a κ)).ennreal_ofReal
-  rw [setLIntegral_pos_iff hm]
-  have he : Function.support (fun θ : ℝ => ENNReal.ofReal (Real.sin θ ^ (4*a κ))) ∩
-      Ioo (0:ℝ) Real.pi = Ioo (0:ℝ) Real.pi := by
-    apply inter_eq_right.mpr
-    intro θ hθ
-    exact (ENNReal.ofReal_pos.mpr
-      (Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2) _)).ne'
-  rw [he,Real.volume_Ioo,sub_zero]
-  exact ENNReal.ofReal_pos.mpr Real.pi_pos
-
-lemma angle_normalizer_lt_top {κ : ℝ} (hκ : 0 < κ) :
-    ((volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
-      (fun θ => ENNReal.ofReal (Real.sin θ ^ (4*a κ)))) univ < ∞ := by
-  rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
-  have hpow : 0 ≤ 4*a κ := mul_nonneg (by norm_num) (div_nonneg (by norm_num) hκ.le)
-  have hle : (∫⁻ θ in Ioo (0:ℝ) Real.pi,
-      ENNReal.ofReal (Real.sin θ ^ (4*a κ))) ≤
-      ∫⁻ _ in Ioo (0:ℝ) Real.pi, (1:ℝ≥0∞) := by
-    apply lintegral_mono_ae
-    filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
-    exact (ENNReal.ofReal_le_ofReal
-      (Real.rpow_le_one (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le
-        (Real.sin_le_one θ) hpow)).trans_eq (by norm_num)
-  exact hle.trans_lt (by simp only [lintegral_const,one_mul,
-    Measure.restrict_apply_univ,Real.volume_Ioo,sub_zero]; exact ENNReal.ofReal_lt_top)
-
-theorem angleLaw_isProbability {κ : ℝ} (hκ : 0 < κ) : IsProbabilityMeasure (angleLaw κ) := by
-  constructor
-  change _⁻¹ * _ = (1:ℝ≥0∞)
-  exact ENNReal.inv_mul_cancel (angle_normalizer_pos κ).ne'
-    (angle_normalizer_lt_top hκ).ne
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.MomentGauge
+
+namespace OAI
+
+/-! The angle law in the prescribed coefficient formula is a probability
+measure supported on the nondegenerate disk charts. -/
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma angleLaw_ae_mem (κ : ℝ) : ∀ᵐ θ ∂angleLaw κ, θ ∈ Ioo (0:ℝ) Real.pi := by
+  have h : ∀ᵐ θ ∂(volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
+      (fun θ => ENNReal.ofReal (Real.sin θ ^ (4*a κ))), θ ∈ Ioo (0:ℝ) Real.pi :=
+    (ae_restrict_mem measurableSet_Ioo).filter_mono
+      (withDensity_absolutelyContinuous _ _).ae_le
+  exact Measure.ae_smul_measure h _
+
+lemma angle_normalizer_pos (κ : ℝ) :
+    0 < ((volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
+      (fun θ => ENNReal.ofReal (Real.sin θ ^ (4*a κ)))) univ := by
+  rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
+  have hm : Measurable (fun θ : ℝ => ENNReal.ofReal (Real.sin θ ^ (4*a κ))) :=
+    (measurable_id.sin.pow_const (4*a κ)).ennreal_ofReal
+  rw [setLIntegral_pos_iff hm]
+  have he : Function.support (fun θ : ℝ => ENNReal.ofReal (Real.sin θ ^ (4*a κ))) ∩
+      Ioo (0:ℝ) Real.pi = Ioo (0:ℝ) Real.pi := by
+    apply inter_eq_right.mpr
+    intro θ hθ
+    exact (ENNReal.ofReal_pos.mpr
+      (Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2) _)).ne'
+  rw [he,Real.volume_Ioo,sub_zero]
+  exact ENNReal.ofReal_pos.mpr Real.pi_pos
+
+lemma angle_normalizer_lt_top {κ : ℝ} (hκ : 0 < κ) :
+    ((volume.restrict (Ioo (0:ℝ) Real.pi)).withDensity
+      (fun θ => ENNReal.ofReal (Real.sin θ ^ (4*a κ)))) univ < ∞ := by
+  rw [withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
+  have hpow : 0 ≤ 4*a κ := mul_nonneg (by norm_num) (div_nonneg (by norm_num) hκ.le)
+  have hle : (∫⁻ θ in Ioo (0:ℝ) Real.pi,
+      ENNReal.ofReal (Real.sin θ ^ (4*a κ))) ≤
+      ∫⁻ _ in Ioo (0:ℝ) Real.pi, (1:ℝ≥0∞) := by
+    apply lintegral_mono_ae
+    filter_upwards [ae_restrict_mem measurableSet_Ioo] with θ hθ
+    exact (ENNReal.ofReal_le_ofReal
+      (Real.rpow_le_one (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le
+        (Real.sin_le_one θ) hpow)).trans_eq (by norm_num)
+  exact hle.trans_lt (by simp only [lintegral_const,one_mul,
+    Measure.restrict_apply_univ,Real.volume_Ioo,sub_zero]; exact ENNReal.ofReal_lt_top)
+
+theorem angleLaw_isProbability {κ : ℝ} (hκ : 0 < κ) : IsProbabilityMeasure (angleLaw κ) := by
+  constructor
+  change _⁻¹ * _ = (1:ℝ≥0∞)
+  exact ENNReal.inv_mul_cancel (angle_normalizer_pos κ).ne'
+    (angle_normalizer_lt_top hκ).ne
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetBrownianNormalization.lean b/lean/OAI/Probability/SLE/Angle/TargetBrownianNormalization.lean
index 06aac61d0a31cf49bc207c4be4b310eda3ef57db..6d36a56a05be3dc9d66b34272c966d14f2c05890 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetBrownianNormalization.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetBrownianNormalization.lean
@@ -1,97 +1,97 @@
-import OAI.Probability.SLE.Loewner.TargetDriverContinuation
-
-namespace OAI
-
-/-! The Brownian time scaling in the prescribed flow, and uniqueness of
-the based Wiener law used by its deterministic coefficient formula. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-local instance normalizationPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance normalizationPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-def normalizedDriver {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (ω : Ω) : Driver where
-  toFun t := Real.sqrt κ*B (t/κ.toNNReal) ω
-  continuous_toFun := continuous_const.mul ((hc ω).comp (continuous_id.div_const _))
-  map_zero' := by simp [h0]
-
-lemma normalizedDriver_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) :
-    Measurable (normalizedDriver κ B hc h0) := by
-  apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp
-  exact source_path_measurable _ (fun t => measurable_const.mul (hm _))
-
-lemma normalizedDriver_matches {Ω : Type*} (κ : ℝ) (hκ : 0<κ)
-    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (ω : Ω) (s : ℝ) (_hs : 0 ≤ s) :
-    Real.sqrt κ*B s.toNNReal ω=driverValue (normalizedDriver κ B hc h0 ω) (κ*s) := by
-  change _=Real.sqrt κ*B ((κ*s).toNNReal/κ.toNNReal) ω
-  rw [Real.toNNReal_mul hκ.le,mul_div_cancel_left₀ _ (Real.toNNReal_pos.mpr hκ).ne']
-
-theorem normalizedDriver_isBrownian {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} (κ : ℝ) (hκ : 0<κ) (B : ℝ≥0 → Ω → ℝ)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) :
-    IsBrownianReal (fun t ω => normalizedDriver κ B hc h0 ω t) P := by
-  have h := hB.smul (c := (κ.toNNReal)⁻¹)
-    (inv_ne_zero (Real.toNNReal_pos.mpr hκ).ne')
-  change IsBrownianReal (fun t ω => Real.sqrt κ*B (t/κ.toNNReal) ω) P
-  simpa only [NNReal.coe_inv,
-    Real.coe_toNNReal _ hκ.le,Real.sqrt_inv,inv_inv,div_eq_mul_inv,mul_comm] using h
-
-theorem normalizedDriver_isWiener {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} (κ : ℝ) (hκ : 0<κ) (B : ℝ≥0 → Ω → ℝ)
-    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) :
-    IsWiener (P.map (normalizedDriver κ B hc h0)) := by
-  have hbp := normalizedDriver_measurable κ B hm hc h0
-  have hBrownian := normalizedDriver_isBrownian κ hκ B hB hc h0
-  refine ⟨⟨fun I => ?_⟩,Eventually.of_forall (fun b => map_continuous b)⟩
-  have hI : Measurable (fun b : Driver => I.restrict (fun t => b t)) :=
-    Measurable.of_eval (fun i => (continuous_eval_const (i:ℝ≥0)).measurable)
-  refine ⟨hI.aemeasurable,?_⟩
-  rw [Measure.map_map hI hbp]
-  exact (hBrownian.hasLaw I).map_eq
-
-theorem isWiener_unique (W V : WienerLaw) (hW : IsWiener W) (hV : IsWiener V) : W=V := by
-  have hm : Measurable (fun b : Driver => fun t => b t) :=
-    Measurable.of_eval (fun t => (continuous_eval_const t).measurable)
-  have hleft : IsProjectiveLimit (W.map (fun b : Driver => fun t => b t))
-      BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map (Finset.measurable_restrict I) hm]
-    exact (hW.hasLaw I).map_eq
-  have hright : IsProjectiveLimit (V.map (fun b : Driver => fun t => b t))
-      BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map (Finset.measurable_restrict I) hm]
-    exact (hV.hasLaw I).map_eq
-  have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)))
-    (hleft.unique hright)
-  have hd : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) :=
-    Measurable.of_eval (fun _ => measurable_pi_apply _)
-  rw [Measure.map_map hd hm,Measure.map_map hd hm] at he
-  have hemb := (source_path_evaluation_embedding (E := ℝ)).comp
-    ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding
-  exact hemb.map_injective he
-
-theorem normalizedDriver_law {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} (κ : ℝ) (hκ : 0<κ) (B : ℝ≥0 → Ω → ℝ)
-    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (W : WienerLaw) (hW : IsWiener W) :
-    HasLaw (normalizedDriver κ B hc h0) W P :=
-  ⟨(normalizedDriver_measurable κ B hm hc h0).aemeasurable,
-    isWiener_unique _ _ (normalizedDriver_isWiener κ hκ B hB hm hc h0) hW⟩
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.TargetDriverContinuation
+
+namespace OAI
+
+/-! The Brownian time scaling in the prescribed flow, and uniqueness of
+the based Wiener law used by its deterministic coefficient formula. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+local instance normalizationPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance normalizationPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+def normalizedDriver {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (ω : Ω) : Driver where
+  toFun t := Real.sqrt κ*B (t/κ.toNNReal) ω
+  continuous_toFun := continuous_const.mul ((hc ω).comp (continuous_id.div_const _))
+  map_zero' := by simp [h0]
+
+lemma normalizedDriver_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) :
+    Measurable (normalizedDriver κ B hc h0) := by
+  apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp
+  exact source_path_measurable _ (fun t => measurable_const.mul (hm _))
+
+lemma normalizedDriver_matches {Ω : Type*} (κ : ℝ) (hκ : 0<κ)
+    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (ω : Ω) (s : ℝ) (_hs : 0 ≤ s) :
+    Real.sqrt κ*B s.toNNReal ω=driverValue (normalizedDriver κ B hc h0 ω) (κ*s) := by
+  change _=Real.sqrt κ*B ((κ*s).toNNReal/κ.toNNReal) ω
+  rw [Real.toNNReal_mul hκ.le,mul_div_cancel_left₀ _ (Real.toNNReal_pos.mpr hκ).ne']
+
+theorem normalizedDriver_isBrownian {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} (κ : ℝ) (hκ : 0<κ) (B : ℝ≥0 → Ω → ℝ)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) :
+    IsBrownianReal (fun t ω => normalizedDriver κ B hc h0 ω t) P := by
+  have h := hB.smul (c := (κ.toNNReal)⁻¹)
+    (inv_ne_zero (Real.toNNReal_pos.mpr hκ).ne')
+  change IsBrownianReal (fun t ω => Real.sqrt κ*B (t/κ.toNNReal) ω) P
+  simpa only [NNReal.coe_inv,
+    Real.coe_toNNReal _ hκ.le,Real.sqrt_inv,inv_inv,div_eq_mul_inv,mul_comm] using h
+
+theorem normalizedDriver_isWiener {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} (κ : ℝ) (hκ : 0<κ) (B : ℝ≥0 → Ω → ℝ)
+    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) :
+    IsWiener (P.map (normalizedDriver κ B hc h0)) := by
+  have hbp := normalizedDriver_measurable κ B hm hc h0
+  have hBrownian := normalizedDriver_isBrownian κ hκ B hB hc h0
+  refine ⟨⟨fun I => ?_⟩,Eventually.of_forall (fun b => map_continuous b)⟩
+  have hI : Measurable (fun b : Driver => I.restrict (fun t => b t)) :=
+    Measurable.of_eval (fun i => (continuous_eval_const (i:ℝ≥0)).measurable)
+  refine ⟨hI.aemeasurable,?_⟩
+  rw [Measure.map_map hI hbp]
+  exact (hBrownian.hasLaw I).map_eq
+
+theorem isWiener_unique (W V : WienerLaw) (hW : IsWiener W) (hV : IsWiener V) : W=V := by
+  have hm : Measurable (fun b : Driver => fun t => b t) :=
+    Measurable.of_eval (fun t => (continuous_eval_const t).measurable)
+  have hleft : IsProjectiveLimit (W.map (fun b : Driver => fun t => b t))
+      BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map (Finset.measurable_restrict I) hm]
+    exact (hW.hasLaw I).map_eq
+  have hright : IsProjectiveLimit (V.map (fun b : Driver => fun t => b t))
+      BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map (Finset.measurable_restrict I) hm]
+    exact (hV.hasLaw I).map_eq
+  have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)))
+    (hleft.unique hright)
+  have hd : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) :=
+    Measurable.of_eval (fun _ => measurable_pi_apply _)
+  rw [Measure.map_map hd hm,Measure.map_map hd hm] at he
+  have hemb := (source_path_evaluation_embedding (E := ℝ)).comp
+    ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding
+  exact hemb.map_injective he
+
+theorem normalizedDriver_law {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} (κ : ℝ) (hκ : 0<κ) (B : ℝ≥0 → Ω → ℝ)
+    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (W : WienerLaw) (hW : IsWiener W) :
+    HasLaw (normalizedDriver κ B hc h0) W P :=
+  ⟨(normalizedDriver_measurable κ B hm hc h0).aemeasurable,
+    isWiener_unique _ _ (normalizedDriver_isWiener κ hκ B hB hm hc h0) hW⟩
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetClockIdentification.lean b/lean/OAI/Probability/SLE/Angle/TargetClockIdentification.lean
index 803061868b1590bca867988a5ea74099d77b7b79..a95d109d0873fbe7550cd29b8240b83f0d406f6c 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetClockIdentification.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetClockIdentification.lean
@@ -1,137 +1,137 @@
-import OAI.Probability.SLE.Angle.TargetFlowIdentification
-import OAI.Probability.SLE.Multipoint.MultipointRegulatorTransport
-
-namespace OAI
-
-/-! Exact clock and radius normalization for the source-prescribed flow. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Target
-
-lemma clock_eq_activeRadiusClock
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (x : ℂ) (T : ℝ≥0)
-    (hz : diskMapInv θ x ∈ survivingDomain γ T) :
-    clock κ θ b x (κ*T)=κ*Multipoint.activeRadiusClock G (diskMapInv θ x) 0 T := by
-  let z := diskMapInv θ x
-  let f (s : ℝ) := chartPoissonRate U G (s/κ).toNNReal z ^ 2
-  let C (s : ℝ) := κ*Multipoint.activeRadiusClock G z 0 (s/κ)
-  have hT : 0 ≤ κ*(T:ℝ) := mul_nonneg hκ.le T.coe_nonneg
-  have hmaps : MapsTo (fun s : ℝ => s/κ) (Icc 0 (κ*T)) (Icc 0 (T:ℝ)) := by
-    intro s hs
-    exact ⟨div_nonneg hs.1 hκ.le,(div_le_iff₀ hκ).2 (by simpa only [mul_comm] using hs.2)⟩
-  have hfc : ContinuousOn f (Icc 0 (κ*T)) :=
-    ((hw.poissonRate_continuousOn hUc T z hz).pow 2).comp
-      (continuous_id.div_const κ).continuousOn hmaps
-  have hCc : ContinuousOn C (Icc 0 (κ*T)) := continuous_const.continuousOn.mul
-    ((hw.activeRadiusClock_continuousOn hUc 0 T z hz).comp
-      (continuous_id.div_const κ).continuousOn hmaps)
-  have hCd (s : ℝ) (hs : s ∈ Ioo 0 (κ*T)) : HasDerivAt C (f s) s := by
-    have hinner : HasDerivWithinAt (fun r : ℝ => r/κ) (1/κ)
-        (Icc 0 (κ*T)) s := ((hasDerivAt_id s).div_const κ).hasDerivWithinAt
-    have hd := (hw.activeRadiusClock_hasDerivWithinAt hUc 0 T z hz (s/κ)
-      (hmaps (Ioo_subset_Icc_self hs))).scomp s hinner hmaps
-    have hd' : HasDerivWithinAt C (f s) (Icc 0 (κ*T)) s := by
-      apply (hd.const_mul κ).congr_deriv
-      dsimp only [f]
-      simp only [smul_eq_mul]
-      field_simp [hκ.ne']
-    exact hd'.hasDerivAt (Icc_mem_nhds hs.1 hs.2)
-  have hi := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hT hCc hCd
-    (hfc.intervalIntegrable_of_Icc hT)
-  have hpoint (s : ℝ) (hs : s ∈ Icc 0 (κ*T)) :
-      (flow κ b z s).im^2/‖centeredFlow κ b z s‖^4=f s := by
-    have hst : (s/κ).toNNReal ≤ T :=
-      (Real.toNNReal_le_iff_le_coe).2 (hmaps hs).2
-    have hlive := survivingDomain_antitone γ hst hz
-    have hf := flow_eq_witness_real_time κ hκ b U γ G F hw hU s hs.1 z hlive
-    have hu : U (s/κ).toNNReal=driverValue b s := by
-      rw [hU _ (hmaps hs).1,mul_div_cancel₀ _ hκ.ne']
-    rw [centeredFlow,hf]
-    simp only [f,chartPoissonRate,hu,div_pow]
-    norm_num only [← pow_mul]
-  have hzero : Multipoint.activeRadiusClock G z 0 (0:ℝ)=0 := by
-    simp [Multipoint.activeRadiusClock]
-  calc
-    clock κ θ b x (κ*T) = ∫ s in Icc 0 (κ*T), f s :=
-      setIntegral_congr_fun measurableSet_Icc hpoint
-    _ = ∫ s in 0..(κ*T), f s := by
-      rw [intervalIntegral.integral_of_le hT,integral_Icc_eq_integral_Ioc]
-    _ = C (κ*T)-C 0 := hi
-    _ = κ*Multipoint.activeRadiusClock G z 0 T := by
-      simp only [C,mul_div_cancel_left₀ _ hκ.ne',zero_div,hzero,mul_zero,sub_zero]
-
-lemma radius_eq_physicalConformalRadius
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (x : ℂ) (T : ℝ≥0)
-    (hz : diskMapInv θ x ∈ survivingDomain γ T) :
-    radius κ θ b x (κ*T)=physicalConformalRadius (diskMap θ) G T (diskMapInv θ x) := by
-  let z := diskMapInv θ x
-  have hz0 := survivingDomain_antitone γ (show 0 ≤ T from zero_le) hz
-  have hp := hw.physical_radius_active_clock hUc (diskMap θ) 0 T z hz0 hz
-  have hi : physicalConformalRadius (diskMap θ) G 0 z=initialRadius θ x := by
-    rw [physicalConformalRadius,hw.fullRadius_initial hUc z hz0]
-    dsimp only [initialRadius,z]
-    ring
-  rw [hp,hi,radius,clock_eq_activeRadiusClock κ hκ b U hUc γ G F hw hU θ x T hz]
-  congr 2
-  dsimp only [a,z]
-  field_simp [hκ.ne']
-  ring
-
-lemma angle_eq_witness
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (x : ℂ) (T : ℝ≥0)
-    (hz : diskMapInv θ x ∈ survivingDomain γ T) :
-    angle κ θ b x (κ*T)=Complex.arg (G T (diskMapInv θ x)-(U T:ℂ)) := by
-  have hu : driverValue b (κ*T)=U T := by
-    simpa only [Real.toNNReal_coe] using (hU T T.coe_nonneg).symm
-  rw [angle,centeredFlow,flow_eq_witness κ hκ b U γ G F hw hU T _ hz,hu]
-
-private lemma alpha_eq_codimension (κ : ℝ) (_hκ : 0 < κ) : α κ=codimension κ := by
-  dsimp only [α,d,a,codimension,dimension]
-  field_simp
-  ring
-
-private lemma p_eq_greenExponent (κ : ℝ) : p κ=greenExponent κ := by
-  dsimp only [p,a,greenExponent]
-  ring
-
-lemma green_eq_transported_density
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (x : ℂ) (T : ℝ≥0) :
-    green κ θ b x (κ*T)=‖deriv (diskMap θ) (diskMapInv θ x)‖^(-codimension κ)*
-      greenDensity κ U γ G T (diskMapInv θ x) := by
-  classical
-  by_cases hz : diskMapInv θ x ∈ survivingDomain γ T
-  · have hs := (surviving_iff_witness κ hκ b U γ G F hw hU T _).2 hz
-    rw [green,ite_eq_left hs,greenDensity,ite_eq_left hz,
-      radius_eq_physicalConformalRadius κ hκ b U hUc γ G F hw hU θ x T hz,
-      angle_eq_witness κ hκ b U γ G F hw hU θ x T hz,
-      alpha_eq_codimension κ hκ,p_eq_greenExponent κ,physicalConformalRadius,
-      Real.mul_rpow (norm_nonneg _) (hw.fullRadius_positive hUc T _ hz).le]
-    ring
-  · have hs : ¬Surviving κ b (diskMapInv θ x) (κ*T) :=
-      fun h => hz ((surviving_iff_witness κ hκ b U γ G F hw hU T _).1 h)
-    rw [green,ite_eq_right hs,greenDensity,ite_eq_right hz,mul_zero]
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetFlowIdentification
+import OAI.Probability.SLE.Multipoint.MultipointRegulatorTransport
+
+namespace OAI
+
+/-! Exact clock and radius normalization for the source-prescribed flow. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Target
+
+lemma clock_eq_activeRadiusClock
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (x : ℂ) (T : ℝ≥0)
+    (hz : diskMapInv θ x ∈ survivingDomain γ T) :
+    clock κ θ b x (κ*T)=κ*Multipoint.activeRadiusClock G (diskMapInv θ x) 0 T := by
+  let z := diskMapInv θ x
+  let f (s : ℝ) := chartPoissonRate U G (s/κ).toNNReal z ^ 2
+  let C (s : ℝ) := κ*Multipoint.activeRadiusClock G z 0 (s/κ)
+  have hT : 0 ≤ κ*(T:ℝ) := mul_nonneg hκ.le T.coe_nonneg
+  have hmaps : MapsTo (fun s : ℝ => s/κ) (Icc 0 (κ*T)) (Icc 0 (T:ℝ)) := by
+    intro s hs
+    exact ⟨div_nonneg hs.1 hκ.le,(div_le_iff₀ hκ).2 (by simpa only [mul_comm] using hs.2)⟩
+  have hfc : ContinuousOn f (Icc 0 (κ*T)) :=
+    ((hw.poissonRate_continuousOn hUc T z hz).pow 2).comp
+      (continuous_id.div_const κ).continuousOn hmaps
+  have hCc : ContinuousOn C (Icc 0 (κ*T)) := continuous_const.continuousOn.mul
+    ((hw.activeRadiusClock_continuousOn hUc 0 T z hz).comp
+      (continuous_id.div_const κ).continuousOn hmaps)
+  have hCd (s : ℝ) (hs : s ∈ Ioo 0 (κ*T)) : HasDerivAt C (f s) s := by
+    have hinner : HasDerivWithinAt (fun r : ℝ => r/κ) (1/κ)
+        (Icc 0 (κ*T)) s := ((hasDerivAt_id s).div_const κ).hasDerivWithinAt
+    have hd := (hw.activeRadiusClock_hasDerivWithinAt hUc 0 T z hz (s/κ)
+      (hmaps (Ioo_subset_Icc_self hs))).scomp s hinner hmaps
+    have hd' : HasDerivWithinAt C (f s) (Icc 0 (κ*T)) s := by
+      apply (hd.const_mul κ).congr_deriv
+      dsimp only [f]
+      simp only [smul_eq_mul]
+      field_simp [hκ.ne']
+    exact hd'.hasDerivAt (Icc_mem_nhds hs.1 hs.2)
+  have hi := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hT hCc hCd
+    (hfc.intervalIntegrable_of_Icc hT)
+  have hpoint (s : ℝ) (hs : s ∈ Icc 0 (κ*T)) :
+      (flow κ b z s).im^2/‖centeredFlow κ b z s‖^4=f s := by
+    have hst : (s/κ).toNNReal ≤ T :=
+      (Real.toNNReal_le_iff_le_coe).2 (hmaps hs).2
+    have hlive := survivingDomain_antitone γ hst hz
+    have hf := flow_eq_witness_real_time κ hκ b U γ G F hw hU s hs.1 z hlive
+    have hu : U (s/κ).toNNReal=driverValue b s := by
+      rw [hU _ (hmaps hs).1,mul_div_cancel₀ _ hκ.ne']
+    rw [centeredFlow,hf]
+    simp only [f,chartPoissonRate,hu,div_pow]
+    norm_num only [← pow_mul]
+  have hzero : Multipoint.activeRadiusClock G z 0 (0:ℝ)=0 := by
+    simp [Multipoint.activeRadiusClock]
+  calc
+    clock κ θ b x (κ*T) = ∫ s in Icc 0 (κ*T), f s :=
+      setIntegral_congr_fun measurableSet_Icc hpoint
+    _ = ∫ s in 0..(κ*T), f s := by
+      rw [intervalIntegral.integral_of_le hT,integral_Icc_eq_integral_Ioc]
+    _ = C (κ*T)-C 0 := hi
+    _ = κ*Multipoint.activeRadiusClock G z 0 T := by
+      simp only [C,mul_div_cancel_left₀ _ hκ.ne',zero_div,hzero,mul_zero,sub_zero]
+
+lemma radius_eq_physicalConformalRadius
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (x : ℂ) (T : ℝ≥0)
+    (hz : diskMapInv θ x ∈ survivingDomain γ T) :
+    radius κ θ b x (κ*T)=physicalConformalRadius (diskMap θ) G T (diskMapInv θ x) := by
+  let z := diskMapInv θ x
+  have hz0 := survivingDomain_antitone γ (show 0 ≤ T from zero_le) hz
+  have hp := hw.physical_radius_active_clock hUc (diskMap θ) 0 T z hz0 hz
+  have hi : physicalConformalRadius (diskMap θ) G 0 z=initialRadius θ x := by
+    rw [physicalConformalRadius,hw.fullRadius_initial hUc z hz0]
+    dsimp only [initialRadius,z]
+    ring
+  rw [hp,hi,radius,clock_eq_activeRadiusClock κ hκ b U hUc γ G F hw hU θ x T hz]
+  congr 2
+  dsimp only [a,z]
+  field_simp [hκ.ne']
+  ring
+
+lemma angle_eq_witness
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (x : ℂ) (T : ℝ≥0)
+    (hz : diskMapInv θ x ∈ survivingDomain γ T) :
+    angle κ θ b x (κ*T)=Complex.arg (G T (diskMapInv θ x)-(U T:ℂ)) := by
+  have hu : driverValue b (κ*T)=U T := by
+    simpa only [Real.toNNReal_coe] using (hU T T.coe_nonneg).symm
+  rw [angle,centeredFlow,flow_eq_witness κ hκ b U γ G F hw hU T _ hz,hu]
+
+private lemma alpha_eq_codimension (κ : ℝ) (_hκ : 0 < κ) : α κ=codimension κ := by
+  dsimp only [α,d,a,codimension,dimension]
+  field_simp
+  ring
+
+private lemma p_eq_greenExponent (κ : ℝ) : p κ=greenExponent κ := by
+  dsimp only [p,a,greenExponent]
+  ring
+
+lemma green_eq_transported_density
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (x : ℂ) (T : ℝ≥0) :
+    green κ θ b x (κ*T)=‖deriv (diskMap θ) (diskMapInv θ x)‖^(-codimension κ)*
+      greenDensity κ U γ G T (diskMapInv θ x) := by
+  classical
+  by_cases hz : diskMapInv θ x ∈ survivingDomain γ T
+  · have hs := (surviving_iff_witness κ hκ b U γ G F hw hU T _).2 hz
+    rw [green,ite_eq_left hs,greenDensity,ite_eq_left hz,
+      radius_eq_physicalConformalRadius κ hκ b U hUc γ G F hw hU θ x T hz,
+      angle_eq_witness κ hκ b U γ G F hw hU θ x T hz,
+      alpha_eq_codimension κ hκ,p_eq_greenExponent κ,physicalConformalRadius,
+      Real.mul_rpow (norm_nonneg _) (hw.fullRadius_positive hUc T _ hz).le]
+    ring
+  · have hs : ¬Surviving κ b (diskMapInv θ x) (κ*T) :=
+      fun h => hz ((surviving_iff_witness κ hκ b U γ G F hw hU T _).1 h)
+    rw [green,ite_eq_right hs,greenDensity,ite_eq_right hz,mul_zero]
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetDiskMap.lean b/lean/OAI/Probability/SLE/Angle/TargetDiskMap.lean
index 5b3e939d1842c6eed2e5e12fbcaea3eac0c80174..b84229bf6dbb8ac943c343d8bca9703633fb660f 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetDiskMap.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetDiskMap.lean
@@ -1,108 +1,108 @@
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLE.DiskHalfPlane
-
-namespace OAI
-
-/-! The exact disk chart appearing in the prescribed coefficient formula. -/
-
-noncomputable section
-open Set Filter Metric Complex
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma disk_exp_conj (θ : ℝ) :
-    Complex.exp (-θ*Complex.I) = star (Complex.exp (θ*Complex.I)) := by
-  rw [Complex.star_def,← Complex.exp_conj]
-  congr 1
-  simp
-
-lemma disk_center_height {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    0 < (Complex.exp (θ*Complex.I)).im := by
-  rw [Complex.exp_ofReal_mul_I_im]
-  exact Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-
-lemma diskMap_denominator_ne {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {z : ℂ} (hz : z ∈ upperHalfPlane) : z-Complex.exp (-θ*Complex.I) ≠ 0 := by
-  intro he
-  have hh := congrArg Complex.im (sub_eq_zero.mp he)
-  rw [disk_exp_conj,Complex.star_def,Complex.conj_im] at hh
-  have hw := disk_center_height hθ
-  change 0 < z.im at hz
-  linarith
-
-lemma diskMap_normSq_gap (θ : ℝ) (z : ℂ) :
-    Complex.normSq (z-Complex.exp (-θ*Complex.I))-
-      Complex.normSq (z-Complex.exp (θ*Complex.I)) =
-      4*z.im*(Complex.exp (θ*Complex.I)).im := by
-  rw [disk_exp_conj]
-  simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
-    Complex.star_def,Complex.conj_re,Complex.conj_im]
-  ring
-
-lemma diskMap_maps {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    MapsTo (diskMap θ) upperHalfPlane (ball 0 1) := by
-  intro z hz
-  rw [mem_ball_zero_iff,diskMap,norm_div]
-  have hd := diskMap_denominator_ne hθ hz
-  apply (div_lt_one (norm_pos_iff.mpr hd)).mpr
-  have hgap := diskMap_normSq_gap θ z
-  rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at hgap
-  have hp : 0 < 4*z.im*(Complex.exp (θ*Complex.I)).im :=
-    mul_pos (mul_pos (by norm_num) hz) (disk_center_height hθ)
-  nlinarith [norm_nonneg (z-Complex.exp (-θ*Complex.I)),
-    norm_nonneg (z-Complex.exp (θ*Complex.I))]
-
-lemma diskMapInv_eq_diskToHalfPlane (θ : ℝ) (x : ℂ) :
-    diskMapInv θ x = diskToHalfPlane (Complex.exp (θ*Complex.I)) x := by
-  rw [diskMapInv,diskToHalfPlane,disk_exp_conj]
-  congr 2
-  ring
-
-lemma diskMapInv_maps {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    MapsTo (diskMapInv θ) (ball 0 1) upperHalfPlane := by
-  intro x hx
-  rw [diskMapInv_eq_diskToHalfPlane]
-  exact diskToHalfPlane_maps (disk_center_height hθ) hx
-
-lemma diskMapInv_diskMap {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {z : ℂ} (hz : z ∈ upperHalfPlane) : diskMapInv θ (diskMap θ z)=z := by
-  have hd := diskMap_denominator_ne hθ hz
-  have hx := disk_denominator_ne (diskMap_maps hθ hz)
-  dsimp only [diskMapInv]
-  apply (div_eq_iff hx).mpr
-  dsimp only [diskMap]
-  have hd' : z-Complex.exp (-(θ*Complex.I)) ≠ 0 := by simpa only [neg_mul] using hd
-  field_simp [hd']
-  ring
-
-lemma diskMap_diskMapInv {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {x : ℂ} (hx : x ∈ ball 0 1) : diskMap θ (diskMapInv θ x)=x := by
-  have hd := disk_denominator_ne hx
-  have hz := diskMap_denominator_ne hθ (diskMapInv_maps hθ hx)
-  dsimp only [diskMap]
-  apply (div_eq_iff hz).mpr
-  dsimp only [diskMapInv]
-  field_simp
-  ring
-
-def diskHomeomorph (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    upperHalfPlane ≃ₜ (ball (0:ℂ) 1) where
-  toFun z := ⟨diskMap θ z,diskMap_maps hθ z.property⟩
-  invFun x := ⟨diskMapInv θ x,diskMapInv_maps hθ x.property⟩
-  left_inv z := Subtype.ext (diskMapInv_diskMap hθ z.property)
-  right_inv x := Subtype.ext (diskMap_diskMapInv hθ x.property)
-  continuous_toFun := by
-    apply Continuous.subtype_mk
-    exact (continuous_subtype_val.sub continuous_const).div
-      (continuous_subtype_val.sub continuous_const) (fun z => diskMap_denominator_ne hθ z.property)
-  continuous_invFun := by
-    apply Continuous.subtype_mk
-    exact (continuous_const.sub (continuous_subtype_val.mul continuous_const)).div
-      (continuous_const.sub continuous_subtype_val) (fun x => disk_denominator_ne x.property)
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLE.DiskHalfPlane
+
+namespace OAI
+
+/-! The exact disk chart appearing in the prescribed coefficient formula. -/
+
+noncomputable section
+open Set Filter Metric Complex
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma disk_exp_conj (θ : ℝ) :
+    Complex.exp (-θ*Complex.I) = star (Complex.exp (θ*Complex.I)) := by
+  rw [Complex.star_def,← Complex.exp_conj]
+  congr 1
+  simp
+
+lemma disk_center_height {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    0 < (Complex.exp (θ*Complex.I)).im := by
+  rw [Complex.exp_ofReal_mul_I_im]
+  exact Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+
+lemma diskMap_denominator_ne {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {z : ℂ} (hz : z ∈ upperHalfPlane) : z-Complex.exp (-θ*Complex.I) ≠ 0 := by
+  intro he
+  have hh := congrArg Complex.im (sub_eq_zero.mp he)
+  rw [disk_exp_conj,Complex.star_def,Complex.conj_im] at hh
+  have hw := disk_center_height hθ
+  change 0 < z.im at hz
+  linarith
+
+lemma diskMap_normSq_gap (θ : ℝ) (z : ℂ) :
+    Complex.normSq (z-Complex.exp (-θ*Complex.I))-
+      Complex.normSq (z-Complex.exp (θ*Complex.I)) =
+      4*z.im*(Complex.exp (θ*Complex.I)).im := by
+  rw [disk_exp_conj]
+  simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
+    Complex.star_def,Complex.conj_re,Complex.conj_im]
+  ring
+
+lemma diskMap_maps {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    MapsTo (diskMap θ) upperHalfPlane (ball 0 1) := by
+  intro z hz
+  rw [mem_ball_zero_iff,diskMap,norm_div]
+  have hd := diskMap_denominator_ne hθ hz
+  apply (div_lt_one (norm_pos_iff.mpr hd)).mpr
+  have hgap := diskMap_normSq_gap θ z
+  rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at hgap
+  have hp : 0 < 4*z.im*(Complex.exp (θ*Complex.I)).im :=
+    mul_pos (mul_pos (by norm_num) hz) (disk_center_height hθ)
+  nlinarith [norm_nonneg (z-Complex.exp (-θ*Complex.I)),
+    norm_nonneg (z-Complex.exp (θ*Complex.I))]
+
+lemma diskMapInv_eq_diskToHalfPlane (θ : ℝ) (x : ℂ) :
+    diskMapInv θ x = diskToHalfPlane (Complex.exp (θ*Complex.I)) x := by
+  rw [diskMapInv,diskToHalfPlane,disk_exp_conj]
+  congr 2
+  ring
+
+lemma diskMapInv_maps {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    MapsTo (diskMapInv θ) (ball 0 1) upperHalfPlane := by
+  intro x hx
+  rw [diskMapInv_eq_diskToHalfPlane]
+  exact diskToHalfPlane_maps (disk_center_height hθ) hx
+
+lemma diskMapInv_diskMap {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {z : ℂ} (hz : z ∈ upperHalfPlane) : diskMapInv θ (diskMap θ z)=z := by
+  have hd := diskMap_denominator_ne hθ hz
+  have hx := disk_denominator_ne (diskMap_maps hθ hz)
+  dsimp only [diskMapInv]
+  apply (div_eq_iff hx).mpr
+  dsimp only [diskMap]
+  have hd' : z-Complex.exp (-(θ*Complex.I)) ≠ 0 := by simpa only [neg_mul] using hd
+  field_simp [hd']
+  ring
+
+lemma diskMap_diskMapInv {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {x : ℂ} (hx : x ∈ ball 0 1) : diskMap θ (diskMapInv θ x)=x := by
+  have hd := disk_denominator_ne hx
+  have hz := diskMap_denominator_ne hθ (diskMapInv_maps hθ hx)
+  dsimp only [diskMap]
+  apply (div_eq_iff hz).mpr
+  dsimp only [diskMapInv]
+  field_simp
+  ring
+
+def diskHomeomorph (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    upperHalfPlane ≃ₜ (ball (0:ℂ) 1) where
+  toFun z := ⟨diskMap θ z,diskMap_maps hθ z.property⟩
+  invFun x := ⟨diskMapInv θ x,diskMapInv_maps hθ x.property⟩
+  left_inv z := Subtype.ext (diskMapInv_diskMap hθ z.property)
+  right_inv x := Subtype.ext (diskMap_diskMapInv hθ x.property)
+  continuous_toFun := by
+    apply Continuous.subtype_mk
+    exact (continuous_subtype_val.sub continuous_const).div
+      (continuous_subtype_val.sub continuous_const) (fun z => diskMap_denominator_ne hθ z.property)
+  continuous_invFun := by
+    apply Continuous.subtype_mk
+    exact (continuous_const.sub (continuous_subtype_val.mul continuous_const)).div
+      (continuous_const.sub continuous_subtype_val) (fun x => disk_denominator_ne x.property)
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetDiskMass.lean b/lean/OAI/Probability/SLE/Angle/TargetDiskMass.lean
index 634335c1860b7ac56769f1bcf3331dbf6aa7f83e..72b7c6285cd611c273a5e3a381768aeb4518b34f 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetDiskMass.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetDiskMass.lean
@@ -1,105 +1,105 @@
-import OAI.Probability.SLE.Angle.TargetDiskPotential
-import OAI.Probability.SLE.Geometry.WeightedCompactTerminal
-import OAI.Probability.SLE.Geometry.RestartHalfPlaneGreen
-
-namespace OAI
-
-/-! The exact compact and continuous conformal weight whose canonical
-mass has the source-prescribed disk Green potential. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-def diskSourceHalfPlane (θ : ℝ) : Set upperHalfPlane :=
-  Subtype.val ⁻¹' diskSourceCompact θ
-
-lemma diskSourceHalfPlane_compact {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    IsCompact (diskSourceHalfPlane θ) := by
-  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
-  have he : Subtype.val '' diskSourceHalfPlane θ=diskSourceCompact θ := by
-    ext z
-    constructor
-    · rintro ⟨x,hx,rfl⟩
-      exact hx
-    · intro hz
-      exact ⟨⟨z,diskSourceCompact_subset hθ hz⟩,hz,rfl⟩
-  rw [he]
-  exact diskSourceCompact_compact θ
-
-lemma diskMap_deriv_continuous {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    Continuous (fun z : upperHalfPlane => deriv (diskMap θ) z) := by
-  have he : (fun z : upperHalfPlane => deriv (diskMap θ) z)=
-      (fun z : upperHalfPlane => (Complex.exp (θ*Complex.I)-Complex.exp (-θ*Complex.I))/
-        ((z:ℂ)-Complex.exp (-θ*Complex.I))^2) :=
-    funext (fun z => (diskMap_hasDerivAt hθ z.property).deriv)
-  rw [he]
-  exact continuous_const.div ((continuous_subtype_val.sub continuous_const).pow 2)
-    (fun z => pow_ne_zero _ (diskMap_denominator_ne hθ z.property))
-
-def diskMassWeight (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) : C(upperHalfPlane,ℝ≥0) where
-  toFun z := (‖deriv (diskMap θ) z‖^dimension κ).toNNReal
-  continuous_toFun := continuous_real_toNNReal.comp
-    ((diskMap_deriv_continuous hθ).norm.rpow_const
-      (fun z => Or.inl (diskMap_deriv_pos hθ z.property).ne'))
-
-@[simp] lemma diskMassWeight_coe (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    (z : upperHalfPlane) :
-    (diskMassWeight κ θ hθ z:ℝ)=‖deriv (diskMap θ) z‖^dimension κ := by
-  exact Real.coe_toNNReal _ (Real.rpow_nonneg (norm_nonneg _) _)
-
-lemma diskMassWeight_ennreal (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    (z : upperHalfPlane) :
-    (diskMassWeight κ θ hθ z:ℝ≥0∞)=ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ) := by
-  rw [← ENNReal.ofReal_coe_nnreal,diskMassWeight_coe]
-
-theorem potential_eq_weighted_green
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : ℝ≥0) :
-    potential κ θ b (κ*T)=
-      ∫⁻ z in diskSourceHalfPlane θ, (diskMassWeight κ θ hθ z:ℝ≥0∞)
-        ∂(volume.withDensity (fun z => ENNReal.ofReal (greenDensity κ U γ G T z))).comap Subtype.val := by
-  have hK := diskSourceHalfPlane_compact hθ
-  rw [← lintegral_indicator hK.measurableSet,
-    Multipoint.halfPlane_realDensity_lintegral _ (Multipoint.currentGreen_measurable hw hUc hκ T)]
-  rw [potential_eq_source_green_integral κ hκ b U hUc γ G F hw hU θ hθ T]
-  symm
-  calc
-    _ = ∫⁻ z in upperHalfPlane, (diskSourceCompact θ).indicator
-        (fun z => ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ)*
-          ENNReal.ofReal (greenDensity κ U γ G T z)) z := by
-      apply setLIntegral_congr_fun isOpen_upperHalfPlane.measurableSet
-      intro z hz
-      simp only [Multipoint.extendHalfPlaneENN,dite_eq_left hz]
-      by_cases hzk : z ∈ diskSourceCompact θ
-      · rw [Set.indicator_of_mem (show (⟨z,hz⟩:upperHalfPlane) ∈ diskSourceHalfPlane θ from hzk),
-          Set.indicator_of_mem hzk,diskMassWeight_ennreal]
-      · rw [Set.indicator_of_notMem (show (⟨z,hz⟩:upperHalfPlane) ∉ diskSourceHalfPlane θ from hzk),
-          Set.indicator_of_notMem hzk,zero_mul]
-    _ = _ := by
-      rw [lintegral_indicator (diskSourceCompact_compact θ).measurableSet,
-        Measure.restrict_restrict (diskSourceCompact_compact θ).measurableSet,
-        inter_eq_left.mpr (diskSourceCompact_subset hθ)]
-
-theorem potential_toReal_eq_compactWeightedMass
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : ℝ≥0) :
-    (potential κ θ b (κ*T)).toReal = Multipoint.compactWeightedMass
-      (diskMassWeight κ θ hθ) (diskSourceHalfPlane θ)
-      ((volume.withDensity (fun z => ENNReal.ofReal (greenDensity κ U γ G T z))).comap Subtype.val) := by
-  rw [potential_eq_weighted_green κ hκ b U hUc γ G F hw hU θ hθ T,
-    Multipoint.compactWeightedMass_eq_lintegral]
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetDiskPotential
+import OAI.Probability.SLE.Geometry.WeightedCompactTerminal
+import OAI.Probability.SLE.Geometry.RestartHalfPlaneGreen
+
+namespace OAI
+
+/-! The exact compact and continuous conformal weight whose canonical
+mass has the source-prescribed disk Green potential. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+def diskSourceHalfPlane (θ : ℝ) : Set upperHalfPlane :=
+  Subtype.val ⁻¹' diskSourceCompact θ
+
+lemma diskSourceHalfPlane_compact {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    IsCompact (diskSourceHalfPlane θ) := by
+  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
+  have he : Subtype.val '' diskSourceHalfPlane θ=diskSourceCompact θ := by
+    ext z
+    constructor
+    · rintro ⟨x,hx,rfl⟩
+      exact hx
+    · intro hz
+      exact ⟨⟨z,diskSourceCompact_subset hθ hz⟩,hz,rfl⟩
+  rw [he]
+  exact diskSourceCompact_compact θ
+
+lemma diskMap_deriv_continuous {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    Continuous (fun z : upperHalfPlane => deriv (diskMap θ) z) := by
+  have he : (fun z : upperHalfPlane => deriv (diskMap θ) z)=
+      (fun z : upperHalfPlane => (Complex.exp (θ*Complex.I)-Complex.exp (-θ*Complex.I))/
+        ((z:ℂ)-Complex.exp (-θ*Complex.I))^2) :=
+    funext (fun z => (diskMap_hasDerivAt hθ z.property).deriv)
+  rw [he]
+  exact continuous_const.div ((continuous_subtype_val.sub continuous_const).pow 2)
+    (fun z => pow_ne_zero _ (diskMap_denominator_ne hθ z.property))
+
+def diskMassWeight (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) : C(upperHalfPlane,ℝ≥0) where
+  toFun z := (‖deriv (diskMap θ) z‖^dimension κ).toNNReal
+  continuous_toFun := continuous_real_toNNReal.comp
+    ((diskMap_deriv_continuous hθ).norm.rpow_const
+      (fun z => Or.inl (diskMap_deriv_pos hθ z.property).ne'))
+
+@[simp] lemma diskMassWeight_coe (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    (z : upperHalfPlane) :
+    (diskMassWeight κ θ hθ z:ℝ)=‖deriv (diskMap θ) z‖^dimension κ := by
+  exact Real.coe_toNNReal _ (Real.rpow_nonneg (norm_nonneg _) _)
+
+lemma diskMassWeight_ennreal (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    (z : upperHalfPlane) :
+    (diskMassWeight κ θ hθ z:ℝ≥0∞)=ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ) := by
+  rw [← ENNReal.ofReal_coe_nnreal,diskMassWeight_coe]
+
+theorem potential_eq_weighted_green
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : ℝ≥0) :
+    potential κ θ b (κ*T)=
+      ∫⁻ z in diskSourceHalfPlane θ, (diskMassWeight κ θ hθ z:ℝ≥0∞)
+        ∂(volume.withDensity (fun z => ENNReal.ofReal (greenDensity κ U γ G T z))).comap Subtype.val := by
+  have hK := diskSourceHalfPlane_compact hθ
+  rw [← lintegral_indicator hK.measurableSet,
+    Multipoint.halfPlane_realDensity_lintegral _ (Multipoint.currentGreen_measurable hw hUc hκ T)]
+  rw [potential_eq_source_green_integral κ hκ b U hUc γ G F hw hU θ hθ T]
+  symm
+  calc
+    _ = ∫⁻ z in upperHalfPlane, (diskSourceCompact θ).indicator
+        (fun z => ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ)*
+          ENNReal.ofReal (greenDensity κ U γ G T z)) z := by
+      apply setLIntegral_congr_fun isOpen_upperHalfPlane.measurableSet
+      intro z hz
+      simp only [Multipoint.extendHalfPlaneENN,dite_eq_left hz]
+      by_cases hzk : z ∈ diskSourceCompact θ
+      · rw [Set.indicator_of_mem (show (⟨z,hz⟩:upperHalfPlane) ∈ diskSourceHalfPlane θ from hzk),
+          Set.indicator_of_mem hzk,diskMassWeight_ennreal]
+      · rw [Set.indicator_of_notMem (show (⟨z,hz⟩:upperHalfPlane) ∉ diskSourceHalfPlane θ from hzk),
+          Set.indicator_of_notMem hzk,zero_mul]
+    _ = _ := by
+      rw [lintegral_indicator (diskSourceCompact_compact θ).measurableSet,
+        Measure.restrict_restrict (diskSourceCompact_compact θ).measurableSet,
+        inter_eq_left.mpr (diskSourceCompact_subset hθ)]
+
+theorem potential_toReal_eq_compactWeightedMass
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : ℝ≥0) :
+    (potential κ θ b (κ*T)).toReal = Multipoint.compactWeightedMass
+      (diskMassWeight κ θ hθ) (diskSourceHalfPlane θ)
+      ((volume.withDensity (fun z => ENNReal.ofReal (greenDensity κ U γ G T z))).comap Subtype.val) := by
+  rw [potential_eq_weighted_green κ hκ b U hUc γ G F hw hU θ hθ T,
+    Multipoint.compactWeightedMass_eq_lintegral]
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetDiskPotential.lean b/lean/OAI/Probability/SLE/Angle/TargetDiskPotential.lean
index 358dda41b2897aab3ea8dd85f4f02b67a5dd81d9..43a1ed0b390653d199a117e9eebbe44af13ee4e0 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetDiskPotential.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetDiskPotential.lean
@@ -1,90 +1,90 @@
-import OAI.Probability.SLE.Angle.TargetClockIdentification
-import OAI.Probability.SLE.Angle.TargetDiskRadius
-import OAI.Probability.SLE.Geometry.RestartGreenIntegral
-
-namespace OAI
-
-/-! The prescribed disk potential is exactly the conformally weighted
-current Green mass of the compact disk preimage. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-def diskSourceCompact (θ : ℝ) : Set ℂ := diskMapInv θ '' diskK
-
-lemma diskK_subset_unit : diskK ⊆ ball (0:ℂ) 1 := by
-  intro x hx
-  exact (closedBall_subset_ball (by norm_num : (1:ℝ)/4<1)) hx
-
-lemma diskSourceCompact_subset {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    diskSourceCompact θ ⊆ upperHalfPlane := by
-  rintro z ⟨x,hx,rfl⟩
-  exact diskMapInv_maps hθ (diskK_subset_unit hx)
-
-lemma diskSourceCompact_compact (θ : ℝ) : IsCompact (diskSourceCompact θ) := by
-  change IsCompact (diskMapInv θ '' closedBall (0:ℂ) (1/4))
-  apply (isCompact_closedBall (0:ℂ) (1/4)).image_of_continuousOn
-  change ContinuousOn (fun x : ℂ =>
-    (Complex.exp (θ*Complex.I)-x*Complex.exp (-θ*Complex.I))/(1-x)) _
-  apply ContinuousOn.div
-  · exact continuous_const.continuousOn.sub (continuous_id.mul continuous_const).continuousOn
-  · exact (continuous_const.sub continuous_id).continuousOn
-  · intro x hx
-    exact disk_denominator_ne (diskK_subset_unit hx)
-
-lemma diskSourceCompact_image {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    diskMap θ '' diskSourceCompact θ=diskK := by
-  ext x
-  constructor
-  · rintro ⟨z,⟨y,hy,rfl⟩,rfl⟩
-    simpa only [diskMap_diskMapInv hθ (diskK_subset_unit hy)] using hy
-  · intro hx
-    exact ⟨diskMapInv θ x,⟨x,hx,rfl⟩,diskMap_diskMapInv hθ (diskK_subset_unit hx)⟩
-
-lemma diskMap_deriv_pos {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {z : ℂ} (hz : z ∈ upperHalfPlane) : 0<‖deriv (diskMap θ) z‖ := by
-  rw [diskMap_deriv_norm hθ hz]
-  exact div_pos (mul_pos (by norm_num) (disk_center_height hθ))
-    (sq_pos_of_pos (norm_pos_iff.mpr (diskMap_denominator_ne hθ hz)))
-
-theorem potential_eq_source_green_integral
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : ℝ≥0) :
-    potential κ θ b (κ*T) =
-      ∫⁻ z in diskSourceCompact θ,
-        ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ) *
-          ENNReal.ofReal (greenDensity κ U γ G T z) := by
-  have hK := diskSourceCompact_compact θ
-  have hsub := diskSourceCompact_subset hθ
-  have hd (z : ℂ) (hz : z ∈ diskSourceCompact θ) := diskMap_hasDerivAt hθ (hsub hz)
-  rw [potential,← diskSourceCompact_image hθ,
-    lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hK.measurableSet
-      (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt)
-      ((diskMap_injOn hθ).mono hsub)]
-  apply setLIntegral_congr_fun hK.measurableSet
-  intro z hz
-  dsimp only
-  rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _),
-    green_eq_transported_density κ hκ b U hUc γ G F hw hU θ (diskMap θ z) T,
-    diskMapInv_diskMap hθ (hsub hz),
-    ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _),← mul_assoc]
-  congr 1
-  rw [← ENNReal.ofReal_mul (sq_nonneg _)]
-  congr 1
-  rw [← (hd z hz).deriv,← Real.rpow_natCast,
-    ← Real.rpow_add (diskMap_deriv_pos hθ (hsub hz))]
-  congr 1
-  simp only [codimension,dimension]
-  ring
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetClockIdentification
+import OAI.Probability.SLE.Angle.TargetDiskRadius
+import OAI.Probability.SLE.Geometry.RestartGreenIntegral
+
+namespace OAI
+
+/-! The prescribed disk potential is exactly the conformally weighted
+current Green mass of the compact disk preimage. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+def diskSourceCompact (θ : ℝ) : Set ℂ := diskMapInv θ '' diskK
+
+lemma diskK_subset_unit : diskK ⊆ ball (0:ℂ) 1 := by
+  intro x hx
+  exact (closedBall_subset_ball (by norm_num : (1:ℝ)/4<1)) hx
+
+lemma diskSourceCompact_subset {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    diskSourceCompact θ ⊆ upperHalfPlane := by
+  rintro z ⟨x,hx,rfl⟩
+  exact diskMapInv_maps hθ (diskK_subset_unit hx)
+
+lemma diskSourceCompact_compact (θ : ℝ) : IsCompact (diskSourceCompact θ) := by
+  change IsCompact (diskMapInv θ '' closedBall (0:ℂ) (1/4))
+  apply (isCompact_closedBall (0:ℂ) (1/4)).image_of_continuousOn
+  change ContinuousOn (fun x : ℂ =>
+    (Complex.exp (θ*Complex.I)-x*Complex.exp (-θ*Complex.I))/(1-x)) _
+  apply ContinuousOn.div
+  · exact continuous_const.continuousOn.sub (continuous_id.mul continuous_const).continuousOn
+  · exact (continuous_const.sub continuous_id).continuousOn
+  · intro x hx
+    exact disk_denominator_ne (diskK_subset_unit hx)
+
+lemma diskSourceCompact_image {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    diskMap θ '' diskSourceCompact θ=diskK := by
+  ext x
+  constructor
+  · rintro ⟨z,⟨y,hy,rfl⟩,rfl⟩
+    simpa only [diskMap_diskMapInv hθ (diskK_subset_unit hy)] using hy
+  · intro hx
+    exact ⟨diskMapInv θ x,⟨x,hx,rfl⟩,diskMap_diskMapInv hθ (diskK_subset_unit hx)⟩
+
+lemma diskMap_deriv_pos {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {z : ℂ} (hz : z ∈ upperHalfPlane) : 0<‖deriv (diskMap θ) z‖ := by
+  rw [diskMap_deriv_norm hθ hz]
+  exact div_pos (mul_pos (by norm_num) (disk_center_height hθ))
+    (sq_pos_of_pos (norm_pos_iff.mpr (diskMap_denominator_ne hθ hz)))
+
+theorem potential_eq_source_green_integral
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : ℝ≥0) :
+    potential κ θ b (κ*T) =
+      ∫⁻ z in diskSourceCompact θ,
+        ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ) *
+          ENNReal.ofReal (greenDensity κ U γ G T z) := by
+  have hK := diskSourceCompact_compact θ
+  have hsub := diskSourceCompact_subset hθ
+  have hd (z : ℂ) (hz : z ∈ diskSourceCompact θ) := diskMap_hasDerivAt hθ (hsub hz)
+  rw [potential,← diskSourceCompact_image hθ,
+    lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hK.measurableSet
+      (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt)
+      ((diskMap_injOn hθ).mono hsub)]
+  apply setLIntegral_congr_fun hK.measurableSet
+  intro z hz
+  dsimp only
+  rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _),
+    green_eq_transported_density κ hκ b U hUc γ G F hw hU θ (diskMap θ z) T,
+    diskMapInv_diskMap hθ (hsub hz),
+    ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _),← mul_assoc]
+  congr 1
+  rw [← ENNReal.ofReal_mul (sq_nonneg _)]
+  congr 1
+  rw [← (hd z hz).deriv,← Real.rpow_natCast,
+    ← Real.rpow_add (diskMap_deriv_pos hθ (hsub hz))]
+  congr 1
+  simp only [codimension,dimension]
+  ring
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetDiskRadius.lean b/lean/OAI/Probability/SLE/Angle/TargetDiskRadius.lean
index 3eaee87dc01b9b35c5622290361ebada90b51f3c..f726ecea5981e9ce68dc5d36eaa08f7642b3c286 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetDiskRadius.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetDiskRadius.lean
@@ -1,74 +1,74 @@
-import OAI.Probability.SLE.Angle.TargetDiskMap
-
-namespace OAI
-
-/-! The prescribed disk chart has the angle-independent initial
-conformal radius needed for the centered moment bounds. -/
-
-noncomputable section
-open Set Filter Metric Complex
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma diskMap_hasDerivAt {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {z : ℂ} (hz : z ∈ upperHalfPlane) :
-    HasDerivAt (diskMap θ)
-      ((Complex.exp (θ*Complex.I)-Complex.exp (-θ*Complex.I))/
-        (z-Complex.exp (-θ*Complex.I))^2) z := by
-  have hh := ((hasDerivAt_id z).sub_const (Complex.exp (θ*Complex.I))).div
-    ((hasDerivAt_id z).sub_const (Complex.exp (-θ*Complex.I)))
-    (diskMap_denominator_ne hθ hz)
-  convert hh using 1 <;> first | rfl | (dsimp; ring)
-
-lemma diskMap_differentiableOn {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    DifferentiableOn ℂ (diskMap θ) upperHalfPlane := by
-  intro z hz
-  exact (diskMap_hasDerivAt hθ hz).differentiableAt.differentiableWithinAt
-
-lemma diskMap_injOn {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    InjOn (diskMap θ) upperHalfPlane := by
-  intro z hz v hv he
-  have h := congrArg (diskMapInv θ) he
-  simpa only [diskMapInv_diskMap hθ hz,diskMapInv_diskMap hθ hv] using h
-
-lemma diskMap_deriv_norm {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {z : ℂ} (hz : z ∈ upperHalfPlane) :
-    ‖deriv (diskMap θ) z‖ =
-      2*(Complex.exp (θ*Complex.I)).im/‖z-Complex.exp (-θ*Complex.I)‖^2 := by
-  rw [(diskMap_hasDerivAt hθ hz).deriv,norm_div,norm_pow,disk_exp_conj,
-    norm_sub_conj_of_im_pos (disk_center_height hθ)]
-
-lemma diskMap_initial_radius {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {z : ℂ} (hz : z ∈ upperHalfPlane) :
-    2*z.im*‖deriv (diskMap θ) z‖ = 1-‖diskMap θ z‖^2 := by
-  have hd := norm_ne_zero_iff.mpr (diskMap_denominator_ne hθ hz)
-  have hg := diskMap_normSq_gap θ z
-  rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at hg
-  rw [diskMap_deriv_norm hθ hz,diskMap,norm_div]
-  have hd' : ‖z-Complex.exp (-(θ*Complex.I))‖ ≠ 0 := by simpa only [neg_mul] using hd
-  field_simp [hd']
-  simp only [neg_mul] at hg
-  nlinarith
-
-theorem initialRadius_disk {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {x : ℂ} (hx : x ∈ ball 0 1) : initialRadius θ x = 1-‖x‖^2 := by
-  have h := diskMap_initial_radius hθ (diskMapInv_maps hθ hx)
-  simpa only [initialRadius,diskMap_diskMapInv hθ hx] using h
-
-lemma initialRadius_zero {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
-    initialRadius θ 0=1 := by
-  simpa only [norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0),sub_zero] using
-    initialRadius_disk hθ (mem_ball_zero_iff.mpr (by norm_num : ‖(0:ℂ)‖<1))
-
-lemma initialRadius_pos {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
-    {x : ℂ} (hx : x ∈ ball 0 1) : 0 < initialRadius θ x := by
-  rw [initialRadius_disk hθ hx]
-  have hn := mem_ball_zero_iff.mp hx
-  nlinarith [norm_nonneg x]
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetDiskMap
+
+namespace OAI
+
+/-! The prescribed disk chart has the angle-independent initial
+conformal radius needed for the centered moment bounds. -/
+
+noncomputable section
+open Set Filter Metric Complex
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma diskMap_hasDerivAt {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {z : ℂ} (hz : z ∈ upperHalfPlane) :
+    HasDerivAt (diskMap θ)
+      ((Complex.exp (θ*Complex.I)-Complex.exp (-θ*Complex.I))/
+        (z-Complex.exp (-θ*Complex.I))^2) z := by
+  have hh := ((hasDerivAt_id z).sub_const (Complex.exp (θ*Complex.I))).div
+    ((hasDerivAt_id z).sub_const (Complex.exp (-θ*Complex.I)))
+    (diskMap_denominator_ne hθ hz)
+  convert hh using 1 <;> first | rfl | (dsimp; ring)
+
+lemma diskMap_differentiableOn {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    DifferentiableOn ℂ (diskMap θ) upperHalfPlane := by
+  intro z hz
+  exact (diskMap_hasDerivAt hθ hz).differentiableAt.differentiableWithinAt
+
+lemma diskMap_injOn {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    InjOn (diskMap θ) upperHalfPlane := by
+  intro z hz v hv he
+  have h := congrArg (diskMapInv θ) he
+  simpa only [diskMapInv_diskMap hθ hz,diskMapInv_diskMap hθ hv] using h
+
+lemma diskMap_deriv_norm {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {z : ℂ} (hz : z ∈ upperHalfPlane) :
+    ‖deriv (diskMap θ) z‖ =
+      2*(Complex.exp (θ*Complex.I)).im/‖z-Complex.exp (-θ*Complex.I)‖^2 := by
+  rw [(diskMap_hasDerivAt hθ hz).deriv,norm_div,norm_pow,disk_exp_conj,
+    norm_sub_conj_of_im_pos (disk_center_height hθ)]
+
+lemma diskMap_initial_radius {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {z : ℂ} (hz : z ∈ upperHalfPlane) :
+    2*z.im*‖deriv (diskMap θ) z‖ = 1-‖diskMap θ z‖^2 := by
+  have hd := norm_ne_zero_iff.mpr (diskMap_denominator_ne hθ hz)
+  have hg := diskMap_normSq_gap θ z
+  rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at hg
+  rw [diskMap_deriv_norm hθ hz,diskMap,norm_div]
+  have hd' : ‖z-Complex.exp (-(θ*Complex.I))‖ ≠ 0 := by simpa only [neg_mul] using hd
+  field_simp [hd']
+  simp only [neg_mul] at hg
+  nlinarith
+
+theorem initialRadius_disk {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {x : ℂ} (hx : x ∈ ball 0 1) : initialRadius θ x = 1-‖x‖^2 := by
+  have h := diskMap_initial_radius hθ (diskMapInv_maps hθ hx)
+  simpa only [initialRadius,diskMap_diskMapInv hθ hx] using h
+
+lemma initialRadius_zero {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) :
+    initialRadius θ 0=1 := by
+  simpa only [norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0),sub_zero] using
+    initialRadius_disk hθ (mem_ball_zero_iff.mpr (by norm_num : ‖(0:ℂ)‖<1))
+
+lemma initialRadius_pos {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi)
+    {x : ℂ} (hx : x ∈ ball 0 1) : 0 < initialRadius θ x := by
+  rw [initialRadius_disk hθ hx]
+  have hn := mem_ball_zero_iff.mp hx
+  nlinarith [norm_nonneg x]
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetDiskVisit.lean b/lean/OAI/Probability/SLE/Angle/TargetDiskVisit.lean
index eab93a2ae4c2d4fe6296e831ab797208aa1f17c3..515751b3d101e8c75333faa42037e7c1d4925848 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetDiskVisit.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetDiskVisit.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Angle.TargetDiskMass
-import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
-
-namespace OAI
-
-/-! A fixed relative half-plane ball lies inside the prescribed disk compact.
-Reaching a sufficiently small radius level forces a positive-time visit there. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma diskMapInv_zero (θ : ℝ) : diskMapInv θ 0=Complex.exp (θ*Complex.I) := by
-  simp [diskMapInv]
-
-lemma marked_ball_subset_diskSource {θ : ℝ} (hθ : θ∈Ioo (0:ℝ) Real.pi) :
-    ball (diskMapInv θ 0) ((diskMapInv θ 0).im/4) ⊆ diskSourceCompact θ := by
-  intro z hz
-  rw [diskMapInv_zero] at hz
-  let c := Complex.exp (θ*Complex.I)
-  have hc : 0<c.im := disk_center_height hθ
-  have hnorm : ‖z-c‖<c.im/4 := mem_ball_iff_norm.mp hz
-  have hzH : z∈upperHalfPlane := ball_subset_upperHalfPlane (by linarith) hz
-  have htri : 2*c.im≤‖z-c‖+‖z-star c‖ := by
-    calc
-      2*c.im=‖c-star c‖ := (norm_sub_conj_of_im_pos hc).symm
-      _=‖(c-z)+(z-star c)‖ := by congr 1; abel
-      _≤‖c-z‖+‖z-star c‖ := norm_add_le _ _
-      _=‖z-c‖+‖z-star c‖ := by rw [norm_sub_rev c z]
-  have hden := norm_pos_iff.mpr (diskMap_denominator_ne hθ hzH)
-  have hsmall : ‖diskMap θ z‖<1/4 := by
-    rw [diskMap,norm_div]
-    apply (div_lt_iff₀ hden).2
-    rw [disk_exp_conj]
-    change ‖z-c‖ < 1/4*‖z-star c‖
-    nlinarith
-  refine ⟨diskMap θ z,?_,diskMapInv_diskMap hθ hzH⟩
-  exact ball_subset_closedBall (mem_ball_zero_iff.mpr hsmall)
-
-lemma radius_level_visits_diskSource
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (u : ℝ)
-    (hu : Real.exp (-u)<1/8)
-    (hf : radiusLevelTime γ G (diskMapInv θ 0) u≠∞) :
-    ∃ s : ℝ≥0, 0<s ∧ s≤(radiusLevelTime γ G (diskMapInv θ 0) u).toNNReal ∧
-      γ s∈ball (diskMapInv θ 0) ((diskMapInv θ 0).im/4) := by
-  let z := diskMapInv θ 0
-  have hz : 0<z.im := diskMapInv_maps hθ (show (0:ℂ)∈ball 0 1 by simp)
-  obtain ⟨hlive,hR⟩ := actual_radius_level_finite_state hw hU z u hf
-  have hsmall : fullRadius G (radiusLevelTime γ G z u).toNNReal z<z.im/4 := by
-    rw [hR]
-    nlinarith
-  obtain ⟨s,hs,hnear⟩ := hw.trace_approach_of_radius_lt hU _ z hlive (z.im/4)
-    (by positivity) (by linarith) hsmall
-  have hmem : γ s∈ball z (z.im/4) := hnear
-  refine ⟨s,?_,hs.2,hmem⟩
-  apply pos_iff_ne_zero.mpr
-  intro he
-  rw [he,hw.2.1] at hmem
-  have hzero := ball_subset_upperHalfPlane (by linarith : z.im/4≤z.im) hmem
-  exact (lt_irrefl (0:ℝ)) hzero
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetDiskMass
+import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
+
+namespace OAI
+
+/-! A fixed relative half-plane ball lies inside the prescribed disk compact.
+Reaching a sufficiently small radius level forces a positive-time visit there. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma diskMapInv_zero (θ : ℝ) : diskMapInv θ 0=Complex.exp (θ*Complex.I) := by
+  simp [diskMapInv]
+
+lemma marked_ball_subset_diskSource {θ : ℝ} (hθ : θ∈Ioo (0:ℝ) Real.pi) :
+    ball (diskMapInv θ 0) ((diskMapInv θ 0).im/4) ⊆ diskSourceCompact θ := by
+  intro z hz
+  rw [diskMapInv_zero] at hz
+  let c := Complex.exp (θ*Complex.I)
+  have hc : 0<c.im := disk_center_height hθ
+  have hnorm : ‖z-c‖<c.im/4 := mem_ball_iff_norm.mp hz
+  have hzH : z∈upperHalfPlane := ball_subset_upperHalfPlane (by linarith) hz
+  have htri : 2*c.im≤‖z-c‖+‖z-star c‖ := by
+    calc
+      2*c.im=‖c-star c‖ := (norm_sub_conj_of_im_pos hc).symm
+      _=‖(c-z)+(z-star c)‖ := by congr 1; abel
+      _≤‖c-z‖+‖z-star c‖ := norm_add_le _ _
+      _=‖z-c‖+‖z-star c‖ := by rw [norm_sub_rev c z]
+  have hden := norm_pos_iff.mpr (diskMap_denominator_ne hθ hzH)
+  have hsmall : ‖diskMap θ z‖<1/4 := by
+    rw [diskMap,norm_div]
+    apply (div_lt_iff₀ hden).2
+    rw [disk_exp_conj]
+    change ‖z-c‖ < 1/4*‖z-star c‖
+    nlinarith
+  refine ⟨diskMap θ z,?_,diskMapInv_diskMap hθ hzH⟩
+  exact ball_subset_closedBall (mem_ball_zero_iff.mpr hsmall)
+
+lemma radius_level_visits_diskSource
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (u : ℝ)
+    (hu : Real.exp (-u)<1/8)
+    (hf : radiusLevelTime γ G (diskMapInv θ 0) u≠∞) :
+    ∃ s : ℝ≥0, 0<s ∧ s≤(radiusLevelTime γ G (diskMapInv θ 0) u).toNNReal ∧
+      γ s∈ball (diskMapInv θ 0) ((diskMapInv θ 0).im/4) := by
+  let z := diskMapInv θ 0
+  have hz : 0<z.im := diskMapInv_maps hθ (show (0:ℂ)∈ball 0 1 by simp)
+  obtain ⟨hlive,hR⟩ := actual_radius_level_finite_state hw hU z u hf
+  have hsmall : fullRadius G (radiusLevelTime γ G z u).toNNReal z<z.im/4 := by
+    rw [hR]
+    nlinarith
+  obtain ⟨s,hs,hnear⟩ := hw.trace_approach_of_radius_lt hU _ z hlive (z.im/4)
+    (by positivity) (by linarith) hsmall
+  have hmem : γ s∈ball z (z.im/4) := hnear
+  refine ⟨s,?_,hs.2,hmem⟩
+  apply pos_iff_ne_zero.mpr
+  intro he
+  rw [he,hw.2.1] at hmem
+  have hzero := ball_subset_upperHalfPlane (by linarith : z.im/4≤z.im) hmem
+  exact (lt_irrefl (0:ℝ)) hzero
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetFlowEvolution.lean b/lean/OAI/Probability/SLE/Angle/TargetFlowEvolution.lean
index 3acf87ad772eca1199ab4be99792cac90414db26..b4725870a6f4166581f7a32fa254b29c41f75733 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetFlowEvolution.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetFlowEvolution.lean
@@ -1,92 +1,92 @@
-import OAI.Probability.SLE.Angle.TargetPositiveFlow
-import OAI.Probability.SLE.Loewner.TargetInverseProperties
-
-namespace OAI
-
-/-! On each surviving time interval the selected raw flow is a single
-positive cutoff trajectory. -/
-noncomputable section
-open Set Filter
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Target
-
-lemma flow_eq_cutoff_on_interval
-    (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (_hT : 0 ≤ T)
-    (j : ℕ) (hj : 1 ≤ j)
-    (hmin : (j:ℝ)⁻¹ < sInf ((fun s : ℝ => (cutoffFlow κ b j z s).im) '' Icc 0 T))
-    (u : ℝ) (hu : u ∈ Icc 0 T) :
-    flow κ b z u=cutoffFlow κ b j z u := by
-  have hheight := surviving_cutoff_height κ b z T j hj hmin
-  have hj0 : (0:ℝ)<j := by exact_mod_cast (Nat.succ_le_iff.mp hj)
-  apply (flow_eq_of_positive_trajectory κ b z u hu.1 (cutoffFlow κ b j z)
-    ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self)
-    (cutoffFlow_zero κ b j hj z) ((j:ℝ)⁻¹) (inv_pos.mpr hj0)
-    (fun s hs => (hheight s ⟨hs.1,hs.2.trans hu.2⟩).le) ?_).2
-  intro s hs
-  have hd := (cutoffFlow_hasDerivWithinAt κ b j hj z s hs.1).mono
-    (Ici_subset_Ici.mpr hs.1)
-  rw [cutoffField_eq_reciprocal κ b j s _
-    (cutoff_inactive_of_height j hj _ _
-      (hheight s ⟨hs.1,hs.2.le.trans hu.2⟩).le)] at hd
-  exact hd
-
-lemma flow_hasDerivWithinAt_of_later_survival
-    (κ : ℝ) (b : Driver) (z : ℂ) (T S : ℝ) (hT : 0 ≤ T) (hTS : T<S)
-    (hs : Surviving κ b z S) :
-    HasDerivWithinAt (flow κ b z)
-      ((a κ:ℂ)/(flow κ b z T-driverValue b T)) (Ici 0) T := by
-  obtain ⟨j,hj,hmin⟩ := hs
-  have hS : 0 ≤ S := hT.trans hTS.le
-  have he : flow κ b z =ᶠ[𝓝[Ici 0] T] cutoffFlow κ b j z := by
-    filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (gt_mem_nhds hTS)] with u hu huS
-    exact flow_eq_cutoff_on_interval κ b z S hS j hj hmin u ⟨hu,huS.le⟩
-  have hval := flow_eq_cutoff_on_interval κ b z S hS j hj hmin T ⟨hT,hTS.le⟩
-  have hd := cutoffFlow_hasDerivWithinAt κ b j hj z T hT
-  rw [cutoffField_eq_reciprocal κ b j T _
-    (cutoff_inactive_of_height j hj _ _
-      (surviving_cutoff_height κ b z S j hj hmin T ⟨hT,hTS.le⟩).le)] at hd
-  rw [← hval] at hd
-  exact hd.congr_of_eventuallyEq he hval
-
-lemma flow_zero (κ : ℝ) (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    flow κ b z 0=z := by
-  obtain ⟨j,hj,hjz⟩ := Real.exists_nat_pos_inv_lt (show 0 < z.im from hz)
-  have hmin : (j:ℝ)⁻¹ < sInf
-      ((fun s : ℝ => (cutoffFlow κ b j z s).im) '' Icc 0 0) := by
-    simpa only [Icc_self,image_singleton,cutoffFlow_zero κ b j hj,csInf_singleton] using hjz
-  exact (flow_eq_cutoff_on_interval κ b z 0 le_rfl j hj hmin 0 ⟨le_rfl,le_rfl⟩).trans
-    (cutoffFlow_zero κ b j hj z)
-
-lemma physical_flow_hasDerivWithinAt
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (z : ℂ) (T S : ℝ≥0) (hTS : T<S)
-    (hs : Surviving κ b z (κ*S)) :
-    HasDerivWithinAt (fun u : ℝ => flow κ b z (κ*u.toNNReal))
-      (2/(flow κ b z (κ*T)-(b (κ.toNNReal*T):ℂ))) (Ici 0) T := by
-  have hd := flow_hasDerivWithinAt_of_later_survival κ b z (κ*T) (κ*S)
-    (mul_nonneg hκ.le T.coe_nonneg) (mul_lt_mul_of_pos_left hTS hκ) hs
-  have hi : HasDerivWithinAt (fun u : ℝ => κ*u) κ (Ici 0) T := by
-    simpa using ((hasDerivAt_id (T:ℝ)).const_mul κ).hasDerivWithinAt (s := Ici 0)
-  have hmaps : MapsTo (fun u : ℝ => κ*u) (Ici 0) (Ici 0) :=
-    fun u hu => mul_nonneg hκ.le hu
-  have hc := hd.scomp (T:ℝ) hi hmaps
-  have he : (fun u : ℝ => flow κ b z (κ*u.toNNReal)) =ᶠ[𝓝[Ici 0] (T:ℝ)]
-      (fun u : ℝ => flow κ b z (κ*u)) := by
-    filter_upwards [self_mem_nhdsWithin] with u hu
-    rw [Real.coe_toNNReal u hu]
-  have hc' := hc.congr_of_eventuallyEq he (by simp only [Real.toNNReal_coe,Function.comp_apply])
-  convert hc' using 1
-  simp only [driverValue,Real.toNNReal_mul hκ.le,Real.toNNReal_coe,Complex.real_smul]
-  have hcoef : (κ:ℂ)*(a κ:ℂ)=2 := by
-    rw [← Complex.ofReal_mul]
-    have ha : κ*a κ=2 := by dsimp [a]; field_simp [hκ.ne']
-    rw [ha]
-    norm_num
-  simp only [div_eq_mul_inv]
-  rw [← mul_assoc,hcoef]
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetPositiveFlow
+import OAI.Probability.SLE.Loewner.TargetInverseProperties
+
+namespace OAI
+
+/-! On each surviving time interval the selected raw flow is a single
+positive cutoff trajectory. -/
+noncomputable section
+open Set Filter
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Target
+
+lemma flow_eq_cutoff_on_interval
+    (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (_hT : 0 ≤ T)
+    (j : ℕ) (hj : 1 ≤ j)
+    (hmin : (j:ℝ)⁻¹ < sInf ((fun s : ℝ => (cutoffFlow κ b j z s).im) '' Icc 0 T))
+    (u : ℝ) (hu : u ∈ Icc 0 T) :
+    flow κ b z u=cutoffFlow κ b j z u := by
+  have hheight := surviving_cutoff_height κ b z T j hj hmin
+  have hj0 : (0:ℝ)<j := by exact_mod_cast (Nat.succ_le_iff.mp hj)
+  apply (flow_eq_of_positive_trajectory κ b z u hu.1 (cutoffFlow κ b j z)
+    ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self)
+    (cutoffFlow_zero κ b j hj z) ((j:ℝ)⁻¹) (inv_pos.mpr hj0)
+    (fun s hs => (hheight s ⟨hs.1,hs.2.trans hu.2⟩).le) ?_).2
+  intro s hs
+  have hd := (cutoffFlow_hasDerivWithinAt κ b j hj z s hs.1).mono
+    (Ici_subset_Ici.mpr hs.1)
+  rw [cutoffField_eq_reciprocal κ b j s _
+    (cutoff_inactive_of_height j hj _ _
+      (hheight s ⟨hs.1,hs.2.le.trans hu.2⟩).le)] at hd
+  exact hd
+
+lemma flow_hasDerivWithinAt_of_later_survival
+    (κ : ℝ) (b : Driver) (z : ℂ) (T S : ℝ) (hT : 0 ≤ T) (hTS : T<S)
+    (hs : Surviving κ b z S) :
+    HasDerivWithinAt (flow κ b z)
+      ((a κ:ℂ)/(flow κ b z T-driverValue b T)) (Ici 0) T := by
+  obtain ⟨j,hj,hmin⟩ := hs
+  have hS : 0 ≤ S := hT.trans hTS.le
+  have he : flow κ b z =ᶠ[𝓝[Ici 0] T] cutoffFlow κ b j z := by
+    filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (gt_mem_nhds hTS)] with u hu huS
+    exact flow_eq_cutoff_on_interval κ b z S hS j hj hmin u ⟨hu,huS.le⟩
+  have hval := flow_eq_cutoff_on_interval κ b z S hS j hj hmin T ⟨hT,hTS.le⟩
+  have hd := cutoffFlow_hasDerivWithinAt κ b j hj z T hT
+  rw [cutoffField_eq_reciprocal κ b j T _
+    (cutoff_inactive_of_height j hj _ _
+      (surviving_cutoff_height κ b z S j hj hmin T ⟨hT,hTS.le⟩).le)] at hd
+  rw [← hval] at hd
+  exact hd.congr_of_eventuallyEq he hval
+
+lemma flow_zero (κ : ℝ) (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    flow κ b z 0=z := by
+  obtain ⟨j,hj,hjz⟩ := Real.exists_nat_pos_inv_lt (show 0 < z.im from hz)
+  have hmin : (j:ℝ)⁻¹ < sInf
+      ((fun s : ℝ => (cutoffFlow κ b j z s).im) '' Icc 0 0) := by
+    simpa only [Icc_self,image_singleton,cutoffFlow_zero κ b j hj,csInf_singleton] using hjz
+  exact (flow_eq_cutoff_on_interval κ b z 0 le_rfl j hj hmin 0 ⟨le_rfl,le_rfl⟩).trans
+    (cutoffFlow_zero κ b j hj z)
+
+lemma physical_flow_hasDerivWithinAt
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (z : ℂ) (T S : ℝ≥0) (hTS : T<S)
+    (hs : Surviving κ b z (κ*S)) :
+    HasDerivWithinAt (fun u : ℝ => flow κ b z (κ*u.toNNReal))
+      (2/(flow κ b z (κ*T)-(b (κ.toNNReal*T):ℂ))) (Ici 0) T := by
+  have hd := flow_hasDerivWithinAt_of_later_survival κ b z (κ*T) (κ*S)
+    (mul_nonneg hκ.le T.coe_nonneg) (mul_lt_mul_of_pos_left hTS hκ) hs
+  have hi : HasDerivWithinAt (fun u : ℝ => κ*u) κ (Ici 0) T := by
+    simpa using ((hasDerivAt_id (T:ℝ)).const_mul κ).hasDerivWithinAt (s := Ici 0)
+  have hmaps : MapsTo (fun u : ℝ => κ*u) (Ici 0) (Ici 0) :=
+    fun u hu => mul_nonneg hκ.le hu
+  have hc := hd.scomp (T:ℝ) hi hmaps
+  have he : (fun u : ℝ => flow κ b z (κ*u.toNNReal)) =ᶠ[𝓝[Ici 0] (T:ℝ)]
+      (fun u : ℝ => flow κ b z (κ*u)) := by
+    filter_upwards [self_mem_nhdsWithin] with u hu
+    rw [Real.coe_toNNReal u hu]
+  have hc' := hc.congr_of_eventuallyEq he (by simp only [Real.toNNReal_coe,Function.comp_apply])
+  convert hc' using 1
+  simp only [driverValue,Real.toNNReal_mul hκ.le,Real.toNNReal_coe,Complex.real_smul]
+  have hcoef : (κ:ℂ)*(a κ:ℂ)=2 := by
+    rw [← Complex.ofReal_mul]
+    have ha : κ*a κ=2 := by dsimp [a]; field_simp [hκ.ne']
+    rw [ha]
+    norm_num
+  simp only [div_eq_mul_inv]
+  rw [← mul_assoc,hcoef]
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetFlowIdentification.lean b/lean/OAI/Probability/SLE/Angle/TargetFlowIdentification.lean
index 4b9943c06199fb4ca903418c956fe714d49d1cb4..34c5613f7bae3b8a5e6dd7f71426e602d889ed25 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetFlowIdentification.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetFlowIdentification.lean
@@ -1,216 +1,216 @@
-import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory
-import OAI.Probability.SLE.LoewnerReflection
-
-namespace OAI
-
-/-! Identification of the prescribed Euler flow with an actual capacity-two
-Loewner witness, including its geometric surviving domain. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Target
-
-private lemma negative_clock_coefficient (κ : ℝ) (hκ : 0 < κ) (w : ℂ) :
-    ((-κ:ℝ):ℂ) * ((a κ:ℂ)/w) = -2*w⁻¹ := by
-  have hc : (κ:ℂ)*(a κ:ℂ)=2 := by
-    rw [← Complex.ofReal_mul]
-    have hr : κ*a κ=2 := by dsimp [a]; field_simp [hκ.ne']
-    rw [hr]
-    norm_num
-  calc
-    ((-κ:ℝ):ℂ) * ((a κ:ℂ)/w) = -((κ:ℂ)*(a κ:ℂ))*w⁻¹ := by
-      simp only [Complex.ofReal_neg,div_eq_mul_inv]
-      ring
-    _ = -2*w⁻¹ := by rw [hc]
-
-lemma surviving_cutoff_identifies
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (T : ℝ≥0) (z : ℂ) (j : ℕ) (hj : 1 ≤ j)
-    (hmin : (j:ℝ)⁻¹ < sInf
-      ((fun s : ℝ => (cutoffFlow κ b j z s).im) '' Icc 0 (κ*T))) :
-    z ∈ survivingDomain γ T ∧ G T z=cutoffFlow κ b j z (κ*T) := by
-  let Q (s : ℝ) := cutoffFlow κ b j z (κ*((T:ℝ)-s))
-  let w := cutoffFlow κ b j z (κ*T)
-  have hj0 : (0:ℝ)<j := by exact_mod_cast (Nat.succ_le_iff.mp hj)
-  have ha : 0 < (j:ℝ)⁻¹ := inv_pos.mpr hj0
-  have hheight := surviving_cutoff_height κ b z (κ*T) j hj hmin
-  have hwheight : (j:ℝ)⁻¹ ≤ w.im :=
-    (hheight (κ*T) ⟨mul_nonneg hκ.le T.coe_nonneg,le_rfl⟩).le
-  have hQ0 : Q 0=w := by simp only [Q,sub_zero,w]
-  have hQd (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) :
-      HasDerivWithinAt Q (reverseField (j:ℝ)⁻¹ (U ((T:ℝ)-s).toNNReal) (Q s))
-        (Icc 0 (T:ℝ)) s := by
-    have hs0 : 0 ≤ (T:ℝ)-s := sub_nonneg.mpr hs.2
-    have ht : κ*((T:ℝ)-s) ∈ Icc 0 (κ*T) := by
-      constructor
-      · exact mul_nonneg hκ.le hs0
-      · exact mul_le_mul_of_nonneg_left (sub_le_self _ hs.1) hκ.le
-    have hmaps : MapsTo (fun r : ℝ => κ*((T:ℝ)-r)) (Icc 0 (T:ℝ)) (Ici 0) := by
-      intro r hr
-      exact mul_nonneg hκ.le (sub_nonneg.mpr hr.2)
-    have hinner : HasDerivWithinAt (fun r : ℝ => κ*((T:ℝ)-r)) (-κ)
-        (Icc 0 (T:ℝ)) s := by
-      simpa using (((hasDerivAt_id s).const_sub (T:ℝ)).const_mul κ).hasDerivWithinAt
-    have hd := (cutoffFlow_hasDerivWithinAt κ b j hj z _ ht.1).scomp s hinner hmaps
-    have hd' : HasDerivWithinAt Q
-        ((-κ) • cutoffField κ b j (κ*((T:ℝ)-s)) (Q s)) (Icc 0 (T:ℝ)) s := hd
-    apply hd'.congr_deriv
-    have hh : (j:ℝ)⁻¹ ≤ (Q s).im := (hheight _ ht).le
-    rw [cutoffField_eq_reciprocal κ b j _ (Q s)
-      (cutoff_inactive_of_height j hj _ _ hh)]
-    simp only [Complex.real_smul]
-    rw [negative_clock_coefficient κ hκ,reverseField,reverseCenter_eq hh,hU _ hs0]
-  have hback := hw.reverse_flow_identifies T w ha hwheight Q hQ0 hQd
-  have hzF : z=F T w := by
-    simpa only [Q,sub_self,mul_zero,cutoffFlow_zero κ b j hj z] using hback
-  have hwH : w ∈ upperHalfPlane := ha.trans_le hwheight
-  constructor
-  · rw [hzF]
-    exact hw.2.2.2.2.2.2.1 T hwH
-  · change G T z=w
-    exact (congrArg (G T) hzF).trans (hw.2.2.2.2.2.2.2.2.1 T w hwH)
-
-private lemma rescaled_witness_trajectory
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
-    let Y := fun s : ℝ => G (s/κ).toNNReal z
-    ContinuousOn Y (Icc 0 (κ*T)) ∧ Y 0=z ∧
-      (∀ s ∈ Icc 0 (κ*T), (G T z).im ≤ (Y s).im) ∧
-      (∀ s ∈ Ico 0 (κ*T), HasDerivWithinAt Y
-        ((a κ:ℂ)/(Y s-driverValue b s)) (Ici s) s) := by
-  let Y := fun s : ℝ => G (s/κ).toNNReal z
-  have hmaps : MapsTo (fun s : ℝ => s/κ) (Icc 0 (κ*T)) (Icc 0 (T:ℝ)) := by
-    intro s hs
-    exact ⟨div_nonneg hs.1 hκ.le,(div_le_iff₀ hκ).2 (by simpa only [mul_comm] using hs.2)⟩
-  have hY : ContinuousOn Y (Icc 0 (κ*T)) :=
-    (hw.time_continuousOn T z hz).comp (continuous_id.div_const κ).continuousOn hmaps
-  refine ⟨hY,?_,?_,?_⟩
-  · simpa only [Y,zero_div,Real.toNNReal_zero] using hw.initial_value z hz.1.1
-  · intro s hs
-    have hi := hw.imag_antitoneOn T z hz (hmaps hs) ⟨T.coe_nonneg,le_rfl⟩ (hmaps hs).2
-    simpa only [Y,Real.toNNReal_coe] using hi
-  · intro s hs
-    have hs' := hmaps (Ico_subset_Icc_self hs)
-    have hinner : HasDerivWithinAt (fun r : ℝ => r/κ) (1/κ)
-        (Icc 0 (κ*T)) s := ((hasDerivAt_id s).div_const κ).hasDerivWithinAt
-    have hd := (hw.time_derivative T z hz (s/κ) hs').scomp s hinner hmaps
-    have hu : U (s/κ).toNNReal=driverValue b s := by
-      rw [hU _ hs'.1,mul_div_cancel₀ _ hκ.ne']
-    have hd' : HasDerivWithinAt Y
-        ((1/κ) • (2/(Y s-(U (s/κ).toNNReal:ℂ)))) (Icc 0 (κ*T)) s := hd
-    have hcoef : (1/κ) • (2/(Y s-(U (s/κ).toNNReal:ℂ))) =
-        (a κ:ℂ)/(Y s-driverValue b s) := by
-      rw [hu]
-      simp only [Complex.real_smul,a,div_eq_mul_inv]
-      push_cast
-      ring
-    exact hasDerivWithinAt_Ici_of_Icc hs (hd'.congr_deriv hcoef)
-
-lemma surviving_iff_witness
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (T : ℝ≥0) (z : ℂ) : Surviving κ b z (κ*T) ↔ z ∈ survivingDomain γ T := by
-  constructor
-  · rintro ⟨j,hj,hmin⟩
-    exact (surviving_cutoff_identifies κ hκ b U γ G F hw hU T z j hj hmin).1
-  · intro hz
-    obtain ⟨hY,hY0,hh,hYd⟩ := rescaled_witness_trajectory κ hκ b U γ G F hw hU T z hz
-    exact surviving_of_positive_trajectory κ b z (κ*T) (mul_nonneg hκ.le T.coe_nonneg)
-      _ hY hY0 (G T z).im (hw.imag_positive T z hz) hh hYd
-
-lemma flow_eq_witness
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
-    flow κ b z (κ*T)=G T z := by
-  have hs := (surviving_iff_witness κ hκ b U γ G F hw hU T z).2 hz
-  rw [flow,dite_eq_left hs]
-  exact (surviving_cutoff_identifies κ hκ b U γ G F hw hU T z
-    (Nat.find hs) (Nat.find_spec hs).1 (Nat.find_spec hs).2).2.symm
-
-lemma surviving_iff_witness_real_time
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (t : ℝ) (ht : 0 ≤ t) (z : ℂ) :
-    Surviving κ b z t ↔ z ∈ survivingDomain γ (t/κ).toNNReal := by
-  have he : κ*((t/κ).toNNReal:ℝ)=t := by
-    rw [Real.coe_toNNReal _ (div_nonneg ht hκ.le),mul_div_cancel₀ _ hκ.ne']
-  simpa only [he] using surviving_iff_witness κ hκ b U γ G F hw hU (t/κ).toNNReal z
-
-lemma flow_eq_witness_real_time
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (t : ℝ) (ht : 0 ≤ t) (z : ℂ)
-    (hz : z ∈ survivingDomain γ (t/κ).toNNReal) :
-    flow κ b z t=G (t/κ).toNNReal z := by
-  have he : κ*((t/κ).toNNReal:ℝ)=t := by
-    rw [Real.coe_toNNReal _ (div_nonneg ht hκ.le),mul_div_cancel₀ _ hκ.ne']
-  simpa only [he] using flow_eq_witness κ hκ b U γ G F hw hU (t/κ).toNNReal z hz
-
-lemma inverseFlow_eq_witness
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (T : ℝ≥0) (w : ℂ) (hwH : w ∈ upperHalfPlane) :
-    inverseFlow κ b (κ*T) w=F T w := by
-  classical
-  have hF : F T w ∈ survivingDomain γ T := hw.2.2.2.2.2.2.1 T hwH
-  have hex : ∃ z : ℂ, Surviving κ b z (κ*T) ∧ flow κ b z (κ*T)=w :=
-    ⟨F T w,(surviving_iff_witness κ hκ b U γ G F hw hU T _).2 hF,
-      (flow_eq_witness κ hκ b U γ G F hw hU T _ hF).trans
-        (hw.2.2.2.2.2.2.2.2.1 T w hwH)⟩
-  rw [inverseFlow,ite_eq_left hex]
-  let z := Classical.epsilon (fun z => Surviving κ b z (κ*T) ∧ flow κ b z (κ*T)=w)
-  have hz := Classical.epsilon_spec hex
-  have hzH : z ∈ survivingDomain γ T :=
-    (surviving_iff_witness κ hκ b U γ G F hw hU T z).1 hz.1
-  have he : G T z=w :=
-    (flow_eq_witness κ hκ b U γ G F hw hU T z hzH).symm.trans hz.2
-  change z=F T w
-  calc
-    z=F T (G T z) := (hw.2.2.2.2.2.2.2.1 T z hzH).symm
-    _=F T w := congrArg (F T) he
-
-lemma trace_eq_witness
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (T : ℝ≥0) : trace κ b T=γ T := by
-  have hUT : driverValue b (κ*T)=U T := by
-    simpa only [Real.toNNReal_coe] using (hU T T.coe_nonneg).symm
-  have he : (fun ε : ℝ => inverseFlow κ b (κ*T)
-        (driverValue b (κ*T)+ε*Complex.I)) =ᶠ[𝓝[>] 0]
-      (fun ε : ℝ => F T ((U T:ℂ)+ε*Complex.I)) := by
-    filter_upwards [self_mem_nhdsWithin] with ε hε
-    change 0 < ε at hε
-    have hp : ((driverValue b (κ*T):ℂ)+ε*Complex.I) ∈ upperHalfPlane := by
-      simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,
-        Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,
-        mul_zero,zero_add,add_zero] using hε
-    rw [inverseFlow_eq_witness κ hκ b U γ G F hw hU T _ hp,hUT]
-  have htip := hw.2.2.2.2.2.2.2.2.2.2.2.2 T
-  exact (htip.congr' he.symm).limUnder_eq
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory
+import OAI.Probability.SLE.LoewnerReflection
+
+namespace OAI
+
+/-! Identification of the prescribed Euler flow with an actual capacity-two
+Loewner witness, including its geometric surviving domain. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Target
+
+private lemma negative_clock_coefficient (κ : ℝ) (hκ : 0 < κ) (w : ℂ) :
+    ((-κ:ℝ):ℂ) * ((a κ:ℂ)/w) = -2*w⁻¹ := by
+  have hc : (κ:ℂ)*(a κ:ℂ)=2 := by
+    rw [← Complex.ofReal_mul]
+    have hr : κ*a κ=2 := by dsimp [a]; field_simp [hκ.ne']
+    rw [hr]
+    norm_num
+  calc
+    ((-κ:ℝ):ℂ) * ((a κ:ℂ)/w) = -((κ:ℂ)*(a κ:ℂ))*w⁻¹ := by
+      simp only [Complex.ofReal_neg,div_eq_mul_inv]
+      ring
+    _ = -2*w⁻¹ := by rw [hc]
+
+lemma surviving_cutoff_identifies
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (T : ℝ≥0) (z : ℂ) (j : ℕ) (hj : 1 ≤ j)
+    (hmin : (j:ℝ)⁻¹ < sInf
+      ((fun s : ℝ => (cutoffFlow κ b j z s).im) '' Icc 0 (κ*T))) :
+    z ∈ survivingDomain γ T ∧ G T z=cutoffFlow κ b j z (κ*T) := by
+  let Q (s : ℝ) := cutoffFlow κ b j z (κ*((T:ℝ)-s))
+  let w := cutoffFlow κ b j z (κ*T)
+  have hj0 : (0:ℝ)<j := by exact_mod_cast (Nat.succ_le_iff.mp hj)
+  have ha : 0 < (j:ℝ)⁻¹ := inv_pos.mpr hj0
+  have hheight := surviving_cutoff_height κ b z (κ*T) j hj hmin
+  have hwheight : (j:ℝ)⁻¹ ≤ w.im :=
+    (hheight (κ*T) ⟨mul_nonneg hκ.le T.coe_nonneg,le_rfl⟩).le
+  have hQ0 : Q 0=w := by simp only [Q,sub_zero,w]
+  have hQd (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) :
+      HasDerivWithinAt Q (reverseField (j:ℝ)⁻¹ (U ((T:ℝ)-s).toNNReal) (Q s))
+        (Icc 0 (T:ℝ)) s := by
+    have hs0 : 0 ≤ (T:ℝ)-s := sub_nonneg.mpr hs.2
+    have ht : κ*((T:ℝ)-s) ∈ Icc 0 (κ*T) := by
+      constructor
+      · exact mul_nonneg hκ.le hs0
+      · exact mul_le_mul_of_nonneg_left (sub_le_self _ hs.1) hκ.le
+    have hmaps : MapsTo (fun r : ℝ => κ*((T:ℝ)-r)) (Icc 0 (T:ℝ)) (Ici 0) := by
+      intro r hr
+      exact mul_nonneg hκ.le (sub_nonneg.mpr hr.2)
+    have hinner : HasDerivWithinAt (fun r : ℝ => κ*((T:ℝ)-r)) (-κ)
+        (Icc 0 (T:ℝ)) s := by
+      simpa using (((hasDerivAt_id s).const_sub (T:ℝ)).const_mul κ).hasDerivWithinAt
+    have hd := (cutoffFlow_hasDerivWithinAt κ b j hj z _ ht.1).scomp s hinner hmaps
+    have hd' : HasDerivWithinAt Q
+        ((-κ) • cutoffField κ b j (κ*((T:ℝ)-s)) (Q s)) (Icc 0 (T:ℝ)) s := hd
+    apply hd'.congr_deriv
+    have hh : (j:ℝ)⁻¹ ≤ (Q s).im := (hheight _ ht).le
+    rw [cutoffField_eq_reciprocal κ b j _ (Q s)
+      (cutoff_inactive_of_height j hj _ _ hh)]
+    simp only [Complex.real_smul]
+    rw [negative_clock_coefficient κ hκ,reverseField,reverseCenter_eq hh,hU _ hs0]
+  have hback := hw.reverse_flow_identifies T w ha hwheight Q hQ0 hQd
+  have hzF : z=F T w := by
+    simpa only [Q,sub_self,mul_zero,cutoffFlow_zero κ b j hj z] using hback
+  have hwH : w ∈ upperHalfPlane := ha.trans_le hwheight
+  constructor
+  · rw [hzF]
+    exact hw.2.2.2.2.2.2.1 T hwH
+  · change G T z=w
+    exact (congrArg (G T) hzF).trans (hw.2.2.2.2.2.2.2.2.1 T w hwH)
+
+private lemma rescaled_witness_trajectory
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
+    let Y := fun s : ℝ => G (s/κ).toNNReal z
+    ContinuousOn Y (Icc 0 (κ*T)) ∧ Y 0=z ∧
+      (∀ s ∈ Icc 0 (κ*T), (G T z).im ≤ (Y s).im) ∧
+      (∀ s ∈ Ico 0 (κ*T), HasDerivWithinAt Y
+        ((a κ:ℂ)/(Y s-driverValue b s)) (Ici s) s) := by
+  let Y := fun s : ℝ => G (s/κ).toNNReal z
+  have hmaps : MapsTo (fun s : ℝ => s/κ) (Icc 0 (κ*T)) (Icc 0 (T:ℝ)) := by
+    intro s hs
+    exact ⟨div_nonneg hs.1 hκ.le,(div_le_iff₀ hκ).2 (by simpa only [mul_comm] using hs.2)⟩
+  have hY : ContinuousOn Y (Icc 0 (κ*T)) :=
+    (hw.time_continuousOn T z hz).comp (continuous_id.div_const κ).continuousOn hmaps
+  refine ⟨hY,?_,?_,?_⟩
+  · simpa only [Y,zero_div,Real.toNNReal_zero] using hw.initial_value z hz.1.1
+  · intro s hs
+    have hi := hw.imag_antitoneOn T z hz (hmaps hs) ⟨T.coe_nonneg,le_rfl⟩ (hmaps hs).2
+    simpa only [Y,Real.toNNReal_coe] using hi
+  · intro s hs
+    have hs' := hmaps (Ico_subset_Icc_self hs)
+    have hinner : HasDerivWithinAt (fun r : ℝ => r/κ) (1/κ)
+        (Icc 0 (κ*T)) s := ((hasDerivAt_id s).div_const κ).hasDerivWithinAt
+    have hd := (hw.time_derivative T z hz (s/κ) hs').scomp s hinner hmaps
+    have hu : U (s/κ).toNNReal=driverValue b s := by
+      rw [hU _ hs'.1,mul_div_cancel₀ _ hκ.ne']
+    have hd' : HasDerivWithinAt Y
+        ((1/κ) • (2/(Y s-(U (s/κ).toNNReal:ℂ)))) (Icc 0 (κ*T)) s := hd
+    have hcoef : (1/κ) • (2/(Y s-(U (s/κ).toNNReal:ℂ))) =
+        (a κ:ℂ)/(Y s-driverValue b s) := by
+      rw [hu]
+      simp only [Complex.real_smul,a,div_eq_mul_inv]
+      push_cast
+      ring
+    exact hasDerivWithinAt_Ici_of_Icc hs (hd'.congr_deriv hcoef)
+
+lemma surviving_iff_witness
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (T : ℝ≥0) (z : ℂ) : Surviving κ b z (κ*T) ↔ z ∈ survivingDomain γ T := by
+  constructor
+  · rintro ⟨j,hj,hmin⟩
+    exact (surviving_cutoff_identifies κ hκ b U γ G F hw hU T z j hj hmin).1
+  · intro hz
+    obtain ⟨hY,hY0,hh,hYd⟩ := rescaled_witness_trajectory κ hκ b U γ G F hw hU T z hz
+    exact surviving_of_positive_trajectory κ b z (κ*T) (mul_nonneg hκ.le T.coe_nonneg)
+      _ hY hY0 (G T z).im (hw.imag_positive T z hz) hh hYd
+
+lemma flow_eq_witness
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
+    flow κ b z (κ*T)=G T z := by
+  have hs := (surviving_iff_witness κ hκ b U γ G F hw hU T z).2 hz
+  rw [flow,dite_eq_left hs]
+  exact (surviving_cutoff_identifies κ hκ b U γ G F hw hU T z
+    (Nat.find hs) (Nat.find_spec hs).1 (Nat.find_spec hs).2).2.symm
+
+lemma surviving_iff_witness_real_time
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (t : ℝ) (ht : 0 ≤ t) (z : ℂ) :
+    Surviving κ b z t ↔ z ∈ survivingDomain γ (t/κ).toNNReal := by
+  have he : κ*((t/κ).toNNReal:ℝ)=t := by
+    rw [Real.coe_toNNReal _ (div_nonneg ht hκ.le),mul_div_cancel₀ _ hκ.ne']
+  simpa only [he] using surviving_iff_witness κ hκ b U γ G F hw hU (t/κ).toNNReal z
+
+lemma flow_eq_witness_real_time
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (t : ℝ) (ht : 0 ≤ t) (z : ℂ)
+    (hz : z ∈ survivingDomain γ (t/κ).toNNReal) :
+    flow κ b z t=G (t/κ).toNNReal z := by
+  have he : κ*((t/κ).toNNReal:ℝ)=t := by
+    rw [Real.coe_toNNReal _ (div_nonneg ht hκ.le),mul_div_cancel₀ _ hκ.ne']
+  simpa only [he] using flow_eq_witness κ hκ b U γ G F hw hU (t/κ).toNNReal z hz
+
+lemma inverseFlow_eq_witness
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (T : ℝ≥0) (w : ℂ) (hwH : w ∈ upperHalfPlane) :
+    inverseFlow κ b (κ*T) w=F T w := by
+  classical
+  have hF : F T w ∈ survivingDomain γ T := hw.2.2.2.2.2.2.1 T hwH
+  have hex : ∃ z : ℂ, Surviving κ b z (κ*T) ∧ flow κ b z (κ*T)=w :=
+    ⟨F T w,(surviving_iff_witness κ hκ b U γ G F hw hU T _).2 hF,
+      (flow_eq_witness κ hκ b U γ G F hw hU T _ hF).trans
+        (hw.2.2.2.2.2.2.2.2.1 T w hwH)⟩
+  rw [inverseFlow,ite_eq_left hex]
+  let z := Classical.epsilon (fun z => Surviving κ b z (κ*T) ∧ flow κ b z (κ*T)=w)
+  have hz := Classical.epsilon_spec hex
+  have hzH : z ∈ survivingDomain γ T :=
+    (surviving_iff_witness κ hκ b U γ G F hw hU T z).1 hz.1
+  have he : G T z=w :=
+    (flow_eq_witness κ hκ b U γ G F hw hU T z hzH).symm.trans hz.2
+  change z=F T w
+  calc
+    z=F T (G T z) := (hw.2.2.2.2.2.2.2.1 T z hzH).symm
+    _=F T w := congrArg (F T) he
+
+lemma trace_eq_witness
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (T : ℝ≥0) : trace κ b T=γ T := by
+  have hUT : driverValue b (κ*T)=U T := by
+    simpa only [Real.toNNReal_coe] using (hU T T.coe_nonneg).symm
+  have he : (fun ε : ℝ => inverseFlow κ b (κ*T)
+        (driverValue b (κ*T)+ε*Complex.I)) =ᶠ[𝓝[>] 0]
+      (fun ε : ℝ => F T ((U T:ℂ)+ε*Complex.I)) := by
+    filter_upwards [self_mem_nhdsWithin] with ε hε
+    change 0 < ε at hε
+    have hp : ((driverValue b (κ*T):ℂ)+ε*Complex.I) ∈ upperHalfPlane := by
+      simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,
+        Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,
+        mul_zero,zero_add,add_zero] using hε
+    rw [inverseFlow_eq_witness κ hκ b U γ G F hw hU T _ hp,hUT]
+  have htip := hw.2.2.2.2.2.2.2.2.2.2.2.2 T
+  exact (htip.congr' he.symm).limUnder_eq
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetFlowInjectivity.lean b/lean/OAI/Probability/SLE/Angle/TargetFlowInjectivity.lean
index 1a8555aa762629f7d6d46b98fb975e7563d777c3..e28bf2a3a5c118f89ed49139789eafe2e880991d 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetFlowInjectivity.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetFlowInjectivity.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Angle.TargetFlowMeasurability
-
-namespace OAI
-
-/-! The raw cutoff construction retains injectivity on the surviving
-domain. This is used to construct its inverse as a measurable map. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma cutoffFlow_injective (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j)
-    (T : ℝ) (hT : 0 ≤ T) : Function.Injective (fun z => cutoffFlow κ b j z T) := by
-  intro z w he
-  obtain ⟨L,K,hL,hK⟩ := cutoffField_bounds κ j hj
-  have hd (z : ℂ) (t : ℝ) (ht : t ∈ Ioc 0 T) :
-      HasDerivWithinAt (cutoffFlow κ b j z)
-        (cutoffField κ b j t (cutoffFlow κ b j z t)) (Iic t) t :=
-    ((cutoffFlow_hasDerivWithinAt κ b j hj z t ht.1.le).hasDerivAt
-      (Ici_mem_nhds ht.1)).hasDerivWithinAt
-  have heq := ODE_solution_unique_of_mem_Icc_left (s := fun _ => univ)
-    (fun t _ => (hK b t).lipschitzOnWith)
-    ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self) (hd z)
-    (fun _ _ => mem_univ _) ((cutoffFlow_continuousOn κ b j hj w).mono Icc_subset_Ici_self)
-    (hd w) (fun _ _ => mem_univ _) he
-  have hz := heq (show (0:ℝ) ∈ Icc 0 T from ⟨le_rfl,hT⟩)
-  simpa only [cutoffFlow_zero κ b j hj] using hz
-
-lemma surviving_common_cutoff
-    (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (hT : 0 ≤ T)
-    (hs : Surviving κ b z T) (j : ℕ) (hj : Nat.find hs ≤ j) :
-    cutoffFlow κ b j z T = flow κ b z T := by
-  let k := Nat.find hs
-  have hk : 1 ≤ k := (Nat.find_spec hs).1
-  have hj' : 1 ≤ j := hk.trans hj
-  have hinv : (j:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ :=
-    inv_anti₀ (by exact_mod_cast (Nat.succ_le_iff.mp hk) : (0:ℝ)<k)
-      (by exact_mod_cast hj)
-  have hheight := surviving_cutoff_height κ b z T k hk (Nat.find_spec hs).2
-  have hderiv (t : ℝ) (ht : t ∈ Ico 0 T) :
-      HasDerivWithinAt (cutoffFlow κ b k z)
-        ((a κ:ℂ)/(cutoffFlow κ b k z t-driverValue b t)) (Ici t) t := by
-    have h := (cutoffFlow_hasDerivWithinAt κ b k hk z t ht.1).mono
-      (Ici_subset_Ici.mpr ht.1)
-    rw [cutoffField_eq_reciprocal κ b k t _
-      (cutoff_inactive_of_height k hk _ _ (hheight t (Ico_subset_Icc_self ht)).le)] at h
-    exact h
-  have he := cutoffFlow_eq_of_positive_trajectory κ b j hj' z T hT
-    (cutoffFlow κ b k z) ((cutoffFlow_continuousOn κ b k hk z).mono Icc_subset_Ici_self)
-    (cutoffFlow_zero κ b k hk z)
-    (fun t ht => hinv.trans (hheight t ht).le) hderiv T ⟨hT,le_rfl⟩
-  simpa only [flow,dite_eq_left hs] using he
-
-lemma flow_injOn_surviving (κ : ℝ) (b : Driver) (T : ℝ) (hT : 0 ≤ T) :
-    InjOn (fun z => flow κ b z T) {z | Surviving κ b z T} := by
-  intro z hz w hw he
-  let j := max (Nat.find hz) (Nat.find hw)
-  have hj : 1 ≤ j := (Nat.find_spec hz).1.trans (le_max_left _ _)
-  apply cutoffFlow_injective κ b j hj T hT
-  change cutoffFlow κ b j z T = cutoffFlow κ b j w T
-  rw [surviving_common_cutoff κ b z T hT hz j (le_max_left _ _),
-    surviving_common_cutoff κ b w T hT hw j (le_max_right _ _)]
-  exact he
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetFlowMeasurability
+
+namespace OAI
+
+/-! The raw cutoff construction retains injectivity on the surviving
+domain. This is used to construct its inverse as a measurable map. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma cutoffFlow_injective (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j)
+    (T : ℝ) (hT : 0 ≤ T) : Function.Injective (fun z => cutoffFlow κ b j z T) := by
+  intro z w he
+  obtain ⟨L,K,hL,hK⟩ := cutoffField_bounds κ j hj
+  have hd (z : ℂ) (t : ℝ) (ht : t ∈ Ioc 0 T) :
+      HasDerivWithinAt (cutoffFlow κ b j z)
+        (cutoffField κ b j t (cutoffFlow κ b j z t)) (Iic t) t :=
+    ((cutoffFlow_hasDerivWithinAt κ b j hj z t ht.1.le).hasDerivAt
+      (Ici_mem_nhds ht.1)).hasDerivWithinAt
+  have heq := ODE_solution_unique_of_mem_Icc_left (s := fun _ => univ)
+    (fun t _ => (hK b t).lipschitzOnWith)
+    ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self) (hd z)
+    (fun _ _ => mem_univ _) ((cutoffFlow_continuousOn κ b j hj w).mono Icc_subset_Ici_self)
+    (hd w) (fun _ _ => mem_univ _) he
+  have hz := heq (show (0:ℝ) ∈ Icc 0 T from ⟨le_rfl,hT⟩)
+  simpa only [cutoffFlow_zero κ b j hj] using hz
+
+lemma surviving_common_cutoff
+    (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (hT : 0 ≤ T)
+    (hs : Surviving κ b z T) (j : ℕ) (hj : Nat.find hs ≤ j) :
+    cutoffFlow κ b j z T = flow κ b z T := by
+  let k := Nat.find hs
+  have hk : 1 ≤ k := (Nat.find_spec hs).1
+  have hj' : 1 ≤ j := hk.trans hj
+  have hinv : (j:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ :=
+    inv_anti₀ (by exact_mod_cast (Nat.succ_le_iff.mp hk) : (0:ℝ)<k)
+      (by exact_mod_cast hj)
+  have hheight := surviving_cutoff_height κ b z T k hk (Nat.find_spec hs).2
+  have hderiv (t : ℝ) (ht : t ∈ Ico 0 T) :
+      HasDerivWithinAt (cutoffFlow κ b k z)
+        ((a κ:ℂ)/(cutoffFlow κ b k z t-driverValue b t)) (Ici t) t := by
+    have h := (cutoffFlow_hasDerivWithinAt κ b k hk z t ht.1).mono
+      (Ici_subset_Ici.mpr ht.1)
+    rw [cutoffField_eq_reciprocal κ b k t _
+      (cutoff_inactive_of_height k hk _ _ (hheight t (Ico_subset_Icc_self ht)).le)] at h
+    exact h
+  have he := cutoffFlow_eq_of_positive_trajectory κ b j hj' z T hT
+    (cutoffFlow κ b k z) ((cutoffFlow_continuousOn κ b k hk z).mono Icc_subset_Ici_self)
+    (cutoffFlow_zero κ b k hk z)
+    (fun t ht => hinv.trans (hheight t ht).le) hderiv T ⟨hT,le_rfl⟩
+  simpa only [flow,dite_eq_left hs] using he
+
+lemma flow_injOn_surviving (κ : ℝ) (b : Driver) (T : ℝ) (hT : 0 ≤ T) :
+    InjOn (fun z => flow κ b z T) {z | Surviving κ b z T} := by
+  intro z hz w hw he
+  let j := max (Nat.find hz) (Nat.find hw)
+  have hj : 1 ≤ j := (Nat.find_spec hz).1.trans (le_max_left _ _)
+  apply cutoffFlow_injective κ b j hj T hT
+  change cutoffFlow κ b j z T = cutoffFlow κ b j w T
+  rw [surviving_common_cutoff κ b z T hT hz j (le_max_left _ _),
+    surviving_common_cutoff κ b w T hT hw j (le_max_right _ _)]
+  exact he
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetFlowMeasurability.lean b/lean/OAI/Probability/SLE/Angle/TargetFlowMeasurability.lean
index c9e0d5ec0fe854fda4f3ad09807ff3f3824cbd2d..dc1137c68f3cc021e50a74d27601718aafb9bde4 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetFlowMeasurability.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetFlowMeasurability.lean
@@ -1,160 +1,160 @@
-import OAI.Probability.SLE.Loewner.TargetCutoffMeasurability
-
-namespace OAI
-
-/-! A measurable compact minimum gives the exact countable survival
-criterion, and therefore measurability of the prescribed uncut flow. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-def cutoffMinimum (κ : ℝ) (j : ℕ) (b : Driver) (z : ℂ) (T : ℝ≥0) : ℝ≥0∞ :=
-  ⨅ r : unitInterval, ENNReal.ofReal (cutoffFlow κ b j z ((T : ℝ)*r)).im
-
-lemma cutoffMinimum_eq (κ : ℝ) (j : ℕ) (b : Driver) (z : ℂ) (T : ℝ≥0) :
-    cutoffMinimum κ j b z T =
-      ENNReal.ofReal (sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 (T : ℝ))) := by
-  have hrange : Set.range (fun r : unitInterval => (T : ℝ)*r) = Icc 0 (T : ℝ) := by
-    ext u
-    constructor
-    · rintro ⟨r,rfl⟩
-      exact ⟨mul_nonneg T.coe_nonneg r.2.1,
-        by simpa only [mul_one] using mul_le_mul_of_nonneg_left r.2.2 T.coe_nonneg⟩
-    · intro hu
-      by_cases hT : (T : ℝ)=0
-      · refine ⟨0,?_⟩
-        have hu0 : u=0 := le_antisymm (by simpa only [hT] using hu.2) hu.1
-        simp only [hT,zero_mul,hu0]
-      · have hT0 : 0 < (T : ℝ) := lt_of_le_of_ne T.coe_nonneg (Ne.symm hT)
-        refine ⟨⟨u/T,div_nonneg hu.1 T.coe_nonneg,(div_le_one hT0).mpr hu.2⟩,?_⟩
-        field_simp [hT]
-  have himage : Set.range (fun r : unitInterval =>
-      (cutoffFlow κ b j z ((T : ℝ)*r)).im) =
-      (fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 (T : ℝ) := by
-    rw [← hrange]
-    exact Set.range_comp (fun u : ℝ => (cutoffFlow κ b j z u).im)
-      (fun r : unitInterval => (T : ℝ)*r)
-  unfold cutoffMinimum
-  rw [← ENNReal.ofReal_iInf]
-  congr 1
-  change sInf _ = sInf _
-  rw [himage]
-
-lemma measurable_cutoffMinimum (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) :
-    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => cutoffMinimum κ j q.2.1 q.2.2 q.1) := by
-  have hm : Measurable (⨅ r : unitInterval, fun q : ℝ≥0 × (Driver × ℂ) =>
-      ENNReal.ofReal (cutoffFlow κ q.2.1 j q.2.2 ((q.1 : ℝ)*r)).im) := by
-    apply measurable_iInf_of_upperSemicontinuous
-    · intro r
-      let s : ℝ≥0 := ⟨(r : ℝ),r.2.1⟩
-      have hm := (measurable_cutoffFlow κ j hj).comp
-        ((measurable_fst.mul_const s).prodMk measurable_snd)
-      exact (Complex.measurable_im.comp hm).ennreal_ofReal
-    · intro q
-      have hc : Continuous (fun r : unitInterval => cutoffFlow κ q.2.1 j q.2.2
-          ((q.1 : ℝ)*r)) :=
-        (cutoffFlow_continuousOn κ q.2.1 j hj q.2.2).comp_continuous
-          (continuous_const.mul continuous_subtype_val)
-          (fun r => mul_nonneg q.1.coe_nonneg r.2.1)
-      exact (ENNReal.continuous_ofReal.comp (Complex.continuous_im.comp hc)).upperSemicontinuous
-  change Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
-    (⨅ r : unitInterval, fun p : ℝ≥0 × (Driver × ℂ) =>
-      ENNReal.ofReal (cutoffFlow κ p.2.1 j p.2.2 ((p.1 : ℝ)*r)).im) q) at hm
-  simpa only [iInf_apply,cutoffMinimum] using hm
-
-lemma cutoff_survival_iff_minimum (κ : ℝ) (j : ℕ) (_hj : 1 ≤ j)
-    (b : Driver) (z : ℂ) (T : ℝ≥0) :
-    (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 (T : ℝ)) ↔
-      ENNReal.ofReal ((j : ℝ)⁻¹) < cutoffMinimum κ j b z T := by
-  rw [cutoffMinimum_eq]
-  exact (ENNReal.ofReal_lt_ofReal_iff_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg j))).symm
-
-lemma measurableSet_cutoff_survival (κ : ℝ) (j : ℕ) :
-    MeasurableSet {q : ℝ≥0 × (Driver × ℂ) | 1 ≤ j ∧
-      (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.2.1 j q.2.2 u).im) ''
-        Icc 0 (q.1 : ℝ))} := by
-  by_cases hj : 1 ≤ j
-  · have he : {q : ℝ≥0 × (Driver × ℂ) | 1 ≤ j ∧
-        (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.2.1 j q.2.2 u).im) ''
-          Icc 0 (q.1 : ℝ))} =
-        {q | ENNReal.ofReal ((j : ℝ)⁻¹) < cutoffMinimum κ j q.2.1 q.2.2 q.1} := by
-      ext q
-      simp only [mem_ofPred_eq,and_iff_right hj,cutoff_survival_iff_minimum κ j hj]
-    rw [he]
-    exact measurableSet_lt measurable_const (measurable_cutoffMinimum κ j hj)
-  · simp only [hj,false_and,ofPred_false,MeasurableSet.empty]
-
-lemma measurableSet_surviving (κ : ℝ) :
-    MeasurableSet {q : ℝ≥0 × (Driver × ℂ) | Surviving κ q.2.1 q.2.2 q.1} := by
-  change MeasurableSet {q : ℝ≥0 × (Driver × ℂ) | ∃ j : ℕ, 1 ≤ j ∧
-    (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.2.1 j q.2.2 u).im) '' Icc 0 (q.1 : ℝ))}
-  simp only [ofPred_exists]
-  exact MeasurableSet.iUnion (measurableSet_cutoff_survival κ)
-
-lemma measurable_flow (κ : ℝ) :
-    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => flow κ q.2.1 q.2.2 q.1) := by
-  let S : Set (ℝ≥0 × (Driver × ℂ)) := {q | Surviving κ q.2.1 q.2.2 q.1}
-  let f (j : ℕ) (q : S) : ℂ :=
-    if 1 ≤ j then cutoffFlow κ q.1.2.1 j q.1.2.2 q.1.1 else 0
-  have hf (j : ℕ) : Measurable (f j) := by
-    by_cases hj : 1 ≤ j
-    · simpa only [f,ite_eq_left hj,Function.comp_def] using
-        (measurable_cutoffFlow κ j hj).comp measurable_subtype_coe
-    · simp only [f,ite_eq_right hj]
-      exact measurable_const
-  have hp (j : ℕ) : MeasurableSet {q : S | 1 ≤ j ∧
-      (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.1.2.1 j q.1.2.2 u).im) ''
-        Icc 0 (q.1.1 : ℝ))} :=
-    measurable_subtype_coe (measurableSet_cutoff_survival κ j)
-  have hm := (Measurable.find hf hp (fun q : S => q.2)).dite
-    (g := fun _ : (Sᶜ : Set (ℝ≥0 × (Driver × ℂ))) => (0 : ℂ))
-      measurable_const (measurableSet_surviving κ)
-  convert hm using 1
-  funext q
-  by_cases hq : Surviving κ q.2.1 q.2.2 q.1
-  · simp only [flow,S,mem_ofPred_eq,dite_eq_left hq,f,ite_eq_left (Nat.find_spec hq).1]
-  · simp only [flow,S,mem_ofPred_eq,dite_eq_right hq]
-
-lemma surviving_flow_eq_of_driver_prefix (κ : ℝ) (b c : Driver) (z : ℂ)
-    (T : ℝ) (hT : 0 ≤ T)
-    (he : ∀ s ∈ Icc 0 T, driverValue b s = driverValue c s) :
-    (Surviving κ b z T ↔ Surviving κ c z T) ∧ flow κ b z T = flow κ c z T := by
-  have hp (j : ℕ) :
-      (1 ≤ j ∧ (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)) ↔
-      (1 ≤ j ∧ (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ c j z u).im) '' Icc 0 T)) := by
-    by_cases hj : 1 ≤ j
-    · have hi : (fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T =
-          (fun u : ℝ => (cutoffFlow κ c j z u).im) '' Icc 0 T := by
-        apply image_congr
-        intro u hu
-        rw [cutoffFlow_eq_of_driver_prefix κ b c j hj z T hT he u hu]
-      rw [hi]
-    · simp only [hj,false_and]
-  have hs : Surviving κ b z T ↔ Surviving κ c z T := exists_congr hp
-  refine ⟨hs,?_⟩
-  by_cases hb : Surviving κ b z T
-  · have hc := hs.mp hb
-    have hn : Nat.find hb = Nat.find hc := Nat.find_congr' (fun {j} => hp j)
-    rw [flow,dite_eq_left hb,flow,dite_eq_left hc,← hn]
-    exact cutoffFlow_eq_of_driver_prefix κ b c (Nat.find hb) (Nat.find_spec hb).1
-      z T hT he T ⟨hT,le_rfl⟩
-  · have hc : ¬Surviving κ c z T := fun hc => hb (hs.mpr hc)
-    simp only [flow,dite_eq_right hb,dite_eq_right hc]
-
-lemma surviving_flow_stopDriver (κ : ℝ) (b : Driver) (z : ℂ) (T t : ℝ≥0)
-    (ht : t ≤ T) :
-    (Surviving κ (stopDriver b T) z t ↔ Surviving κ b z t) ∧
-      flow κ (stopDriver b T) z t = flow κ b z t := by
-  apply surviving_flow_eq_of_driver_prefix κ _ _ z t t.coe_nonneg
-  intro s hs
-  simp only [driverValue,stopDriver_apply,
-    min_eq_left (Real.toNNReal_le_iff_le_coe.mpr (hs.2.trans (show (t : ℝ) ≤ T from ht)))]
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.TargetCutoffMeasurability
+
+namespace OAI
+
+/-! A measurable compact minimum gives the exact countable survival
+criterion, and therefore measurability of the prescribed uncut flow. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+def cutoffMinimum (κ : ℝ) (j : ℕ) (b : Driver) (z : ℂ) (T : ℝ≥0) : ℝ≥0∞ :=
+  ⨅ r : unitInterval, ENNReal.ofReal (cutoffFlow κ b j z ((T : ℝ)*r)).im
+
+lemma cutoffMinimum_eq (κ : ℝ) (j : ℕ) (b : Driver) (z : ℂ) (T : ℝ≥0) :
+    cutoffMinimum κ j b z T =
+      ENNReal.ofReal (sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 (T : ℝ))) := by
+  have hrange : Set.range (fun r : unitInterval => (T : ℝ)*r) = Icc 0 (T : ℝ) := by
+    ext u
+    constructor
+    · rintro ⟨r,rfl⟩
+      exact ⟨mul_nonneg T.coe_nonneg r.2.1,
+        by simpa only [mul_one] using mul_le_mul_of_nonneg_left r.2.2 T.coe_nonneg⟩
+    · intro hu
+      by_cases hT : (T : ℝ)=0
+      · refine ⟨0,?_⟩
+        have hu0 : u=0 := le_antisymm (by simpa only [hT] using hu.2) hu.1
+        simp only [hT,zero_mul,hu0]
+      · have hT0 : 0 < (T : ℝ) := lt_of_le_of_ne T.coe_nonneg (Ne.symm hT)
+        refine ⟨⟨u/T,div_nonneg hu.1 T.coe_nonneg,(div_le_one hT0).mpr hu.2⟩,?_⟩
+        field_simp [hT]
+  have himage : Set.range (fun r : unitInterval =>
+      (cutoffFlow κ b j z ((T : ℝ)*r)).im) =
+      (fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 (T : ℝ) := by
+    rw [← hrange]
+    exact Set.range_comp (fun u : ℝ => (cutoffFlow κ b j z u).im)
+      (fun r : unitInterval => (T : ℝ)*r)
+  unfold cutoffMinimum
+  rw [← ENNReal.ofReal_iInf]
+  congr 1
+  change sInf _ = sInf _
+  rw [himage]
+
+lemma measurable_cutoffMinimum (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) :
+    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => cutoffMinimum κ j q.2.1 q.2.2 q.1) := by
+  have hm : Measurable (⨅ r : unitInterval, fun q : ℝ≥0 × (Driver × ℂ) =>
+      ENNReal.ofReal (cutoffFlow κ q.2.1 j q.2.2 ((q.1 : ℝ)*r)).im) := by
+    apply measurable_iInf_of_upperSemicontinuous
+    · intro r
+      let s : ℝ≥0 := ⟨(r : ℝ),r.2.1⟩
+      have hm := (measurable_cutoffFlow κ j hj).comp
+        ((measurable_fst.mul_const s).prodMk measurable_snd)
+      exact (Complex.measurable_im.comp hm).ennreal_ofReal
+    · intro q
+      have hc : Continuous (fun r : unitInterval => cutoffFlow κ q.2.1 j q.2.2
+          ((q.1 : ℝ)*r)) :=
+        (cutoffFlow_continuousOn κ q.2.1 j hj q.2.2).comp_continuous
+          (continuous_const.mul continuous_subtype_val)
+          (fun r => mul_nonneg q.1.coe_nonneg r.2.1)
+      exact (ENNReal.continuous_ofReal.comp (Complex.continuous_im.comp hc)).upperSemicontinuous
+  change Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
+    (⨅ r : unitInterval, fun p : ℝ≥0 × (Driver × ℂ) =>
+      ENNReal.ofReal (cutoffFlow κ p.2.1 j p.2.2 ((p.1 : ℝ)*r)).im) q) at hm
+  simpa only [iInf_apply,cutoffMinimum] using hm
+
+lemma cutoff_survival_iff_minimum (κ : ℝ) (j : ℕ) (_hj : 1 ≤ j)
+    (b : Driver) (z : ℂ) (T : ℝ≥0) :
+    (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 (T : ℝ)) ↔
+      ENNReal.ofReal ((j : ℝ)⁻¹) < cutoffMinimum κ j b z T := by
+  rw [cutoffMinimum_eq]
+  exact (ENNReal.ofReal_lt_ofReal_iff_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg j))).symm
+
+lemma measurableSet_cutoff_survival (κ : ℝ) (j : ℕ) :
+    MeasurableSet {q : ℝ≥0 × (Driver × ℂ) | 1 ≤ j ∧
+      (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.2.1 j q.2.2 u).im) ''
+        Icc 0 (q.1 : ℝ))} := by
+  by_cases hj : 1 ≤ j
+  · have he : {q : ℝ≥0 × (Driver × ℂ) | 1 ≤ j ∧
+        (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.2.1 j q.2.2 u).im) ''
+          Icc 0 (q.1 : ℝ))} =
+        {q | ENNReal.ofReal ((j : ℝ)⁻¹) < cutoffMinimum κ j q.2.1 q.2.2 q.1} := by
+      ext q
+      simp only [mem_ofPred_eq,and_iff_right hj,cutoff_survival_iff_minimum κ j hj]
+    rw [he]
+    exact measurableSet_lt measurable_const (measurable_cutoffMinimum κ j hj)
+  · simp only [hj,false_and,ofPred_false,MeasurableSet.empty]
+
+lemma measurableSet_surviving (κ : ℝ) :
+    MeasurableSet {q : ℝ≥0 × (Driver × ℂ) | Surviving κ q.2.1 q.2.2 q.1} := by
+  change MeasurableSet {q : ℝ≥0 × (Driver × ℂ) | ∃ j : ℕ, 1 ≤ j ∧
+    (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.2.1 j q.2.2 u).im) '' Icc 0 (q.1 : ℝ))}
+  simp only [ofPred_exists]
+  exact MeasurableSet.iUnion (measurableSet_cutoff_survival κ)
+
+lemma measurable_flow (κ : ℝ) :
+    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => flow κ q.2.1 q.2.2 q.1) := by
+  let S : Set (ℝ≥0 × (Driver × ℂ)) := {q | Surviving κ q.2.1 q.2.2 q.1}
+  let f (j : ℕ) (q : S) : ℂ :=
+    if 1 ≤ j then cutoffFlow κ q.1.2.1 j q.1.2.2 q.1.1 else 0
+  have hf (j : ℕ) : Measurable (f j) := by
+    by_cases hj : 1 ≤ j
+    · simpa only [f,ite_eq_left hj,Function.comp_def] using
+        (measurable_cutoffFlow κ j hj).comp measurable_subtype_coe
+    · simp only [f,ite_eq_right hj]
+      exact measurable_const
+  have hp (j : ℕ) : MeasurableSet {q : S | 1 ≤ j ∧
+      (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ q.1.2.1 j q.1.2.2 u).im) ''
+        Icc 0 (q.1.1 : ℝ))} :=
+    measurable_subtype_coe (measurableSet_cutoff_survival κ j)
+  have hm := (Measurable.find hf hp (fun q : S => q.2)).dite
+    (g := fun _ : (Sᶜ : Set (ℝ≥0 × (Driver × ℂ))) => (0 : ℂ))
+      measurable_const (measurableSet_surviving κ)
+  convert hm using 1
+  funext q
+  by_cases hq : Surviving κ q.2.1 q.2.2 q.1
+  · simp only [flow,S,mem_ofPred_eq,dite_eq_left hq,f,ite_eq_left (Nat.find_spec hq).1]
+  · simp only [flow,S,mem_ofPred_eq,dite_eq_right hq]
+
+lemma surviving_flow_eq_of_driver_prefix (κ : ℝ) (b c : Driver) (z : ℂ)
+    (T : ℝ) (hT : 0 ≤ T)
+    (he : ∀ s ∈ Icc 0 T, driverValue b s = driverValue c s) :
+    (Surviving κ b z T ↔ Surviving κ c z T) ∧ flow κ b z T = flow κ c z T := by
+  have hp (j : ℕ) :
+      (1 ≤ j ∧ (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)) ↔
+      (1 ≤ j ∧ (j : ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ c j z u).im) '' Icc 0 T)) := by
+    by_cases hj : 1 ≤ j
+    · have hi : (fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T =
+          (fun u : ℝ => (cutoffFlow κ c j z u).im) '' Icc 0 T := by
+        apply image_congr
+        intro u hu
+        rw [cutoffFlow_eq_of_driver_prefix κ b c j hj z T hT he u hu]
+      rw [hi]
+    · simp only [hj,false_and]
+  have hs : Surviving κ b z T ↔ Surviving κ c z T := exists_congr hp
+  refine ⟨hs,?_⟩
+  by_cases hb : Surviving κ b z T
+  · have hc := hs.mp hb
+    have hn : Nat.find hb = Nat.find hc := Nat.find_congr' (fun {j} => hp j)
+    rw [flow,dite_eq_left hb,flow,dite_eq_left hc,← hn]
+    exact cutoffFlow_eq_of_driver_prefix κ b c (Nat.find hb) (Nat.find_spec hb).1
+      z T hT he T ⟨hT,le_rfl⟩
+  · have hc : ¬Surviving κ c z T := fun hc => hb (hs.mpr hc)
+    simp only [flow,dite_eq_right hb,dite_eq_right hc]
+
+lemma surviving_flow_stopDriver (κ : ℝ) (b : Driver) (z : ℂ) (T t : ℝ≥0)
+    (ht : t ≤ T) :
+    (Surviving κ (stopDriver b T) z t ↔ Surviving κ b z t) ∧
+      flow κ (stopDriver b T) z t = flow κ b z t := by
+  apply surviving_flow_eq_of_driver_prefix κ _ _ z t t.coe_nonneg
+  intro s hs
+  simp only [driverValue,stopDriver_apply,
+    min_eq_left (Real.toNNReal_le_iff_le_coe.mpr (hs.2.trans (show (t : ℝ) ≤ T from ht)))]
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetLevelDensity.lean b/lean/OAI/Probability/SLE/Angle/TargetLevelDensity.lean
index c63b09b0eba3dc6301576108613691ffe72d3d61..a175f2769198e296bd6d8ebb7c4a818309597f1a 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetLevelDensity.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetLevelDensity.lean
@@ -1,134 +1,134 @@
-import OAI.Probability.SLE.Angle.TargetLevelTime
-import OAI.Probability.SLE.Angle.TargetDiskRadius
-import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel
-
-namespace OAI
-
-/-! The prescribed finite-level density is the actual normalized stopped
-Green density, with no change to its finite/infinite-time convention. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Target
-
-private lemma alpha_eq (κ : ℝ) (_hκ : 0 < κ) : α κ=codimension κ := by
-  dsimp only [α,d,a,codimension,dimension]
-  field_simp [_hκ.ne']
-  ring
-
-private lemma p_eq (κ : ℝ) : p κ=greenExponent κ := by
-  dsimp only [p,a,greenExponent]
-  ring
-
-lemma disk_center_sin_arg (θ : ℝ) :
-    Real.sin (Complex.arg (diskMapInv θ 0))=Real.sin θ := by
-  have hz : diskMapInv θ 0=Complex.exp (θ*Complex.I) := by simp [diskMapInv]
-  rw [hz,Complex.sin_arg,Complex.norm_exp_ofReal_mul_I,div_one,
-    Complex.exp_ofReal_mul_I_im]
-
-lemma levelDensity_eq_stoppedGreen
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
-    levelDensity κ θ b ℓ =
-      stoppedGreen κ U γ G (diskMapInv θ 0) (4*ℓ/κ)/initialGreen κ (diskMapInv θ 0) := by
-  let z := diskMapInv θ 0
-  let u := 4*ℓ/κ
-  let τ := radiusLevelTime γ G z u
-  have hzH : z ∈ upperHalfPlane := diskMapInv_maps hθ (by simp)
-  have hc : 0 < 2*z.im := mul_pos (by norm_num) hzH
-  have hsin : 0 < Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  have hA : (2*z.im)^(-codimension κ) ≠ 0 := (Real.rpow_pos_of_pos hc _).ne'
-  have hD : (Real.sin θ)^greenExponent κ ≠ 0 := (Real.rpow_pos_of_pos hsin _).ne'
-  have htime := levelTime_eq_radiusLevelTime κ hκ b U hUc γ G F hw hU θ ℓ hℓ
-  change levelTime κ θ b ℓ=ENNReal.ofReal κ*τ at htime
-  by_cases hfin : τ<∞
-  · obtain ⟨T,hT,hTR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u hfin
-    have hτT : τ=(T:ℝ≥0∞) := radiusLevelTime_eq_of_live_level hUc hw z u T hT hTR
-    have htarget : levelTime κ θ b ℓ=ENNReal.ofReal (κ*(T:ℝ)) := by
-      rw [htime,hτT,ENNReal.ofReal_mul hκ.le,ENNReal.ofReal_coe_nnreal]
-    have htargetfin : levelTime κ θ b ℓ<∞ := by rw [htarget]; exact ENNReal.ofReal_lt_top
-    have htargreal : (levelTime κ θ b ℓ).toReal=κ*(T:ℝ) := by
-      rw [htarget,ENNReal.toReal_ofReal (mul_nonneg hκ.le T.coe_nonneg)]
-    have hsg : stoppedGreen κ U γ G z u =
-        ((2*z.im)*Real.exp (-u))^(-codimension κ)*
-          (Real.sin (Complex.arg (G T z-(U T:ℂ))))^greenExponent κ := by
-      change (if τ<∞ then greenDensity κ U γ G τ.toNNReal z else 0)=_
-      rw [ite_eq_left hfin,hτT,ENNReal.toNNReal_coe,greenDensity,ite_eq_left hT,hTR]
-    have hexp : 2*a κ*α κ*ℓ=(-u)*(-codimension κ) := by
-      rw [alpha_eq κ hκ]
-      dsimp only [a,u]
-      field_simp [hκ.ne']
-      ring
-    have hinit : initialGreen κ z=(2*z.im)^(-codimension κ)*(Real.sin θ)^greenExponent κ := by
-      rw [initialGreen]
-      exact congrArg (fun s : ℝ => (2*z.im)^(-codimension κ)*s^greenExponent κ)
-        (disk_center_sin_arg θ)
-    rw [levelDensity,ite_eq_left htargetfin,htargreal,
-      angle_eq_witness κ hκ b U γ G F hw hU θ 0 T hT,p_eq κ,hexp]
-    change _=stoppedGreen κ U γ G z u/initialGreen κ z
-    rw [hsg,hinit,Real.mul_rpow hc.le (Real.exp_pos _).le,← Real.exp_mul]
-    field_simp [hA,hD]
-    rfl
-  · have htop : τ=∞ := top_le_iff.mp (not_lt.mp hfin)
-    have htoptarget : levelTime κ θ b ℓ=∞ := by
-      rw [htime,htop,ENNReal.mul_top (ENNReal.ofReal_pos.mpr hκ).ne']
-    rw [levelDensity,htoptarget,ite_eq_right (not_lt_of_ge le_rfl)]
-    change 0=(if τ<∞ then _ else 0)/_
-    rw [ite_eq_right hfin,zero_div]
-
-lemma levelDensity_eq_canonicalFiniteDensity
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (p : C(ℝ≥0,ℝ))
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*p t) γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → Real.sqrt κ*p s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
-    ENNReal.ofReal (levelDensity κ θ b ℓ)=
-      canonicalFiniteDensity κ (4*ℓ/κ,diskMapInv θ 0,p) := by
-  rw [levelDensity_eq_stoppedGreen κ hκ b _ (p.continuous.const_mul _) γ G F hw hU θ hθ ℓ hℓ,
-    canonicalFiniteDensity_eq κ p hw _ (div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le)]
-
-theorem levelDensity_integrable_mean
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ<8) (B : ℝ≥0 → Ω → ℝ)
-    (hB : IsBrownianReal B P) (b : Ω → Driver)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hU : ∀ᵐ ω ∂P, ∀ s : ℝ, 0 ≤ s →
-      Real.sqrt κ*B s.toNNReal ω=driverValue (b ω) (κ*s))
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
-    Integrable (fun ω => levelDensity κ θ (b ω) ℓ) P ∧
-      (∫ ω, levelDensity κ θ (b ω) ℓ ∂P)=1 := by
-  let z := diskMapInv θ 0
-  let u := 4*ℓ/κ
-  have hz : z ∈ upperHalfPlane := diskMapInv_maps hθ (by simp)
-  have hu : 0 ≤ u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
-  have hp : 0 < initialGreen κ z := initialGreen_pos κ z hz
-  obtain ⟨hi,he⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
-  have heq : (fun ω => stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-      (γ ω) (G ω) z u/initialGreen κ z) =ᵐ[P]
-      (fun ω => levelDensity κ θ (b ω) ℓ) := by
-    filter_upwards [hw,hU] with ω hω hmatch
-    have heU : (fun t : ℝ≥0 => Real.sqrt κ*B t ω)=
-        (fun t : ℝ≥0 => driverValue (b ω) (κ*(t:ℝ))) := by
-      funext t
-      simpa only [Real.toNNReal_coe] using hmatch t t.coe_nonneg
-    have hc : Continuous (fun t : ℝ≥0 => Real.sqrt κ*B t ω) := by
-      rw [heU]
-      exact (b ω).continuous.comp (continuous_real_toNNReal.comp
-        (continuous_const.mul continuous_subtype_val))
-    exact (levelDensity_eq_stoppedGreen κ hκ (b ω) _ hc (γ ω)
-      (G ω) (F ω) hω hmatch θ hθ ℓ hℓ).symm
-  refine ⟨(hi.div_const _).congr heq,?_⟩
-  rw [← integral_congr_ae heq,integral_div,he,div_self hp.ne']
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetLevelTime
+import OAI.Probability.SLE.Angle.TargetDiskRadius
+import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel
+
+namespace OAI
+
+/-! The prescribed finite-level density is the actual normalized stopped
+Green density, with no change to its finite/infinite-time convention. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Target
+
+private lemma alpha_eq (κ : ℝ) (_hκ : 0 < κ) : α κ=codimension κ := by
+  dsimp only [α,d,a,codimension,dimension]
+  field_simp [_hκ.ne']
+  ring
+
+private lemma p_eq (κ : ℝ) : p κ=greenExponent κ := by
+  dsimp only [p,a,greenExponent]
+  ring
+
+lemma disk_center_sin_arg (θ : ℝ) :
+    Real.sin (Complex.arg (diskMapInv θ 0))=Real.sin θ := by
+  have hz : diskMapInv θ 0=Complex.exp (θ*Complex.I) := by simp [diskMapInv]
+  rw [hz,Complex.sin_arg,Complex.norm_exp_ofReal_mul_I,div_one,
+    Complex.exp_ofReal_mul_I_im]
+
+lemma levelDensity_eq_stoppedGreen
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
+    levelDensity κ θ b ℓ =
+      stoppedGreen κ U γ G (diskMapInv θ 0) (4*ℓ/κ)/initialGreen κ (diskMapInv θ 0) := by
+  let z := diskMapInv θ 0
+  let u := 4*ℓ/κ
+  let τ := radiusLevelTime γ G z u
+  have hzH : z ∈ upperHalfPlane := diskMapInv_maps hθ (by simp)
+  have hc : 0 < 2*z.im := mul_pos (by norm_num) hzH
+  have hsin : 0 < Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  have hA : (2*z.im)^(-codimension κ) ≠ 0 := (Real.rpow_pos_of_pos hc _).ne'
+  have hD : (Real.sin θ)^greenExponent κ ≠ 0 := (Real.rpow_pos_of_pos hsin _).ne'
+  have htime := levelTime_eq_radiusLevelTime κ hκ b U hUc γ G F hw hU θ ℓ hℓ
+  change levelTime κ θ b ℓ=ENNReal.ofReal κ*τ at htime
+  by_cases hfin : τ<∞
+  · obtain ⟨T,hT,hTR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u hfin
+    have hτT : τ=(T:ℝ≥0∞) := radiusLevelTime_eq_of_live_level hUc hw z u T hT hTR
+    have htarget : levelTime κ θ b ℓ=ENNReal.ofReal (κ*(T:ℝ)) := by
+      rw [htime,hτT,ENNReal.ofReal_mul hκ.le,ENNReal.ofReal_coe_nnreal]
+    have htargetfin : levelTime κ θ b ℓ<∞ := by rw [htarget]; exact ENNReal.ofReal_lt_top
+    have htargreal : (levelTime κ θ b ℓ).toReal=κ*(T:ℝ) := by
+      rw [htarget,ENNReal.toReal_ofReal (mul_nonneg hκ.le T.coe_nonneg)]
+    have hsg : stoppedGreen κ U γ G z u =
+        ((2*z.im)*Real.exp (-u))^(-codimension κ)*
+          (Real.sin (Complex.arg (G T z-(U T:ℂ))))^greenExponent κ := by
+      change (if τ<∞ then greenDensity κ U γ G τ.toNNReal z else 0)=_
+      rw [ite_eq_left hfin,hτT,ENNReal.toNNReal_coe,greenDensity,ite_eq_left hT,hTR]
+    have hexp : 2*a κ*α κ*ℓ=(-u)*(-codimension κ) := by
+      rw [alpha_eq κ hκ]
+      dsimp only [a,u]
+      field_simp [hκ.ne']
+      ring
+    have hinit : initialGreen κ z=(2*z.im)^(-codimension κ)*(Real.sin θ)^greenExponent κ := by
+      rw [initialGreen]
+      exact congrArg (fun s : ℝ => (2*z.im)^(-codimension κ)*s^greenExponent κ)
+        (disk_center_sin_arg θ)
+    rw [levelDensity,ite_eq_left htargetfin,htargreal,
+      angle_eq_witness κ hκ b U γ G F hw hU θ 0 T hT,p_eq κ,hexp]
+    change _=stoppedGreen κ U γ G z u/initialGreen κ z
+    rw [hsg,hinit,Real.mul_rpow hc.le (Real.exp_pos _).le,← Real.exp_mul]
+    field_simp [hA,hD]
+    rfl
+  · have htop : τ=∞ := top_le_iff.mp (not_lt.mp hfin)
+    have htoptarget : levelTime κ θ b ℓ=∞ := by
+      rw [htime,htop,ENNReal.mul_top (ENNReal.ofReal_pos.mpr hκ).ne']
+    rw [levelDensity,htoptarget,ite_eq_right (not_lt_of_ge le_rfl)]
+    change 0=(if τ<∞ then _ else 0)/_
+    rw [ite_eq_right hfin,zero_div]
+
+lemma levelDensity_eq_canonicalFiniteDensity
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (p : C(ℝ≥0,ℝ))
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*p t) γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → Real.sqrt κ*p s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
+    ENNReal.ofReal (levelDensity κ θ b ℓ)=
+      canonicalFiniteDensity κ (4*ℓ/κ,diskMapInv θ 0,p) := by
+  rw [levelDensity_eq_stoppedGreen κ hκ b _ (p.continuous.const_mul _) γ G F hw hU θ hθ ℓ hℓ,
+    canonicalFiniteDensity_eq κ p hw _ (div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le)]
+
+theorem levelDensity_integrable_mean
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ<8) (B : ℝ≥0 → Ω → ℝ)
+    (hB : IsBrownianReal B P) (b : Ω → Driver)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hU : ∀ᵐ ω ∂P, ∀ s : ℝ, 0 ≤ s →
+      Real.sqrt κ*B s.toNNReal ω=driverValue (b ω) (κ*s))
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
+    Integrable (fun ω => levelDensity κ θ (b ω) ℓ) P ∧
+      (∫ ω, levelDensity κ θ (b ω) ℓ ∂P)=1 := by
+  let z := diskMapInv θ 0
+  let u := 4*ℓ/κ
+  have hz : z ∈ upperHalfPlane := diskMapInv_maps hθ (by simp)
+  have hu : 0 ≤ u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
+  have hp : 0 < initialGreen κ z := initialGreen_pos κ z hz
+  obtain ⟨hi,he⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
+  have heq : (fun ω => stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+      (γ ω) (G ω) z u/initialGreen κ z) =ᵐ[P]
+      (fun ω => levelDensity κ θ (b ω) ℓ) := by
+    filter_upwards [hw,hU] with ω hω hmatch
+    have heU : (fun t : ℝ≥0 => Real.sqrt κ*B t ω)=
+        (fun t : ℝ≥0 => driverValue (b ω) (κ*(t:ℝ))) := by
+      funext t
+      simpa only [Real.toNNReal_coe] using hmatch t t.coe_nonneg
+    have hc : Continuous (fun t : ℝ≥0 => Real.sqrt κ*B t ω) := by
+      rw [heU]
+      exact (b ω).continuous.comp (continuous_real_toNNReal.comp
+        (continuous_const.mul continuous_subtype_val))
+    exact (levelDensity_eq_stoppedGreen κ hκ (b ω) _ hc (γ ω)
+      (G ω) (F ω) hω hmatch θ hθ ℓ hℓ).symm
+  refine ⟨(hi.div_const _).congr heq,?_⟩
+  rw [← integral_congr_ae heq,integral_div,he,div_self hp.ne']
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetLevelMeasurability.lean b/lean/OAI/Probability/SLE/Angle/TargetLevelMeasurability.lean
index 5ad28a98dde667252268309347ac4faaa247094e..d042e51bb6a6e344201c357b44995a38bd07c5d5 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetLevelMeasurability.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetLevelMeasurability.lean
@@ -1,107 +1,107 @@
-import OAI.Probability.SLE.Loewner.TargetJointMeasurability
-
-namespace OAI
-
-/-! Joint angle, level and driver measurability of the exact rational
-level stop and its prescribed change-of-law density. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma levelTime_eq_iInf (κ θ : ℝ) (b : Driver) (ℓ : ℝ) :
-    levelTime κ θ b ℓ = ⨅ q : ℚ,
-      if 0 < q ∧ Surviving κ b (diskMapInv θ 0) q ∧ ℓ < clock κ θ b 0 q
-      then ENNReal.ofReal (q : ℝ) else ∞ := by
-  apply le_antisymm
-  · apply le_iInf
-    intro q
-    split_ifs with hq
-    · exact sInf_le ⟨q,hq.1,rfl,hq.2.1,hq.2.2⟩
-    · exact le_top
-  · apply le_sInf
-    rintro t ⟨q,hq,rfl,hs,hc⟩
-    exact iInf_le_of_le q (by simp only [hq,hs,hc,and_self,ite_true,le_refl])
-
-lemma measurable_levelTime_joint (κ : ℝ) :
-    Measurable (fun p : ℝ × (ℝ × Driver) => levelTime κ p.2.1 p.2.2 p.1) := by
-  have he : (fun p : ℝ × (ℝ × Driver) => levelTime κ p.2.1 p.2.2 p.1) =
-      (fun p => ⨅ q : ℚ,
-        if 0 < q ∧ Surviving κ p.2.2 (diskMapInv p.2.1 0) q ∧
-          p.1 < clock κ p.2.1 p.2.2 0 q
-        then ENNReal.ofReal (q : ℝ) else ∞) :=
-    funext (fun p => levelTime_eq_iInf κ p.2.1 p.2.2 p.1)
-  rw [he]
-  apply Measurable.iInf
-  intro q
-  by_cases hq : (0:ℚ) < q
-  · let t : ℝ≥0 := NNReal.mk (q:ℝ) (Rat.cast_nonneg.mpr hq.le)
-    have hp : Measurable (fun p : ℝ × (ℝ × Driver) =>
-        (t,(p.2.1,(p.2.2,(0:ℂ))))) :=
-      measurable_const.prodMk (measurable_snd.fst.prodMk
-        (measurable_snd.snd.prodMk measurable_const))
-    have hs : MeasurableSet {p : ℝ × (ℝ × Driver) |
-        Surviving κ p.2.2 (diskMapInv p.2.1 0) (q:ℝ)} :=
-      hp (measurableSet_disk_surviving_joint κ)
-    have hc : Measurable (fun p : ℝ × (ℝ × Driver) => clock κ p.2.1 p.2.2 0 (q:ℝ)) :=
-      @Measurable.comp (ℝ × (ℝ × Driver)) (ℝ≥0 × (ℝ × (Driver × ℂ))) ℝ
-        inferInstance inferInstance inferInstance
-        (fun p => clock κ p.2.1 p.2.2.1 p.2.2.2 p.1)
-        (fun p => (t,(p.2.1,(p.2.2,(0:ℂ))))) (measurable_clock_joint κ) hp
-    have hA := hs.inter (measurableSet_lt measurable_fst hc)
-    simpa only [hq,true_and] using
-      (measurable_const.ite hA measurable_const : Measurable
-        (fun p : ℝ × (ℝ × Driver) =>
-          if Surviving κ p.2.2 (diskMapInv p.2.1 0) (q:ℝ) ∧
-            p.1 < clock κ p.2.1 p.2.2 0 (q:ℝ)
-          then ENNReal.ofReal (q:ℝ) else ∞))
-  · simp only [hq,false_and,ite_false]
-    exact measurable_const
-
-lemma measurable_levelTime_angle (κ ℓ : ℝ) :
-    Measurable (fun p : ℝ × Driver => levelTime κ p.1 p.2 ℓ) :=
-  @Measurable.comp (ℝ × Driver) (ℝ × (ℝ × Driver)) ℝ≥0∞
-    inferInstance inferInstance inferInstance
-    (fun p => levelTime κ p.2.1 p.2.2 p.1) (fun p => (ℓ,p))
-    (measurable_levelTime_joint κ) (measurable_const.prodMk measurable_id)
-
-lemma measurable_levelDensity_joint (κ : ℝ) :
-    Measurable (fun p : ℝ × (ℝ × Driver) => levelDensity κ p.2.1 p.2.2 p.1) := by
-  have hτ := measurable_levelTime_joint κ
-  have ht : Measurable (fun p : ℝ × (ℝ × Driver) =>
-      (levelTime κ p.2.1 p.2.2 p.1).toReal.toNNReal) :=
-    continuous_real_toNNReal.measurable.comp hτ.ennreal_toReal
-  have hp : Measurable (fun p : ℝ × (ℝ × Driver) =>
-      ((levelTime κ p.2.1 p.2.2 p.1).toReal.toNNReal,
-        (p.2.1,(p.2.2,(0:ℂ))))) :=
-    ht.prodMk (measurable_snd.fst.prodMk (measurable_snd.snd.prodMk measurable_const))
-  have ha : Measurable (fun p : ℝ × (ℝ × Driver) =>
-      angle κ p.2.1 p.2.2 0 (levelTime κ p.2.1 p.2.2 p.1).toReal) := by
-    have hm := @Measurable.comp (ℝ × (ℝ × Driver)) (ℝ≥0 × (ℝ × (Driver × ℂ))) ℝ
-      inferInstance inferInstance inferInstance
-      (fun p => angle κ p.2.1 p.2.2.1 p.2.2.2 p.1)
-      (fun p => ((levelTime κ p.2.1 p.2.2 p.1).toReal.toNNReal,
-        (p.2.1,(p.2.2,(0:ℂ))))) (measurable_angle_joint κ) hp
-    simpa only [Function.comp_def,Real.toNNReal_of_nonneg ENNReal.toReal_nonneg,NNReal.coe_mk] using hm
-  have hb : Measurable (fun p : ℝ × (ℝ × Driver) =>
-      Real.exp (2*a κ*α κ*p.1)*
-        Real.sin (angle κ p.2.1 p.2.2 0 (levelTime κ p.2.1 p.2.2 p.1).toReal)^(SLEExactGauge.Target.p κ) /
-          Real.sin p.2.1^(SLEExactGauge.Target.p κ)) :=
-    ((measurable_const.mul measurable_fst).exp.mul (ha.sin.pow_const (SLEExactGauge.Target.p κ))).div
-      (measurable_snd.fst.sin.pow_const (SLEExactGauge.Target.p κ))
-  exact hb.ite (measurableSet_lt hτ measurable_const) measurable_const
-
-lemma measurable_levelDensity_angle (κ ℓ : ℝ) :
-    Measurable (fun p : ℝ × Driver => levelDensity κ p.1 p.2 ℓ) :=
-  @Measurable.comp (ℝ × Driver) (ℝ × (ℝ × Driver)) ℝ
-    inferInstance inferInstance inferInstance
-    (fun p => levelDensity κ p.2.1 p.2.2 p.1) (fun p => (ℓ,p))
-    (measurable_levelDensity_joint κ) (measurable_const.prodMk measurable_id)
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.TargetJointMeasurability
+
+namespace OAI
+
+/-! Joint angle, level and driver measurability of the exact rational
+level stop and its prescribed change-of-law density. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma levelTime_eq_iInf (κ θ : ℝ) (b : Driver) (ℓ : ℝ) :
+    levelTime κ θ b ℓ = ⨅ q : ℚ,
+      if 0 < q ∧ Surviving κ b (diskMapInv θ 0) q ∧ ℓ < clock κ θ b 0 q
+      then ENNReal.ofReal (q : ℝ) else ∞ := by
+  apply le_antisymm
+  · apply le_iInf
+    intro q
+    split_ifs with hq
+    · exact sInf_le ⟨q,hq.1,rfl,hq.2.1,hq.2.2⟩
+    · exact le_top
+  · apply le_sInf
+    rintro t ⟨q,hq,rfl,hs,hc⟩
+    exact iInf_le_of_le q (by simp only [hq,hs,hc,and_self,ite_true,le_refl])
+
+lemma measurable_levelTime_joint (κ : ℝ) :
+    Measurable (fun p : ℝ × (ℝ × Driver) => levelTime κ p.2.1 p.2.2 p.1) := by
+  have he : (fun p : ℝ × (ℝ × Driver) => levelTime κ p.2.1 p.2.2 p.1) =
+      (fun p => ⨅ q : ℚ,
+        if 0 < q ∧ Surviving κ p.2.2 (diskMapInv p.2.1 0) q ∧
+          p.1 < clock κ p.2.1 p.2.2 0 q
+        then ENNReal.ofReal (q : ℝ) else ∞) :=
+    funext (fun p => levelTime_eq_iInf κ p.2.1 p.2.2 p.1)
+  rw [he]
+  apply Measurable.iInf
+  intro q
+  by_cases hq : (0:ℚ) < q
+  · let t : ℝ≥0 := NNReal.mk (q:ℝ) (Rat.cast_nonneg.mpr hq.le)
+    have hp : Measurable (fun p : ℝ × (ℝ × Driver) =>
+        (t,(p.2.1,(p.2.2,(0:ℂ))))) :=
+      measurable_const.prodMk (measurable_snd.fst.prodMk
+        (measurable_snd.snd.prodMk measurable_const))
+    have hs : MeasurableSet {p : ℝ × (ℝ × Driver) |
+        Surviving κ p.2.2 (diskMapInv p.2.1 0) (q:ℝ)} :=
+      hp (measurableSet_disk_surviving_joint κ)
+    have hc : Measurable (fun p : ℝ × (ℝ × Driver) => clock κ p.2.1 p.2.2 0 (q:ℝ)) :=
+      @Measurable.comp (ℝ × (ℝ × Driver)) (ℝ≥0 × (ℝ × (Driver × ℂ))) ℝ
+        inferInstance inferInstance inferInstance
+        (fun p => clock κ p.2.1 p.2.2.1 p.2.2.2 p.1)
+        (fun p => (t,(p.2.1,(p.2.2,(0:ℂ))))) (measurable_clock_joint κ) hp
+    have hA := hs.inter (measurableSet_lt measurable_fst hc)
+    simpa only [hq,true_and] using
+      (measurable_const.ite hA measurable_const : Measurable
+        (fun p : ℝ × (ℝ × Driver) =>
+          if Surviving κ p.2.2 (diskMapInv p.2.1 0) (q:ℝ) ∧
+            p.1 < clock κ p.2.1 p.2.2 0 (q:ℝ)
+          then ENNReal.ofReal (q:ℝ) else ∞))
+  · simp only [hq,false_and,ite_false]
+    exact measurable_const
+
+lemma measurable_levelTime_angle (κ ℓ : ℝ) :
+    Measurable (fun p : ℝ × Driver => levelTime κ p.1 p.2 ℓ) :=
+  @Measurable.comp (ℝ × Driver) (ℝ × (ℝ × Driver)) ℝ≥0∞
+    inferInstance inferInstance inferInstance
+    (fun p => levelTime κ p.2.1 p.2.2 p.1) (fun p => (ℓ,p))
+    (measurable_levelTime_joint κ) (measurable_const.prodMk measurable_id)
+
+lemma measurable_levelDensity_joint (κ : ℝ) :
+    Measurable (fun p : ℝ × (ℝ × Driver) => levelDensity κ p.2.1 p.2.2 p.1) := by
+  have hτ := measurable_levelTime_joint κ
+  have ht : Measurable (fun p : ℝ × (ℝ × Driver) =>
+      (levelTime κ p.2.1 p.2.2 p.1).toReal.toNNReal) :=
+    continuous_real_toNNReal.measurable.comp hτ.ennreal_toReal
+  have hp : Measurable (fun p : ℝ × (ℝ × Driver) =>
+      ((levelTime κ p.2.1 p.2.2 p.1).toReal.toNNReal,
+        (p.2.1,(p.2.2,(0:ℂ))))) :=
+    ht.prodMk (measurable_snd.fst.prodMk (measurable_snd.snd.prodMk measurable_const))
+  have ha : Measurable (fun p : ℝ × (ℝ × Driver) =>
+      angle κ p.2.1 p.2.2 0 (levelTime κ p.2.1 p.2.2 p.1).toReal) := by
+    have hm := @Measurable.comp (ℝ × (ℝ × Driver)) (ℝ≥0 × (ℝ × (Driver × ℂ))) ℝ
+      inferInstance inferInstance inferInstance
+      (fun p => angle κ p.2.1 p.2.2.1 p.2.2.2 p.1)
+      (fun p => ((levelTime κ p.2.1 p.2.2 p.1).toReal.toNNReal,
+        (p.2.1,(p.2.2,(0:ℂ))))) (measurable_angle_joint κ) hp
+    simpa only [Function.comp_def,Real.toNNReal_of_nonneg ENNReal.toReal_nonneg,NNReal.coe_mk] using hm
+  have hb : Measurable (fun p : ℝ × (ℝ × Driver) =>
+      Real.exp (2*a κ*α κ*p.1)*
+        Real.sin (angle κ p.2.1 p.2.2 0 (levelTime κ p.2.1 p.2.2 p.1).toReal)^(SLEExactGauge.Target.p κ) /
+          Real.sin p.2.1^(SLEExactGauge.Target.p κ)) :=
+    ((measurable_const.mul measurable_fst).exp.mul (ha.sin.pow_const (SLEExactGauge.Target.p κ))).div
+      (measurable_snd.fst.sin.pow_const (SLEExactGauge.Target.p κ))
+  exact hb.ite (measurableSet_lt hτ measurable_const) measurable_const
+
+lemma measurable_levelDensity_angle (κ ℓ : ℝ) :
+    Measurable (fun p : ℝ × Driver => levelDensity κ p.1 p.2 ℓ) :=
+  @Measurable.comp (ℝ × Driver) (ℝ × (ℝ × Driver)) ℝ
+    inferInstance inferInstance inferInstance
+    (fun p => levelDensity κ p.2.1 p.2.2 p.1) (fun p => (ℓ,p))
+    (measurable_levelDensity_joint κ) (measurable_const.prodMk measurable_id)
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetLevelTime.lean b/lean/OAI/Probability/SLE/Angle/TargetLevelTime.lean
index ef328a2fba31177cf30326350777f4c2a31f0d51..cc654bee7c1907b5636dfc7ff1ff71986c0e0e74 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetLevelTime.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetLevelTime.lean
@@ -1,153 +1,153 @@
-import OAI.Probability.SLE.Angle.TargetClockIdentification
-import OAI.Probability.SLE.Geometry.JointRadiusKernel
-
-namespace OAI
-
-/-! The strict rational infimum in the prescribed construction is the
-actual geometric radius stop, with the source time normalization. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Target
-
-private lemma clock_gt_iff_radius_lt
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ ℓ : ℝ) (T : ℝ≥0) (hz : diskMapInv θ 0 ∈ survivingDomain γ T) :
-    ℓ < clock κ θ b 0 (κ*T) ↔ fullRadius G T (diskMapInv θ 0) <
-      (2*(diskMapInv θ 0).im)*Real.exp (-(4*ℓ/κ)) := by
-  let z := diskMapInv θ 0
-  let L := Real.log ((2*z.im)/fullRadius G T z)
-  have hz0 := survivingDomain_antitone γ (show 0 ≤ T from zero_le) hz
-  have hy : 0 < 2*z.im := mul_pos (by norm_num) hz.1.1
-  have he : clock κ θ b 0 (κ*T)=κ*(L/4) := by
-    rw [clock_eq_activeRadiusClock κ hκ b U hUc γ G F hw hU θ 0 T hz]
-    simp only [Multipoint.activeRadiusClock,Real.toNNReal_coe]
-    change κ*((Real.log ((2*z.im)/fullRadius G T z)-
-      Real.log ((2*z.im)/fullRadius G 0 z))/4)=κ*(L/4)
-    rw [hw.fullRadius_initial hUc z hz0,div_self hy.ne',Real.log_one,sub_zero]
-  rw [he,fullRadius_eq_logClock hUc hw z T hz,
-    logRadiusClock_logCoordinates hUc hw z T hz,mul_lt_mul_iff_right₀ hy,Real.exp_lt_exp]
-  change ℓ < κ*(L/4) ↔ -L < -(4*ℓ/κ)
-  rw [neg_lt_neg_iff,div_lt_iff₀ hκ]
-  constructor <;> intro h <;> nlinarith
-
-lemma levelTime_eq_radiusLevelTime
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
-    levelTime κ θ b ℓ=ENNReal.ofReal κ * radiusLevelTime γ G (diskMapInv θ 0) (4*ℓ/κ) := by
-  let z := diskMapInv θ 0
-  let u := 4*ℓ/κ
-  let τ := radiusLevelTime γ G z u
-  have hu : 0 ≤ u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
-  have hκe : ENNReal.ofReal κ ≠ 0 := (ENNReal.ofReal_pos.mpr hκ).ne'
-  have hscale (t : ℝ≥0) : ENNReal.ofReal (κ*(t:ℝ))=ENNReal.ofReal κ*(t:ℝ≥0∞) := by
-    rw [ENNReal.ofReal_mul hκ.le,ENNReal.ofReal_coe_nnreal]
-  unfold levelTime
-  apply sInf_eq_of_forall_ge_of_forall_gt_exists_lt
-  · rintro r ⟨q,hq,rfl,hqLive,hqClock⟩
-    have hq0 : 0 ≤ (q:ℝ)/κ := div_nonneg (Rat.cast_nonneg.mpr hq.le) hκ.le
-    let t : ℝ≥0 := NNReal.mk ((q:ℝ)/κ) hq0
-    have ht : (t:ℝ)=(q:ℝ)/κ := rfl
-    have htime : κ*(t:ℝ)=(q:ℝ) := by rw [ht,mul_div_cancel₀ _ hκ.ne']
-    have htlive : z ∈ survivingDomain γ t := by
-      have hh := (surviving_iff_witness_real_time κ hκ b U γ G F hw hU (q:ℝ)
-        (Rat.cast_nonneg.mpr hq.le) z).1 hqLive
-      simpa only [← ht,Real.toNNReal_coe] using hh
-    have hR : fullRadius G t z < (2*z.im)*Real.exp (-u) := by
-      apply (clock_gt_iff_radius_lt κ hκ b U hUc γ G F hw hU θ ℓ t htlive).1
-      simpa only [htime] using hqClock
-    have hτt : τ < (t:ℝ≥0∞) := radiusLevelTime_lt_of_live_radius_lt hUc hw z u hu t htlive hR
-    calc
-      ENNReal.ofReal κ*τ ≤ ENNReal.ofReal κ*(t:ℝ≥0∞) :=
-        mul_le_mul_of_nonneg_left hτt.le (show (0:ℝ≥0∞) ≤ ENNReal.ofReal κ from zero_le)
-      _ = ENNReal.ofReal (q:ℝ) := by rw [← hscale t,htime]
-  · intro w hwlt
-    change ENNReal.ofReal κ*τ<w at hwlt
-    have hτfinite : τ < ∞ := by
-      by_contra hn
-      have htop : τ=∞ := top_le_iff.mp (not_lt.mp hn)
-      rw [htop,ENNReal.mul_top hκe] at hwlt
-      exact not_lt_of_ge le_top hwlt
-    obtain ⟨T,hT,hTR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u hτfinite
-    have hτT : τ=(T:ℝ≥0∞) := radiusLevelTime_eq_of_live_level hUc hw z u T hT hTR
-    have hTw : ENNReal.ofReal (κ*(T:ℝ))<w := by rwa [hτT,← hscale T] at hwlt
-    let V : Set ℝ≥0 := {t | z ∈ survivingDomain γ t ∧ ENNReal.ofReal (κ*(t:ℝ))<w}
-    have hVo : IsOpen V := by
-      have hLive : IsOpen {t : ℝ≥0 | z ∈ survivingDomain γ t} :=
-        source_survival_open.preimage
-          (((continuous_const : Continuous (fun _ : ℝ≥0 =>
-            (⟨γ,hw.1⟩ : C(ℝ≥0,ℂ)))).prodMk continuous_id).prodMk continuous_const)
-      have hc : Continuous (fun t : ℝ≥0 => ENNReal.ofReal (κ*(t:ℝ))) :=
-        ENNReal.continuous_ofReal.comp (continuous_const.mul continuous_subtype_val)
-      exact hLive.inter (isOpen_lt hc continuous_const)
-    obtain ⟨t,hTt,htV⟩ := exists_Ico_subset_of_mem_nhds
-      (hVo.mem_nhds (show T ∈ V from ⟨hT,hTw⟩)) (exists_gt T)
-    have hTtR : κ*(T:ℝ)<κ*(t:ℝ) := mul_lt_mul_of_pos_left (by exact_mod_cast hTt) hκ
-    obtain ⟨q,hTq,hqt⟩ := exists_rat_btwn hTtR
-    have hqpos : (0:ℚ)<q := by
-      have hh := (mul_nonneg hκ.le T.coe_nonneg).trans_lt hTq
-      exact_mod_cast hh
-    have hq0 : 0 ≤ (q:ℝ)/κ := div_nonneg (Rat.cast_nonneg.mpr hqpos.le) hκ.le
-    let s : ℝ≥0 := NNReal.mk ((q:ℝ)/κ) hq0
-    have hs : (s:ℝ)=(q:ℝ)/κ := rfl
-    have hTs : T<s := by
-      apply NNReal.coe_lt_coe.mp
-      rw [hs,lt_div_iff₀ hκ]
-      simpa only [mul_comm] using hTq
-    have hst : s<t := by
-      apply NNReal.coe_lt_coe.mp
-      rw [hs,div_lt_iff₀ hκ]
-      simpa only [mul_comm] using hqt
-    have hVs := htV ⟨hTs.le,hst⟩
-    have hsTime : κ*(s:ℝ)=(q:ℝ) := by rw [hs,mul_div_cancel₀ _ hκ.ne']
-    have hR : fullRadius G s z < (2*z.im)*Real.exp (-u) := by
-      have hh := fullRadius_strictAnti_on_live hUc hw z hTs hVs.1
-      rwa [hTR] at hh
-    have hqClock : ℓ < clock κ θ b 0 (q:ℝ) := by
-      rw [← hsTime]
-      exact (clock_gt_iff_radius_lt κ hκ b U hUc γ G F hw hU θ ℓ s hVs.1).2 hR
-    refine ⟨ENNReal.ofReal (q:ℝ),⟨q,hqpos,rfl,?_,hqClock⟩,?_⟩
-    · rw [← hsTime]
-      exact (surviving_iff_witness κ hκ b U γ G F hw hU s z).2 hVs.1
-    · simpa only [hsTime] using hVs.2
-
-lemma levelTime_toReal
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
-    (levelTime κ θ b ℓ).toReal=κ*(radiusLevelTime γ G (diskMapInv θ 0) (4*ℓ/κ)).toReal := by
-  rw [levelTime_eq_radiusLevelTime κ hκ b U hUc γ G F hw hU θ ℓ hℓ,
-    ENNReal.toReal_mul,ENNReal.toReal_ofReal hκ.le]
-
-lemma measurable_levelTime_of_radius_kernel
-    {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (hκ : 0 < κ) (b : Ω → Driver) (U : Ω → ℝ≥0 → ℝ)
-    (hUc : ∀ ω, Continuous (U ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (U ω) (γ ω) (G ω) (F ω))
-    (hU : ∀ ω (s : ℝ), 0 ≤ s → U ω s.toNNReal=driverValue (b ω) (κ*s))
-    (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ)
-    (hm : Measurable (fun ω => radiusLevelTime (γ ω) (G ω) (diskMapInv θ 0) (4*ℓ/κ))) :
-    Measurable (fun ω => levelTime κ θ (b ω) ℓ) := by
-  have he : (fun ω => levelTime κ θ (b ω) ℓ)=
-      (fun ω => ENNReal.ofReal κ*radiusLevelTime (γ ω) (G ω) (diskMapInv θ 0) (4*ℓ/κ)) :=
-    funext (fun ω => levelTime_eq_radiusLevelTime κ hκ (b ω) (U ω) (hUc ω)
-      (γ ω) (G ω) (F ω) (hw ω) (hU ω) θ ℓ hℓ)
-  rw [he]
-  exact measurable_const.mul hm
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetClockIdentification
+import OAI.Probability.SLE.Geometry.JointRadiusKernel
+
+namespace OAI
+
+/-! The strict rational infimum in the prescribed construction is the
+actual geometric radius stop, with the source time normalization. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Target
+
+private lemma clock_gt_iff_radius_lt
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ ℓ : ℝ) (T : ℝ≥0) (hz : diskMapInv θ 0 ∈ survivingDomain γ T) :
+    ℓ < clock κ θ b 0 (κ*T) ↔ fullRadius G T (diskMapInv θ 0) <
+      (2*(diskMapInv θ 0).im)*Real.exp (-(4*ℓ/κ)) := by
+  let z := diskMapInv θ 0
+  let L := Real.log ((2*z.im)/fullRadius G T z)
+  have hz0 := survivingDomain_antitone γ (show 0 ≤ T from zero_le) hz
+  have hy : 0 < 2*z.im := mul_pos (by norm_num) hz.1.1
+  have he : clock κ θ b 0 (κ*T)=κ*(L/4) := by
+    rw [clock_eq_activeRadiusClock κ hκ b U hUc γ G F hw hU θ 0 T hz]
+    simp only [Multipoint.activeRadiusClock,Real.toNNReal_coe]
+    change κ*((Real.log ((2*z.im)/fullRadius G T z)-
+      Real.log ((2*z.im)/fullRadius G 0 z))/4)=κ*(L/4)
+    rw [hw.fullRadius_initial hUc z hz0,div_self hy.ne',Real.log_one,sub_zero]
+  rw [he,fullRadius_eq_logClock hUc hw z T hz,
+    logRadiusClock_logCoordinates hUc hw z T hz,mul_lt_mul_iff_right₀ hy,Real.exp_lt_exp]
+  change ℓ < κ*(L/4) ↔ -L < -(4*ℓ/κ)
+  rw [neg_lt_neg_iff,div_lt_iff₀ hκ]
+  constructor <;> intro h <;> nlinarith
+
+lemma levelTime_eq_radiusLevelTime
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
+    levelTime κ θ b ℓ=ENNReal.ofReal κ * radiusLevelTime γ G (diskMapInv θ 0) (4*ℓ/κ) := by
+  let z := diskMapInv θ 0
+  let u := 4*ℓ/κ
+  let τ := radiusLevelTime γ G z u
+  have hu : 0 ≤ u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
+  have hκe : ENNReal.ofReal κ ≠ 0 := (ENNReal.ofReal_pos.mpr hκ).ne'
+  have hscale (t : ℝ≥0) : ENNReal.ofReal (κ*(t:ℝ))=ENNReal.ofReal κ*(t:ℝ≥0∞) := by
+    rw [ENNReal.ofReal_mul hκ.le,ENNReal.ofReal_coe_nnreal]
+  unfold levelTime
+  apply sInf_eq_of_forall_ge_of_forall_gt_exists_lt
+  · rintro r ⟨q,hq,rfl,hqLive,hqClock⟩
+    have hq0 : 0 ≤ (q:ℝ)/κ := div_nonneg (Rat.cast_nonneg.mpr hq.le) hκ.le
+    let t : ℝ≥0 := NNReal.mk ((q:ℝ)/κ) hq0
+    have ht : (t:ℝ)=(q:ℝ)/κ := rfl
+    have htime : κ*(t:ℝ)=(q:ℝ) := by rw [ht,mul_div_cancel₀ _ hκ.ne']
+    have htlive : z ∈ survivingDomain γ t := by
+      have hh := (surviving_iff_witness_real_time κ hκ b U γ G F hw hU (q:ℝ)
+        (Rat.cast_nonneg.mpr hq.le) z).1 hqLive
+      simpa only [← ht,Real.toNNReal_coe] using hh
+    have hR : fullRadius G t z < (2*z.im)*Real.exp (-u) := by
+      apply (clock_gt_iff_radius_lt κ hκ b U hUc γ G F hw hU θ ℓ t htlive).1
+      simpa only [htime] using hqClock
+    have hτt : τ < (t:ℝ≥0∞) := radiusLevelTime_lt_of_live_radius_lt hUc hw z u hu t htlive hR
+    calc
+      ENNReal.ofReal κ*τ ≤ ENNReal.ofReal κ*(t:ℝ≥0∞) :=
+        mul_le_mul_of_nonneg_left hτt.le (show (0:ℝ≥0∞) ≤ ENNReal.ofReal κ from zero_le)
+      _ = ENNReal.ofReal (q:ℝ) := by rw [← hscale t,htime]
+  · intro w hwlt
+    change ENNReal.ofReal κ*τ<w at hwlt
+    have hτfinite : τ < ∞ := by
+      by_contra hn
+      have htop : τ=∞ := top_le_iff.mp (not_lt.mp hn)
+      rw [htop,ENNReal.mul_top hκe] at hwlt
+      exact not_lt_of_ge le_top hwlt
+    obtain ⟨T,hT,hTR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u hτfinite
+    have hτT : τ=(T:ℝ≥0∞) := radiusLevelTime_eq_of_live_level hUc hw z u T hT hTR
+    have hTw : ENNReal.ofReal (κ*(T:ℝ))<w := by rwa [hτT,← hscale T] at hwlt
+    let V : Set ℝ≥0 := {t | z ∈ survivingDomain γ t ∧ ENNReal.ofReal (κ*(t:ℝ))<w}
+    have hVo : IsOpen V := by
+      have hLive : IsOpen {t : ℝ≥0 | z ∈ survivingDomain γ t} :=
+        source_survival_open.preimage
+          (((continuous_const : Continuous (fun _ : ℝ≥0 =>
+            (⟨γ,hw.1⟩ : C(ℝ≥0,ℂ)))).prodMk continuous_id).prodMk continuous_const)
+      have hc : Continuous (fun t : ℝ≥0 => ENNReal.ofReal (κ*(t:ℝ))) :=
+        ENNReal.continuous_ofReal.comp (continuous_const.mul continuous_subtype_val)
+      exact hLive.inter (isOpen_lt hc continuous_const)
+    obtain ⟨t,hTt,htV⟩ := exists_Ico_subset_of_mem_nhds
+      (hVo.mem_nhds (show T ∈ V from ⟨hT,hTw⟩)) (exists_gt T)
+    have hTtR : κ*(T:ℝ)<κ*(t:ℝ) := mul_lt_mul_of_pos_left (by exact_mod_cast hTt) hκ
+    obtain ⟨q,hTq,hqt⟩ := exists_rat_btwn hTtR
+    have hqpos : (0:ℚ)<q := by
+      have hh := (mul_nonneg hκ.le T.coe_nonneg).trans_lt hTq
+      exact_mod_cast hh
+    have hq0 : 0 ≤ (q:ℝ)/κ := div_nonneg (Rat.cast_nonneg.mpr hqpos.le) hκ.le
+    let s : ℝ≥0 := NNReal.mk ((q:ℝ)/κ) hq0
+    have hs : (s:ℝ)=(q:ℝ)/κ := rfl
+    have hTs : T<s := by
+      apply NNReal.coe_lt_coe.mp
+      rw [hs,lt_div_iff₀ hκ]
+      simpa only [mul_comm] using hTq
+    have hst : s<t := by
+      apply NNReal.coe_lt_coe.mp
+      rw [hs,div_lt_iff₀ hκ]
+      simpa only [mul_comm] using hqt
+    have hVs := htV ⟨hTs.le,hst⟩
+    have hsTime : κ*(s:ℝ)=(q:ℝ) := by rw [hs,mul_div_cancel₀ _ hκ.ne']
+    have hR : fullRadius G s z < (2*z.im)*Real.exp (-u) := by
+      have hh := fullRadius_strictAnti_on_live hUc hw z hTs hVs.1
+      rwa [hTR] at hh
+    have hqClock : ℓ < clock κ θ b 0 (q:ℝ) := by
+      rw [← hsTime]
+      exact (clock_gt_iff_radius_lt κ hκ b U hUc γ G F hw hU θ ℓ s hVs.1).2 hR
+    refine ⟨ENNReal.ofReal (q:ℝ),⟨q,hqpos,rfl,?_,hqClock⟩,?_⟩
+    · rw [← hsTime]
+      exact (surviving_iff_witness κ hκ b U γ G F hw hU s z).2 hVs.1
+    · simpa only [hsTime] using hVs.2
+
+lemma levelTime_toReal
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
+    (levelTime κ θ b ℓ).toReal=κ*(radiusLevelTime γ G (diskMapInv θ 0) (4*ℓ/κ)).toReal := by
+  rw [levelTime_eq_radiusLevelTime κ hκ b U hUc γ G F hw hU θ ℓ hℓ,
+    ENNReal.toReal_mul,ENNReal.toReal_ofReal hκ.le]
+
+lemma measurable_levelTime_of_radius_kernel
+    {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (hκ : 0 < κ) (b : Ω → Driver) (U : Ω → ℝ≥0 → ℝ)
+    (hUc : ∀ ω, Continuous (U ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (U ω) (γ ω) (G ω) (F ω))
+    (hU : ∀ ω (s : ℝ), 0 ≤ s → U ω s.toNNReal=driverValue (b ω) (κ*s))
+    (θ ℓ : ℝ) (hℓ : 0 ≤ ℓ)
+    (hm : Measurable (fun ω => radiusLevelTime (γ ω) (G ω) (diskMapInv θ 0) (4*ℓ/κ))) :
+    Measurable (fun ω => levelTime κ θ (b ω) ℓ) := by
+  have he : (fun ω => levelTime κ θ (b ω) ℓ)=
+      (fun ω => ENNReal.ofReal κ*radiusLevelTime (γ ω) (G ω) (diskMapInv θ 0) (4*ℓ/κ)) :=
+    funext (fun ω => levelTime_eq_radiusLevelTime κ hκ (b ω) (U ω) (hUc ω)
+      (γ ω) (G ω) (F ω) (hw ω) (hU ω) θ ℓ hℓ)
+  rw [he]
+  exact measurable_const.mul hm
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetNormalizedFiltration.lean b/lean/OAI/Probability/SLE/Angle/TargetNormalizedFiltration.lean
index ad3eaaaba0cf22092d4a509cac211742f8773b53..992e04c5395b374f1a5f85a97122aa37be353339 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetNormalizedFiltration.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetNormalizedFiltration.lean
@@ -1,67 +1,67 @@
-import OAI.Probability.SLE.Angle.TargetBrownianNormalization
-
-namespace OAI
-
-/-! The exact time change of the natural Brownian past for the
-source-prescribed normalized driver. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-
-def normalizedFiltration (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) : Filtration ℝ≥0 mΩ :=
-  Filtration.natural (fun t ω => normalizedDriver κ B hc h0 ω t)
-    (fun t => ((continuous_eval_const t).measurable.comp
-      (normalizedDriver_measurable κ B hm hc h0)).stronglyMeasurable)
-
-theorem normalizedFiltration_scaled (κ : ℝ) (hκ : 0<κ)
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (t : ℝ≥0) :
-    normalizedFiltration κ B hm hc h0 (κ.toNNReal*t)=
-      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t := by
-  let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable)
-  let 𝒢 := normalizedFiltration κ B hm hc h0
-  have hp : 0<κ.toNNReal := Real.toNNReal_pos.mpr hκ
-  apply le_antisymm
-  · change (⨆ s ≤ κ.toNNReal*t,
-      MeasurableSpace.comap (fun ω => normalizedDriver κ B hc h0 ω s) (borel ℝ)) ≤ ℱ t
-    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
-    have hst : s/κ.toNNReal ≤ t := (div_le_iff₀ hp).mpr (by simpa only [mul_comm] using hs)
-    have hBm : Measurable[ℱ t] (B (s/κ.toNNReal)) :=
-      ((Filtration.stronglyAdapted_natural (fun r => (hm r).stronglyMeasurable)
-        (s/κ.toNNReal)).mono (ℱ.mono hst)).measurable
-    exact measurable_const.mul hBm
-  · change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ 𝒢 (κ.toNNReal*t)
-    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
-    have hst : κ.toNNReal*s ≤ κ.toNNReal*t := mul_le_mul_of_nonneg_left hs zero_le
-    have hYm : Measurable[𝒢 (κ.toNNReal*t)]
-        (fun ω => normalizedDriver κ B hc h0 ω (κ.toNNReal*s)) :=
-      ((Filtration.stronglyAdapted_natural (fun r =>
-        ((continuous_eval_const r).measurable.comp
-          (normalizedDriver_measurable κ B hm hc h0)).stronglyMeasurable)
-        (κ.toNNReal*s)).mono (𝒢.mono hst)).measurable
-    have he : B s=(fun ω => (Real.sqrt κ)⁻¹*normalizedDriver κ B hc h0 ω (κ.toNNReal*s)) := by
-      funext ω
-      change B s ω=(Real.sqrt κ)⁻¹*(Real.sqrt κ*B ((κ.toNNReal*s)/κ.toNNReal) ω)
-      rw [mul_div_cancel_left₀ _ hp.ne',inv_mul_cancel_left₀ (Real.sqrt_pos.mpr hκ).ne']
-    rw [he]
-    exact measurable_const.mul hYm
-
-theorem normalizedFiltration_at (κ : ℝ) (hκ : 0<κ)
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (t : ℝ≥0) :
-    normalizedFiltration κ B hm hc h0 t=
-      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) (t/κ.toNNReal) := by
-  have h := normalizedFiltration_scaled κ hκ B hm hc h0 (t/κ.toNNReal)
-  simpa only [mul_div_cancel₀ _ (Real.toNNReal_pos.mpr hκ).ne'] using h
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetBrownianNormalization
+
+namespace OAI
+
+/-! The exact time change of the natural Brownian past for the
+source-prescribed normalized driver. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+
+def normalizedFiltration (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) : Filtration ℝ≥0 mΩ :=
+  Filtration.natural (fun t ω => normalizedDriver κ B hc h0 ω t)
+    (fun t => ((continuous_eval_const t).measurable.comp
+      (normalizedDriver_measurable κ B hm hc h0)).stronglyMeasurable)
+
+theorem normalizedFiltration_scaled (κ : ℝ) (hκ : 0<κ)
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (t : ℝ≥0) :
+    normalizedFiltration κ B hm hc h0 (κ.toNNReal*t)=
+      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t := by
+  let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable)
+  let 𝒢 := normalizedFiltration κ B hm hc h0
+  have hp : 0<κ.toNNReal := Real.toNNReal_pos.mpr hκ
+  apply le_antisymm
+  · change (⨆ s ≤ κ.toNNReal*t,
+      MeasurableSpace.comap (fun ω => normalizedDriver κ B hc h0 ω s) (borel ℝ)) ≤ ℱ t
+    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
+    have hst : s/κ.toNNReal ≤ t := (div_le_iff₀ hp).mpr (by simpa only [mul_comm] using hs)
+    have hBm : Measurable[ℱ t] (B (s/κ.toNNReal)) :=
+      ((Filtration.stronglyAdapted_natural (fun r => (hm r).stronglyMeasurable)
+        (s/κ.toNNReal)).mono (ℱ.mono hst)).measurable
+    exact measurable_const.mul hBm
+  · change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ 𝒢 (κ.toNNReal*t)
+    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
+    have hst : κ.toNNReal*s ≤ κ.toNNReal*t := mul_le_mul_of_nonneg_left hs zero_le
+    have hYm : Measurable[𝒢 (κ.toNNReal*t)]
+        (fun ω => normalizedDriver κ B hc h0 ω (κ.toNNReal*s)) :=
+      ((Filtration.stronglyAdapted_natural (fun r =>
+        ((continuous_eval_const r).measurable.comp
+          (normalizedDriver_measurable κ B hm hc h0)).stronglyMeasurable)
+        (κ.toNNReal*s)).mono (𝒢.mono hst)).measurable
+    have he : B s=(fun ω => (Real.sqrt κ)⁻¹*normalizedDriver κ B hc h0 ω (κ.toNNReal*s)) := by
+      funext ω
+      change B s ω=(Real.sqrt κ)⁻¹*(Real.sqrt κ*B ((κ.toNNReal*s)/κ.toNNReal) ω)
+      rw [mul_div_cancel_left₀ _ hp.ne',inv_mul_cancel_left₀ (Real.sqrt_pos.mpr hκ).ne']
+    rw [he]
+    exact measurable_const.mul hYm
+
+theorem normalizedFiltration_at (κ : ℝ) (hκ : 0<κ)
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (t : ℝ≥0) :
+    normalizedFiltration κ B hm hc h0 t=
+      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) (t/κ.toNNReal) := by
+  have h := normalizedFiltration_scaled κ hκ B hm hc h0 (t/κ.toNNReal)
+  simpa only [mul_div_cancel₀ _ (Real.toNNReal_pos.mpr hκ).ne'] using h
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetPathNormalization.lean b/lean/OAI/Probability/SLE/Angle/TargetPathNormalization.lean
index 1268788e27beb69074396625781bcf1a579ad09f..6fd3759d1e9c4426ff57cd968401f3d224c6de17 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetPathNormalization.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetPathNormalization.lean
@@ -1,61 +1,61 @@
-import OAI.Probability.SLE.Loewner.CanonicalTraceCandidate
-
-namespace OAI
-
-/-! A measurable normalization of arbitrary continuous paths. On based
-paths it is inverse to the physical-time unnormalization. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance pathNormalizationMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance pathNormalizationBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance pathNormalizationDriverPolish : PolishSpace Target.Driver :=
-  ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace
-
-def normalizePath (κ : ℝ) (p : C(ℝ≥0,ℝ)) : Target.Driver where
-  toFun t := Real.sqrt κ*(p (t/κ.toNNReal)-p 0)
-  continuous_toFun := by fun_prop
-  map_zero' := by simp
-
-lemma measurable_normalizePath (κ : ℝ) : Measurable (normalizePath κ) := by
-  apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp
-  apply source_path_measurable
-  intro t
-  exact measurable_const.mul (((continuous_eval_const _).measurable).sub
-    ((continuous_eval_const 0).measurable))
-
-lemma normalize_unnormalize (κ : ℝ) (hκ : 0 < κ) (b : Target.Driver) :
-    normalizePath κ (unnormalizeDriver κ b)=b := by
-  ext t
-  change Real.sqrt κ*(b (κ.toNNReal*(t/κ.toNNReal))/Real.sqrt κ
-    -b (κ.toNNReal*0)/Real.sqrt κ)=b t
-  rw [mul_div_cancel₀ _ (Real.toNNReal_pos.mpr hκ).ne']
-  simp only [mul_zero,map_zero,zero_div,sub_zero]
-  exact mul_div_cancel₀ _ (Real.sqrt_pos.mpr hκ).ne'
-
-lemma measurableEmbedding_unnormalizeDriver (κ : ℝ) (hκ : 0 < κ) :
-    MeasurableEmbedding (unnormalizeDriver κ) :=
-  (measurable_unnormalizeDriver κ).measurableEmbedding
-    (fun b c h => by simpa only [normalize_unnormalize κ hκ] using congrArg (normalizePath κ) h)
-
-lemma unnormalizeDriver_law
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (W : Target.WienerLaw) (hW : Target.IsWiener W) :
-    W.map (unnormalizeDriver κ)=P.map (sourceDriverPath B hc) := by
-  rw [← (Target.normalizedDriver_law κ hκ B hB hm hc h0 W hW).map_eq,
-    Measure.map_map (measurable_unnormalizeDriver κ)
-      (Target.normalizedDriver_measurable κ B hm hc h0)]
-  congr 1
-  funext ω
-  exact unnormalize_normalizedDriver κ hκ B hc h0 ω
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.CanonicalTraceCandidate
+
+namespace OAI
+
+/-! A measurable normalization of arbitrary continuous paths. On based
+paths it is inverse to the physical-time unnormalization. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance pathNormalizationMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance pathNormalizationBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance pathNormalizationDriverPolish : PolishSpace Target.Driver :=
+  ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace
+
+def normalizePath (κ : ℝ) (p : C(ℝ≥0,ℝ)) : Target.Driver where
+  toFun t := Real.sqrt κ*(p (t/κ.toNNReal)-p 0)
+  continuous_toFun := by fun_prop
+  map_zero' := by simp
+
+lemma measurable_normalizePath (κ : ℝ) : Measurable (normalizePath κ) := by
+  apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp
+  apply source_path_measurable
+  intro t
+  exact measurable_const.mul (((continuous_eval_const _).measurable).sub
+    ((continuous_eval_const 0).measurable))
+
+lemma normalize_unnormalize (κ : ℝ) (hκ : 0 < κ) (b : Target.Driver) :
+    normalizePath κ (unnormalizeDriver κ b)=b := by
+  ext t
+  change Real.sqrt κ*(b (κ.toNNReal*(t/κ.toNNReal))/Real.sqrt κ
+    -b (κ.toNNReal*0)/Real.sqrt κ)=b t
+  rw [mul_div_cancel₀ _ (Real.toNNReal_pos.mpr hκ).ne']
+  simp only [mul_zero,map_zero,zero_div,sub_zero]
+  exact mul_div_cancel₀ _ (Real.sqrt_pos.mpr hκ).ne'
+
+lemma measurableEmbedding_unnormalizeDriver (κ : ℝ) (hκ : 0 < κ) :
+    MeasurableEmbedding (unnormalizeDriver κ) :=
+  (measurable_unnormalizeDriver κ).measurableEmbedding
+    (fun b c h => by simpa only [normalize_unnormalize κ hκ] using congrArg (normalizePath κ) h)
+
+lemma unnormalizeDriver_law
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (W : Target.WienerLaw) (hW : Target.IsWiener W) :
+    W.map (unnormalizeDriver κ)=P.map (sourceDriverPath B hc) := by
+  rw [← (Target.normalizedDriver_law κ hκ B hB hm hc h0 W hW).map_eq,
+    Measure.map_map (measurable_unnormalizeDriver κ)
+      (Target.normalizedDriver_measurable κ B hm hc h0)]
+  congr 1
+  funext ω
+  exact unnormalize_normalizedDriver κ hκ B hc h0 ω
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetPositiveFlow.lean b/lean/OAI/Probability/SLE/Angle/TargetPositiveFlow.lean
index d650a45990d0de866d1a27b915fbb7a080fbf0a8..73c8297ef25de4c739e6d9d283c7fcb8d946724b 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetPositiveFlow.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetPositiveFlow.lean
@@ -1,37 +1,37 @@
-import OAI.Probability.SLE.Angle.TargetFlowInjectivity
-
-namespace OAI
-
-/-! Identification of the raw selected flow with any positive trajectory.
-No geometric trace witness is needed. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma flow_eq_of_positive_trajectory
-    (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (hT : 0 ≤ T)
-    (Y : ℝ → ℂ) (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z)
-    (m : ℝ) (hm : 0 < m) (hheight : ∀ t ∈ Icc 0 T, m ≤ (Y t).im)
-    (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y
-      ((a κ:ℂ)/(Y t-driverValue b t)) (Ici t) t) :
-    Surviving κ b z T ∧ flow κ b z T=Y T := by
-  have hs := surviving_of_positive_trajectory κ b z T hT Y hY hY0 m hm hheight hYd
-  obtain ⟨k,hk,hkm⟩ := Real.exists_nat_pos_inv_lt hm
-  have hk1 : 1 ≤ k := hk
-  let j := max k (Nat.find hs)
-  have hj : 1 ≤ j := hk1.trans (le_max_left _ _)
-  have hinv : (j:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ :=
-    inv_anti₀ (by exact_mod_cast (Nat.succ_le_iff.mp hk) : (0:ℝ)<k)
-      (by exact_mod_cast (show k ≤ j from le_max_left _ _))
-  refine ⟨hs,?_⟩
-  rw [← surviving_common_cutoff κ b z T hT hs j (le_max_right _ _)]
-  exact cutoffFlow_eq_of_positive_trajectory κ b j hj z T hT Y hY hY0
-    (fun t ht => hinv.trans (hkm.le.trans (hheight t ht))) hYd T ⟨hT,le_rfl⟩
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetFlowInjectivity
+
+namespace OAI
+
+/-! Identification of the raw selected flow with any positive trajectory.
+No geometric trace witness is needed. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma flow_eq_of_positive_trajectory
+    (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (hT : 0 ≤ T)
+    (Y : ℝ → ℂ) (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z)
+    (m : ℝ) (hm : 0 < m) (hheight : ∀ t ∈ Icc 0 T, m ≤ (Y t).im)
+    (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y
+      ((a κ:ℂ)/(Y t-driverValue b t)) (Ici t) t) :
+    Surviving κ b z T ∧ flow κ b z T=Y T := by
+  have hs := surviving_of_positive_trajectory κ b z T hT Y hY hY0 m hm hheight hYd
+  obtain ⟨k,hk,hkm⟩ := Real.exists_nat_pos_inv_lt hm
+  have hk1 : 1 ≤ k := hk
+  let j := max k (Nat.find hs)
+  have hj : 1 ≤ j := hk1.trans (le_max_left _ _)
+  have hinv : (j:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ :=
+    inv_anti₀ (by exact_mod_cast (Nat.succ_le_iff.mp hk) : (0:ℝ)<k)
+      (by exact_mod_cast (show k ≤ j from le_max_left _ _))
+  refine ⟨hs,?_⟩
+  rw [← surviving_common_cutoff κ b z T hT hs j (le_max_right _ _)]
+  exact cutoffFlow_eq_of_positive_trajectory κ b j hj z T hT Y hY hY0
+    (fun t ht => hinv.trans (hkm.le.trans (hheight t ht))) hYd T ⟨hT,le_rfl⟩
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetPotentialIncrement.lean b/lean/OAI/Probability/SLE/Angle/TargetPotentialIncrement.lean
index a2e9fc722347b69e18d0ef826888478449922b04..cddc3e7cd55f93e84d1e6b6d4cbb79d5e4141804 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetPotentialIncrement.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetPotentialIncrement.lean
@@ -1,61 +1,61 @@
-import OAI.Probability.SLE.Loewner.TargetDriverContinuation
-import OAI.Probability.SLE.Mass.CanonicalMassCompensator
-import OAI.Probability.SLE.Mass.MassPotentialCompensator
-
-namespace OAI
-
-/-! The exact fresh-driver average in the source's potential difference
-is a conditional expectation, including its finite-value convention. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma mesh_eq_compensatorMeshTime (n i : ℕ) :
-    (i:ℝ)*mesh n=(compensatorMeshTime n i:ℝ) := by
-  simp [compensatorMeshTime,mesh]
-
-lemma potentialDifference_eq_current_sub_condExp
-    (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W) (n i : ℕ)
-    (hm : ∀ j, Measurable (fun b => potential κ θ b (compensatorMeshTime n j)))
-    (hfin : ∀ j, (∫⁻ b, potential κ θ b (compensatorMeshTime n j) ∂W) ≠ ∞) :
-    (fun b => potentialDifference κ θ W b n i) =ᵐ[W]
-      (fun b => (potential κ θ b (compensatorMeshTime n i)).toReal -
-        condExp (driverFiltration (compensatorMeshTime n i)) W
-          (fun b => (potential κ θ b (compensatorMeshTime n (i+1))).toReal) b) := by
-  have h := continuationMean_finite_condExp W hW (compensatorMeshTime n i)
-    (fun b => potential κ θ b (compensatorMeshTime n (i+1))) (hm (i+1)) (hfin (i+1))
-  filter_upwards [ae_lt_top (hm i) (hfin i),h.1,h.2] with b hb hc he
-  have hnext : ((i:ℝ)+1)*mesh n=(compensatorMeshTime n (i+1):ℝ) := by
-    simpa only [Nat.cast_add,Nat.cast_one] using mesh_eq_compensatorMeshTime n (i+1)
-  simp only [potentialDifference,mesh_eq_compensatorMeshTime,hnext]
-  change (if potential κ θ b (compensatorMeshTime n i)<∞ ∧
-      continuationMean W (compensatorMeshTime n i)
-        (fun b => potential κ θ b (compensatorMeshTime n (i+1))) b<∞ then _ else _)=_
-  rw [ite_eq_left ⟨hb,hc⟩]
-  dsimp only [continuationMean] at he
-  rw [he]
-
-theorem potentialDifference_eq_mass_increment
-    (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W) (n : ℕ)
-    (A : ℕ → Driver → ℝ) (L : Driver → ℝ)
-    (hA : StronglyAdapted (compensatorMeshFiltration driverFiltration n) A)
-    (hAI : ∀ i, Integrable (A i) W) (hLI : Integrable L W)
-    (hm : ∀ i, Measurable (fun b => potential κ θ b (compensatorMeshTime n i)))
-    (hfin : ∀ i, (∫⁻ b, potential κ θ b (compensatorMeshTime n i) ∂W) ≠ ∞)
-    (hV : ∀ i, (fun b => (potential κ θ b (compensatorMeshTime n i)).toReal) =ᵐ[W]
-      W[L-A i|compensatorMeshFiltration driverFiltration n i]) (i : ℕ) :
-    (fun b => potentialDifference κ θ W b n i) =ᵐ[W]
-      W[A (i+1)-A i|compensatorMeshFiltration driverFiltration n i] := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  exact (potentialDifference_eq_current_sub_condExp κ θ W hW n i hm hfin).trans
-    (remainingPotential_increment W (compensatorMeshFiltration driverFiltration n) A
-      (fun i b => (potential κ θ b (compensatorMeshTime n i)).toReal) L hA hAI hLI hV i)
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.TargetDriverContinuation
+import OAI.Probability.SLE.Mass.CanonicalMassCompensator
+import OAI.Probability.SLE.Mass.MassPotentialCompensator
+
+namespace OAI
+
+/-! The exact fresh-driver average in the source's potential difference
+is a conditional expectation, including its finite-value convention. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma mesh_eq_compensatorMeshTime (n i : ℕ) :
+    (i:ℝ)*mesh n=(compensatorMeshTime n i:ℝ) := by
+  simp [compensatorMeshTime,mesh]
+
+lemma potentialDifference_eq_current_sub_condExp
+    (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W) (n i : ℕ)
+    (hm : ∀ j, Measurable (fun b => potential κ θ b (compensatorMeshTime n j)))
+    (hfin : ∀ j, (∫⁻ b, potential κ θ b (compensatorMeshTime n j) ∂W) ≠ ∞) :
+    (fun b => potentialDifference κ θ W b n i) =ᵐ[W]
+      (fun b => (potential κ θ b (compensatorMeshTime n i)).toReal -
+        condExp (driverFiltration (compensatorMeshTime n i)) W
+          (fun b => (potential κ θ b (compensatorMeshTime n (i+1))).toReal) b) := by
+  have h := continuationMean_finite_condExp W hW (compensatorMeshTime n i)
+    (fun b => potential κ θ b (compensatorMeshTime n (i+1))) (hm (i+1)) (hfin (i+1))
+  filter_upwards [ae_lt_top (hm i) (hfin i),h.1,h.2] with b hb hc he
+  have hnext : ((i:ℝ)+1)*mesh n=(compensatorMeshTime n (i+1):ℝ) := by
+    simpa only [Nat.cast_add,Nat.cast_one] using mesh_eq_compensatorMeshTime n (i+1)
+  simp only [potentialDifference,mesh_eq_compensatorMeshTime,hnext]
+  change (if potential κ θ b (compensatorMeshTime n i)<∞ ∧
+      continuationMean W (compensatorMeshTime n i)
+        (fun b => potential κ θ b (compensatorMeshTime n (i+1))) b<∞ then _ else _)=_
+  rw [ite_eq_left ⟨hb,hc⟩]
+  dsimp only [continuationMean] at he
+  rw [he]
+
+theorem potentialDifference_eq_mass_increment
+    (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W) (n : ℕ)
+    (A : ℕ → Driver → ℝ) (L : Driver → ℝ)
+    (hA : StronglyAdapted (compensatorMeshFiltration driverFiltration n) A)
+    (hAI : ∀ i, Integrable (A i) W) (hLI : Integrable L W)
+    (hm : ∀ i, Measurable (fun b => potential κ θ b (compensatorMeshTime n i)))
+    (hfin : ∀ i, (∫⁻ b, potential κ θ b (compensatorMeshTime n i) ∂W) ≠ ∞)
+    (hV : ∀ i, (fun b => (potential κ θ b (compensatorMeshTime n i)).toReal) =ᵐ[W]
+      W[L-A i|compensatorMeshFiltration driverFiltration n i]) (i : ℕ) :
+    (fun b => potentialDifference κ θ W b n i) =ᵐ[W]
+      W[A (i+1)-A i|compensatorMeshFiltration driverFiltration n i] := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  exact (potentialDifference_eq_current_sub_condExp κ θ W hW n i hm hfin).trans
+    (remainingPotential_increment W (compensatorMeshFiltration driverFiltration n) A
+      (fun i b => (potential κ θ b (compensatorMeshTime n i)).toReal) L hA hAI hLI hV i)
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Angle/TargetPotentialMeasurability.lean b/lean/OAI/Probability/SLE/Angle/TargetPotentialMeasurability.lean
index 88a62f1691779e61f56e7f3b99ca2767504319c9..298507abe0ce39378ee593c0fbfa85befee08b14 100644
--- a/lean/OAI/Probability/SLE/Angle/TargetPotentialMeasurability.lean
+++ b/lean/OAI/Probability/SLE/Angle/TargetPotentialMeasurability.lean
@@ -1,169 +1,169 @@
-import OAI.Probability.SLE.Angle.TargetFlowMeasurability
-
-namespace OAI
-
-/-! The exact prescribed disk potential is a measurable path functional.
-All definitions, including the killed flow and its time integral, are
-retained on every continuous based driver. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma measurable_diskMapInv (θ : ℝ) : Measurable (diskMapInv θ) := by
-  unfold diskMapInv
-  fun_prop
-
-lemma measurable_initialRadius (θ : ℝ) : Measurable (initialRadius θ) := by
-  exact (measurable_const.mul (Complex.measurable_im.comp (measurable_diskMapInv θ))).mul
-    (((measurable_deriv (diskMap θ)).comp (measurable_diskMapInv θ)).norm)
-
-private lemma measurable_clockIntegrand (κ θ : ℝ) :
-    Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
-      (flow κ q.1.2.1 (diskMapInv θ q.1.2.2) q.2.toNNReal).im ^ 2 /
-        ‖flow κ q.1.2.1 (diskMapInv θ q.1.2.2) q.2.toNNReal -
-          (driverValue q.1.2.1 q.2 : ℂ)‖ ^ 4) := by
-  have hp : Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
-      (q.2.toNNReal,(q.1.2.1,diskMapInv θ q.1.2.2))) :=
-    (continuous_real_toNNReal.measurable.comp measurable_snd).prodMk
-      ((measurable_fst.snd.fst).prodMk
-        ((measurable_diskMapInv θ).comp measurable_fst.snd.snd))
-  have hF : Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
-      flow κ q.1.2.1 (diskMapInv θ q.1.2.2) q.2.toNNReal) :=
-    @Measurable.comp ((ℝ≥0 × (Driver × ℂ)) × ℝ) (ℝ≥0 × (Driver × ℂ)) ℂ
-      inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1)
-      (fun q => (q.2.toNNReal,(q.1.2.1,diskMapInv θ q.1.2.2))) (measurable_flow κ) hp
-  have hB : Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
-      driverValue q.1.2.1 q.2) :=
-    continuous_eval.measurable.comp (measurable_fst.snd.fst.prodMk
-      (continuous_real_toNNReal.measurable.comp measurable_snd))
-  exact ((Complex.measurable_im.comp hF).pow_const 2).div ((hF.sub hB.complex_ofReal).norm.pow_const 4)
-
-lemma measurable_clock (κ θ : ℝ) :
-    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => clock κ θ q.2.1 q.2.2 q.1) := by
-  let A : Set ((ℝ≥0 × (Driver × ℂ)) × ℝ) := {q | 0 ≤ q.2 ∧ q.2 ≤ (q.1.1 : ℝ)}
-  have hA : MeasurableSet A :=
-    (measurableSet_le measurable_const measurable_snd).inter
-      (measurableSet_le measurable_snd measurable_fst.fst.coe_nnreal_real)
-  have hm : StronglyMeasurable (fun q : ℝ≥0 × (Driver × ℂ) => ∫ u : ℝ,
-      A.indicator (fun p : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
-        (flow κ p.1.2.1 (diskMapInv θ p.1.2.2) p.2.toNNReal).im ^ 2 /
-          ‖flow κ p.1.2.1 (diskMapInv θ p.1.2.2) p.2.toNNReal -
-            (driverValue p.1.2.1 p.2 : ℂ)‖ ^ 4) (q,u)) :=
-    ((measurable_clockIntegrand κ θ).indicator hA).stronglyMeasurable.integral_prod_right'
-  convert hm.measurable using 1
-  funext q
-  rw [clock,← integral_indicator measurableSet_Icc]
-  apply integral_congr_ae
-  filter_upwards [] with u
-  by_cases hu : u ∈ Icc 0 (q.1 : ℝ)
-  · have hu' : (q,u) ∈ A := hu
-    simp only [indicator_of_mem hu,indicator_of_mem hu']
-    simp only [Real.toNNReal_of_nonneg hu.1,NNReal.coe_mk,centeredFlow]
-  · have hu' : (q,u) ∉ A := hu
-    simp only [indicator_of_notMem hu,indicator_of_notMem hu']
-
-lemma measurable_angle (κ θ : ℝ) :
-    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => angle κ θ q.2.1 q.2.2 q.1) := by
-  have hpoint : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
-      (q.1,(q.2.1,diskMapInv θ q.2.2))) :=
-    measurable_fst.prodMk (measurable_snd.fst.prodMk
-      ((measurable_diskMapInv θ).comp measurable_snd.snd))
-  have hF : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
-      flow κ q.2.1 (diskMapInv θ q.2.2) q.1) :=
-    @Measurable.comp (ℝ≥0 × (Driver × ℂ)) (ℝ≥0 × (Driver × ℂ)) ℂ
-      inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1)
-      (fun q => (q.1,(q.2.1,diskMapInv θ q.2.2))) (measurable_flow κ) hpoint
-  have hB : Measurable (fun q : ℝ≥0 × (Driver × ℂ) => driverValue q.2.1 q.1) := by
-    simpa only [driverValue,Real.toNNReal_coe,Function.comp_def] using
-      (continuous_eval.measurable.comp (measurable_snd.fst.prodMk measurable_fst) :
-        Measurable (fun q : ℝ≥0 × (Driver × ℂ) => q.2.1 q.1))
-  exact (hF.sub hB.complex_ofReal).carg
-
-lemma measurable_radius (κ θ : ℝ) :
-    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => radius κ θ q.2.1 q.2.2 q.1) := by
-  exact ((measurable_initialRadius θ).comp measurable_snd.snd).mul
-    ((measurable_const.mul (measurable_clock κ θ)).exp)
-
-private lemma measurableSet_disk_surviving (κ θ : ℝ) :
-    MeasurableSet {q : ℝ≥0 × (Driver × ℂ) |
-      Surviving κ q.2.1 (diskMapInv θ q.2.2) q.1} := by
-  have hpoint : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
-      (q.1,(q.2.1,diskMapInv θ q.2.2))) :=
-    measurable_fst.prodMk (measurable_snd.fst.prodMk
-      ((measurable_diskMapInv θ).comp measurable_snd.snd))
-  exact hpoint (measurableSet_surviving κ)
-
-lemma measurable_green (κ θ : ℝ) :
-    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => green κ θ q.2.1 q.2.2 q.1) := by
-  have hbody : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
-      radius κ θ q.2.1 q.2.2 q.1 ^ (-α κ) * Real.sin (angle κ θ q.2.1 q.2.2 q.1) ^ p κ) :=
-    ((measurable_radius κ θ).pow_const (-α κ)).mul
-      ((measurable_angle κ θ).sin.pow_const (p κ))
-  unfold green
-  exact hbody.ite (measurableSet_disk_surviving κ θ) measurable_const
-
-lemma measurable_potential (κ θ : ℝ) (t : ℝ≥0) :
-    Measurable (fun b : Driver => potential κ θ b t) := by
-  have hm : Measurable (fun q : Driver × ℂ => ENNReal.ofReal (green κ θ q.1 q.2 t)) :=
-    ((measurable_green κ θ).comp (measurable_const.prodMk measurable_id)).ennreal_ofReal
-  exact @Measurable.lintegral_prod_right' _ _ _ _ (volume.restrict diskK) _ _ hm
-
-lemma clock_stopDriver (κ θ : ℝ) (b : Driver) (x : ℂ) (T t : ℝ≥0) (ht : t ≤ T) :
-    clock κ θ (stopDriver b T) x t = clock κ θ b x t := by
-  unfold clock
-  apply setIntegral_congr_fun measurableSet_Icc
-  intro u hu
-  let s : ℝ≥0 := ⟨u,hu.1⟩
-  have hs : s ≤ T := (show s ≤ t from hu.2).trans ht
-  have he := (surviving_flow_stopDriver κ b (diskMapInv θ x) T s hs).2
-  change flow κ (stopDriver b T) (diskMapInv θ x) u = flow κ b (diskMapInv θ x) u at he
-  dsimp only
-  rw [he]
-  have hb : driverValue (stopDriver b T) u = driverValue b u := by
-    simp only [driverValue,stopDriver_apply,Real.toNNReal_of_nonneg hu.1]
-    change b (min s T) = b s
-    rw [min_eq_left hs]
-  simp only [centeredFlow,he,hb]
-
-lemma green_stopDriver (κ θ : ℝ) (b : Driver) (x : ℂ) (T t : ℝ≥0) (ht : t ≤ T) :
-    green κ θ (stopDriver b T) x t = green κ θ b x t := by
-  have he := surviving_flow_stopDriver κ b (diskMapInv θ x) T t ht
-  have hb : driverValue (stopDriver b T) t = driverValue b t := by
-    simp only [driverValue,stopDriver_apply,Real.toNNReal_coe,min_eq_left ht]
-  simp only [green,he.1,radius,clock_stopDriver κ θ b x T t ht,angle,centeredFlow,he.2,hb]
-
-lemma potential_stopDriver (κ θ : ℝ) (b : Driver) (T t : ℝ≥0) (ht : t ≤ T) :
-    potential κ θ (stopDriver b T) t = potential κ θ b t := by
-  unfold potential
-  apply lintegral_congr
-  intro x
-  rw [green_stopDriver κ θ b x T t ht]
-
-lemma potential_measurable_past (κ θ : ℝ) (T t : ℝ≥0) (ht : t ≤ T) :
-    Measurable[driverFiltration T] (fun b : Driver => potential κ θ b t) := by
-  have hm := (measurable_potential κ θ t).comp (stopDriver_measurable_past T)
-  simpa only [Function.comp_def,potential_stopDriver κ θ _ T t ht] using hm
-
-theorem potential_continuation_condLExp (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W)
-    (t T : ℝ≥0) :
-    (fun b : Driver => ∫⁻ c, potential κ θ (concat b c t) T ∂W) =ᵐ[W]
-      W⁻[(fun b : Driver => potential κ θ b T) | driverFiltration t] :=
-  continuationMean_condLExp W hW t (fun b => potential κ θ b T)
-    (measurable_potential κ θ T)
-
-theorem potential_continuation_finite_condExp (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W)
-    (t T : ℝ≥0) (hfin : (∫⁻ b, potential κ θ b T ∂W) ≠ ∞) :
-    (∀ᵐ b ∂W, (∫⁻ c, potential κ θ (concat b c t) T ∂W) < ∞) ∧
-    (fun b : Driver => (∫⁻ c, potential κ θ (concat b c t) T ∂W).toReal) =ᵐ[W]
-      W[(fun b : Driver => (potential κ θ b T).toReal) | driverFiltration t] :=
-  continuationMean_finite_condExp W hW t (fun b => potential κ θ b T)
-    (measurable_potential κ θ T) hfin
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetFlowMeasurability
+
+namespace OAI
+
+/-! The exact prescribed disk potential is a measurable path functional.
+All definitions, including the killed flow and its time integral, are
+retained on every continuous based driver. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma measurable_diskMapInv (θ : ℝ) : Measurable (diskMapInv θ) := by
+  unfold diskMapInv
+  fun_prop
+
+lemma measurable_initialRadius (θ : ℝ) : Measurable (initialRadius θ) := by
+  exact (measurable_const.mul (Complex.measurable_im.comp (measurable_diskMapInv θ))).mul
+    (((measurable_deriv (diskMap θ)).comp (measurable_diskMapInv θ)).norm)
+
+private lemma measurable_clockIntegrand (κ θ : ℝ) :
+    Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
+      (flow κ q.1.2.1 (diskMapInv θ q.1.2.2) q.2.toNNReal).im ^ 2 /
+        ‖flow κ q.1.2.1 (diskMapInv θ q.1.2.2) q.2.toNNReal -
+          (driverValue q.1.2.1 q.2 : ℂ)‖ ^ 4) := by
+  have hp : Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
+      (q.2.toNNReal,(q.1.2.1,diskMapInv θ q.1.2.2))) :=
+    (continuous_real_toNNReal.measurable.comp measurable_snd).prodMk
+      ((measurable_fst.snd.fst).prodMk
+        ((measurable_diskMapInv θ).comp measurable_fst.snd.snd))
+  have hF : Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
+      flow κ q.1.2.1 (diskMapInv θ q.1.2.2) q.2.toNNReal) :=
+    @Measurable.comp ((ℝ≥0 × (Driver × ℂ)) × ℝ) (ℝ≥0 × (Driver × ℂ)) ℂ
+      inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1)
+      (fun q => (q.2.toNNReal,(q.1.2.1,diskMapInv θ q.1.2.2))) (measurable_flow κ) hp
+  have hB : Measurable (fun q : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
+      driverValue q.1.2.1 q.2) :=
+    continuous_eval.measurable.comp (measurable_fst.snd.fst.prodMk
+      (continuous_real_toNNReal.measurable.comp measurable_snd))
+  exact ((Complex.measurable_im.comp hF).pow_const 2).div ((hF.sub hB.complex_ofReal).norm.pow_const 4)
+
+lemma measurable_clock (κ θ : ℝ) :
+    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => clock κ θ q.2.1 q.2.2 q.1) := by
+  let A : Set ((ℝ≥0 × (Driver × ℂ)) × ℝ) := {q | 0 ≤ q.2 ∧ q.2 ≤ (q.1.1 : ℝ)}
+  have hA : MeasurableSet A :=
+    (measurableSet_le measurable_const measurable_snd).inter
+      (measurableSet_le measurable_snd measurable_fst.fst.coe_nnreal_real)
+  have hm : StronglyMeasurable (fun q : ℝ≥0 × (Driver × ℂ) => ∫ u : ℝ,
+      A.indicator (fun p : (ℝ≥0 × (Driver × ℂ)) × ℝ =>
+        (flow κ p.1.2.1 (diskMapInv θ p.1.2.2) p.2.toNNReal).im ^ 2 /
+          ‖flow κ p.1.2.1 (diskMapInv θ p.1.2.2) p.2.toNNReal -
+            (driverValue p.1.2.1 p.2 : ℂ)‖ ^ 4) (q,u)) :=
+    ((measurable_clockIntegrand κ θ).indicator hA).stronglyMeasurable.integral_prod_right'
+  convert hm.measurable using 1
+  funext q
+  rw [clock,← integral_indicator measurableSet_Icc]
+  apply integral_congr_ae
+  filter_upwards [] with u
+  by_cases hu : u ∈ Icc 0 (q.1 : ℝ)
+  · have hu' : (q,u) ∈ A := hu
+    simp only [indicator_of_mem hu,indicator_of_mem hu']
+    simp only [Real.toNNReal_of_nonneg hu.1,NNReal.coe_mk,centeredFlow]
+  · have hu' : (q,u) ∉ A := hu
+    simp only [indicator_of_notMem hu,indicator_of_notMem hu']
+
+lemma measurable_angle (κ θ : ℝ) :
+    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => angle κ θ q.2.1 q.2.2 q.1) := by
+  have hpoint : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
+      (q.1,(q.2.1,diskMapInv θ q.2.2))) :=
+    measurable_fst.prodMk (measurable_snd.fst.prodMk
+      ((measurable_diskMapInv θ).comp measurable_snd.snd))
+  have hF : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
+      flow κ q.2.1 (diskMapInv θ q.2.2) q.1) :=
+    @Measurable.comp (ℝ≥0 × (Driver × ℂ)) (ℝ≥0 × (Driver × ℂ)) ℂ
+      inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1)
+      (fun q => (q.1,(q.2.1,diskMapInv θ q.2.2))) (measurable_flow κ) hpoint
+  have hB : Measurable (fun q : ℝ≥0 × (Driver × ℂ) => driverValue q.2.1 q.1) := by
+    simpa only [driverValue,Real.toNNReal_coe,Function.comp_def] using
+      (continuous_eval.measurable.comp (measurable_snd.fst.prodMk measurable_fst) :
+        Measurable (fun q : ℝ≥0 × (Driver × ℂ) => q.2.1 q.1))
+  exact (hF.sub hB.complex_ofReal).carg
+
+lemma measurable_radius (κ θ : ℝ) :
+    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => radius κ θ q.2.1 q.2.2 q.1) := by
+  exact ((measurable_initialRadius θ).comp measurable_snd.snd).mul
+    ((measurable_const.mul (measurable_clock κ θ)).exp)
+
+private lemma measurableSet_disk_surviving (κ θ : ℝ) :
+    MeasurableSet {q : ℝ≥0 × (Driver × ℂ) |
+      Surviving κ q.2.1 (diskMapInv θ q.2.2) q.1} := by
+  have hpoint : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
+      (q.1,(q.2.1,diskMapInv θ q.2.2))) :=
+    measurable_fst.prodMk (measurable_snd.fst.prodMk
+      ((measurable_diskMapInv θ).comp measurable_snd.snd))
+  exact hpoint (measurableSet_surviving κ)
+
+lemma measurable_green (κ θ : ℝ) :
+    Measurable (fun q : ℝ≥0 × (Driver × ℂ) => green κ θ q.2.1 q.2.2 q.1) := by
+  have hbody : Measurable (fun q : ℝ≥0 × (Driver × ℂ) =>
+      radius κ θ q.2.1 q.2.2 q.1 ^ (-α κ) * Real.sin (angle κ θ q.2.1 q.2.2 q.1) ^ p κ) :=
+    ((measurable_radius κ θ).pow_const (-α κ)).mul
+      ((measurable_angle κ θ).sin.pow_const (p κ))
+  unfold green
+  exact hbody.ite (measurableSet_disk_surviving κ θ) measurable_const
+
+lemma measurable_potential (κ θ : ℝ) (t : ℝ≥0) :
+    Measurable (fun b : Driver => potential κ θ b t) := by
+  have hm : Measurable (fun q : Driver × ℂ => ENNReal.ofReal (green κ θ q.1 q.2 t)) :=
+    ((measurable_green κ θ).comp (measurable_const.prodMk measurable_id)).ennreal_ofReal
+  exact @Measurable.lintegral_prod_right' _ _ _ _ (volume.restrict diskK) _ _ hm
+
+lemma clock_stopDriver (κ θ : ℝ) (b : Driver) (x : ℂ) (T t : ℝ≥0) (ht : t ≤ T) :
+    clock κ θ (stopDriver b T) x t = clock κ θ b x t := by
+  unfold clock
+  apply setIntegral_congr_fun measurableSet_Icc
+  intro u hu
+  let s : ℝ≥0 := ⟨u,hu.1⟩
+  have hs : s ≤ T := (show s ≤ t from hu.2).trans ht
+  have he := (surviving_flow_stopDriver κ b (diskMapInv θ x) T s hs).2
+  change flow κ (stopDriver b T) (diskMapInv θ x) u = flow κ b (diskMapInv θ x) u at he
+  dsimp only
+  rw [he]
+  have hb : driverValue (stopDriver b T) u = driverValue b u := by
+    simp only [driverValue,stopDriver_apply,Real.toNNReal_of_nonneg hu.1]
+    change b (min s T) = b s
+    rw [min_eq_left hs]
+  simp only [centeredFlow,he,hb]
+
+lemma green_stopDriver (κ θ : ℝ) (b : Driver) (x : ℂ) (T t : ℝ≥0) (ht : t ≤ T) :
+    green κ θ (stopDriver b T) x t = green κ θ b x t := by
+  have he := surviving_flow_stopDriver κ b (diskMapInv θ x) T t ht
+  have hb : driverValue (stopDriver b T) t = driverValue b t := by
+    simp only [driverValue,stopDriver_apply,Real.toNNReal_coe,min_eq_left ht]
+  simp only [green,he.1,radius,clock_stopDriver κ θ b x T t ht,angle,centeredFlow,he.2,hb]
+
+lemma potential_stopDriver (κ θ : ℝ) (b : Driver) (T t : ℝ≥0) (ht : t ≤ T) :
+    potential κ θ (stopDriver b T) t = potential κ θ b t := by
+  unfold potential
+  apply lintegral_congr
+  intro x
+  rw [green_stopDriver κ θ b x T t ht]
+
+lemma potential_measurable_past (κ θ : ℝ) (T t : ℝ≥0) (ht : t ≤ T) :
+    Measurable[driverFiltration T] (fun b : Driver => potential κ θ b t) := by
+  have hm := (measurable_potential κ θ t).comp (stopDriver_measurable_past T)
+  simpa only [Function.comp_def,potential_stopDriver κ θ _ T t ht] using hm
+
+theorem potential_continuation_condLExp (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W)
+    (t T : ℝ≥0) :
+    (fun b : Driver => ∫⁻ c, potential κ θ (concat b c t) T ∂W) =ᵐ[W]
+      W⁻[(fun b : Driver => potential κ θ b T) | driverFiltration t] :=
+  continuationMean_condLExp W hW t (fun b => potential κ θ b T)
+    (measurable_potential κ θ T)
+
+theorem potential_continuation_finite_condExp (κ θ : ℝ) (W : WienerLaw) (hW : IsWiener W)
+    (t T : ℝ≥0) (hfin : (∫⁻ b, potential κ θ b T ∂W) ≠ ∞) :
+    (∀ᵐ b ∂W, (∫⁻ c, potential κ θ (concat b c t) T ∂W) < ∞) ∧
+    (fun b : Driver => (∫⁻ c, potential κ θ (concat b c t) T ∂W).toReal) =ᵐ[W]
+      W[(fun b : Driver => (potential κ θ b T).toReal) | driverFiltration t] :=
+  continuationMean_finite_condExp W hW t (fun b => potential κ θ b T)
+    (measurable_potential κ θ T) hfin
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/AngularLaws.lean b/lean/OAI/Probability/SLE/AngularLaws.lean
index 5ffbdfa0f49948d77d59658f7cf9b29e99b19256..33a79511cc1e4c1902011bf75a9fdd0be2d792c7 100644
--- a/lean/OAI/Probability/SLE/AngularLaws.lean
+++ b/lean/OAI/Probability/SLE/AngularLaws.lean
@@ -1,668 +1,668 @@
-import Mathlib
-import OAI.Probability.SLE.RadiusClocks
-
-namespace OAI
-
-/-! Angular Laws for chordal SLE. -/
-
-section
-section
-section
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology Laplacian ENNReal
-namespace SLEExactGauge
-noncomputable def greenInner (n : ℕ) : ℝ := (1 / 2) * (1 / ((n : ℝ) + 1))
-noncomputable def greenOuter (n : ℕ) : ℝ := 1 - greenInner n
-
-lemma greenInner_pos (n : ℕ) : 0 < greenInner n := by unfold greenInner; positivity
-
-lemma greenInner_le_half (n : ℕ) : greenInner n ≤ 1 / 2 := by
-  unfold greenInner
-  have hn : (1 : ℝ) ≤ (n : ℝ) + 1 := le_add_of_nonneg_left (Nat.cast_nonneg n)
-  have hh : (1 : ℝ) / ((n : ℝ) + 1) ≤ 1 :=
-    (div_le_one (by positivity)).mpr hn
-  linarith
-
-lemma greenInner_le_outer (n : ℕ) : greenInner n ≤ greenOuter n := by
-  unfold greenOuter
-  linarith [greenInner_le_half n]
-
-lemma greenOuter_pos (n : ℕ) : 0 < greenOuter n :=
-  (greenInner_pos n).trans_le (greenInner_le_outer n)
-
-lemma greenOuter_lt_one (n : ℕ) : greenOuter n < 1 := by
-  unfold greenOuter
-  linarith [greenInner_pos n]
-
-lemma greenInner_tendsto : Tendsto greenInner atTop (𝓝 0) := by
-  change Tendsto (fun n : ℕ => (1 / 2) * (1 / ((n : ℝ) + 1))) atTop (𝓝 0)
-  simpa only [greenInner, mul_zero] using
-    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (1 / 2 : ℝ)
-
-lemma greenOuter_tendsto : Tendsto greenOuter atTop (𝓝 1) := by
-  change Tendsto (fun n => 1 - greenInner n) atTop (𝓝 1)
-  simpa only [greenOuter, sub_zero] using tendsto_const_nhds.sub greenInner_tendsto
-
-lemma greenInner_log_tendsto : Tendsto (fun n => -greenInner n * Real.log (greenInner n)) atTop (𝓝 0) := by
-  simpa only [Function.comp_def, zero_mul, neg_zero, neg_mul] using
-    ((Real.continuous_mul_log.tendsto 0).comp greenInner_tendsto).neg
-
-def polarUnitDisk : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo (-Real.pi) Real.pi
-
-noncomputable def polarGreenTarget (u : ℂ → ℝ) (p : ℝ × ℝ) : ℝ≥0∞ :=
-  polarUnitDisk.indicator (fun p : ℝ × ℝ =>
-    ENNReal.ofReal (p.1 * (-Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2)))) p
-
-lemma greenTruncation_tendsto {u : ℂ → ℝ} {p : ℝ × ℝ} (hp : p ∈ polarUnitDisk) :
-    Tendsto (fun n => ENNReal.ofReal (greenTruncation u (greenInner n) (greenOuter n) p)) atTop
-      (𝓝 (ENNReal.ofReal (p.1 * (-Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2))))) := by
-  have he : ∀ᶠ n in atTop, greenTruncation u (greenInner n) (greenOuter n) p =
-      p.1 * (Real.log (greenOuter n) - Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2)) := by
-    filter_upwards [greenInner_tendsto (Iio_mem_nhds hp.1.1),
-      greenOuter_tendsto (Ioi_mem_nhds hp.1.2)] with n ha hR
-    have hm : p ∈ greenRectangle (greenInner n) (greenOuter n) :=
-      ⟨⟨ha.le, hR.le⟩, ⟨hp.2.1.le, hp.2.2.le⟩⟩
-    simp only [greenTruncation, indicator_of_mem hm, annulusDensity]
-  have hl := (Real.continuousAt_log (by norm_num : (1 : ℝ) ≠ 0)).tendsto.comp greenOuter_tendsto
-  have ht := ENNReal.continuous_ofReal.continuousAt.tendsto.comp
-    (((hl.sub_const (Real.log p.1)).const_mul p.1).mul_const
-      (-(Δ u) (polarPoint p.1 p.2)))
-  simp only [Function.comp_def, Real.log_one, zero_sub] at ht
-  exact ht.congr' (he.mono fun n hn => congrArg ENNReal.ofReal hn.symm)
-
- 
-
-lemma polar_green_le {u : ℂ → ℝ} {C : ℝ}
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
-    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
-    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
-    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0) :
-    (∫⁻ p, polarGreenTarget u p) ≤ ENNReal.ofReal (2 * Real.pi * C) := by
-  let F : ℕ → ℝ × ℝ → ℝ≥0∞ := fun n p =>
-    ENNReal.ofReal (greenTruncation u (greenInner n) (greenOuter n) p)
-  have hmeas (n : ℕ) : AEMeasurable (F n) :=
-    ENNReal.measurable_ofReal.comp_aemeasurable
-      (greenTruncation_integrable (greenInner_pos n) (greenOuter_lt_one n) hu).aemeasurable
-  have hp : polarGreenTarget u ≤ fun p => liminf (fun n => F n p) atTop := by
-    intro p
-    by_cases hp : p ∈ polarUnitDisk
-    · rw [polarGreenTarget, indicator_of_mem hp]
-      exact le_of_eq (greenTruncation_tendsto (u := u) hp).liminf_eq.symm
-    · simp only [polarGreenTarget, indicator_of_notMem hp]
-      exact bot_le
-  rcases derivative_locally_bounded (hu 0 (by simp)) with ⟨δ, M, hδ, hM, hgrad⟩
-  have hbound : ∀ᶠ n in atTop, (∫⁻ p, F n p) ≤
-      ENNReal.ofReal (2 * Real.pi * (C + M * (-greenInner n * Real.log (greenInner n)))) := by
-    filter_upwards [greenInner_tendsto (Iio_mem_nhds hδ)] with n hn
-    change greenInner n < δ at hn
-    exact greenTruncation_lintegral_le (greenInner_pos n) (greenInner_le_outer n)
-      (greenOuter_lt_one n) hu hun hub hΔ hM (fun θ => hgrad _ (by
-        simpa only [polarPoint_norm, abs_of_pos (greenInner_pos n)] using hn))
-  have ht : Tendsto (fun n => ENNReal.ofReal
-      (2 * Real.pi * (C + M * (-greenInner n * Real.log (greenInner n))))) atTop
-      (𝓝 (ENNReal.ofReal (2 * Real.pi * C))) := by
-    simpa only [Function.comp_def, mul_zero, add_zero] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp
-      ((tendsto_const_nhds.add (greenInner_log_tendsto.const_mul M)).const_mul (2 * Real.pi))
-  exact (lintegral_mono hp).trans ((lintegral_liminf_le' hmeas).trans
-    ((liminf_le_liminf hbound).trans_eq ht.liminf_eq))
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge
-
-lemma polarPoint_eq_polarCoord (p : ℝ × ℝ) :
-    polarPoint p.1 p.2 = Complex.polarCoord.symm p := by
-  rw [polarPoint_eq_circleMap]
-  apply Complex.ext
-  · simp [circleMap_zero_re, Complex.polarCoord_symm_apply, Complex.cos_ofReal_re, Complex.sin_ofReal_re]
-  · simp [circleMap_zero_im, Complex.polarCoord_symm_apply, Complex.cos_ofReal_re, Complex.sin_ofReal_re]
-
-lemma disk_log_lintegral_polar (h : ℂ → ℝ) :
-    (∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((-Real.log ‖z‖) * h z)) =
-      ∫⁻ p in Ioo (0 : ℝ) 1 ×ˢ Ioo (-Real.pi) Real.pi,
-        ENNReal.ofReal (p.1 * (-Real.log p.1) * h (polarPoint p.1 p.2)) := by
-  let T : Set (ℝ × ℝ) := Ioi 0 ×ˢ Ioo (-Real.pi) Real.pi
-  let S : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo (-Real.pi) Real.pi
-  have hT : MeasurableSet T := measurableSet_Ioi.prod measurableSet_Ioo
-  have hS : MeasurableSet S := measurableSet_Ioo.prod measurableSet_Ioo
-  have hST : S ⊆ T := prod_mono Ioo_subset_Ioi_self subset_rfl
-  let f : ℂ → ℝ≥0∞ := (ball (0 : ℂ) 1).indicator (fun z => ENNReal.ofReal ((-Real.log ‖z‖) * h z))
-  calc
-    _ = ∫⁻ z, f z := (lintegral_indicator measurableSet_ball _).symm
-    _ = ∫⁻ p in T, ENNReal.ofReal p.1 • f (Complex.polarCoord.symm p) :=
-      (Complex.lintegral_comp_polarCoord_symm f).symm
-    _ = ∫⁻ p in T, S.indicator (fun p : ℝ × ℝ =>
-        ENNReal.ofReal (p.1 * (-Real.log p.1) * h (polarPoint p.1 p.2))) p := by
-      apply setLIntegral_congr_fun hT
-      intro p hp
-      have hn : ‖Complex.polarCoord.symm p‖ = p.1 := by
-        rw [Complex.norm_polarCoord_symm, abs_of_pos hp.1]
-      by_cases hr : p.1 < 1
-      · have hz : Complex.polarCoord.symm p ∈ ball (0 : ℂ) 1 := by
-          rw [mem_ball_zero_iff, hn]
-          exact hr
-        have hs : p ∈ S := ⟨⟨hp.1, hr⟩, hp.2⟩
-        simp only [f, indicator_of_mem hz, indicator_of_mem hs, hn, smul_eq_mul,
-          polarPoint_eq_polarCoord, ← ENNReal.ofReal_mul hp.1.le, mul_assoc]
-      · have hz : Complex.polarCoord.symm p ∉ ball (0 : ℂ) 1 := by
-          simpa only [mem_ball_zero_iff, hn] using hr
-        have hs : p ∉ S := fun hs => hr hs.1.2
-        simp only [f, indicator_of_notMem hz, indicator_of_notMem hs, smul_zero]
-    _ = _ := by
-      rw [lintegral_indicator hS, Measure.restrict_restrict hS, inter_eq_left.mpr hST]
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology Laplacian ENNReal
-namespace SLEExactGauge
-
- 
-lemma disk_green_le {u : ℂ → ℝ} {C : ℝ}
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
-    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
-    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
-    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0) :
-    (∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((-Real.log ‖z‖) * (-(Δ u) z))) ≤
-      ENNReal.ofReal (2 * Real.pi * C) := by
-  rw [disk_log_lintegral_polar]
-  have hh := polar_green_le hu hun hub hΔ
-  simpa only [polarGreenTarget, polarUnitDisk,
-    lintegral_indicator (measurableSet_Ioo.prod measurableSet_Ioo)] using hh
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-namespace SLEExactGauge
-
-lemma second_direction_chain {f : ℝ → ℝ} {g : ℂ → ℝ} {z : ℂ}
-    (hg : ContDiffAt ℝ 2 g z) (hf : ContDiffAt ℝ 2 f (g z)) (v : ℂ) :
-    fderiv ℝ (fderiv ℝ (f ∘ g)) z v v =
-      deriv (deriv f) (g z) * (fderiv ℝ g z v) ^ 2 +
-        deriv f (g z) * fderiv ℝ (fderiv ℝ g) z v v := by
-  have hgd : DifferentiableAt ℝ g z := hg.differentiableAt (by norm_num)
-  have hfdd : DifferentiableAt ℝ (deriv f) (g z) :=
-    (hf.derivWithin (m := 1) (by norm_num)).differentiableAt (by norm_num)
-  have hgdd : DifferentiableAt ℝ (fderiv ℝ g) z :=
-    (hg.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)
-  have hcompdd : DifferentiableAt ℝ (fderiv ℝ (f ∘ g)) z :=
-    ((hf.comp z hg).fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)
-  have heq : (fun w => fderiv ℝ (f ∘ g) w v) =ᶠ[𝓝 z]
-      (fun w => deriv f (g w) * fderiv ℝ g w v) := by
-    filter_upwards [hg.eventually (by norm_num),
-      hgd.continuousAt (hf.eventually (by norm_num))] with w hgw hfw
-    change ContDiffAt ℝ 2 f (g w) at hfw
-    rw [fderiv_comp w (hfw.differentiableAt (by norm_num)) (hgw.differentiableAt (by norm_num)),
-      ContinuousLinearMap.comp_apply, fderiv_eq_smul_deriv, smul_eq_mul, mul_comm]
-  have hleft : fderiv ℝ (fun w => fderiv ℝ (f ∘ g) w v) z v =
-      fderiv ℝ (fderiv ℝ (f ∘ g)) z v v := by
-    rw [fderiv_clm_apply hcompdd (differentiableAt_const v)]
-    simp
-  have hright : fderiv ℝ (fun w => deriv f (g w) * fderiv ℝ g w v) z v =
-      deriv (deriv f) (g z) * (fderiv ℝ g z v) ^ 2 +
-        deriv f (g z) * fderiv ℝ (fderiv ℝ g) z v v := by
-    have hmul := fderiv_fun_mul (hfdd.comp z hgd) (hgdd.clm_apply (differentiableAt_const v))
-    simp only [Function.comp_apply] at hmul
-    rw [hmul]
-    simp only [add_apply, smul_apply, smul_eq_mul]
-    rw [fderiv_comp z hfdd hgd, ContinuousLinearMap.comp_apply, fderiv_eq_smul_deriv,
-      fderiv_clm_apply hgdd (differentiableAt_const v)]
-    simp only [fderiv_fun_const, Pi.zero_apply, ContinuousLinearMap.comp_zero, zero_add,
-      ContinuousLinearMap.flip_apply, smul_eq_mul]
-    ring
-  rw [← hleft, heq.fderiv_eq, hright]
-
- 
-
-lemma plane_laplacian_chain {f : ℝ → ℝ} {g : ℂ → ℝ} {z : ℂ}
-    (hg : ContDiffAt ℝ 2 g z) (hf : ContDiffAt ℝ 2 f (g z)) :
-    (Δ (f ∘ g)) z = deriv f (g z) * (Δ g) z +
-      deriv (deriv f) (g z) * ((fderiv ℝ g z 1) ^ 2 + (fderiv ℝ g z I) ^ 2) := by
-  simp only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
-    Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
-  rw [second_direction_chain hg hf 1, second_direction_chain hg hf I]
-  ring
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian RealInnerProductSpace
-namespace SLEExactGauge
-
-noncomputable def diskWeight (z : ℂ) : ℝ := 1 - ‖z‖ ^ 2
-
-lemma diskWeight_pos {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : 0 < diskWeight z := by
-  have hn := mem_ball_zero_iff.mp hz
-  dsimp [diskWeight]
-  nlinarith [norm_nonneg z]
-
-lemma diskWeight_contDiff : ContDiff ℝ 2 diskWeight :=
-  contDiff_const.sub (contDiff_id.norm_sq ℂ)
-
-lemma diskWeight_fderiv (z : ℂ) :
-    fderiv ℝ diskWeight z = (-2 : ℝ) • innerSL ℝ z := by
-  change fderiv ℝ (fun w : ℂ => 1 - ‖w‖ ^ 2) z = _
-  rw [fderiv_const_sub, fderiv_norm_sq_apply]
-  ext v
-  simp [smul_apply, nsmul_eq_mul]
-  ring
-
-lemma diskWeight_laplacian (z : ℂ) : (Δ diskWeight) z = -4 := by
-  simp only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
-    Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
-  have heq : fderiv ℝ diskWeight = fun z => (-2 : ℝ) • innerSL ℝ z := funext diskWeight_fderiv
-  rw [heq]
-  rw [fderiv_fun_const_smul (𝕜 := ℝ) (x := z)
-    (innerSL ℝ (E := ℂ)).differentiableAt (-2 : ℝ)]
-  simp only [ContinuousLinearMap.fderiv, smul_apply, smul_eq_mul]
-  change -2 * inner ℝ (1 : ℂ) 1 + -2 * inner ℝ (I : ℂ) I = -4
-  norm_num [real_inner_self_eq_norm_sq]
-
-lemma log_diskWeight_laplacian {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) :
-    (Δ (fun w => Real.log (diskWeight w))) z = -4 / diskWeight z ^ 2 := by
-  have hg := diskWeight_contDiff.contDiffAt (x := z)
-  have hf : ContDiffAt ℝ 2 Real.log (diskWeight z) := Real.contDiffAt_log.mpr (diskWeight_pos hz).ne'
-  have hh := plane_laplacian_chain hg hf
-  have hlog2 : deriv (deriv Real.log) (diskWeight z) = -(diskWeight z ^ 2)⁻¹ := by
-    rw [Real.deriv_log']
-    exact (hasDerivAt_inv (diskWeight_pos hz).ne').deriv
-  change (Δ (fun w => Real.log (diskWeight w))) z = _ at hh
-  rw [Real.deriv_log, hlog2, diskWeight_laplacian, diskWeight_fderiv] at hh
-  rw [hh]
-  simp only [smul_apply, innerSL_apply_apply, smul_eq_mul]
-  have he : ‖z‖ ^ 2 = z.re ^ 2 + z.im ^ 2 := by simpa [Complex.normSq_apply, sq] using Complex.sq_norm z
-  have hwne : 1 - ‖z‖ ^ 2 ≠ 0 := (diskWeight_pos hz).ne'
-  simp [diskWeight, Complex.mul_re]
-  field_simp [hwne]
-  nlinarith [he]
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology
-namespace SLEExactGauge
-
-lemma log_norm_direction_bound {f : ℂ → ℂ} {z : ℂ}
-    (hf : DifferentiableAt ℂ f z) (hne : f z ≠ 0) (v : ℂ) :
-    |fderiv ℝ (fun w => Real.log ‖f w‖) z v| ≤ ‖deriv f z / f z‖ * ‖v‖ := by
-  have hnorm : DifferentiableAt ℝ (fun w : ℂ => ‖w‖) (f z) :=
-    differentiableAt_id.norm ℂ hne
-  have hd := ((hnorm.hasFDerivAt.comp z (hf.restrictScalars ℝ).hasFDerivAt).log (norm_ne_zero_iff.mpr hne)).fderiv
-  change fderiv ℝ (fun w => Real.log ‖f w‖) z = _ at hd
-  rw [hd]
-  simp only [smul_apply, ContinuousLinearMap.comp_apply, smul_eq_mul, abs_mul,
-    abs_inv, Function.comp_apply, abs_norm]
-  have hbound := (fderiv ℝ (fun w : ℂ => ‖w‖) (f z)).le_opNorm (fderiv ℝ f z v)
-  rw [norm_fderiv_norm hnorm, one_mul, Real.norm_eq_abs] at hbound
-  have hvalue : fderiv ℝ f z v = v * deriv f z := by
-    rw [hf.fderiv_restrictScalars (𝕜 := ℝ)]
-    change fderiv ℂ f z v = _
-    rw [fderiv_eq_smul_deriv, smul_eq_mul]
-  rw [hvalue, norm_mul] at hbound
-  rw [hvalue]
-  calc
-    ‖f z‖⁻¹ * |(fderiv ℝ (fun w : ℂ => ‖w‖) (f z)) (v * deriv f z)|
-        ≤ ‖f z‖⁻¹ * (‖v‖ * ‖deriv f z‖) := mul_le_mul_of_nonneg_left hbound (inv_nonneg.mpr (norm_nonneg _))
-    _ = ‖deriv f z / f z‖ * ‖v‖ := by rw [norm_div]; ring
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma univalent_omitted_value_bound_ball {f : ℂ → ℂ} {z : ℂ} {r : ℝ}
-    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball z r))
-    (hi : InjOn f (ball z r)) (a : ℂ) (ha : a ∉ f '' ball z r) :
-    r * ‖deriv f z‖ ≤ (128 * Real.pi) * ‖f z - a‖ := by
-  let g : ℂ → ℂ := fun w => z + (r : ℂ) * w
-  have hgmap : MapsTo g (ball 0 1) (ball z r) := by
-    intro w hw
-    rw [mem_ball_iff_norm]
-    dsimp only [g]
-    rw [add_sub_cancel_left, norm_mul, Complex.norm_real,
-      Real.norm_eq_abs, abs_of_pos hr]
-    exact (mul_lt_mul_of_pos_left (by simpa [mem_ball_zero_iff] using hw) hr).trans_eq (mul_one r)
-  have hgd : Differentiable ℂ g := (differentiable_const z).add ((differentiable_const (r : ℂ)).mul differentiable_id)
-  have hgi : Function.Injective g := by
-    intro v w he
-    exact mul_left_cancel₀ (by exact_mod_cast hr.ne') (add_left_cancel he)
-  have hfd : DifferentiableOn ℂ (f ∘ g) (ball 0 1) := hd.comp hgd.differentiableOn hgmap
-  have hfi : InjOn (f ∘ g) (ball 0 1) := by
-    intro v hv w hw he
-    exact hgi (hi (hgmap hv) (hgmap hw) he)
-  have hfa : a ∉ (f ∘ g) '' ball 0 1 := by
-    rintro ⟨w, hw, he⟩
-    exact ha ⟨g w, hgmap hw, he⟩
-  have hg0 : g 0 = z := by simp [g]
-  have hgder : HasDerivAt g (r : ℂ) 0 := by
-    simpa [g] using ((hasDerivAt_id (0 : ℂ)).const_mul (r : ℂ)).const_add z
-  have hfz := (hd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
-  have hder : deriv (f ∘ g) 0 = deriv f z * (r : ℂ) := by
-    have hfg : HasDerivAt f (deriv f z) (g 0) := by rw [hg0]; exact hfz
-    exact (hfg.comp 0 hgder).deriv
-  have hh := univalent_omitted_value_bound hfd hfi a hfa
-  simpa [hder, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr,
-    Function.comp_apply, hg0, mul_comm] using hh
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma logarithmic_lift_deriv_bound {f h : ℂ → ℂ} {a : ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (ha : a ∉ f '' ball 0 1)
-    (hh : ∀ z ∈ ball (0 : ℂ) 1, deriv h z = deriv f z / (f z - a))
-    {z : ℂ} (hz : z ∈ closedBall (0 : ℂ) (1 / 2 : ℝ)) :
-    ‖deriv h z‖ ≤ 256 * Real.pi := by
-  have hsub : ball z (1 / 2 : ℝ) ⊆ ball (0 : ℂ) 1 := by
-    intro w hw
-    apply mem_ball.mpr
-    have hw' := mem_ball.mp hw
-    have hz' := mem_closedBall.mp hz
-    exact (dist_triangle w z 0).trans_lt (by linarith)
-  have hz' : z ∈ ball (0 : ℂ) 1 := (closedBall_subset_ball (by norm_num)) hz
-  have hn : f z - a ≠ 0 := by
-    intro he
-    exact ha ⟨z, hz', sub_eq_zero.mp he⟩
-  have hb := univalent_omitted_value_bound_ball (by norm_num : (0 : ℝ) < 1 / 2)
-    (hd.mono hsub) (hi.mono hsub) a (fun he => ha (image_mono hsub he))
-  rw [hh z hz', norm_div]
-  apply (div_le_iff₀ (norm_pos_iff.mpr hn)).mpr
-  linarith
-
-lemma logarithmic_lift_second_bound {f h : ℂ → ℂ} {a : ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (ha : a ∉ f '' ball 0 1) (hhd : DifferentiableOn ℂ h (ball 0 1))
-    (hh : ∀ z ∈ ball (0 : ℂ) 1, deriv h z = deriv f z / (f z - a)) :
-    ‖deriv (deriv h) 0‖ ≤ 512 * Real.pi := by
-  have hs : closedBall (0 : ℂ) (1 / 2 : ℝ) ⊆ ball 0 1 :=
-    closedBall_subset_ball (by norm_num)
-  have hb := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le
-    (by norm_num : (0 : ℝ) < 1 / 2)
-    ((hhd.deriv isOpen_ball).diffContOnCl_ball hs)
-    (fun z hz => logarithmic_lift_deriv_bound hd hi ha hh (sphere_subset_closedBall hz))
-  convert hb using 1; ring
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma exists_omitted_near_of_inverse {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hf0 : deriv f 0 ≠ 0)
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ z ∈ ball (0 : ℂ) 1, g (f z) = z) :
-    ∃ a : ℂ, a ∉ f '' ball 0 1 ∧ ‖f 0 - a‖ < 2 * ‖deriv f 0‖ := by
-  have hz : (0 : ℂ) ∈ ball 0 1 := mem_ball_self (by norm_num)
-  have hn : 0 < ‖deriv f 0‖ := norm_pos_iff.mpr hf0
-  have hr : 0 < 2 * ‖deriv f 0‖ := by positivity
-  by_contra! hcontra
-  have hsub : ball (f 0) (2 * ‖deriv f 0‖) ⊆ f '' ball 0 1 := by
-    intro a ha
-    by_contra hnot
-    have hh := hcontra a hnot
-    have hn' : ‖f 0 - a‖ < 2 * ‖deriv f 0‖ := by
-      simpa only [mem_ball_iff_norm, norm_sub_rev] using ha
-    linarith
-  have hgd := hg.mono hsub
-  have hgm : MapsTo g (ball (f 0) (2 * ‖deriv f 0‖)) (closedBall (g (f 0)) 1) := by
-    intro w hw
-    obtain ⟨z, hz', rfl⟩ := hsub hw
-    rw [hgf z hz', hgf 0 hz]
-    exact ball_subset_closedBall hz'
-  have hbound := Complex.norm_deriv_le_div_of_mapsTo_ball hgd hgm hr
-  have hdf := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
-  have hdg := (hgd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
-  have heq : (g ∘ f) =ᶠ[𝓝 (0 : ℂ)] id := by
-    filter_upwards [isOpen_ball.mem_nhds hz] with z hz'
-    exact hgf z hz'
-  have hprod : deriv g (f 0) * deriv f 0 = 1 :=
-    ((hdg.comp 0 hdf).congr_of_eventuallyEq heq.symm).unique (hasDerivAt_id 0)
-  have hnormprod : ‖deriv g (f 0)‖ * ‖deriv f 0‖ = 1 := by
-    simpa only [norm_mul, norm_one] using congrArg norm hprod
-  have hh := (le_div_iff₀ hr).mp hbound
-  nlinarith
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma preSchwarzian_bound_of_inverse {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hf0 : deriv f 0 ≠ 0)
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ z ∈ ball (0 : ℂ) 1, g (f z) = z) :
-    ‖deriv (deriv f) 0 / deriv f 0‖ ≤ 1280 * Real.pi := by
-  obtain ⟨a, ha, hnear⟩ := exists_omitted_near_of_inverse hd hf0 hg hgf
-  have hz : (0 : ℂ) ∈ ball 0 1 := mem_ball_self (by norm_num)
-  have hnon (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) : f z - a ≠ 0 := by
-    intro he
-    exact ha ⟨z, hz, sub_eq_zero.mp he⟩
-  obtain ⟨h, hhd, hexp, hh⟩ := exists_holomorphic_log_on_ball (by norm_num : (0 : ℝ) < 1)
-    (hd.sub_const a) hnon
-  have hh' (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) :
-      deriv h z = deriv f z / (f z - a) := by
-    rw [hh z hz]
-    congr 1
-    exact ((hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.sub_const a).deriv
-  have hdf := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
-  have hddf := ((hd.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
-  have heq : deriv h =ᶠ[𝓝 (0 : ℂ)] (fun z => deriv f z / (f z - a)) := by
-    filter_upwards [isOpen_ball.mem_nhds hz] with z hz'
-    exact hh' z hz'
-  have hsecond : deriv (deriv h) 0 =
-      (deriv (deriv f) 0 * (f 0 - a) - deriv f 0 * deriv f 0) / (f 0 - a) ^ 2 :=
-    ((hddf.div (hdf.sub_const a) (hnon 0 hz)).congr_of_eventuallyEq heq).deriv
-  have halg : deriv (deriv f) 0 / deriv f 0 =
-      deriv (deriv h) 0 / deriv h 0 + deriv h 0 := by
-    rw [hsecond, hh' 0 hz]
-    field_simp [hf0, hnon 0 hz]
-    ring
-  have hfirst := logarithmic_lift_deriv_bound hd hi ha hh'
-    (mem_closedBall_self (by norm_num : (0 : ℝ) ≤ 1 / 2))
-  have htwo := logarithmic_lift_second_bound hd hi ha hhd hh'
-  have hlower : (1 / 2 : ℝ) ≤ ‖deriv h 0‖ := by
-    rw [hh' 0 hz, norm_div]
-    apply (le_div_iff₀ (norm_pos_iff.mpr (hnon 0 hz))).mpr
-    linarith
-  have hpos : 0 < ‖deriv h 0‖ := by linarith
-  have hratio : ‖deriv (deriv h) 0‖ / ‖deriv h 0‖ ≤ 1024 * Real.pi := by
-    apply (div_le_iff₀ hpos).mpr
-    nlinarith [Real.pi_pos]
-  rw [halg]
-  calc
-    ‖deriv (deriv h) 0 / deriv h 0 + deriv h 0‖ ≤
-        ‖deriv (deriv h) 0‖ / ‖deriv h 0‖ + ‖deriv h 0‖ := by
-      simpa only [norm_div] using norm_add_le (deriv (deriv h) 0 / deriv h 0) (deriv h 0)
-    _ ≤ 1280 * Real.pi := by linarith
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma preSchwarzian_bound_ball {f g : ℂ → ℂ} {z : ℂ} {r : ℝ}
-    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball z r))
-    (hi : InjOn f (ball z r)) (hf0 : deriv f z ≠ 0)
-    (hg : DifferentiableOn ℂ g (f '' ball z r))
-    (hgf : ∀ w ∈ ball z r, g (f w) = w) :
-    r * ‖deriv (deriv f) z / deriv f z‖ ≤ 1280 * Real.pi := by
-  let A : ℂ → ℂ := fun w => z + (r : ℂ) * w
-  have hrC : (r : ℂ) ≠ 0 := by exact_mod_cast hr.ne'
-  have hAmap : MapsTo A (ball 0 1) (ball z r) := by
-    intro w hw
-    rw [mem_ball_iff_norm]
-    dsimp only [A]
-    rw [add_sub_cancel_left, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr]
-    exact (mul_lt_mul_of_pos_left (mem_ball_zero_iff.mp hw) hr).trans_eq (mul_one r)
-  have hAd : Differentiable ℂ A :=
-    (differentiable_const z).add ((differentiable_const (r : ℂ)).mul differentiable_id)
-  have hAder (w : ℂ) : HasDerivAt A (r : ℂ) w := by
-    simpa [A] using ((hasDerivAt_id w).const_mul (r : ℂ)).const_add z
-  have hAi : Function.Injective A := by
-    intro v w he
-    exact mul_left_cancel₀ hrC (add_left_cancel he)
-  have hA0 : A 0 = z := by simp [A]
-  let F := f ∘ A
-  let G := fun w => (g w - z) / (r : ℂ)
-  have hFd : DifferentiableOn ℂ F (ball 0 1) := hd.comp hAd.differentiableOn hAmap
-  have hFi : InjOn F (ball 0 1) := by
-    intro v hv w hw he
-    exact hAi (hi (hAmap hv) (hAmap hw) he)
-  have hsub : F '' ball 0 1 ⊆ f '' ball z r := by
-    rintro w ⟨v, hv, rfl⟩
-    exact ⟨A v, hAmap hv, rfl⟩
-  have hGd : DifferentiableOn ℂ G (F '' ball 0 1) :=
-    ((hg.mono hsub).sub_const z).div_const (r : ℂ)
-  have hGF : ∀ w ∈ ball (0 : ℂ) 1, G (F w) = w := by
-    intro w hw
-    dsimp [G, F]
-    rw [hgf _ (hAmap hw)]
-    dsimp [A]
-    rw [add_sub_cancel_left, mul_div_cancel_left₀ _ hrC]
-  have hdf := (hd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
-  have hFder : deriv F 0 = deriv f z * (r : ℂ) := by
-    have hh : HasDerivAt f (deriv f z) (A 0) := by rw [hA0]; exact hdf
-    exact (hh.comp 0 (hAder 0)).deriv
-  have hbound := preSchwarzian_bound_of_inverse hFd hFi
-    (by rw [hFder]; exact mul_ne_zero hf0 hrC) hGd hGF
-  have hFsecond : deriv (deriv F) 0 = deriv (deriv f) z * (r : ℂ) ^ 2 := by
-    have heq : deriv F =ᶠ[𝓝 (0 : ℂ)] (fun w => deriv f (A w) * (r : ℂ)) := by
-      filter_upwards [isOpen_ball.mem_nhds (mem_ball_self (by norm_num : (0 : ℝ) < 1))] with w hw
-      exact (((hd.differentiableAt (isOpen_ball.mem_nhds (hAmap hw))).hasDerivAt).comp w (hAder w)).deriv
-    have hdd := ((hd.deriv isOpen_ball).differentiableAt
-      (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
-    have hh : HasDerivAt (deriv f) (deriv (deriv f) z) (A 0) := by rw [hA0]; exact hdd
-    have hh' := (((hh.comp 0 (hAder 0)).mul_const (r : ℂ)).congr_of_eventuallyEq heq).deriv
-    simpa only [pow_two, mul_assoc] using hh'
-  have halg : deriv (deriv F) 0 / deriv F 0 = (r : ℂ) * (deriv (deriv f) z / deriv f z) := by
-    rw [hFsecond, hFder]
-    field_simp
-  rw [halg, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr] at hbound
-  exact hbound
-
- 
-
-lemma disk_preSchwarzian_bound {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hfz : deriv f z ≠ 0) :
-    ‖((1 - ‖z‖ ^ 2 : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z) - 2 * star z‖ ≤
-      2560 * Real.pi + 2 := by
-  have hn := mem_ball_zero_iff.mp hz
-  have hr : 0 < 1 - ‖z‖ := by linarith
-  have hsub : ball z (1 - ‖z‖) ⊆ ball (0 : ℂ) 1 := by
-    intro w hw
-    apply mem_ball.mpr
-    have hh := mem_ball.mp hw
-    have hd0 : dist z 0 = ‖z‖ := dist_zero_right z
-    exact (dist_triangle w z 0).trans_lt (by linarith)
-  have hb := preSchwarzian_bound_ball hr (hd.mono hsub) (hi.mono hsub) hfz
-    (hg.mono (image_mono hsub)) (fun w hw => hgf w (hsub hw))
-  have hrad : 0 ≤ 1 - ‖z‖ ^ 2 := by nlinarith [norm_nonneg z]
-  calc
-    _ ≤ ‖((1 - ‖z‖ ^ 2 : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z)‖ + ‖2 * star z‖ := norm_sub_le _ _
-    _ = (1 - ‖z‖ ^ 2) * ‖deriv (deriv f) z / deriv f z‖ + 2 * ‖z‖ := by
-      simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hrad,
-        norm_ofNat, norm_star]
-    _ ≤ 2560 * Real.pi + 2 := by
-      have hprod := mul_le_mul_of_nonneg_right (show (1 + ‖z‖) ≤ 2 by linarith)
-        (mul_nonneg hr.le (norm_nonneg (deriv (deriv f) z / deriv f z)))
-      nlinarith
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian RealInnerProductSpace
-namespace SLEExactGauge
-
-noncomputable def diskLogRadius (f : ℂ → ℂ) (z : ℂ) : ℝ :=
-  Real.log (diskWeight z) + Real.log ‖deriv f z‖
-
-lemma diskLogRadius_contDiffAt {f : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) {z : ℂ}
-    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
-    ContDiffAt ℝ 2 (diskLogRadius f) z := by
-  have hh := (((hd.deriv isOpen_ball).analyticAt (isOpen_ball.mem_nhds hz))).harmonicAt_log_norm hne
-  exact (diskWeight_contDiff.contDiffAt.log (diskWeight_pos hz).ne').add hh.1
-
-end SLEExactGauge
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.RadiusClocks
+
+namespace OAI
+
+/-! Angular Laws for chordal SLE. -/
+
+section
+section
+section
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology Laplacian ENNReal
+namespace SLEExactGauge
+noncomputable def greenInner (n : ℕ) : ℝ := (1 / 2) * (1 / ((n : ℝ) + 1))
+noncomputable def greenOuter (n : ℕ) : ℝ := 1 - greenInner n
+
+lemma greenInner_pos (n : ℕ) : 0 < greenInner n := by unfold greenInner; positivity
+
+lemma greenInner_le_half (n : ℕ) : greenInner n ≤ 1 / 2 := by
+  unfold greenInner
+  have hn : (1 : ℝ) ≤ (n : ℝ) + 1 := le_add_of_nonneg_left (Nat.cast_nonneg n)
+  have hh : (1 : ℝ) / ((n : ℝ) + 1) ≤ 1 :=
+    (div_le_one (by positivity)).mpr hn
+  linarith
+
+lemma greenInner_le_outer (n : ℕ) : greenInner n ≤ greenOuter n := by
+  unfold greenOuter
+  linarith [greenInner_le_half n]
+
+lemma greenOuter_pos (n : ℕ) : 0 < greenOuter n :=
+  (greenInner_pos n).trans_le (greenInner_le_outer n)
+
+lemma greenOuter_lt_one (n : ℕ) : greenOuter n < 1 := by
+  unfold greenOuter
+  linarith [greenInner_pos n]
+
+lemma greenInner_tendsto : Tendsto greenInner atTop (𝓝 0) := by
+  change Tendsto (fun n : ℕ => (1 / 2) * (1 / ((n : ℝ) + 1))) atTop (𝓝 0)
+  simpa only [greenInner, mul_zero] using
+    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (1 / 2 : ℝ)
+
+lemma greenOuter_tendsto : Tendsto greenOuter atTop (𝓝 1) := by
+  change Tendsto (fun n => 1 - greenInner n) atTop (𝓝 1)
+  simpa only [greenOuter, sub_zero] using tendsto_const_nhds.sub greenInner_tendsto
+
+lemma greenInner_log_tendsto : Tendsto (fun n => -greenInner n * Real.log (greenInner n)) atTop (𝓝 0) := by
+  simpa only [Function.comp_def, zero_mul, neg_zero, neg_mul] using
+    ((Real.continuous_mul_log.tendsto 0).comp greenInner_tendsto).neg
+
+def polarUnitDisk : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo (-Real.pi) Real.pi
+
+noncomputable def polarGreenTarget (u : ℂ → ℝ) (p : ℝ × ℝ) : ℝ≥0∞ :=
+  polarUnitDisk.indicator (fun p : ℝ × ℝ =>
+    ENNReal.ofReal (p.1 * (-Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2)))) p
+
+lemma greenTruncation_tendsto {u : ℂ → ℝ} {p : ℝ × ℝ} (hp : p ∈ polarUnitDisk) :
+    Tendsto (fun n => ENNReal.ofReal (greenTruncation u (greenInner n) (greenOuter n) p)) atTop
+      (𝓝 (ENNReal.ofReal (p.1 * (-Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2))))) := by
+  have he : ∀ᶠ n in atTop, greenTruncation u (greenInner n) (greenOuter n) p =
+      p.1 * (Real.log (greenOuter n) - Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2)) := by
+    filter_upwards [greenInner_tendsto (Iio_mem_nhds hp.1.1),
+      greenOuter_tendsto (Ioi_mem_nhds hp.1.2)] with n ha hR
+    have hm : p ∈ greenRectangle (greenInner n) (greenOuter n) :=
+      ⟨⟨ha.le, hR.le⟩, ⟨hp.2.1.le, hp.2.2.le⟩⟩
+    simp only [greenTruncation, indicator_of_mem hm, annulusDensity]
+  have hl := (Real.continuousAt_log (by norm_num : (1 : ℝ) ≠ 0)).tendsto.comp greenOuter_tendsto
+  have ht := ENNReal.continuous_ofReal.continuousAt.tendsto.comp
+    (((hl.sub_const (Real.log p.1)).const_mul p.1).mul_const
+      (-(Δ u) (polarPoint p.1 p.2)))
+  simp only [Function.comp_def, Real.log_one, zero_sub] at ht
+  exact ht.congr' (he.mono fun n hn => congrArg ENNReal.ofReal hn.symm)
+
+ 
+
+lemma polar_green_le {u : ℂ → ℝ} {C : ℝ}
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
+    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
+    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
+    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0) :
+    (∫⁻ p, polarGreenTarget u p) ≤ ENNReal.ofReal (2 * Real.pi * C) := by
+  let F : ℕ → ℝ × ℝ → ℝ≥0∞ := fun n p =>
+    ENNReal.ofReal (greenTruncation u (greenInner n) (greenOuter n) p)
+  have hmeas (n : ℕ) : AEMeasurable (F n) :=
+    ENNReal.measurable_ofReal.comp_aemeasurable
+      (greenTruncation_integrable (greenInner_pos n) (greenOuter_lt_one n) hu).aemeasurable
+  have hp : polarGreenTarget u ≤ fun p => liminf (fun n => F n p) atTop := by
+    intro p
+    by_cases hp : p ∈ polarUnitDisk
+    · rw [polarGreenTarget, indicator_of_mem hp]
+      exact le_of_eq (greenTruncation_tendsto (u := u) hp).liminf_eq.symm
+    · simp only [polarGreenTarget, indicator_of_notMem hp]
+      exact bot_le
+  rcases derivative_locally_bounded (hu 0 (by simp)) with ⟨δ, M, hδ, hM, hgrad⟩
+  have hbound : ∀ᶠ n in atTop, (∫⁻ p, F n p) ≤
+      ENNReal.ofReal (2 * Real.pi * (C + M * (-greenInner n * Real.log (greenInner n)))) := by
+    filter_upwards [greenInner_tendsto (Iio_mem_nhds hδ)] with n hn
+    change greenInner n < δ at hn
+    exact greenTruncation_lintegral_le (greenInner_pos n) (greenInner_le_outer n)
+      (greenOuter_lt_one n) hu hun hub hΔ hM (fun θ => hgrad _ (by
+        simpa only [polarPoint_norm, abs_of_pos (greenInner_pos n)] using hn))
+  have ht : Tendsto (fun n => ENNReal.ofReal
+      (2 * Real.pi * (C + M * (-greenInner n * Real.log (greenInner n))))) atTop
+      (𝓝 (ENNReal.ofReal (2 * Real.pi * C))) := by
+    simpa only [Function.comp_def, mul_zero, add_zero] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp
+      ((tendsto_const_nhds.add (greenInner_log_tendsto.const_mul M)).const_mul (2 * Real.pi))
+  exact (lintegral_mono hp).trans ((lintegral_liminf_le' hmeas).trans
+    ((liminf_le_liminf hbound).trans_eq ht.liminf_eq))
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge
+
+lemma polarPoint_eq_polarCoord (p : ℝ × ℝ) :
+    polarPoint p.1 p.2 = Complex.polarCoord.symm p := by
+  rw [polarPoint_eq_circleMap]
+  apply Complex.ext
+  · simp [circleMap_zero_re, Complex.polarCoord_symm_apply, Complex.cos_ofReal_re, Complex.sin_ofReal_re]
+  · simp [circleMap_zero_im, Complex.polarCoord_symm_apply, Complex.cos_ofReal_re, Complex.sin_ofReal_re]
+
+lemma disk_log_lintegral_polar (h : ℂ → ℝ) :
+    (∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((-Real.log ‖z‖) * h z)) =
+      ∫⁻ p in Ioo (0 : ℝ) 1 ×ˢ Ioo (-Real.pi) Real.pi,
+        ENNReal.ofReal (p.1 * (-Real.log p.1) * h (polarPoint p.1 p.2)) := by
+  let T : Set (ℝ × ℝ) := Ioi 0 ×ˢ Ioo (-Real.pi) Real.pi
+  let S : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo (-Real.pi) Real.pi
+  have hT : MeasurableSet T := measurableSet_Ioi.prod measurableSet_Ioo
+  have hS : MeasurableSet S := measurableSet_Ioo.prod measurableSet_Ioo
+  have hST : S ⊆ T := prod_mono Ioo_subset_Ioi_self subset_rfl
+  let f : ℂ → ℝ≥0∞ := (ball (0 : ℂ) 1).indicator (fun z => ENNReal.ofReal ((-Real.log ‖z‖) * h z))
+  calc
+    _ = ∫⁻ z, f z := (lintegral_indicator measurableSet_ball _).symm
+    _ = ∫⁻ p in T, ENNReal.ofReal p.1 • f (Complex.polarCoord.symm p) :=
+      (Complex.lintegral_comp_polarCoord_symm f).symm
+    _ = ∫⁻ p in T, S.indicator (fun p : ℝ × ℝ =>
+        ENNReal.ofReal (p.1 * (-Real.log p.1) * h (polarPoint p.1 p.2))) p := by
+      apply setLIntegral_congr_fun hT
+      intro p hp
+      have hn : ‖Complex.polarCoord.symm p‖ = p.1 := by
+        rw [Complex.norm_polarCoord_symm, abs_of_pos hp.1]
+      by_cases hr : p.1 < 1
+      · have hz : Complex.polarCoord.symm p ∈ ball (0 : ℂ) 1 := by
+          rw [mem_ball_zero_iff, hn]
+          exact hr
+        have hs : p ∈ S := ⟨⟨hp.1, hr⟩, hp.2⟩
+        simp only [f, indicator_of_mem hz, indicator_of_mem hs, hn, smul_eq_mul,
+          polarPoint_eq_polarCoord, ← ENNReal.ofReal_mul hp.1.le, mul_assoc]
+      · have hz : Complex.polarCoord.symm p ∉ ball (0 : ℂ) 1 := by
+          simpa only [mem_ball_zero_iff, hn] using hr
+        have hs : p ∉ S := fun hs => hr hs.1.2
+        simp only [f, indicator_of_notMem hz, indicator_of_notMem hs, smul_zero]
+    _ = _ := by
+      rw [lintegral_indicator hS, Measure.restrict_restrict hS, inter_eq_left.mpr hST]
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology Laplacian ENNReal
+namespace SLEExactGauge
+
+ 
+lemma disk_green_le {u : ℂ → ℝ} {C : ℝ}
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
+    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
+    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
+    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0) :
+    (∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((-Real.log ‖z‖) * (-(Δ u) z))) ≤
+      ENNReal.ofReal (2 * Real.pi * C) := by
+  rw [disk_log_lintegral_polar]
+  have hh := polar_green_le hu hun hub hΔ
+  simpa only [polarGreenTarget, polarUnitDisk,
+    lintegral_indicator (measurableSet_Ioo.prod measurableSet_Ioo)] using hh
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+namespace SLEExactGauge
+
+lemma second_direction_chain {f : ℝ → ℝ} {g : ℂ → ℝ} {z : ℂ}
+    (hg : ContDiffAt ℝ 2 g z) (hf : ContDiffAt ℝ 2 f (g z)) (v : ℂ) :
+    fderiv ℝ (fderiv ℝ (f ∘ g)) z v v =
+      deriv (deriv f) (g z) * (fderiv ℝ g z v) ^ 2 +
+        deriv f (g z) * fderiv ℝ (fderiv ℝ g) z v v := by
+  have hgd : DifferentiableAt ℝ g z := hg.differentiableAt (by norm_num)
+  have hfdd : DifferentiableAt ℝ (deriv f) (g z) :=
+    (hf.derivWithin (m := 1) (by norm_num)).differentiableAt (by norm_num)
+  have hgdd : DifferentiableAt ℝ (fderiv ℝ g) z :=
+    (hg.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)
+  have hcompdd : DifferentiableAt ℝ (fderiv ℝ (f ∘ g)) z :=
+    ((hf.comp z hg).fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)
+  have heq : (fun w => fderiv ℝ (f ∘ g) w v) =ᶠ[𝓝 z]
+      (fun w => deriv f (g w) * fderiv ℝ g w v) := by
+    filter_upwards [hg.eventually (by norm_num),
+      hgd.continuousAt (hf.eventually (by norm_num))] with w hgw hfw
+    change ContDiffAt ℝ 2 f (g w) at hfw
+    rw [fderiv_comp w (hfw.differentiableAt (by norm_num)) (hgw.differentiableAt (by norm_num)),
+      ContinuousLinearMap.comp_apply, fderiv_eq_smul_deriv, smul_eq_mul, mul_comm]
+  have hleft : fderiv ℝ (fun w => fderiv ℝ (f ∘ g) w v) z v =
+      fderiv ℝ (fderiv ℝ (f ∘ g)) z v v := by
+    rw [fderiv_clm_apply hcompdd (differentiableAt_const v)]
+    simp
+  have hright : fderiv ℝ (fun w => deriv f (g w) * fderiv ℝ g w v) z v =
+      deriv (deriv f) (g z) * (fderiv ℝ g z v) ^ 2 +
+        deriv f (g z) * fderiv ℝ (fderiv ℝ g) z v v := by
+    have hmul := fderiv_fun_mul (hfdd.comp z hgd) (hgdd.clm_apply (differentiableAt_const v))
+    simp only [Function.comp_apply] at hmul
+    rw [hmul]
+    simp only [add_apply, smul_apply, smul_eq_mul]
+    rw [fderiv_comp z hfdd hgd, ContinuousLinearMap.comp_apply, fderiv_eq_smul_deriv,
+      fderiv_clm_apply hgdd (differentiableAt_const v)]
+    simp only [fderiv_fun_const, Pi.zero_apply, ContinuousLinearMap.comp_zero, zero_add,
+      ContinuousLinearMap.flip_apply, smul_eq_mul]
+    ring
+  rw [← hleft, heq.fderiv_eq, hright]
+
+ 
+
+lemma plane_laplacian_chain {f : ℝ → ℝ} {g : ℂ → ℝ} {z : ℂ}
+    (hg : ContDiffAt ℝ 2 g z) (hf : ContDiffAt ℝ 2 f (g z)) :
+    (Δ (f ∘ g)) z = deriv f (g z) * (Δ g) z +
+      deriv (deriv f) (g z) * ((fderiv ℝ g z 1) ^ 2 + (fderiv ℝ g z I) ^ 2) := by
+  simp only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
+    Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+  rw [second_direction_chain hg hf 1, second_direction_chain hg hf I]
+  ring
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian RealInnerProductSpace
+namespace SLEExactGauge
+
+noncomputable def diskWeight (z : ℂ) : ℝ := 1 - ‖z‖ ^ 2
+
+lemma diskWeight_pos {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : 0 < diskWeight z := by
+  have hn := mem_ball_zero_iff.mp hz
+  dsimp [diskWeight]
+  nlinarith [norm_nonneg z]
+
+lemma diskWeight_contDiff : ContDiff ℝ 2 diskWeight :=
+  contDiff_const.sub (contDiff_id.norm_sq ℂ)
+
+lemma diskWeight_fderiv (z : ℂ) :
+    fderiv ℝ diskWeight z = (-2 : ℝ) • innerSL ℝ z := by
+  change fderiv ℝ (fun w : ℂ => 1 - ‖w‖ ^ 2) z = _
+  rw [fderiv_const_sub, fderiv_norm_sq_apply]
+  ext v
+  simp [smul_apply, nsmul_eq_mul]
+  ring
+
+lemma diskWeight_laplacian (z : ℂ) : (Δ diskWeight) z = -4 := by
+  simp only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
+    Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+  have heq : fderiv ℝ diskWeight = fun z => (-2 : ℝ) • innerSL ℝ z := funext diskWeight_fderiv
+  rw [heq]
+  rw [fderiv_fun_const_smul (𝕜 := ℝ) (x := z)
+    (innerSL ℝ (E := ℂ)).differentiableAt (-2 : ℝ)]
+  simp only [ContinuousLinearMap.fderiv, smul_apply, smul_eq_mul]
+  change -2 * inner ℝ (1 : ℂ) 1 + -2 * inner ℝ (I : ℂ) I = -4
+  norm_num [real_inner_self_eq_norm_sq]
+
+lemma log_diskWeight_laplacian {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) :
+    (Δ (fun w => Real.log (diskWeight w))) z = -4 / diskWeight z ^ 2 := by
+  have hg := diskWeight_contDiff.contDiffAt (x := z)
+  have hf : ContDiffAt ℝ 2 Real.log (diskWeight z) := Real.contDiffAt_log.mpr (diskWeight_pos hz).ne'
+  have hh := plane_laplacian_chain hg hf
+  have hlog2 : deriv (deriv Real.log) (diskWeight z) = -(diskWeight z ^ 2)⁻¹ := by
+    rw [Real.deriv_log']
+    exact (hasDerivAt_inv (diskWeight_pos hz).ne').deriv
+  change (Δ (fun w => Real.log (diskWeight w))) z = _ at hh
+  rw [Real.deriv_log, hlog2, diskWeight_laplacian, diskWeight_fderiv] at hh
+  rw [hh]
+  simp only [smul_apply, innerSL_apply_apply, smul_eq_mul]
+  have he : ‖z‖ ^ 2 = z.re ^ 2 + z.im ^ 2 := by simpa [Complex.normSq_apply, sq] using Complex.sq_norm z
+  have hwne : 1 - ‖z‖ ^ 2 ≠ 0 := (diskWeight_pos hz).ne'
+  simp [diskWeight, Complex.mul_re]
+  field_simp [hwne]
+  nlinarith [he]
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology
+namespace SLEExactGauge
+
+lemma log_norm_direction_bound {f : ℂ → ℂ} {z : ℂ}
+    (hf : DifferentiableAt ℂ f z) (hne : f z ≠ 0) (v : ℂ) :
+    |fderiv ℝ (fun w => Real.log ‖f w‖) z v| ≤ ‖deriv f z / f z‖ * ‖v‖ := by
+  have hnorm : DifferentiableAt ℝ (fun w : ℂ => ‖w‖) (f z) :=
+    differentiableAt_id.norm ℂ hne
+  have hd := ((hnorm.hasFDerivAt.comp z (hf.restrictScalars ℝ).hasFDerivAt).log (norm_ne_zero_iff.mpr hne)).fderiv
+  change fderiv ℝ (fun w => Real.log ‖f w‖) z = _ at hd
+  rw [hd]
+  simp only [smul_apply, ContinuousLinearMap.comp_apply, smul_eq_mul, abs_mul,
+    abs_inv, Function.comp_apply, abs_norm]
+  have hbound := (fderiv ℝ (fun w : ℂ => ‖w‖) (f z)).le_opNorm (fderiv ℝ f z v)
+  rw [norm_fderiv_norm hnorm, one_mul, Real.norm_eq_abs] at hbound
+  have hvalue : fderiv ℝ f z v = v * deriv f z := by
+    rw [hf.fderiv_restrictScalars (𝕜 := ℝ)]
+    change fderiv ℂ f z v = _
+    rw [fderiv_eq_smul_deriv, smul_eq_mul]
+  rw [hvalue, norm_mul] at hbound
+  rw [hvalue]
+  calc
+    ‖f z‖⁻¹ * |(fderiv ℝ (fun w : ℂ => ‖w‖) (f z)) (v * deriv f z)|
+        ≤ ‖f z‖⁻¹ * (‖v‖ * ‖deriv f z‖) := mul_le_mul_of_nonneg_left hbound (inv_nonneg.mpr (norm_nonneg _))
+    _ = ‖deriv f z / f z‖ * ‖v‖ := by rw [norm_div]; ring
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma univalent_omitted_value_bound_ball {f : ℂ → ℂ} {z : ℂ} {r : ℝ}
+    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball z r))
+    (hi : InjOn f (ball z r)) (a : ℂ) (ha : a ∉ f '' ball z r) :
+    r * ‖deriv f z‖ ≤ (128 * Real.pi) * ‖f z - a‖ := by
+  let g : ℂ → ℂ := fun w => z + (r : ℂ) * w
+  have hgmap : MapsTo g (ball 0 1) (ball z r) := by
+    intro w hw
+    rw [mem_ball_iff_norm]
+    dsimp only [g]
+    rw [add_sub_cancel_left, norm_mul, Complex.norm_real,
+      Real.norm_eq_abs, abs_of_pos hr]
+    exact (mul_lt_mul_of_pos_left (by simpa [mem_ball_zero_iff] using hw) hr).trans_eq (mul_one r)
+  have hgd : Differentiable ℂ g := (differentiable_const z).add ((differentiable_const (r : ℂ)).mul differentiable_id)
+  have hgi : Function.Injective g := by
+    intro v w he
+    exact mul_left_cancel₀ (by exact_mod_cast hr.ne') (add_left_cancel he)
+  have hfd : DifferentiableOn ℂ (f ∘ g) (ball 0 1) := hd.comp hgd.differentiableOn hgmap
+  have hfi : InjOn (f ∘ g) (ball 0 1) := by
+    intro v hv w hw he
+    exact hgi (hi (hgmap hv) (hgmap hw) he)
+  have hfa : a ∉ (f ∘ g) '' ball 0 1 := by
+    rintro ⟨w, hw, he⟩
+    exact ha ⟨g w, hgmap hw, he⟩
+  have hg0 : g 0 = z := by simp [g]
+  have hgder : HasDerivAt g (r : ℂ) 0 := by
+    simpa [g] using ((hasDerivAt_id (0 : ℂ)).const_mul (r : ℂ)).const_add z
+  have hfz := (hd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
+  have hder : deriv (f ∘ g) 0 = deriv f z * (r : ℂ) := by
+    have hfg : HasDerivAt f (deriv f z) (g 0) := by rw [hg0]; exact hfz
+    exact (hfg.comp 0 hgder).deriv
+  have hh := univalent_omitted_value_bound hfd hfi a hfa
+  simpa [hder, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr,
+    Function.comp_apply, hg0, mul_comm] using hh
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma logarithmic_lift_deriv_bound {f h : ℂ → ℂ} {a : ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (ha : a ∉ f '' ball 0 1)
+    (hh : ∀ z ∈ ball (0 : ℂ) 1, deriv h z = deriv f z / (f z - a))
+    {z : ℂ} (hz : z ∈ closedBall (0 : ℂ) (1 / 2 : ℝ)) :
+    ‖deriv h z‖ ≤ 256 * Real.pi := by
+  have hsub : ball z (1 / 2 : ℝ) ⊆ ball (0 : ℂ) 1 := by
+    intro w hw
+    apply mem_ball.mpr
+    have hw' := mem_ball.mp hw
+    have hz' := mem_closedBall.mp hz
+    exact (dist_triangle w z 0).trans_lt (by linarith)
+  have hz' : z ∈ ball (0 : ℂ) 1 := (closedBall_subset_ball (by norm_num)) hz
+  have hn : f z - a ≠ 0 := by
+    intro he
+    exact ha ⟨z, hz', sub_eq_zero.mp he⟩
+  have hb := univalent_omitted_value_bound_ball (by norm_num : (0 : ℝ) < 1 / 2)
+    (hd.mono hsub) (hi.mono hsub) a (fun he => ha (image_mono hsub he))
+  rw [hh z hz', norm_div]
+  apply (div_le_iff₀ (norm_pos_iff.mpr hn)).mpr
+  linarith
+
+lemma logarithmic_lift_second_bound {f h : ℂ → ℂ} {a : ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (ha : a ∉ f '' ball 0 1) (hhd : DifferentiableOn ℂ h (ball 0 1))
+    (hh : ∀ z ∈ ball (0 : ℂ) 1, deriv h z = deriv f z / (f z - a)) :
+    ‖deriv (deriv h) 0‖ ≤ 512 * Real.pi := by
+  have hs : closedBall (0 : ℂ) (1 / 2 : ℝ) ⊆ ball 0 1 :=
+    closedBall_subset_ball (by norm_num)
+  have hb := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le
+    (by norm_num : (0 : ℝ) < 1 / 2)
+    ((hhd.deriv isOpen_ball).diffContOnCl_ball hs)
+    (fun z hz => logarithmic_lift_deriv_bound hd hi ha hh (sphere_subset_closedBall hz))
+  convert hb using 1; ring
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma exists_omitted_near_of_inverse {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hf0 : deriv f 0 ≠ 0)
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ z ∈ ball (0 : ℂ) 1, g (f z) = z) :
+    ∃ a : ℂ, a ∉ f '' ball 0 1 ∧ ‖f 0 - a‖ < 2 * ‖deriv f 0‖ := by
+  have hz : (0 : ℂ) ∈ ball 0 1 := mem_ball_self (by norm_num)
+  have hn : 0 < ‖deriv f 0‖ := norm_pos_iff.mpr hf0
+  have hr : 0 < 2 * ‖deriv f 0‖ := by positivity
+  by_contra! hcontra
+  have hsub : ball (f 0) (2 * ‖deriv f 0‖) ⊆ f '' ball 0 1 := by
+    intro a ha
+    by_contra hnot
+    have hh := hcontra a hnot
+    have hn' : ‖f 0 - a‖ < 2 * ‖deriv f 0‖ := by
+      simpa only [mem_ball_iff_norm, norm_sub_rev] using ha
+    linarith
+  have hgd := hg.mono hsub
+  have hgm : MapsTo g (ball (f 0) (2 * ‖deriv f 0‖)) (closedBall (g (f 0)) 1) := by
+    intro w hw
+    obtain ⟨z, hz', rfl⟩ := hsub hw
+    rw [hgf z hz', hgf 0 hz]
+    exact ball_subset_closedBall hz'
+  have hbound := Complex.norm_deriv_le_div_of_mapsTo_ball hgd hgm hr
+  have hdf := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
+  have hdg := (hgd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
+  have heq : (g ∘ f) =ᶠ[𝓝 (0 : ℂ)] id := by
+    filter_upwards [isOpen_ball.mem_nhds hz] with z hz'
+    exact hgf z hz'
+  have hprod : deriv g (f 0) * deriv f 0 = 1 :=
+    ((hdg.comp 0 hdf).congr_of_eventuallyEq heq.symm).unique (hasDerivAt_id 0)
+  have hnormprod : ‖deriv g (f 0)‖ * ‖deriv f 0‖ = 1 := by
+    simpa only [norm_mul, norm_one] using congrArg norm hprod
+  have hh := (le_div_iff₀ hr).mp hbound
+  nlinarith
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma preSchwarzian_bound_of_inverse {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hf0 : deriv f 0 ≠ 0)
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ z ∈ ball (0 : ℂ) 1, g (f z) = z) :
+    ‖deriv (deriv f) 0 / deriv f 0‖ ≤ 1280 * Real.pi := by
+  obtain ⟨a, ha, hnear⟩ := exists_omitted_near_of_inverse hd hf0 hg hgf
+  have hz : (0 : ℂ) ∈ ball 0 1 := mem_ball_self (by norm_num)
+  have hnon (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) : f z - a ≠ 0 := by
+    intro he
+    exact ha ⟨z, hz, sub_eq_zero.mp he⟩
+  obtain ⟨h, hhd, hexp, hh⟩ := exists_holomorphic_log_on_ball (by norm_num : (0 : ℝ) < 1)
+    (hd.sub_const a) hnon
+  have hh' (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) :
+      deriv h z = deriv f z / (f z - a) := by
+    rw [hh z hz]
+    congr 1
+    exact ((hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.sub_const a).deriv
+  have hdf := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
+  have hddf := ((hd.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
+  have heq : deriv h =ᶠ[𝓝 (0 : ℂ)] (fun z => deriv f z / (f z - a)) := by
+    filter_upwards [isOpen_ball.mem_nhds hz] with z hz'
+    exact hh' z hz'
+  have hsecond : deriv (deriv h) 0 =
+      (deriv (deriv f) 0 * (f 0 - a) - deriv f 0 * deriv f 0) / (f 0 - a) ^ 2 :=
+    ((hddf.div (hdf.sub_const a) (hnon 0 hz)).congr_of_eventuallyEq heq).deriv
+  have halg : deriv (deriv f) 0 / deriv f 0 =
+      deriv (deriv h) 0 / deriv h 0 + deriv h 0 := by
+    rw [hsecond, hh' 0 hz]
+    field_simp [hf0, hnon 0 hz]
+    ring
+  have hfirst := logarithmic_lift_deriv_bound hd hi ha hh'
+    (mem_closedBall_self (by norm_num : (0 : ℝ) ≤ 1 / 2))
+  have htwo := logarithmic_lift_second_bound hd hi ha hhd hh'
+  have hlower : (1 / 2 : ℝ) ≤ ‖deriv h 0‖ := by
+    rw [hh' 0 hz, norm_div]
+    apply (le_div_iff₀ (norm_pos_iff.mpr (hnon 0 hz))).mpr
+    linarith
+  have hpos : 0 < ‖deriv h 0‖ := by linarith
+  have hratio : ‖deriv (deriv h) 0‖ / ‖deriv h 0‖ ≤ 1024 * Real.pi := by
+    apply (div_le_iff₀ hpos).mpr
+    nlinarith [Real.pi_pos]
+  rw [halg]
+  calc
+    ‖deriv (deriv h) 0 / deriv h 0 + deriv h 0‖ ≤
+        ‖deriv (deriv h) 0‖ / ‖deriv h 0‖ + ‖deriv h 0‖ := by
+      simpa only [norm_div] using norm_add_le (deriv (deriv h) 0 / deriv h 0) (deriv h 0)
+    _ ≤ 1280 * Real.pi := by linarith
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma preSchwarzian_bound_ball {f g : ℂ → ℂ} {z : ℂ} {r : ℝ}
+    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball z r))
+    (hi : InjOn f (ball z r)) (hf0 : deriv f z ≠ 0)
+    (hg : DifferentiableOn ℂ g (f '' ball z r))
+    (hgf : ∀ w ∈ ball z r, g (f w) = w) :
+    r * ‖deriv (deriv f) z / deriv f z‖ ≤ 1280 * Real.pi := by
+  let A : ℂ → ℂ := fun w => z + (r : ℂ) * w
+  have hrC : (r : ℂ) ≠ 0 := by exact_mod_cast hr.ne'
+  have hAmap : MapsTo A (ball 0 1) (ball z r) := by
+    intro w hw
+    rw [mem_ball_iff_norm]
+    dsimp only [A]
+    rw [add_sub_cancel_left, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr]
+    exact (mul_lt_mul_of_pos_left (mem_ball_zero_iff.mp hw) hr).trans_eq (mul_one r)
+  have hAd : Differentiable ℂ A :=
+    (differentiable_const z).add ((differentiable_const (r : ℂ)).mul differentiable_id)
+  have hAder (w : ℂ) : HasDerivAt A (r : ℂ) w := by
+    simpa [A] using ((hasDerivAt_id w).const_mul (r : ℂ)).const_add z
+  have hAi : Function.Injective A := by
+    intro v w he
+    exact mul_left_cancel₀ hrC (add_left_cancel he)
+  have hA0 : A 0 = z := by simp [A]
+  let F := f ∘ A
+  let G := fun w => (g w - z) / (r : ℂ)
+  have hFd : DifferentiableOn ℂ F (ball 0 1) := hd.comp hAd.differentiableOn hAmap
+  have hFi : InjOn F (ball 0 1) := by
+    intro v hv w hw he
+    exact hAi (hi (hAmap hv) (hAmap hw) he)
+  have hsub : F '' ball 0 1 ⊆ f '' ball z r := by
+    rintro w ⟨v, hv, rfl⟩
+    exact ⟨A v, hAmap hv, rfl⟩
+  have hGd : DifferentiableOn ℂ G (F '' ball 0 1) :=
+    ((hg.mono hsub).sub_const z).div_const (r : ℂ)
+  have hGF : ∀ w ∈ ball (0 : ℂ) 1, G (F w) = w := by
+    intro w hw
+    dsimp [G, F]
+    rw [hgf _ (hAmap hw)]
+    dsimp [A]
+    rw [add_sub_cancel_left, mul_div_cancel_left₀ _ hrC]
+  have hdf := (hd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
+  have hFder : deriv F 0 = deriv f z * (r : ℂ) := by
+    have hh : HasDerivAt f (deriv f z) (A 0) := by rw [hA0]; exact hdf
+    exact (hh.comp 0 (hAder 0)).deriv
+  have hbound := preSchwarzian_bound_of_inverse hFd hFi
+    (by rw [hFder]; exact mul_ne_zero hf0 hrC) hGd hGF
+  have hFsecond : deriv (deriv F) 0 = deriv (deriv f) z * (r : ℂ) ^ 2 := by
+    have heq : deriv F =ᶠ[𝓝 (0 : ℂ)] (fun w => deriv f (A w) * (r : ℂ)) := by
+      filter_upwards [isOpen_ball.mem_nhds (mem_ball_self (by norm_num : (0 : ℝ) < 1))] with w hw
+      exact (((hd.differentiableAt (isOpen_ball.mem_nhds (hAmap hw))).hasDerivAt).comp w (hAder w)).deriv
+    have hdd := ((hd.deriv isOpen_ball).differentiableAt
+      (isOpen_ball.mem_nhds (mem_ball_self hr))).hasDerivAt
+    have hh : HasDerivAt (deriv f) (deriv (deriv f) z) (A 0) := by rw [hA0]; exact hdd
+    have hh' := (((hh.comp 0 (hAder 0)).mul_const (r : ℂ)).congr_of_eventuallyEq heq).deriv
+    simpa only [pow_two, mul_assoc] using hh'
+  have halg : deriv (deriv F) 0 / deriv F 0 = (r : ℂ) * (deriv (deriv f) z / deriv f z) := by
+    rw [hFsecond, hFder]
+    field_simp
+  rw [halg, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr] at hbound
+  exact hbound
+
+ 
+
+lemma disk_preSchwarzian_bound {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hfz : deriv f z ≠ 0) :
+    ‖((1 - ‖z‖ ^ 2 : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z) - 2 * star z‖ ≤
+      2560 * Real.pi + 2 := by
+  have hn := mem_ball_zero_iff.mp hz
+  have hr : 0 < 1 - ‖z‖ := by linarith
+  have hsub : ball z (1 - ‖z‖) ⊆ ball (0 : ℂ) 1 := by
+    intro w hw
+    apply mem_ball.mpr
+    have hh := mem_ball.mp hw
+    have hd0 : dist z 0 = ‖z‖ := dist_zero_right z
+    exact (dist_triangle w z 0).trans_lt (by linarith)
+  have hb := preSchwarzian_bound_ball hr (hd.mono hsub) (hi.mono hsub) hfz
+    (hg.mono (image_mono hsub)) (fun w hw => hgf w (hsub hw))
+  have hrad : 0 ≤ 1 - ‖z‖ ^ 2 := by nlinarith [norm_nonneg z]
+  calc
+    _ ≤ ‖((1 - ‖z‖ ^ 2 : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z)‖ + ‖2 * star z‖ := norm_sub_le _ _
+    _ = (1 - ‖z‖ ^ 2) * ‖deriv (deriv f) z / deriv f z‖ + 2 * ‖z‖ := by
+      simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hrad,
+        norm_ofNat, norm_star]
+    _ ≤ 2560 * Real.pi + 2 := by
+      have hprod := mul_le_mul_of_nonneg_right (show (1 + ‖z‖) ≤ 2 by linarith)
+        (mul_nonneg hr.le (norm_nonneg (deriv (deriv f) z / deriv f z)))
+      nlinarith
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian RealInnerProductSpace
+namespace SLEExactGauge
+
+noncomputable def diskLogRadius (f : ℂ → ℂ) (z : ℂ) : ℝ :=
+  Real.log (diskWeight z) + Real.log ‖deriv f z‖
+
+lemma diskLogRadius_contDiffAt {f : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) {z : ℂ}
+    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
+    ContDiffAt ℝ 2 (diskLogRadius f) z := by
+  have hh := (((hd.deriv isOpen_ball).analyticAt (isOpen_ball.mem_nhds hz))).harmonicAt_log_norm hne
+  exact (diskWeight_contDiff.contDiffAt.log (diskWeight_pos hz).ne').add hh.1
+
+end SLEExactGauge
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/AngularMoments.lean b/lean/OAI/Probability/SLE/AngularMoments.lean
index 1afd79bccead0594ea9c3036bea421d36dc446c5..62b7ff0f80a309de15db539bbc0e1fc840eb1730 100644
--- a/lean/OAI/Probability/SLE/AngularMoments.lean
+++ b/lean/OAI/Probability/SLE/AngularMoments.lean
@@ -1,667 +1,667 @@
-import Mathlib
-import OAI.Probability.SLE.KernelConsistency
-
-namespace OAI
-
-/-! Angular Moments for chordal SLE. -/
-
-section
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-noncomputable def physicalGreenLimit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (t : ℝ≥0) (ω : Ω) : ℝ :=
-  limUnder atTop (fun n => (a n).physicalWeight t ω)
-
-lemma physicalWeight_exists_limit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) (ω : Ω) :
-    ∃ l : ℝ, Tendsto (fun n => (a n).physicalWeight t ω) atTop (𝓝 l) := by
-  by_cases ht : ∃ n, t ≤ (a n).τ ω
-  · obtain ⟨n, hn⟩ := ht
-    refine ⟨(a n).physicalWeight t ω, tendsto_const_nhds.congr' ?_⟩
-    filter_upwards [eventually_ge_atTop n] with m hnm
-    have hm' := hn.trans ((a n).stopping_le (a m) le_rfl (hε hnm) ω)
-    simp only [AngularLocalization.physicalWeight, ← NNReal.coe_min, min_eq_left hn, min_eq_left hm']
-    rw [(a n).synchronize (a m) ω t hn hm']
-  · by_cases hh : ∃ n, ω ∈ (a n).hit
-    · obtain ⟨n, hn⟩ := hh
-      refine ⟨(a n).physicalWeight t ω, tendsto_const_nhds.congr' ?_⟩
-      filter_upwards [eventually_ge_atTop n] with m hnm
-      have heτ : (a n).τ ω = (a m).τ ω := le_antisymm
-        ((a n).stopping_le (a m) le_rfl (hε hnm) ω)
-        ((a m).stopping_le_of_hit (a n) le_rfl ω hn)
-      have hnt : (a n).τ ω ≤ t := (lt_of_not_ge (fun h => ht ⟨n, h⟩)).le
-      have hmt : (a m).τ ω ≤ t := heτ ▸ hnt
-      simp only [AngularLocalization.physicalWeight, ← NNReal.coe_min, min_eq_right hnt, min_eq_right hmt]
-      rw [(a n).synchronize (a m) ω ((a n).τ ω) le_rfl heτ.le, heτ]
-    · refine ⟨0, ?_⟩
-      have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
-      let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
-      have hpow : Tendsto (fun n => K * ε n ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
-        have hx := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
-        simpa only [Real.zero_rpow hp.ne', mul_zero, Function.comp_def] using hx
-      apply squeeze_zero (fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).1) _ hpow
-      intro n
-      have hnt : (a n).τ ω ≤ t := (lt_of_not_ge (fun h => ht ⟨n, h⟩)).le
-      change greenLogDensity κ _ _ ≤ _
-      rw [← NNReal.coe_min, min_eq_right hnt]
-      exact (a n).weight_small_of_not_hit hκ hκ8 ω (fun hn => hh ⟨n, hn⟩)
-
-lemma physicalWeight_tendsto_limit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) (ω : Ω) :
-    Tendsto (fun n => (a n).physicalWeight t ω) atTop (𝓝 (physicalGreenLimit a t ω)) := by
-  obtain ⟨l, hl⟩ := physicalWeight_exists_limit a hε hεlim hκ hκ8 t ω
-  simpa only [physicalGreenLimit, hl.limUnder_eq] using hl
-
-lemma physicalGreenLimit_bound
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) (ω : Ω) :
-    0 ≤ physicalGreenLimit a t ω ∧ physicalGreenLimit a t ω ≤
-      Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) := by
-  have ht := physicalWeight_tendsto_limit a hε hεlim hκ hκ8 t ω
-  exact ⟨ge_of_tendsto ht (Eventually.of_forall fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).1),
-    le_of_tendsto ht (Eventually.of_forall fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).2)⟩
-
-lemma physicalGreenLimit_adapted
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) :
-    StronglyAdapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) (physicalGreenLimit a) := by
-  intro t
-  have hs (n : ℕ) := (a n).physicalWeight_adapted hBc t
-  let : MeasurableSpace Ω := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t
-  exact StronglyMeasurable.limUnder hs
-
-lemma physicalGreenLimit_integrable
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} [IsFiniteMeasure P] (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) : Integrable (physicalGreenLimit a t) P := by
-  apply Integrable.of_bound ((physicalGreenLimit_adapted a hBc t).mono
-    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le t)).aestronglyMeasurable
-    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-  filter_upwards [] with ω
-  rw [Real.norm_eq_abs, abs_of_nonneg (physicalGreenLimit_bound a hε hεlim hκ hκ8 t ω).1]
-  exact (physicalGreenLimit_bound a hε hεlim hκ hκ8 t ω).2
-
- 
-
-theorem physicalGreenLimit_martingale
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) :
-    Martingale (physicalGreenLimit a) (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) P := by
-  let _ := hB.isGaussianProcess.isProbabilityMeasure
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have had := physicalGreenLimit_adapted a hBc
-  have hi := physicalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8
-  have hmar (n : ℕ) := (a n).physicalWeight_martingale hB hBm hBc hB0 hκ hκ8
-  refine ⟨had, fun s t hst => ?_⟩
-  apply EventuallyEq.symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq (f.le s) (hi t)
-    (fun _ _ _ => (hi s).integrableOn) _ (had s).aestronglyMeasurable
-  intro A hA _
-  have hlim (v : ℝ≥0) : Tendsto (fun n => ∫ ω in A, (a n).physicalWeight v ω ∂P) atTop
-      (𝓝 (∫ ω in A, physicalGreenLimit a v ω ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence
-      (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-      (fun n => ((a n).physicalWeight_integrable (P := P) hBc hκ hκ8 v).integrableOn.aestronglyMeasurable)
-      (integrable_const _) _ (Eventually.of_forall (physicalWeight_tendsto_limit a hε hεlim hκ hκ8 v))
-    intro n
-    filter_upwards [] with ω
-    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).physicalWeight_bound hκ hκ8 v ω).1]
-    exact ((a n).physicalWeight_bound hκ hκ8 v ω).2
-  apply tendsto_nhds_unique (hlim s)
-  convert hlim t using 1
-  ext n
-  exact (hmar n).setIntegral_eq hst hA
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-noncomputable def physicalStoppedGreen (κ : ℝ) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (u : ℝ) (t : ℝ≥0) : ℝ :=
-  if radiusLevelTime γ G z u ≤ (t : ℝ≥0∞) then stoppedGreen κ U γ G z u
-    else greenDensity κ U γ G t z
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
-lemma physicalWeight_tendsto_source
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hεlim : Tendsto ε atTop (𝓝 0))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) γ G F) (t : ℝ≥0) :
-    Tendsto (fun n => (a n).physicalWeight t ω) atTop
-      (𝓝 (physicalStoppedGreen κ (fun s => Real.sqrt κ * B s ω) γ G z u t)) := by
-  have hU := (hBc ω).const_mul (Real.sqrt κ)
-  have hρ := localizedRadiusStop_eq_source a hεlim hBc hz ω γ G F hw
-  by_cases hrt : radiusLevelTime γ G z u ≤ (t : ℝ≥0∞)
-  · rw [physicalStoppedGreen, ite_eq_left hrt]
-    have hρt : localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0) := by
-      rw [hρ]
-      exact hrt
-    obtain ⟨n, hn, hnt⟩ := (localizedRadiusStop_le_iff a ω t).mp hρt
-    obtain ⟨hl, hR⟩ := (a n).radius_at_hit hBc hz ω γ G F hw hn
-    apply tendsto_const_nhds.congr'
-    filter_upwards [eventually_hit_of_live_level a hεlim hBc hz ω γ G F hw ((a n).τ ω) hl hR]
-      with m hm'
-    have ht' : (a m).τ ω ≤ t := hm'.2.le.trans hnt
-    simpa only [AngularLocalization.physicalWeight, ← NNReal.coe_min, min_eq_right ht', AngularLocalization.weight]
-      using ((a m).weight_eq_stoppedGreen_of_hit hBc hz ω γ G F hw hm'.1).symm
-  · rw [physicalStoppedGreen, ite_eq_right hrt]
-    have hno (n : ℕ) (hnt : (a n).τ ω ≤ t) : ω ∉ (a n).hit := by
-      intro hn
-      apply hrt
-      have hcan := (localizedRadiusStop_le_iff a ω t).mpr ⟨n, hn, hnt⟩
-      rw [hρ] at hcan
-      exact hcan
-    by_cases hl : z ∈ survivingDomain γ t
-    · obtain ⟨δ, hδ, hδb⟩ := live_angleSq_lower_bound hU hw z t hl
-      apply tendsto_const_nhds.congr'
-      filter_upwards [(tendsto_order.mp hεlim).2 δ hδ] with n hn
-      have hbefore : t ≤ (a n).τ ω := by
-        by_contra hnt
-        have hnt' := lt_of_not_ge hnt
-        have hexit := ((a n).boundary ω).resolve_left (hno n hnt'.le)
-        obtain ⟨hla, heq⟩ := (a n).chart_identification hBc hz ω γ G F hw
-        have ha : logAngleSq κ ((a n).solution.Q ((a n).τ ω) ω) (B ((a n).τ ω) ω) =
-          (Real.sin (Complex.arg (G ((a n).τ ω) z - ((Real.sqrt κ * B ((a n).τ ω) ω : ℝ) : ℂ)))) ^ 2 := by
-          rw [heq _ le_rfl]
-          exact logAngleSq_logCoordinates κ _ G z _ (hw.imag_positive _ z hla)
-        have hb := hδb ((a n).τ ω) hnt'.le
-        rw [← ha, hexit] at hb
-        exact (not_lt_of_ge hb) hn
-      dsimp only [AngularLocalization.physicalWeight]
-      rw [← NNReal.coe_min, min_eq_left hbefore, ((a n).chart_identification hBc hz ω γ G F hw).2 t hbefore]
-      exact (greenLogDensity_logCoordinates hU hw κ _ z t hl rfl).symm
-    · have hnt (n : ℕ) : (a n).τ ω < t := by
-        apply lt_of_not_ge
-        intro hn
-        exact hl (survivingDomain_antitone γ hn ((a n).chart_identification hBc hz ω γ G F hw).1)
-      rw [greenDensity, ite_eq_right hl]
-      have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
-      let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
-      have hpow : Tendsto (fun n => K * ε n ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
-        have hx := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
-        simpa only [Real.zero_rpow hp.ne', mul_zero, Function.comp_def] using hx
-      apply squeeze_zero (fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).1) _ hpow
-      intro n
-      change greenLogDensity κ _ _ ≤ _
-      rw [← NNReal.coe_min, min_eq_right (hnt n).le]
-      exact (a n).weight_small_of_not_hit hκ hκ8 ω (hno n (hnt n).le)
-
-lemma physicalGreenLimit_eq_source
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hεlim : Tendsto ε atTop (𝓝 0))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) γ G F) (t : ℝ≥0) :
-    physicalGreenLimit a t ω = physicalStoppedGreen κ (fun s => Real.sqrt κ * B s ω) γ G z u t :=
-  (physicalWeight_tendsto_source a hεlim hBc hκ hκ8 hz ω γ G F hw t).limUnder_eq
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
-noncomputable def terminalGreenLimit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) : ℝ :=
-  limUnder atTop (fun n => (a n).weight ω)
-
-lemma localization_weight_tendsto_limit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hκ : 0 < κ) (hκ8 : κ < 8) (ω : Ω) :
-    Tendsto (fun n => (a n).weight ω) atTop (𝓝 (terminalGreenLimit a ω)) := by
-  suffices ∃ l : ℝ, Tendsto (fun n => (a n).weight ω) atTop (𝓝 l) by
-    obtain ⟨l, hl⟩ := this
-    simpa only [terminalGreenLimit, hl.limUnder_eq] using hl
-  by_cases hh : ∃ n, ω ∈ (a n).hit
-  · obtain ⟨n, hn⟩ := hh
-    refine ⟨(a n).weight ω, tendsto_const_nhds.congr' ?_⟩
-    filter_upwards [eventually_ge_atTop n] with m hnm
-    have ht : (a n).τ ω = (a m).τ ω := le_antisymm
-      ((a n).stopping_le (a m) le_rfl (hε hnm) ω)
-      ((a m).stopping_le_of_hit (a n) le_rfl ω hn)
-    have he := (a n).synchronize (a m) ω ((a n).τ ω) le_rfl ht.le
-    simp only [AngularLocalization.weight]
-    rw [he, ht]
-  · refine ⟨0, ?_⟩
-    have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
-    let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
-    have ht : Tendsto (fun n => K * ε n ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
-      have hx := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
-      simpa only [Real.zero_rpow hp.ne', mul_zero, Function.comp_def] using hx
-    apply squeeze_zero (fun n => ((a n).weight_bound hκ hκ8 ω).1) _ ht
-    intro n
-    exact (a n).weight_small_of_not_hit hκ hκ8 ω (fun hn => hh ⟨n, hn⟩)
-
-lemma terminalGreenLimit_bound
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hκ : 0 < κ) (hκ8 : κ < 8) (ω : Ω) :
-    0 ≤ terminalGreenLimit a ω ∧ terminalGreenLimit a ω ≤
-      Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) := by
-  have ht := localization_weight_tendsto_limit a hε hεlim hκ hκ8 ω
-  exact ⟨ge_of_tendsto ht (Eventually.of_forall fun n => ((a n).weight_bound hκ hκ8 ω).1),
-    le_of_tendsto ht (Eventually.of_forall fun n => ((a n).weight_bound hκ hκ8 ω).2)⟩
-
-lemma terminalGreenLimit_stronglyMeasurable
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) : StronglyMeasurable (terminalGreenLimit a) := by
-  have hs (n : ℕ) := ((a n).physicalWeight_adapted hBc (a n).T).mono
-    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le (a n).T)
-  have hs' (n : ℕ) : StronglyMeasurable ((a n).weight) := by
-    convert hs n using 1
-    ext ω
-    simp only [AngularLocalization.physicalWeight, AngularLocalization.weight, ← NNReal.coe_min,
-      min_eq_right ((a n).bounded ω)]
-  exact StronglyMeasurable.limUnder hs'
-
-lemma terminalGreenLimit_integrable
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} [IsFiniteMeasure P] (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) : Integrable (terminalGreenLimit a) P := by
-  apply Integrable.of_bound (terminalGreenLimit_stronglyMeasurable a hBc).aestronglyMeasurable
-    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-  filter_upwards [] with ω
-  rw [Real.norm_eq_abs, abs_of_nonneg (terminalGreenLimit_bound a hε hεlim hκ hκ8 ω).1]
-  exact (terminalGreenLimit_bound a hε hεlim hκ hκ8 ω).2
-
- 
-
-theorem physicalGreenLimit_condExp_terminal
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ωω, Continuous (fun t => B t ωω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) :
-    physicalGreenLimit a t =ᵐ[P]
-      P[terminalGreenLimit a | (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t] := by
-  let _ := hB.isGaussianProcess.isProbabilityMeasure
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have hi := terminalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8
-  apply ae_eq_condExp_of_forall_setIntegral_eq (f.le t) hi
-    (fun _ _ _ => (physicalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8 t).integrableOn)
-    _ (physicalGreenLimit_adapted a hBc t).aestronglyMeasurable
-  intro A hA _
-  have hlim₁ : Tendsto (fun n => ∫ ω in A, (a n).physicalWeight t ω ∂P) atTop
-      (𝓝 (∫ ω in A, physicalGreenLimit a t ω ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence
-      (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-      (fun n => ((a n).physicalWeight_integrable (P := P) hBc hκ hκ8 t).integrableOn.aestronglyMeasurable)
-      (integrable_const _) _ (Eventually.of_forall (physicalWeight_tendsto_limit a hε hεlim hκ hκ8 t))
-    intro n
-    filter_upwards [] with ω
-    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).physicalWeight_bound hκ hκ8 t ω).1]
-    exact ((a n).physicalWeight_bound hκ hκ8 t ω).2
-  have hw (n : ℕ) : (a n).physicalWeight (max t (a n).T) = (a n).weight := by
-    ext ω
-    simp only [AngularLocalization.physicalWeight, AngularLocalization.weight, ← NNReal.coe_min,
-      min_eq_right (((a n).bounded ω).trans (le_max_right _ _))]
-  have hwi (n : ℕ) : Integrable ((a n).weight) P := by
-    rw [← hw n]
-    exact (a n).physicalWeight_integrable (P := P) hBc hκ hκ8 _
-  have hlim₂ : Tendsto (fun n => ∫ ω in A, (a n).weight ω ∂P) atTop
-      (𝓝 (∫ ω in A, terminalGreenLimit a ω ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence
-      (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-      (fun n => (hwi n).integrableOn.aestronglyMeasurable) (integrable_const _) _
-      (Eventually.of_forall (localization_weight_tendsto_limit a hε hεlim hκ hκ8))
-    intro n
-    filter_upwards [] with ω
-    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).weight_bound hκ hκ8 ω).1]
-    exact ((a n).weight_bound hκ hκ8 ω).2
-  apply tendsto_nhds_unique hlim₁
-  convert hlim₂ using 1
-  ext n
-  rw [← hw n]
-  exact ((a n).physicalWeight_martingale hB hBm hBc hB0 hκ hκ8).setIntegral_eq (le_max_left _ _) hA
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
-def naturalNatFiltration (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) : Filtration ℕ mΩ where
-  seq n := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) n
-  mono' := fun n m h => (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono (by exact_mod_cast h)
-  le' n := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le n
-
-lemma terminalGreenLimit_measurable_at_infinity
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) :
-    StronglyMeasurable[⨆ n, naturalNatFiltration B hm n] (terminalGreenLimit a) := by
-  have hs (n : ℕ) : StronglyMeasurable[⨆ n, naturalNatFiltration B hm n] ((a n).weight) := by
-    obtain ⟨m, hm'⟩ := exists_nat_ge (a n).T
-    have hme := ((a n).physicalWeight_adapted hBc (a n).T).mono
-      (((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono hm').trans
-        (show naturalNatFiltration B hm m ≤ ⨆ n, naturalNatFiltration B hm n from le_iSup _ m))
-    convert hme using 1
-    ext ω
-    simp only [AngularLocalization.physicalWeight, AngularLocalization.weight, ← NNReal.coe_min,
-      min_eq_right ((a n).bounded ω)]
-  let : MeasurableSpace Ω := ⨆ n, naturalNatFiltration B hm n
-  exact StronglyMeasurable.limUnder hs
-
-theorem physicalGreenLimit_tendsto_at_infinity
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => physicalGreenLimit a n ω) atTop (𝓝 (terminalGreenLimit a ω)) := by
-  let _ := hB.isGaussianProcess.isProbabilityMeasure
-  have hi := terminalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8
-  have ht := hi.tendsto_ae_condExp (ℱ := naturalNatFiltration B hm)
-    (terminalGreenLimit_measurable_at_infinity a hBc)
-  have he : ∀ᵐ ω ∂P, ∀ n : ℕ, physicalGreenLimit a n ω =
-      P[terminalGreenLimit a | naturalNatFiltration B hm n] ω := by
-    rw [ae_all_iff]
-    intro n
-    exact physicalGreenLimit_condExp_terminal a hε hεlim hB hBm hBc hB0 hκ hκ8 n
-  filter_upwards [ht, he] with ω hω heω
-  simpa only [heω] using hω
-
-lemma terminalGreenLimit_eq_source
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hεlim : Tendsto ε atTop (𝓝 0))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) γ G F) :
-    terminalGreenLimit a ω = stoppedGreen κ (fun s => Real.sqrt κ * B s ω) γ G z u :=
-  (localization_weight_tendsto a hεlim hBc hκ hκ8 hz ω γ G F hw).limUnder_eq
-
- 
-
-theorem killedGreen_tendsto_zero_of_no_radius
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) :
-    ∀ᵐ ω ∂P, radiusLevelTime (γ ω) (G ω) z u = ∞ →
-      Tendsto (fun n : ℕ => greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) n z)
-        atTop (𝓝 0) := by
-  filter_upwards [hw, physicalGreenLimit_tendsto_at_infinity a hε hεlim hB hBm hBc hB0 hκ hκ8]
-    with ω hω ht
-  intro hn
-  have ht' : terminalGreenLimit a ω = 0 := by
-    rw [terminalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω]
-    simp only [stoppedGreen, hn, lt_irrefl, ite_false]
-  rw [ht'] at ht
-  convert ht using 1
-  ext n
-  rw [physicalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω n,
-    physicalStoppedGreen, hn, ite_eq_right (by exact not_le.mpr (ENNReal.coe_lt_top (r := (n : ℝ≥0))))]
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem exists_physical_green_martingale_regular
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
-    let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-    ∃ (M : ℝ≥0 → Ω → ℝ) (Z : Ω → ℝ),
-      Martingale M (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) P ∧
-      Integrable Z P ∧
-      (∀ t ω, 0 ≤ M t ω ∧ M t ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))) ∧
-      (∀ ω, 0 ≤ Z ω ∧ Z ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))) ∧
-      (∀ t, M t =ᵐ[P] P[Z | (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t]) ∧
-      (∀ᵐ ω ∂P, (∀ t, M t ω = physicalStoppedGreen κ (fun s => Real.sqrt κ * B s ω)
-          (γ ω) (G ω) z u t) ∧
-        Z ω = stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u) ∧
-      (∀ᵐ ω ∂P, radiusLevelTime (γ ω) (G ω) z u = ∞ →
-        Tendsto (fun n : ℕ => greenDensity κ (fun s => Real.sqrt κ * B s ω)
-          (γ ω) (G ω) n z) atTop (𝓝 0)) := by
-  dsimp only
-  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
-  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
-  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
-  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
-  have hεA (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
-  have hεlim : Tendsto ε atTop (𝓝 0) := by
-    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
-    simpa only [mul_one_div, mul_zero] using hh
-  have hεanti : Antitone ε := by
-    intro n m hnm
-    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
-  let a (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
-    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
-      hu (hε0 n) (hεA n))
-  refine ⟨physicalGreenLimit a, terminalGreenLimit a,
-    physicalGreenLimit_martingale a hεanti hεlim hB hBm hBc hB0 hκ hκ8,
-    terminalGreenLimit_integrable a hεanti hεlim hBc hκ hκ8,
-    physicalGreenLimit_bound a hεanti hεlim hκ hκ8,
-    terminalGreenLimit_bound a hεanti hεlim hκ hκ8,
-    physicalGreenLimit_condExp_terminal a hεanti hεlim hB hBm hBc hB0 hκ hκ8, ?_,
-    killedGreen_tendsto_zero_of_no_radius a hεanti hεlim hB hBm hBc hB0 hκ hκ8 hz γ G F hw⟩
-  filter_upwards [hw] with ω hω
-  exact ⟨fun t => physicalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω t,
-    terminalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω⟩
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ}
-
-lemma physicalWeight_continuous (a : AngularLocalization B hm κ z u ε)
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
-    Continuous (fun t => a.physicalWeight t ω) := by
-  have hc : Continuous (fun t : ℝ≥0 => min t (a.τ ω)) := continuous_id.min continuous_const
-  exact (greenLogDensity_contDiff κ).continuous.comp
-    ((((a.solution.lipschitz ω).continuous.comp NNReal.continuous_coe).comp hc).prodMk
-      ((hBc ω).comp hc))
-
-lemma physicalWeight_stopping_measurable (a : AngularLocalization B hm κ z u ε)
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0))) :
-    Measurable[hσ.measurableSpace] (fun ω => a.physicalWeight (σ ω) ω) := by
-  have hp := (a.physicalWeight_adapted hBc).isStronglyProgressive_of_continuous
-    (a.physicalWeight_continuous hBc)
-  exact measurable_stoppedValue hp hσ
-
-lemma physicalWeight_stopping_integrable (a : AngularLocalization B hm κ z u ε)
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0))) :
-    Integrable (fun ω => a.physicalWeight (σ ω) ω) P := by
-  apply Integrable.of_bound
-    ((a.physicalWeight_stopping_measurable hBc σ hσ).mono hσ.measurableSpace_le le_rfl).aestronglyMeasurable
-    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-  filter_upwards [] with ω
-  rw [Real.norm_eq_abs, abs_of_nonneg (a.physicalWeight_bound hκ hκ8 (σ ω) ω).1]
-  exact (a.physicalWeight_bound hκ hκ8 (σ ω) ω).2
-
- 
-
-lemma integral_physicalWeight_stopping (a : AngularLocalization B hm κ z u ε)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0))) :
-    (∫ ω, a.physicalWeight (σ ω) ω ∂P) = greenLogDensity κ (initialLogCoordinates z) 0 := by
-  let ρ (ω : Ω) : ℝ≥0 := min (σ ω) (a.τ ω)
-  have hρ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (ρ ω : WithTop ℝ≥0)) := by
-    simpa only [ρ, WithTop.coe_min] using hσ.min a.stopping
-  have hin (ω : Ω) (v : ℝ≥0) (hv : v ≤ ρ ω) :
-      logRadiusClock (Real.log z.im) (a.solution.Q v ω) ≤ u ∧
-        a.lo ≤ (a.solution.Q v ω).1.im ∧ (a.solution.Q v ω).1.im ≤ a.hi := by
-    have hh := a.inside ω v (hv.trans (min_le_right _ _))
-    exact ⟨hh.1, hh.2.2⟩
-  have hh := (a.solution.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
-    (Real.log z.im) u ρ hρ a.T (fun ω => (min_le_right _ _).trans (a.bounded ω)) hin).2
-  simpa only [ρ, physicalWeight, NNReal.coe_min] using hh
-
-lemma physicalWeight_setIntegral_stopping (a : AngularLocalization B hm κ z u ε)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (σ τ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hστ : ∀ ω, σ ω ≤ τ ω) {A : Set Ω} (hA : MeasurableSet[hσ.measurableSpace] A) :
-    (∫ ω in A, a.physicalWeight (σ ω) ω ∂P) =
-      ∫ ω in A, a.physicalWeight (τ ω) ω ∂P := by
-  let _ := hB.isGaussianProcess.isProbabilityMeasure
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let ρ (ω : Ω) : ℝ≥0 := if ω ∈ A then τ ω else σ ω
-  have hρ : IsStoppingTime f (fun ω => (ρ ω : WithTop ℝ≥0)) :=
-    stopping_paste_after f hσ hτ hστ hA
-  have hiσ := a.physicalWeight_stopping_integrable (P := P) hBc hκ hκ8 σ hσ
-  have hiρ := a.physicalWeight_stopping_integrable (P := P) hBc hκ hκ8 ρ hρ
-  have hiτ := a.physicalWeight_stopping_integrable (P := P) hBc hκ hκ8 τ hτ
-  have he (ω : Ω) : a.physicalWeight (ρ ω) ω - a.physicalWeight (σ ω) ω =
-      A.indicator (fun ω => a.physicalWeight (τ ω) ω - a.physicalWeight (σ ω) ω) ω := by
-    by_cases hω : ω ∈ A <;> simp [ρ, hω]
-  have heI : (∫ ω, A.indicator (fun ω => a.physicalWeight (τ ω) ω -
-      a.physicalWeight (σ ω) ω) ω ∂P) = 0 := by
-    simp_rw [← he]
-    rw [integral_sub hiρ hiσ,
-      a.integral_physicalWeight_stopping hB hBm hBc hB0 hκ hκ8 ρ hρ,
-      a.integral_physicalWeight_stopping hB hBm hBc hB0 hκ hκ8 σ hσ, sub_self]
-  rw [integral_indicator (hσ.measurableSpace_le _ hA),
-    integral_sub hiτ.integrableOn hiσ.integrableOn] at heI
-  exact (sub_eq_zero.mp heI).symm
-
-end AngularLocalization
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
-lemma physicalGreenLimit_stopping_measurable
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0))) :
-    StronglyMeasurable[hσ.measurableSpace] (fun ω => physicalGreenLimit a (σ ω) ω) := by
-  have hs (n : ℕ) : StronglyMeasurable[hσ.measurableSpace]
-      (fun ω => (a n).physicalWeight (σ ω) ω) :=
-    ((a n).physicalWeight_stopping_measurable hBc σ hσ).stronglyMeasurable
-  let : MeasurableSpace Ω := hσ.measurableSpace
-  exact StronglyMeasurable.limUnder hs
-
-lemma physicalGreenLimit_stopping_integrable
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0))) :
-    Integrable (fun ω => physicalGreenLimit a (σ ω) ω) P := by
-  apply Integrable.of_bound
-    ((physicalGreenLimit_stopping_measurable a hBc σ hσ).mono hσ.measurableSpace_le).aestronglyMeasurable
-    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-  filter_upwards [] with ω
-  rw [Real.norm_eq_abs, abs_of_nonneg (physicalGreenLimit_bound a hε hεlim hκ hκ8 (σ ω) ω).1]
-  exact (physicalGreenLimit_bound a hε hεlim hκ hκ8 (σ ω) ω).2
-
- 
-
-end SLEExactGauge
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.KernelConsistency
+
+namespace OAI
+
+/-! Angular Moments for chordal SLE. -/
+
+section
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+noncomputable def physicalGreenLimit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (t : ℝ≥0) (ω : Ω) : ℝ :=
+  limUnder atTop (fun n => (a n).physicalWeight t ω)
+
+lemma physicalWeight_exists_limit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) (ω : Ω) :
+    ∃ l : ℝ, Tendsto (fun n => (a n).physicalWeight t ω) atTop (𝓝 l) := by
+  by_cases ht : ∃ n, t ≤ (a n).τ ω
+  · obtain ⟨n, hn⟩ := ht
+    refine ⟨(a n).physicalWeight t ω, tendsto_const_nhds.congr' ?_⟩
+    filter_upwards [eventually_ge_atTop n] with m hnm
+    have hm' := hn.trans ((a n).stopping_le (a m) le_rfl (hε hnm) ω)
+    simp only [AngularLocalization.physicalWeight, ← NNReal.coe_min, min_eq_left hn, min_eq_left hm']
+    rw [(a n).synchronize (a m) ω t hn hm']
+  · by_cases hh : ∃ n, ω ∈ (a n).hit
+    · obtain ⟨n, hn⟩ := hh
+      refine ⟨(a n).physicalWeight t ω, tendsto_const_nhds.congr' ?_⟩
+      filter_upwards [eventually_ge_atTop n] with m hnm
+      have heτ : (a n).τ ω = (a m).τ ω := le_antisymm
+        ((a n).stopping_le (a m) le_rfl (hε hnm) ω)
+        ((a m).stopping_le_of_hit (a n) le_rfl ω hn)
+      have hnt : (a n).τ ω ≤ t := (lt_of_not_ge (fun h => ht ⟨n, h⟩)).le
+      have hmt : (a m).τ ω ≤ t := heτ ▸ hnt
+      simp only [AngularLocalization.physicalWeight, ← NNReal.coe_min, min_eq_right hnt, min_eq_right hmt]
+      rw [(a n).synchronize (a m) ω ((a n).τ ω) le_rfl heτ.le, heτ]
+    · refine ⟨0, ?_⟩
+      have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
+      let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
+      have hpow : Tendsto (fun n => K * ε n ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
+        have hx := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
+        simpa only [Real.zero_rpow hp.ne', mul_zero, Function.comp_def] using hx
+      apply squeeze_zero (fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).1) _ hpow
+      intro n
+      have hnt : (a n).τ ω ≤ t := (lt_of_not_ge (fun h => ht ⟨n, h⟩)).le
+      change greenLogDensity κ _ _ ≤ _
+      rw [← NNReal.coe_min, min_eq_right hnt]
+      exact (a n).weight_small_of_not_hit hκ hκ8 ω (fun hn => hh ⟨n, hn⟩)
+
+lemma physicalWeight_tendsto_limit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) (ω : Ω) :
+    Tendsto (fun n => (a n).physicalWeight t ω) atTop (𝓝 (physicalGreenLimit a t ω)) := by
+  obtain ⟨l, hl⟩ := physicalWeight_exists_limit a hε hεlim hκ hκ8 t ω
+  simpa only [physicalGreenLimit, hl.limUnder_eq] using hl
+
+lemma physicalGreenLimit_bound
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) (ω : Ω) :
+    0 ≤ physicalGreenLimit a t ω ∧ physicalGreenLimit a t ω ≤
+      Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) := by
+  have ht := physicalWeight_tendsto_limit a hε hεlim hκ hκ8 t ω
+  exact ⟨ge_of_tendsto ht (Eventually.of_forall fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).1),
+    le_of_tendsto ht (Eventually.of_forall fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).2)⟩
+
+lemma physicalGreenLimit_adapted
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) :
+    StronglyAdapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) (physicalGreenLimit a) := by
+  intro t
+  have hs (n : ℕ) := (a n).physicalWeight_adapted hBc t
+  let : MeasurableSpace Ω := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t
+  exact StronglyMeasurable.limUnder hs
+
+lemma physicalGreenLimit_integrable
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} [IsFiniteMeasure P] (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) : Integrable (physicalGreenLimit a t) P := by
+  apply Integrable.of_bound ((physicalGreenLimit_adapted a hBc t).mono
+    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le t)).aestronglyMeasurable
+    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+  filter_upwards [] with ω
+  rw [Real.norm_eq_abs, abs_of_nonneg (physicalGreenLimit_bound a hε hεlim hκ hκ8 t ω).1]
+  exact (physicalGreenLimit_bound a hε hεlim hκ hκ8 t ω).2
+
+ 
+
+theorem physicalGreenLimit_martingale
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) :
+    Martingale (physicalGreenLimit a) (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) P := by
+  let _ := hB.isGaussianProcess.isProbabilityMeasure
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have had := physicalGreenLimit_adapted a hBc
+  have hi := physicalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8
+  have hmar (n : ℕ) := (a n).physicalWeight_martingale hB hBm hBc hB0 hκ hκ8
+  refine ⟨had, fun s t hst => ?_⟩
+  apply EventuallyEq.symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq (f.le s) (hi t)
+    (fun _ _ _ => (hi s).integrableOn) _ (had s).aestronglyMeasurable
+  intro A hA _
+  have hlim (v : ℝ≥0) : Tendsto (fun n => ∫ ω in A, (a n).physicalWeight v ω ∂P) atTop
+      (𝓝 (∫ ω in A, physicalGreenLimit a v ω ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence
+      (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+      (fun n => ((a n).physicalWeight_integrable (P := P) hBc hκ hκ8 v).integrableOn.aestronglyMeasurable)
+      (integrable_const _) _ (Eventually.of_forall (physicalWeight_tendsto_limit a hε hεlim hκ hκ8 v))
+    intro n
+    filter_upwards [] with ω
+    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).physicalWeight_bound hκ hκ8 v ω).1]
+    exact ((a n).physicalWeight_bound hκ hκ8 v ω).2
+  apply tendsto_nhds_unique (hlim s)
+  convert hlim t using 1
+  ext n
+  exact (hmar n).setIntegral_eq hst hA
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+noncomputable def physicalStoppedGreen (κ : ℝ) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (u : ℝ) (t : ℝ≥0) : ℝ :=
+  if radiusLevelTime γ G z u ≤ (t : ℝ≥0∞) then stoppedGreen κ U γ G z u
+    else greenDensity κ U γ G t z
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+lemma physicalWeight_tendsto_source
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hεlim : Tendsto ε atTop (𝓝 0))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) γ G F) (t : ℝ≥0) :
+    Tendsto (fun n => (a n).physicalWeight t ω) atTop
+      (𝓝 (physicalStoppedGreen κ (fun s => Real.sqrt κ * B s ω) γ G z u t)) := by
+  have hU := (hBc ω).const_mul (Real.sqrt κ)
+  have hρ := localizedRadiusStop_eq_source a hεlim hBc hz ω γ G F hw
+  by_cases hrt : radiusLevelTime γ G z u ≤ (t : ℝ≥0∞)
+  · rw [physicalStoppedGreen, ite_eq_left hrt]
+    have hρt : localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0) := by
+      rw [hρ]
+      exact hrt
+    obtain ⟨n, hn, hnt⟩ := (localizedRadiusStop_le_iff a ω t).mp hρt
+    obtain ⟨hl, hR⟩ := (a n).radius_at_hit hBc hz ω γ G F hw hn
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_hit_of_live_level a hεlim hBc hz ω γ G F hw ((a n).τ ω) hl hR]
+      with m hm'
+    have ht' : (a m).τ ω ≤ t := hm'.2.le.trans hnt
+    simpa only [AngularLocalization.physicalWeight, ← NNReal.coe_min, min_eq_right ht', AngularLocalization.weight]
+      using ((a m).weight_eq_stoppedGreen_of_hit hBc hz ω γ G F hw hm'.1).symm
+  · rw [physicalStoppedGreen, ite_eq_right hrt]
+    have hno (n : ℕ) (hnt : (a n).τ ω ≤ t) : ω ∉ (a n).hit := by
+      intro hn
+      apply hrt
+      have hcan := (localizedRadiusStop_le_iff a ω t).mpr ⟨n, hn, hnt⟩
+      rw [hρ] at hcan
+      exact hcan
+    by_cases hl : z ∈ survivingDomain γ t
+    · obtain ⟨δ, hδ, hδb⟩ := live_angleSq_lower_bound hU hw z t hl
+      apply tendsto_const_nhds.congr'
+      filter_upwards [(tendsto_order.mp hεlim).2 δ hδ] with n hn
+      have hbefore : t ≤ (a n).τ ω := by
+        by_contra hnt
+        have hnt' := lt_of_not_ge hnt
+        have hexit := ((a n).boundary ω).resolve_left (hno n hnt'.le)
+        obtain ⟨hla, heq⟩ := (a n).chart_identification hBc hz ω γ G F hw
+        have ha : logAngleSq κ ((a n).solution.Q ((a n).τ ω) ω) (B ((a n).τ ω) ω) =
+          (Real.sin (Complex.arg (G ((a n).τ ω) z - ((Real.sqrt κ * B ((a n).τ ω) ω : ℝ) : ℂ)))) ^ 2 := by
+          rw [heq _ le_rfl]
+          exact logAngleSq_logCoordinates κ _ G z _ (hw.imag_positive _ z hla)
+        have hb := hδb ((a n).τ ω) hnt'.le
+        rw [← ha, hexit] at hb
+        exact (not_lt_of_ge hb) hn
+      dsimp only [AngularLocalization.physicalWeight]
+      rw [← NNReal.coe_min, min_eq_left hbefore, ((a n).chart_identification hBc hz ω γ G F hw).2 t hbefore]
+      exact (greenLogDensity_logCoordinates hU hw κ _ z t hl rfl).symm
+    · have hnt (n : ℕ) : (a n).τ ω < t := by
+        apply lt_of_not_ge
+        intro hn
+        exact hl (survivingDomain_antitone γ hn ((a n).chart_identification hBc hz ω γ G F hw).1)
+      rw [greenDensity, ite_eq_right hl]
+      have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
+      let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
+      have hpow : Tendsto (fun n => K * ε n ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
+        have hx := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
+        simpa only [Real.zero_rpow hp.ne', mul_zero, Function.comp_def] using hx
+      apply squeeze_zero (fun n => ((a n).physicalWeight_bound hκ hκ8 t ω).1) _ hpow
+      intro n
+      change greenLogDensity κ _ _ ≤ _
+      rw [← NNReal.coe_min, min_eq_right (hnt n).le]
+      exact (a n).weight_small_of_not_hit hκ hκ8 ω (hno n (hnt n).le)
+
+lemma physicalGreenLimit_eq_source
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hεlim : Tendsto ε atTop (𝓝 0))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) γ G F) (t : ℝ≥0) :
+    physicalGreenLimit a t ω = physicalStoppedGreen κ (fun s => Real.sqrt κ * B s ω) γ G z u t :=
+  (physicalWeight_tendsto_source a hεlim hBc hκ hκ8 hz ω γ G F hw t).limUnder_eq
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+noncomputable def terminalGreenLimit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) : ℝ :=
+  limUnder atTop (fun n => (a n).weight ω)
+
+lemma localization_weight_tendsto_limit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hκ : 0 < κ) (hκ8 : κ < 8) (ω : Ω) :
+    Tendsto (fun n => (a n).weight ω) atTop (𝓝 (terminalGreenLimit a ω)) := by
+  suffices ∃ l : ℝ, Tendsto (fun n => (a n).weight ω) atTop (𝓝 l) by
+    obtain ⟨l, hl⟩ := this
+    simpa only [terminalGreenLimit, hl.limUnder_eq] using hl
+  by_cases hh : ∃ n, ω ∈ (a n).hit
+  · obtain ⟨n, hn⟩ := hh
+    refine ⟨(a n).weight ω, tendsto_const_nhds.congr' ?_⟩
+    filter_upwards [eventually_ge_atTop n] with m hnm
+    have ht : (a n).τ ω = (a m).τ ω := le_antisymm
+      ((a n).stopping_le (a m) le_rfl (hε hnm) ω)
+      ((a m).stopping_le_of_hit (a n) le_rfl ω hn)
+    have he := (a n).synchronize (a m) ω ((a n).τ ω) le_rfl ht.le
+    simp only [AngularLocalization.weight]
+    rw [he, ht]
+  · refine ⟨0, ?_⟩
+    have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
+    let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
+    have ht : Tendsto (fun n => K * ε n ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
+      have hx := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
+      simpa only [Real.zero_rpow hp.ne', mul_zero, Function.comp_def] using hx
+    apply squeeze_zero (fun n => ((a n).weight_bound hκ hκ8 ω).1) _ ht
+    intro n
+    exact (a n).weight_small_of_not_hit hκ hκ8 ω (fun hn => hh ⟨n, hn⟩)
+
+lemma terminalGreenLimit_bound
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hκ : 0 < κ) (hκ8 : κ < 8) (ω : Ω) :
+    0 ≤ terminalGreenLimit a ω ∧ terminalGreenLimit a ω ≤
+      Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) := by
+  have ht := localization_weight_tendsto_limit a hε hεlim hκ hκ8 ω
+  exact ⟨ge_of_tendsto ht (Eventually.of_forall fun n => ((a n).weight_bound hκ hκ8 ω).1),
+    le_of_tendsto ht (Eventually.of_forall fun n => ((a n).weight_bound hκ hκ8 ω).2)⟩
+
+lemma terminalGreenLimit_stronglyMeasurable
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) : StronglyMeasurable (terminalGreenLimit a) := by
+  have hs (n : ℕ) := ((a n).physicalWeight_adapted hBc (a n).T).mono
+    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le (a n).T)
+  have hs' (n : ℕ) : StronglyMeasurable ((a n).weight) := by
+    convert hs n using 1
+    ext ω
+    simp only [AngularLocalization.physicalWeight, AngularLocalization.weight, ← NNReal.coe_min,
+      min_eq_right ((a n).bounded ω)]
+  exact StronglyMeasurable.limUnder hs'
+
+lemma terminalGreenLimit_integrable
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} [IsFiniteMeasure P] (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) : Integrable (terminalGreenLimit a) P := by
+  apply Integrable.of_bound (terminalGreenLimit_stronglyMeasurable a hBc).aestronglyMeasurable
+    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+  filter_upwards [] with ω
+  rw [Real.norm_eq_abs, abs_of_nonneg (terminalGreenLimit_bound a hε hεlim hκ hκ8 ω).1]
+  exact (terminalGreenLimit_bound a hε hεlim hκ hκ8 ω).2
+
+ 
+
+theorem physicalGreenLimit_condExp_terminal
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ωω, Continuous (fun t => B t ωω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (t : ℝ≥0) :
+    physicalGreenLimit a t =ᵐ[P]
+      P[terminalGreenLimit a | (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t] := by
+  let _ := hB.isGaussianProcess.isProbabilityMeasure
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have hi := terminalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8
+  apply ae_eq_condExp_of_forall_setIntegral_eq (f.le t) hi
+    (fun _ _ _ => (physicalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8 t).integrableOn)
+    _ (physicalGreenLimit_adapted a hBc t).aestronglyMeasurable
+  intro A hA _
+  have hlim₁ : Tendsto (fun n => ∫ ω in A, (a n).physicalWeight t ω ∂P) atTop
+      (𝓝 (∫ ω in A, physicalGreenLimit a t ω ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence
+      (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+      (fun n => ((a n).physicalWeight_integrable (P := P) hBc hκ hκ8 t).integrableOn.aestronglyMeasurable)
+      (integrable_const _) _ (Eventually.of_forall (physicalWeight_tendsto_limit a hε hεlim hκ hκ8 t))
+    intro n
+    filter_upwards [] with ω
+    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).physicalWeight_bound hκ hκ8 t ω).1]
+    exact ((a n).physicalWeight_bound hκ hκ8 t ω).2
+  have hw (n : ℕ) : (a n).physicalWeight (max t (a n).T) = (a n).weight := by
+    ext ω
+    simp only [AngularLocalization.physicalWeight, AngularLocalization.weight, ← NNReal.coe_min,
+      min_eq_right (((a n).bounded ω).trans (le_max_right _ _))]
+  have hwi (n : ℕ) : Integrable ((a n).weight) P := by
+    rw [← hw n]
+    exact (a n).physicalWeight_integrable (P := P) hBc hκ hκ8 _
+  have hlim₂ : Tendsto (fun n => ∫ ω in A, (a n).weight ω ∂P) atTop
+      (𝓝 (∫ ω in A, terminalGreenLimit a ω ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence
+      (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+      (fun n => (hwi n).integrableOn.aestronglyMeasurable) (integrable_const _) _
+      (Eventually.of_forall (localization_weight_tendsto_limit a hε hεlim hκ hκ8))
+    intro n
+    filter_upwards [] with ω
+    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).weight_bound hκ hκ8 ω).1]
+    exact ((a n).weight_bound hκ hκ8 ω).2
+  apply tendsto_nhds_unique hlim₁
+  convert hlim₂ using 1
+  ext n
+  rw [← hw n]
+  exact ((a n).physicalWeight_martingale hB hBm hBc hB0 hκ hκ8).setIntegral_eq (le_max_left _ _) hA
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+def naturalNatFiltration (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) : Filtration ℕ mΩ where
+  seq n := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) n
+  mono' := fun n m h => (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono (by exact_mod_cast h)
+  le' n := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le n
+
+lemma terminalGreenLimit_measurable_at_infinity
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) :
+    StronglyMeasurable[⨆ n, naturalNatFiltration B hm n] (terminalGreenLimit a) := by
+  have hs (n : ℕ) : StronglyMeasurable[⨆ n, naturalNatFiltration B hm n] ((a n).weight) := by
+    obtain ⟨m, hm'⟩ := exists_nat_ge (a n).T
+    have hme := ((a n).physicalWeight_adapted hBc (a n).T).mono
+      (((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono hm').trans
+        (show naturalNatFiltration B hm m ≤ ⨆ n, naturalNatFiltration B hm n from le_iSup _ m))
+    convert hme using 1
+    ext ω
+    simp only [AngularLocalization.physicalWeight, AngularLocalization.weight, ← NNReal.coe_min,
+      min_eq_right ((a n).bounded ω)]
+  let : MeasurableSpace Ω := ⨆ n, naturalNatFiltration B hm n
+  exact StronglyMeasurable.limUnder hs
+
+theorem physicalGreenLimit_tendsto_at_infinity
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => physicalGreenLimit a n ω) atTop (𝓝 (terminalGreenLimit a ω)) := by
+  let _ := hB.isGaussianProcess.isProbabilityMeasure
+  have hi := terminalGreenLimit_integrable a hε hεlim (P := P) hBc hκ hκ8
+  have ht := hi.tendsto_ae_condExp (ℱ := naturalNatFiltration B hm)
+    (terminalGreenLimit_measurable_at_infinity a hBc)
+  have he : ∀ᵐ ω ∂P, ∀ n : ℕ, physicalGreenLimit a n ω =
+      P[terminalGreenLimit a | naturalNatFiltration B hm n] ω := by
+    rw [ae_all_iff]
+    intro n
+    exact physicalGreenLimit_condExp_terminal a hε hεlim hB hBm hBc hB0 hκ hκ8 n
+  filter_upwards [ht, he] with ω hω heω
+  simpa only [heω] using hω
+
+lemma terminalGreenLimit_eq_source
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hεlim : Tendsto ε atTop (𝓝 0))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) γ G F) :
+    terminalGreenLimit a ω = stoppedGreen κ (fun s => Real.sqrt κ * B s ω) γ G z u :=
+  (localization_weight_tendsto a hεlim hBc hκ hκ8 hz ω γ G F hw).limUnder_eq
+
+ 
+
+theorem killedGreen_tendsto_zero_of_no_radius
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) :
+    ∀ᵐ ω ∂P, radiusLevelTime (γ ω) (G ω) z u = ∞ →
+      Tendsto (fun n : ℕ => greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) n z)
+        atTop (𝓝 0) := by
+  filter_upwards [hw, physicalGreenLimit_tendsto_at_infinity a hε hεlim hB hBm hBc hB0 hκ hκ8]
+    with ω hω ht
+  intro hn
+  have ht' : terminalGreenLimit a ω = 0 := by
+    rw [terminalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω]
+    simp only [stoppedGreen, hn, lt_irrefl, ite_false]
+  rw [ht'] at ht
+  convert ht using 1
+  ext n
+  rw [physicalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω n,
+    physicalStoppedGreen, hn, ite_eq_right (by exact not_le.mpr (ENNReal.coe_lt_top (r := (n : ℝ≥0))))]
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem exists_physical_green_martingale_regular
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
+    let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+    ∃ (M : ℝ≥0 → Ω → ℝ) (Z : Ω → ℝ),
+      Martingale M (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) P ∧
+      Integrable Z P ∧
+      (∀ t ω, 0 ≤ M t ω ∧ M t ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))) ∧
+      (∀ ω, 0 ≤ Z ω ∧ Z ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))) ∧
+      (∀ t, M t =ᵐ[P] P[Z | (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t]) ∧
+      (∀ᵐ ω ∂P, (∀ t, M t ω = physicalStoppedGreen κ (fun s => Real.sqrt κ * B s ω)
+          (γ ω) (G ω) z u t) ∧
+        Z ω = stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u) ∧
+      (∀ᵐ ω ∂P, radiusLevelTime (γ ω) (G ω) z u = ∞ →
+        Tendsto (fun n : ℕ => greenDensity κ (fun s => Real.sqrt κ * B s ω)
+          (γ ω) (G ω) n z) atTop (𝓝 0)) := by
+  dsimp only
+  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
+  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
+  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
+  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
+  have hεA (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
+  have hεlim : Tendsto ε atTop (𝓝 0) := by
+    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
+    simpa only [mul_one_div, mul_zero] using hh
+  have hεanti : Antitone ε := by
+    intro n m hnm
+    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
+  let a (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
+    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
+      hu (hε0 n) (hεA n))
+  refine ⟨physicalGreenLimit a, terminalGreenLimit a,
+    physicalGreenLimit_martingale a hεanti hεlim hB hBm hBc hB0 hκ hκ8,
+    terminalGreenLimit_integrable a hεanti hεlim hBc hκ hκ8,
+    physicalGreenLimit_bound a hεanti hεlim hκ hκ8,
+    terminalGreenLimit_bound a hεanti hεlim hκ hκ8,
+    physicalGreenLimit_condExp_terminal a hεanti hεlim hB hBm hBc hB0 hκ hκ8, ?_,
+    killedGreen_tendsto_zero_of_no_radius a hεanti hεlim hB hBm hBc hB0 hκ hκ8 hz γ G F hw⟩
+  filter_upwards [hw] with ω hω
+  exact ⟨fun t => physicalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω t,
+    terminalGreenLimit_eq_source a hεlim hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω⟩
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ}
+
+lemma physicalWeight_continuous (a : AngularLocalization B hm κ z u ε)
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
+    Continuous (fun t => a.physicalWeight t ω) := by
+  have hc : Continuous (fun t : ℝ≥0 => min t (a.τ ω)) := continuous_id.min continuous_const
+  exact (greenLogDensity_contDiff κ).continuous.comp
+    ((((a.solution.lipschitz ω).continuous.comp NNReal.continuous_coe).comp hc).prodMk
+      ((hBc ω).comp hc))
+
+lemma physicalWeight_stopping_measurable (a : AngularLocalization B hm κ z u ε)
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0))) :
+    Measurable[hσ.measurableSpace] (fun ω => a.physicalWeight (σ ω) ω) := by
+  have hp := (a.physicalWeight_adapted hBc).isStronglyProgressive_of_continuous
+    (a.physicalWeight_continuous hBc)
+  exact measurable_stoppedValue hp hσ
+
+lemma physicalWeight_stopping_integrable (a : AngularLocalization B hm κ z u ε)
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0))) :
+    Integrable (fun ω => a.physicalWeight (σ ω) ω) P := by
+  apply Integrable.of_bound
+    ((a.physicalWeight_stopping_measurable hBc σ hσ).mono hσ.measurableSpace_le le_rfl).aestronglyMeasurable
+    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+  filter_upwards [] with ω
+  rw [Real.norm_eq_abs, abs_of_nonneg (a.physicalWeight_bound hκ hκ8 (σ ω) ω).1]
+  exact (a.physicalWeight_bound hκ hκ8 (σ ω) ω).2
+
+ 
+
+lemma integral_physicalWeight_stopping (a : AngularLocalization B hm κ z u ε)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0))) :
+    (∫ ω, a.physicalWeight (σ ω) ω ∂P) = greenLogDensity κ (initialLogCoordinates z) 0 := by
+  let ρ (ω : Ω) : ℝ≥0 := min (σ ω) (a.τ ω)
+  have hρ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (ρ ω : WithTop ℝ≥0)) := by
+    simpa only [ρ, WithTop.coe_min] using hσ.min a.stopping
+  have hin (ω : Ω) (v : ℝ≥0) (hv : v ≤ ρ ω) :
+      logRadiusClock (Real.log z.im) (a.solution.Q v ω) ≤ u ∧
+        a.lo ≤ (a.solution.Q v ω).1.im ∧ (a.solution.Q v ω).1.im ≤ a.hi := by
+    have hh := a.inside ω v (hv.trans (min_le_right _ _))
+    exact ⟨hh.1, hh.2.2⟩
+  have hh := (a.solution.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
+    (Real.log z.im) u ρ hρ a.T (fun ω => (min_le_right _ _).trans (a.bounded ω)) hin).2
+  simpa only [ρ, physicalWeight, NNReal.coe_min] using hh
+
+lemma physicalWeight_setIntegral_stopping (a : AngularLocalization B hm κ z u ε)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (σ τ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hστ : ∀ ω, σ ω ≤ τ ω) {A : Set Ω} (hA : MeasurableSet[hσ.measurableSpace] A) :
+    (∫ ω in A, a.physicalWeight (σ ω) ω ∂P) =
+      ∫ ω in A, a.physicalWeight (τ ω) ω ∂P := by
+  let _ := hB.isGaussianProcess.isProbabilityMeasure
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let ρ (ω : Ω) : ℝ≥0 := if ω ∈ A then τ ω else σ ω
+  have hρ : IsStoppingTime f (fun ω => (ρ ω : WithTop ℝ≥0)) :=
+    stopping_paste_after f hσ hτ hστ hA
+  have hiσ := a.physicalWeight_stopping_integrable (P := P) hBc hκ hκ8 σ hσ
+  have hiρ := a.physicalWeight_stopping_integrable (P := P) hBc hκ hκ8 ρ hρ
+  have hiτ := a.physicalWeight_stopping_integrable (P := P) hBc hκ hκ8 τ hτ
+  have he (ω : Ω) : a.physicalWeight (ρ ω) ω - a.physicalWeight (σ ω) ω =
+      A.indicator (fun ω => a.physicalWeight (τ ω) ω - a.physicalWeight (σ ω) ω) ω := by
+    by_cases hω : ω ∈ A <;> simp [ρ, hω]
+  have heI : (∫ ω, A.indicator (fun ω => a.physicalWeight (τ ω) ω -
+      a.physicalWeight (σ ω) ω) ω ∂P) = 0 := by
+    simp_rw [← he]
+    rw [integral_sub hiρ hiσ,
+      a.integral_physicalWeight_stopping hB hBm hBc hB0 hκ hκ8 ρ hρ,
+      a.integral_physicalWeight_stopping hB hBm hBc hB0 hκ hκ8 σ hσ, sub_self]
+  rw [integral_indicator (hσ.measurableSpace_le _ hA),
+    integral_sub hiτ.integrableOn hiσ.integrableOn] at heI
+  exact (sub_eq_zero.mp heI).symm
+
+end AngularLocalization
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+lemma physicalGreenLimit_stopping_measurable
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0))) :
+    StronglyMeasurable[hσ.measurableSpace] (fun ω => physicalGreenLimit a (σ ω) ω) := by
+  have hs (n : ℕ) : StronglyMeasurable[hσ.measurableSpace]
+      (fun ω => (a n).physicalWeight (σ ω) ω) :=
+    ((a n).physicalWeight_stopping_measurable hBc σ hσ).stronglyMeasurable
+  let : MeasurableSpace Ω := hσ.measurableSpace
+  exact StronglyMeasurable.limUnder hs
+
+lemma physicalGreenLimit_stopping_integrable
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0))) :
+    Integrable (fun ω => physicalGreenLimit a (σ ω) ω) P := by
+  apply Integrable.of_bound
+    ((physicalGreenLimit_stopping_measurable a hBc σ hσ).mono hσ.measurableSpace_le).aestronglyMeasurable
+    (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+  filter_upwards [] with ω
+  rw [Real.norm_eq_abs, abs_of_nonneg (physicalGreenLimit_bound a hε hεlim hκ hκ8 (σ ω) ω).1]
+  exact (physicalGreenLimit_bound a hε hεlim hκ hκ8 (σ ω) ω).2
+
+ 
+
+end SLEExactGauge
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/AngularPowerDynkin.lean b/lean/OAI/Probability/SLE/AngularPowerDynkin.lean
index 04846eb6a973e4b4c76b38affe68dbf5b23efba0..6c4c6a2e93992c79f166de2bfee179dd4225f564 100644
--- a/lean/OAI/Probability/SLE/AngularPowerDynkin.lean
+++ b/lean/OAI/Probability/SLE/AngularPowerDynkin.lean
@@ -1,806 +1,806 @@
-import Mathlib
-import OAI.Probability.SLE.DiskHalfPlane
-
-namespace OAI
-
-/-! Angular Power Dynkin for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma powerLogGenerator_lyapunov (κ b : ℝ) (hκ : 0 < κ)
-    (q : ℂ × ℝ) (x : ℝ) :
-    powerLogGenerator κ (codimension κ) (greenExponent κ - b) q x =
-      powerLogDensity κ (codimension κ) (greenExponent κ - b) q x *
-        logClockRate κ q x * (κ * b / 8) *
-          (1 - (greenExponent κ - b) * ((logAngleSq κ q x)⁻¹ - 1)) := by
-  unfold powerLogGenerator
-  rw [← power_physical_generator κ _ _ hκ.le]
-  exact green_weighted_power_generator κ b hκ q x
-
-lemma powerLogDensity_div_angle (κ p α : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    powerLogDensity κ p α q x / logAngleSq κ q x =
-      powerLogDensity κ p (α - 2) q x := by
-  rw [powerLogDensity_radius_angle κ p α 0, powerLogDensity_radius_angle κ p (α - 2) 0,
-    mul_div_assoc]
-  congr 1
-  rw [show (α - 2) / 2 = α / 2 - 1 by ring,
-    Real.rpow_sub (logAngleSq_pos κ q x), Real.rpow_one]
-
-lemma powerLogGenerator_two_powers (κ b : ℝ) (hκ : 0 < κ)
-    (q : ℂ × ℝ) (x : ℝ) :
-    powerLogGenerator κ (codimension κ) (greenExponent κ - b) q x =
-      logClockRate κ q x * (κ * b / 8) *
-        ((1 + (greenExponent κ - b)) *
-            powerLogDensity κ (codimension κ) (greenExponent κ - b) q x -
-          (greenExponent κ - b) *
-            powerLogDensity κ (codimension κ) (greenExponent κ - b - 2) q x) := by
-  rw [powerLogGenerator_lyapunov κ b hκ,
-    ← powerLogDensity_div_angle κ (codimension κ) (greenExponent κ - b)]
-  ring
-
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ}
-  (a : AngularLocalization B hm κ z u ε)
-
- 
-
-theorem mean_power
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 ≤ κ) (p α : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ α) :
-    Integrable (fun ω => powerLogDensity κ p α (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
-    Integrable (fun ω => ∫ t in (0 : ℝ)..(a.τ ω : ℝ),
-      powerLogGenerator κ p α (a.solution.Q t ω) (B t.toNNReal ω)) P ∧
-      (∫ ω, powerLogDensity κ p α (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) =
-        powerLogDensity κ p α (initialLogCoordinates z) 0 +
-        ∫ ω, (∫ t in (0 : ℝ)..(a.τ ω : ℝ),
-          powerLogGenerator κ p α (a.solution.Q t ω) (B t.toNNReal ω)) ∂P := by
-  let K := Real.exp (p * (u - Real.log 2 - Real.log z.im))
-  let C := K * (|(-4 * α + κ * α * (α + 1) / 2)| + |(4 * p - κ * α / 2)|) /
-    Real.exp (2 * a.lo)
-  apply a.solution.stopped_power_dynkin_bounded hB hBm hBc hB0 hκ p α a.τ a.stopping
-    a.T a.bounded (fun ω t ht => (a.inside ω t ht).2.2) K C (by dsimp [C, K]; positivity)
-  · intro ω t ht
-    rw [abs_of_pos (powerLogDensity_pos _ _ _ _ _)]
-    exact powerLogDensity_le_radius κ p α (Real.log z.im) u hp ha _ _ (a.inside ω t ht).1
-  · intro ω t ht
-    exact powerLogGenerator_le_radius_height κ p α (Real.log z.im) u a.lo hp ha _ _
-      (a.inside ω t ht).1 (a.inside ω t ht).2.2.1
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace AngularLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u v ε δ : ℝ}
-
- 
-
-lemma synchronize (a : AngularLocalization B hm κ z u ε)
-    (b : AngularLocalization B hm κ z v δ) (ω : Ω)
-    (t : ℝ≥0) (ha : t ≤ a.τ ω) (hb : t ≤ b.τ ω) :
-    a.solution.Q t ω = b.solution.Q t ω := by
-  let lo := min a.lo b.lo
-  let hi := max a.hi b.hi
-  have hle : lo ≤ hi := (min_le_left _ _).trans (a.le.trans (le_max_left _ _))
-  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (s : ℝ) := clippedLogLoewnerField lo hi hle (Real.sqrt κ * B s.toNNReal ω)
-  have hd (c : AngularLocalization B hm κ z u ε) (hc : t ≤ c.τ ω)
-      (hlo : lo ≤ c.lo) (hhi : c.hi ≤ hi) (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
-      HasDerivWithinAt (fun s => c.solution.Q s ω) (V s (c.solution.Q s ω)) (Icc 0 (t : ℝ)) s := by
-    have hsc : s.toNNReal ≤ c.τ ω := (Real.toNNReal_le_iff_le_coe.mpr hs.2).trans hc
-    have hcbar := (c.inside ω s.toNNReal hsc).2.2
-    rw [Real.coe_toNNReal _ hs.1] at hcbar
-    have hder := (c.solution.derivative ω s hs.1).mono
-      (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-    rwa [clippedLogLoewnerField_eq_physical _ _ _ _ _ hcbar,
-      ← clippedLogLoewnerField_eq_physical lo hi hle _ _ ⟨hlo.trans hcbar.1, hcbar.2.trans hhi⟩] at hder
-  have hda := hd a ha (min_le_left _ _) (le_max_left _ _)
-  have hdb (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
-      HasDerivWithinAt (fun s => b.solution.Q s ω) (V s (b.solution.Q s ω)) (Icc 0 (t : ℝ)) s := by
-    have hsb : s.toNNReal ≤ b.τ ω := (Real.toNNReal_le_iff_le_coe.mpr hs.2).trans hb
-    have hcbar := (b.inside ω s.toNNReal hsb).2.2
-    rw [Real.coe_toNNReal _ hs.1] at hcbar
-    have hder := (b.solution.derivative ω s hs.1).mono
-      (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-    rwa [clippedLogLoewnerField_eq_physical _ _ _ _ _ hcbar,
-      ← clippedLogLoewnerField_eq_physical lo hi hle _ _
-        ⟨(min_le_right _ _).trans hcbar.1, hcbar.2.trans (le_max_right _ _)⟩] at hder
-  have he : EqOn (fun s => a.solution.Q s ω) (fun s => b.solution.Q s ω) (Icc 0 (t : ℝ)) := by
-    apply ODE_solution_unique (fun s => hlip (Real.sqrt κ * B s.toNNReal ω))
-      (fun s hs => (hda s hs).continuousWithinAt)
-      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hda s ⟨hs.1, hs.2.le⟩))
-      (fun s hs => (hdb s hs).continuousWithinAt)
-      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hdb s ⟨hs.1, hs.2.le⟩))
-    exact (a.solution.initial ω).trans (b.solution.initial ω).symm
-  exact he ⟨t.coe_nonneg, le_rfl⟩
-
-lemma stopping_le (a : AngularLocalization B hm κ z u ε)
-    (b : AngularLocalization B hm κ z v δ) (huv : u ≤ v) (hδε : δ ≤ ε)
-    (ω : Ω) : a.τ ω ≤ b.τ ω := by
-  by_contra hn
-  have hlt := lt_of_not_ge hn
-  have he := a.synchronize b ω (b.τ ω) hlt.le le_rfl
-  have hh := a.before ω _ hlt
-  rw [he] at hh
-  rcases b.boundary ω with hc | hθ
-  · linarith [hh.1]
-  · linarith [hh.2]
-
-lemma hit_time_unique (a : AngularLocalization B hm κ z u ε)
-    (b : AngularLocalization B hm κ z u δ) (ω : Ω)
-    (ha : logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω) = u)
-    (hb : logRadiusClock (Real.log z.im) (b.solution.Q (b.τ ω) ω) = u) :
-    a.τ ω = b.τ ω := by
-  apply le_antisymm
-  · by_contra hn
-    have hlt := lt_of_not_ge hn
-    have he := a.synchronize b ω (b.τ ω) hlt.le le_rfl
-    have hh := (a.before ω _ hlt).1
-    rw [he, hb] at hh
-    exact lt_irrefl _ hh
-  · by_contra hn
-    have hlt := lt_of_not_ge hn
-    have he := b.synchronize a ω (a.τ ω) hlt.le le_rfl
-    have hh := (b.before ω _ hlt).1
-    rw [he, ha] at hh
-    exact lt_irrefl _ hh
-
-lemma hit_finer (a : AngularLocalization B hm κ z u ε)
-    (b : AngularLocalization B hm κ z u δ) (hδε : δ ≤ ε) (ω : Ω)
-    (ha : logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω) = u) :
-    b.τ ω = a.τ ω ∧ logRadiusClock (Real.log z.im) (b.solution.Q (b.τ ω) ω) = u := by
-  have hab := a.stopping_le b le_rfl hδε ω
-  have he : b.τ ω = a.τ ω := by
-    apply le_antisymm _ hab
-    by_contra hn
-    have hlt := lt_of_not_ge hn
-    have hq := a.synchronize b ω (a.τ ω) le_rfl hlt.le
-    have hh := (b.before ω _ hlt).1
-    rw [← hq, ha] at hh
-    exact lt_irrefl _ hh
-  refine ⟨he, ?_⟩
-  rw [he, ← a.synchronize b ω (a.τ ω) le_rfl hab]
-  exact ha
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma continuous_logClockRate (κ : ℝ) :
-    Continuous (fun p : (ℂ × ℝ) × ℝ => logClockRate κ p.1 p.2) := by
-  have hN : Continuous (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) := by
-    unfold greenLogNormSq
-    fun_prop
-  unfold logClockRate
-  exact (continuous_const.mul (Real.continuous_exp.comp (by fun_prop))).div (hN.pow 2)
-    (fun p => pow_ne_zero 2 (greenLogNormSq_pos κ p.1 p.2).ne')
-
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ}
-  (a : AngularLocalization B hm κ z u ε)
-
-lemma radius_clock_derivative (ω : Ω) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
-    HasDerivWithinAt (fun t => logRadiusClock (Real.log z.im) (a.solution.Q t ω))
-      (logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)) (Icc (0 : ℝ) (a.τ ω : ℝ)) t := by
-  have hd := (a.solution.derivative ω t ht.1).mono
-    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-  have hb := (a.inside ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)).2.2
-  rw [Real.coe_toNNReal t ht.1] at hb
-  rw [clippedLogLoewnerField_eq_physical _ _ _ _ _ hb] at hd
-  exact log_ode_clock_derivative (Real.log z.im) hd
-
-noncomputable def clock : FiniteIncreasingClock Ω where
-  S t ω := logRadiusClock (Real.log z.im) (a.solution.Q t ω)
-  T := a.τ
-  zero ω := by rw [a.solution.initial]; simp [logRadiusClock, initialLogCoordinates]
-  continuous ω := (continuous_logRadiusClock _).comp (a.solution.lipschitz ω).continuous
-  strict ω := by
-    apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc (0 : ℝ) (a.τ ω : ℝ))
-      ((continuous_logRadiusClock _).comp (a.solution.lipschitz ω).continuous).continuousOn
-    · intro t ht
-      exact (a.radius_clock_derivative ω t (interior_subset ht)).mono interior_subset
-    · intro t ht
-      exact logClockRate_pos κ (a.solution.Q t ω) (B t.toNNReal ω)
-
-lemma clock_measurable : Measurable (Function.uncurry a.clock.S) :=
-  (continuous_logRadiusClock _).measurable.comp a.solution.measurable
-
-lemma inverse_measurable : Measurable (Function.uncurry a.clock.inverse) :=
-  a.clock.measurable_inverse a.clock_measurable (stopping_time_measurable_nnreal a.stopping)
-
-lemma clock_upper (ω : Ω) : a.clock.S (a.τ ω) ω ≤ u := (a.inside ω _ le_rfl).1
-
-noncomputable def radiusValue (p α s : ℝ) (ω : Ω) : ℝ :=
-  powerLogDensity κ p α (a.solution.Q (a.clock.inverse s ω) ω)
-    (B (a.clock.inverse s ω).toNNReal ω)
-
-lemma radiusValue_continuous
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω) :
-    Continuous (fun s => a.radiusValue p α s ω) := by
-  exact (powerLogDensity_contDiff κ p α).continuous.comp
-    (((a.solution.lipschitz ω).continuous.comp (a.clock.continuous_inverse ω)).prodMk
-      ((hBc ω).comp (continuous_real_toNNReal.comp (a.clock.continuous_inverse ω))))
-
-lemma radiusValue_measurable
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (p α : ℝ) :
-    Measurable (Function.uncurry (a.radiusValue p α)) := by
-  exact (powerLogDensity_contDiff κ p α).continuous.measurable.comp
-    ((a.solution.measurable.comp (a.inverse_measurable.prodMk measurable_snd)).prodMk
-      (hBm.comp ((continuous_real_toNNReal.measurable.comp a.inverse_measurable).prodMk measurable_snd)))
-
-lemma radiusValue_at_clock (p α : ℝ) (ω : Ω) (t : ℝ)
-    (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
-    a.radiusValue p α (a.clock.S t ω) ω =
-      powerLogDensity κ p α (a.solution.Q t ω) (B t.toNNReal ω) := by
-  unfold radiusValue
-  rw [a.clock.inverse_clock ω t ht]
-
-lemma radius_integral_substitution
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω) :
-    (∫ t in (0 : ℝ)..(a.τ ω : ℝ),
-      powerLogDensity κ p α (a.solution.Q t ω) (B t.toNNReal ω) *
-        logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)) =
-      ∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), a.radiusValue p α s ω := by
-  have hh := intervalIntegral.integral_comp_mul_deriv''
-    (a.clock.continuous ω).continuousOn
-    (f' := fun t => logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω))
-    (a := (0 : ℝ)) (b := (a.τ ω : ℝ)) (g := fun s => a.radiusValue p α s ω)
-    (fun t ht => by
-      have ht' : t ∈ Ioo (0 : ℝ) (a.τ ω : ℝ) := by
-        simpa only [min_eq_left (a.τ ω).coe_nonneg, max_eq_right (a.τ ω).coe_nonneg] using ht
-      exact ((a.radius_clock_derivative ω t ⟨ht'.1.le, ht'.2.le⟩).hasDerivAt
-        (Icc_mem_nhds ht'.1 ht'.2)).hasDerivWithinAt)
-    ((continuous_logClockRate κ).comp
-      ((a.solution.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal))).continuousOn
-    (a.radiusValue_continuous hBc p α ω).continuousOn
-  rw [a.clock.zero ω] at hh
-  rw [← hh]
-  apply intervalIntegral.integral_congr
-  intro t ht
-  have ht' : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ) := by
-    simpa only [uIcc_of_le (a.τ ω).coe_nonneg] using ht
-  dsimp only [Function.comp_def]
-  rw [a.radiusValue_at_clock p α ω t ht']
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma powerLogDensity_le_radius_angle
-    (κ p α η₀ u ε : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (q : ℂ × ℝ) (x : ℝ)
-    (hs : logRadiusClock η₀ q ≤ u) (ha : ε ≤ logAngleSq κ q x) :
-    powerLogDensity κ p α q x ≤
-      Real.exp (p * (u - Real.log 2 - η₀)) * max 1 (ε ^ (α / 2)) := by
-  rw [powerLogDensity_radius_angle κ p α η₀ q x]
-  have hpow : (logAngleSq κ q x) ^ (α / 2) ≤ max 1 (ε ^ (α / 2)) := by
-    by_cases hp' : 0 ≤ α / 2
-    · have hh : (logAngleSq κ q x) ^ (α / 2) ≤ 1 := by
-        simpa only [Real.one_rpow] using
-          (Real.rpow_le_rpow (logAngleSq_pos κ q x).le (logAngleSq_le_one κ q x) hp')
-      exact hh.trans (le_max_left _ _)
-    · exact (Real.rpow_le_rpow_of_nonpos hε ha (le_of_not_ge hp')).trans (le_max_right _ _)
-  exact mul_le_mul (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (by linarith) hp))
-    hpow (Real.rpow_nonneg (logAngleSq_pos κ q x).le _) (Real.exp_pos _).le
-
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε v δ : ℝ}
-  (a : AngularLocalization B hm κ z u ε)
-
-noncomputable def radiusKernel (p α s : ℝ) (ω : Ω) : ℝ :=
-  if s ≤ a.clock.S (a.τ ω) ω then a.radiusValue p α s ω else 0
-
-lemma radiusKernel_nonneg (p α s : ℝ) (ω : Ω) : 0 ≤ a.radiusKernel p α s ω := by
-  unfold radiusKernel
-  split_ifs
-  · exact (powerLogDensity_pos _ _ _ _ _).le
-  · rfl
-
-lemma radiusKernel_le (p α : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (s : ℝ) (ω : Ω) :
-    a.radiusKernel p α s ω ≤
-      Real.exp (p * (u - Real.log 2 - Real.log z.im)) * max 1 (ε ^ (α / 2)) := by
-  have ht := a.clock.inverse_mem s ω
-  have htn := Real.toNNReal_le_iff_le_coe.mpr ht.2
-  have hh := a.inside ω (a.clock.inverse s ω).toNNReal htn
-  rw [Real.coe_toNNReal _ ht.1] at hh
-  unfold radiusKernel
-  split_ifs
-  · exact powerLogDensity_le_radius_angle κ p α (Real.log z.im) u ε hp hε _ _ hh.1 hh.2.1
-  · exact mul_nonneg (Real.exp_pos _).le ((by norm_num : (0 : ℝ) ≤ 1).trans (le_max_left _ _))
-
-lemma radiusKernel_measurable
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (p α : ℝ) :
-    Measurable (Function.uncurry (a.radiusKernel p α)) := by
-  have ht : Measurable (fun q : ℝ × Ω => a.clock.S (a.τ q.2) q.2) :=
-    a.clock_measurable.comp (((measurable_coe_nnreal_real.comp
-      (stopping_time_measurable_nnreal a.stopping)).comp measurable_snd).prodMk measurable_snd)
-  exact Measurable.ite (measurableSet_le measurable_fst ht) (a.radiusValue_measurable hBm p α) measurable_const
-
-lemma radiusKernel_integral (p α : ℝ) (ω : Ω) :
-    (∫ s in (0 : ℝ)..u, a.radiusKernel p α s ω) =
-      ∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), a.radiusValue p α s ω := by
-  change (∫ s in (0 : ℝ)..u,
-    Set.indicator {s | s ≤ a.clock.S (a.τ ω) ω} (fun s => a.radiusValue p α s ω) s) = _
-  exact intervalIntegral.integral_indicator ⟨a.clock.terminal_nonneg ω, a.clock_upper ω⟩
-
-lemma synchronize_real (b : AngularLocalization B hm κ z v δ)
-    (ω : Ω) (t : ℝ) (h0 : 0 ≤ t) (ha : t ≤ (a.τ ω : ℝ)) (hb : t ≤ (b.τ ω : ℝ)) :
-    a.solution.Q t ω = b.solution.Q t ω := by
-  simpa only [Real.coe_toNNReal t h0] using a.synchronize b ω t.toNNReal
-    (Real.toNNReal_le_iff_le_coe.mpr ha) (Real.toNNReal_le_iff_le_coe.mpr hb)
-
-lemma terminal_clock_le (b : AngularLocalization B hm κ z v δ)
-    (huv : u ≤ v) (hδε : δ ≤ ε) (ω : Ω) :
-    a.clock.S (a.τ ω) ω ≤ b.clock.S (b.τ ω) ω := by
-  have ht := a.stopping_le b huv hδε ω
-  have he := a.synchronize b ω (a.τ ω) le_rfl ht
-  change logRadiusClock _ _ ≤ logRadiusClock _ _
-  rw [he]
-  exact (b.clock.strict ω).monotoneOn ⟨(a.τ ω).coe_nonneg, by exact_mod_cast ht⟩
-    ⟨(b.τ ω).coe_nonneg, le_rfl⟩ (by exact_mod_cast ht)
-
-lemma inverse_synchronize (b : AngularLocalization B hm κ z v δ)
-    (huv : u ≤ v) (hδε : δ ≤ ε) (ω : Ω) (s : ℝ)
-    (hs0 : 0 ≤ s) (hs : s ≤ a.clock.S (a.τ ω) ω) :
-    a.clock.inverse s ω = b.clock.inverse s ω := by
-  let t := a.clock.inverse s ω
-  have htm := a.clock.inverse_mem s ω
-  have hab : (a.τ ω : ℝ) ≤ (b.τ ω : ℝ) := by exact_mod_cast a.stopping_le b huv hδε ω
-  have he : b.clock.S t ω = s := by
-    change logRadiusClock _ (b.solution.Q t ω) = s
-    rw [← a.synchronize_real b ω t htm.1 htm.2 (htm.2.trans hab)]
-    change a.clock.S (a.clock.inverse s ω) ω = s
-    rw [a.clock.clock_inverse, max_eq_left hs0]
-    exact min_eq_left hs
-  exact (b.clock.inverse_clock ω t ⟨htm.1, htm.2.trans hab⟩).symm.trans (by rw [he])
-
-lemma radiusKernel_mono (b : AngularLocalization B hm κ z v δ)
-    (huv : u ≤ v) (hδε : δ ≤ ε) (p α s : ℝ) (hs0 : 0 ≤ s) (ω : Ω) :
-    a.radiusKernel p α s ω ≤ b.radiusKernel p α s ω := by
-  by_cases hs : s ≤ a.clock.S (a.τ ω) ω
-  · have hsb := hs.trans (a.terminal_clock_le b huv hδε ω)
-    rw [radiusKernel, ite_eq_left hs, radiusKernel, ite_eq_left hsb]
-    have he := a.inverse_synchronize b huv hδε ω s hs0 hs
-    have htm := a.clock.inverse_mem s ω
-    have hab : (a.τ ω : ℝ) ≤ (b.τ ω : ℝ) := by exact_mod_cast a.stopping_le b huv hδε ω
-    unfold radiusValue
-    rw [← he, a.synchronize_real b ω _ htm.1 htm.2 (htm.2.trans hab)]
-  · rw [radiusKernel, ite_eq_right hs]
-    exact b.radiusKernel_nonneg p α s ω
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ}
-  (a : AngularLocalization B hm κ z u ε)
-
-lemma physical_power_continuous
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω) :
-    Continuous (fun t : ℝ =>
-      powerLogDensity κ p α (a.solution.Q t ω) (B t.toNNReal ω) *
-        logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)) := by
-  have hq := (a.solution.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal)
-  exact ((powerLogDensity_contDiff κ p α).continuous.comp hq).mul
-    ((continuous_logClockRate κ).comp hq)
-
-lemma generator_radial_integral
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (b : ℝ) (ω : Ω) :
-    (∫ t in (0 : ℝ)..(a.τ ω : ℝ),
-      powerLogGenerator κ (codimension κ) (greenExponent κ - b)
-        (a.solution.Q t ω) (B t.toNNReal ω)) =
-      (κ * b / 8) * ((1 + (greenExponent κ - b)) *
-          (∫ s in (0 : ℝ)..u, a.radiusKernel (codimension κ) (greenExponent κ - b) s ω) -
-        (greenExponent κ - b) *
-          (∫ s in (0 : ℝ)..u, a.radiusKernel (codimension κ) (greenExponent κ - b - 2) s ω)) := by
-  let f (α : ℝ) (t : ℝ) :=
-    powerLogDensity κ (codimension κ) α (a.solution.Q t ω) (B t.toNNReal ω) *
-      logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)
-  have hfi (α : ℝ) : IntervalIntegrable (f α) volume (0 : ℝ) (a.τ ω : ℝ) :=
-    (a.physical_power_continuous hBc (codimension κ) α ω).intervalIntegrable _ _
-  have he (t : ℝ) : powerLogGenerator κ (codimension κ) (greenExponent κ - b)
-      (a.solution.Q t ω) (B t.toNNReal ω) =
-        (κ * b / 8) * ((1 + (greenExponent κ - b)) * f (greenExponent κ - b) t -
-          (greenExponent κ - b) * f (greenExponent κ - b - 2) t) := by
-    rw [powerLogGenerator_two_powers κ b hκ]
-    dsimp only [f]
-    ring
-  simp_rw [he]
-  rw [intervalIntegral.integral_const_mul,
-    intervalIntegral.integral_sub ((hfi _).const_mul _) ((hfi _).const_mul _),
-    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul]
-  rw [a.radiusKernel_integral, a.radiusKernel_integral,
-    ← a.radius_integral_substitution hBc, ← a.radius_integral_substitution hBc]
-
-lemma radiusKernel_cumulative_integrable
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (p α : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (hu : 0 ≤ u) :
-    Integrable (fun ω => ∫ s in (0 : ℝ)..u, a.radiusKernel p α s ω) P := by
-  let C := Real.exp (p * (u - Real.log 2 - Real.log z.im)) * max 1 (ε ^ (α / 2))
-  have hC : 0 ≤ C := mul_nonneg (Real.exp_pos _).le
-    ((by norm_num : (0 : ℝ) ≤ 1).trans (le_max_left _ _))
-  have hh := bounded_stopped_cumulative_integrable (P := P) (a.radiusKernel p α)
-    (a.radiusKernel_measurable hBm p α) (fun _ => u.toNNReal) (fun _ => u.toNNReal)
-    measurable_const (fun _ => le_rfl) u.toNNReal (fun _ => le_rfl) C hC
-    (fun ω s _ _ => by
-      rw [abs_of_nonneg (a.radiusKernel_nonneg p α s ω)]
-      exact a.radiusKernel_le p α hp hε s ω)
-  simpa only [Real.coe_toNNReal u hu] using hh.1
-
-lemma radiusKernel_fubini
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (p α : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (hu : 0 ≤ u) :
-    (∫ ω, (∫ s in (0 : ℝ)..u, a.radiusKernel p α s ω) ∂P) =
-      ∫ s in (0 : ℝ)..u, ∫ ω, a.radiusKernel p α s ω ∂P := by
-  have hi : Integrable (Function.uncurry (a.radiusKernel p α)) ((volume.restrict (Ioc 0 u)).prod P) :=
-    Integrable.of_bound (a.radiusKernel_measurable hBm p α).aestronglyMeasurable
-      (Real.exp (p * (u - Real.log 2 - Real.log z.im)) * max 1 (ε ^ (α / 2)))
-      (Eventually.of_forall fun q => by
-        rw [Real.norm_eq_abs]
-        change |a.radiusKernel p α q.1 q.2| ≤ _
-        rw [abs_of_nonneg (a.radiusKernel_nonneg p α q.1 q.2)]
-        exact a.radiusKernel_le p α hp hε q.1 q.2)
-  simp_rw [intervalIntegral.integral_of_le hu]
-  exact (integral_integral_swap hi).symm
-
- 
-
-theorem power_balance
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (b : ℝ) (hb : b ≤ greenExponent κ) (hε : 0 < ε) (hu : 0 ≤ u) :
-    (∫ ω, powerLogDensity κ (codimension κ) (greenExponent κ - b)
-      (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) +
-      (κ * b / 8) * (greenExponent κ - b) *
-        (∫ ω, (∫ s in (0 : ℝ)..u,
-          a.radiusKernel (codimension κ) (greenExponent κ - b - 2) s ω) ∂P) =
-      powerLogDensity κ (codimension κ) (greenExponent κ - b) (initialLogCoordinates z) 0 +
-        (κ * b / 8) * (1 + (greenExponent κ - b)) *
-          (∫ ω, (∫ s in (0 : ℝ)..u,
-            a.radiusKernel (codimension κ) (greenExponent κ - b) s ω) ∂P) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨_, _, he⟩ := a.mean_power hB hBm hBc hB0 hκ.le (codimension κ)
-    (greenExponent κ - b) (codimension_pos κ hκ8).le (sub_nonneg.mpr hb)
-  simp_rw [a.generator_radial_integral hBc hκ b] at he
-  rw [integral_const_mul,
-    integral_sub ((a.radiusKernel_cumulative_integrable hBm _ _ (codimension_pos κ hκ8).le hε hu).const_mul _)
-      ((a.radiusKernel_cumulative_integrable hBm _ _ (codimension_pos κ hκ8).le hε hu).const_mul _),
-    integral_const_mul, integral_const_mul] at he
-  linarith
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u v ε δ : ℝ}
-
-lemma radiusKernel_eq_of_live (a : AngularLocalization B hm κ z u ε)
-    (b : AngularLocalization B hm κ z v δ) (huv : u ≤ v) (hδε : δ ≤ ε)
-    (p α s : ℝ) (hs0 : 0 ≤ s) (ω : Ω) (hs : s ≤ a.clock.S (a.τ ω) ω) :
-    a.radiusKernel p α s ω = b.radiusKernel p α s ω := by
-  have hsb := hs.trans (a.terminal_clock_le b huv hδε ω)
-  rw [radiusKernel, ite_eq_left hs, radiusKernel, ite_eq_left hsb]
-  have he := a.inverse_synchronize b huv hδε ω s hs0 hs
-  have htm := a.clock.inverse_mem s ω
-  have hab : (a.τ ω : ℝ) ≤ (b.τ ω : ℝ) := by exact_mod_cast a.stopping_le b huv hδε ω
-  unfold radiusValue
-  rw [← he, a.synchronize_real b ω _ htm.1 htm.2 (htm.2.trans hab)]
-
-lemma terminal_clock_min (a : AngularLocalization B hm κ z u ε)
-    (b : AngularLocalization B hm κ z v ε) (huv : u ≤ v) (ω : Ω) :
-    a.clock.S (a.τ ω) ω = min u (b.clock.S (b.τ ω) ω) := by
-  have ht := a.stopping_le b huv le_rfl ω
-  apply le_antisymm
-  · exact le_min (a.clock_upper ω) (a.terminal_clock_le b huv le_rfl ω)
-  · rcases a.boundary ω with hc | ha
-    · exact (min_le_left _ _).trans_eq hc.symm
-    · have he : a.τ ω = b.τ ω := by
-        apply le_antisymm ht
-        by_contra hn
-        have hh := (b.before ω _ (lt_of_not_ge hn)).2
-        rw [← a.synchronize b ω (a.τ ω) le_rfl ht, ha] at hh
-        exact lt_irrefl _ hh
-      have hq := a.synchronize b ω (a.τ ω) le_rfl ht
-      change min u (logRadiusClock _ _) ≤ logRadiusClock _ _
-      rw [hq, he]
-      exact min_le_right _ _
-
-lemma radiusKernel_eq_below (a : AngularLocalization B hm κ z u ε)
-    (b : AngularLocalization B hm κ z v ε) (huv : u ≤ v)
-    (p α s : ℝ) (hs0 : 0 ≤ s) (hsu : s ≤ u) (ω : Ω) :
-    a.radiusKernel p α s ω = b.radiusKernel p α s ω := by
-  by_cases hs : s ≤ a.clock.S (a.τ ω) ω
-  · exact a.radiusKernel_eq_of_live b huv le_rfl p α s hs0 ω hs
-  · have hsb : ¬ s ≤ b.clock.S (b.τ ω) ω := by
-      intro h
-      exact hs (by rw [a.terminal_clock_min b huv ω]; exact le_min hsu h)
-    rw [radiusKernel, ite_eq_right hs, radiusKernel, ite_eq_right hsb]
-
-lemma radiusKernel_uniform_bound (a : AngularLocalization B hm κ z u ε)
-    (p α : ℝ) (hp : 0 ≤ p) (hα : 0 ≤ α) (s : ℝ) (ω : Ω) :
-    a.radiusKernel p α s ω ≤ Real.exp (p * (u - Real.log 2 - Real.log z.im)) := by
-  unfold radiusKernel
-  split_ifs
-  · have ht := a.clock.inverse_mem s ω
-    have hh := a.inside ω (a.clock.inverse s ω).toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)
-    rw [Real.coe_toNNReal _ ht.1] at hh
-    exact powerLogDensity_le_radius κ p α (Real.log z.im) u hp hα _ _ hh.1
-  · exact (Real.exp_pos _).le
-
-end AngularLocalization
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
- 
-
-noncomputable def limitRadiusKernel (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (p α s : ℝ) (ω : Ω) : ℝ :=
-  if h : ∃ n, max s 0 ≤ (a n).clock.S ((a n).τ ω) ω then
-    (a (Nat.find h)).radiusKernel p α (max s 0) ω else 0
-
-lemma radiusKernel_eventually_limit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (p α s : ℝ) (ω : Ω) :
-    ∀ᶠ n in atTop, (a n).radiusKernel p α (max s 0) ω = limitRadiusKernel a p α s ω := by
-  unfold limitRadiusKernel
-  split_ifs with hh
-  · filter_upwards [eventually_ge_atTop (Nat.find hh)] with n hn
-    exact ((a (Nat.find hh)).radiusKernel_eq_of_live (a n) le_rfl (hε hn)
-      p α _ (le_max_right _ _) ω (Nat.find_spec hh)).symm
-  · apply Eventually.of_forall
-    intro n
-    have hn : ¬ max s 0 ≤ (a n).clock.S ((a n).τ ω) ω := fun h => hh ⟨n, h⟩
-    rw [AngularLocalization.radiusKernel, ite_eq_right hn]
-
-lemma limitRadiusKernel_nonneg
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (p α s : ℝ) (ω : Ω) :
-    0 ≤ limitRadiusKernel a p α s ω := by
-  unfold limitRadiusKernel
-  split_ifs
-  · exact AngularLocalization.radiusKernel_nonneg _ _ _ _ _
-  · rfl
-
-lemma limitRadiusKernel_measurable
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (p α : ℝ) :
-    Measurable (Function.uncurry (limitRadiusKernel a p α)) := by
-  apply measurable_of_tendsto_metrizable
-    (fun n => ((a n).radiusKernel_measurable hBm p α).comp
-      ((measurable_fst.max measurable_const).prodMk measurable_snd))
-  apply tendsto_pi_nhds.mpr
-  intro q
-  apply tendsto_const_nhds.congr'
-  filter_upwards [radiusKernel_eventually_limit a hε p α q.1 q.2] with n hn
-  exact hn.symm
-
-lemma limitRadiusKernel_uniform_bound
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (p α : ℝ) (hp : 0 ≤ p) (hα : 0 ≤ α) (s : ℝ) (ω : Ω) :
-    limitRadiusKernel a p α s ω ≤ Real.exp (p * (u - Real.log 2 - Real.log z.im)) := by
-  unfold limitRadiusKernel
-  split_ifs
-  · exact AngularLocalization.radiusKernel_uniform_bound _ _ _ hp hα _ _
-  · exact (Real.exp_pos _).le
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ}
-
-def hit (a : AngularLocalization B hm κ z u ε) : Set Ω :=
-  {ω | logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω) = u}
-
-lemma measurable_hit (a : AngularLocalization B hm κ z u ε) :
-    MeasurableSet[a.stopping.measurableSpace] a.hit :=
-  measurableSet_eq_fun a.clock_stopped_measurable measurable_const
-
-noncomputable def hitTime (a : AngularLocalization B hm κ z u ε) (ω : Ω) : WithTop ℝ≥0 :=
-  if ω ∈ a.hit then (a.τ ω : WithTop ℝ≥0) else ⊤
-
-lemma hitTime_stopping (a : AngularLocalization B hm κ z u ε) :
-    IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) a.hitTime := by
-  intro t
-  have he : {ω | a.hitTime ω ≤ (t : WithTop ℝ≥0)} = a.hit ∩ {ω | (a.τ ω : WithTop ℝ≥0) ≤ t} := by
-    ext ω
-    simp only [hitTime, mem_ofPred_eq, mem_inter_iff]
-    split_ifs with h
-    · simp only [h, true_and]
-    · simp only [h, false_and, iff_false]
-      exact WithTop.not_top_le_coe t
-  rw [he]
-  exact a.measurable_hit.2 t
-
-end AngularLocalization
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
- 
-
-noncomputable def localizedRadiusStop
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) : WithTop ℝ≥0 :=
-  ⨅ n, (a n).hitTime ω
-
-lemma localizedRadiusStop_eq_hit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) (n : ℕ)
-    (hn : ω ∈ (a n).hit) : localizedRadiusStop a ω = ((a n).τ ω : WithTop ℝ≥0) := by
-  apply le_antisymm
-  · exact (iInf_le _ n).trans_eq (by rw [AngularLocalization.hitTime, ite_eq_left hn])
-  · apply le_iInf
-    intro m
-    by_cases h : ω ∈ (a m).hit
-    · have he := (a m).hit_time_unique (a n) ω h hn
-      rw [AngularLocalization.hitTime, ite_eq_left h, he]
-    · rw [AngularLocalization.hitTime, ite_eq_right h]
-      exact le_top
-
-lemma localizedRadiusStop_eq_top
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω)
-    (hn : ∀ n, ω ∉ (a n).hit) : localizedRadiusStop a ω = ⊤ := by
-  simp [localizedRadiusStop, AngularLocalization.hitTime, hn]
-
-lemma localizedRadiusStop_le_iff
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) (t : ℝ≥0) :
-    localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0) ↔ ∃ n, ω ∈ (a n).hit ∧ (a n).τ ω ≤ t := by
-  constructor
-  · intro h
-    by_cases he : ∃ n, ω ∈ (a n).hit
-    · obtain ⟨n, hn⟩ := he
-      rw [localizedRadiusStop_eq_hit a ω n hn, WithTop.coe_le_coe] at h
-      exact ⟨n, hn, h⟩
-    · rw [localizedRadiusStop_eq_top a ω (not_exists.mp he)] at h
-      exact False.elim (WithTop.coe_ne_top (a := t) (top_le_iff.mp h))
-  · rintro ⟨n, hn, ht⟩
-    rw [localizedRadiusStop_eq_hit a ω n hn, WithTop.coe_le_coe]
-    exact ht
-
-lemma localizedRadiusStop_stopping
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) :
-    IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (localizedRadiusStop a) := by
-  intro t
-  have he : {ω | localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0)} =
-      ⋃ n, ((a n).hit ∩ {ω | ((a n).τ ω : WithTop ℝ≥0) ≤ t}) := by
-    ext ω
-    simp only [mem_ofPred_eq, mem_iUnion, mem_inter_iff, WithTop.coe_le_coe]
-    exact localizedRadiusStop_le_iff a ω t
-  rw [he]
-  exact MeasurableSet.iUnion fun n => (a n).measurable_hit.2 t
-
-lemma localizedRadiusStop_history_hit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (n : ℕ) (A : Set Ω)
-    (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
-    MeasurableSet[(a n).stopping.measurableSpace] (A ∩ (a n).hit) := by
-  refine ⟨hA.1.inter (a n).measurable_hit.1, ?_⟩
-  intro t
-  have he : (A ∩ (a n).hit) ∩ {ω | ((a n).τ ω : WithTop ℝ≥0) ≤ t} =
-      (A ∩ {ω | localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0)}) ∩
-        ((a n).hit ∩ {ω | ((a n).τ ω : WithTop ℝ≥0) ≤ t}) := by
-    ext ω
-    simp only [mem_inter_iff, mem_ofPred_eq]
-    constructor
-    · rintro ⟨⟨hωA, hωn⟩, hωt⟩
-      exact ⟨⟨hωA, (localizedRadiusStop_eq_hit a ω n hωn) ▸ hωt⟩, hωn, hωt⟩
-    · rintro ⟨⟨hωA, _⟩, hωn, hωt⟩
-      exact ⟨⟨hωA, hωn⟩, hωt⟩
-  rw [he]
-  exact (hA.2 t).inter ((a n).measurable_hit.2 t)
-
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma integrable_monotone_limit_of_integral_le
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {f : ℕ → Ω → ℝ} {F : Ω → ℝ}
-    (hf : ∀ n, Integrable (f n) P)
-    (hF : AEStronglyMeasurable F P)
-    (hn : ∀ n ω, 0 ≤ f n ω) (hFn : ∀ ω, 0 ≤ F ω)
-    (hmono : ∀ ω, Monotone (fun n => f n ω))
-    (hlim : ∀ ω, Tendsto (fun n => f n ω) atTop (𝓝 (F ω)))
-    (C : ℝ) (hbound : ∀ n, (∫ ω, f n ω ∂P) ≤ C) :
-    Integrable F P ∧
-      Tendsto (fun n => ∫ ω, f n ω ∂P) atTop (𝓝 (∫ ω, F ω ∂P)) := by
-  have hl : Tendsto (fun n => ∫⁻ ω, ENNReal.ofReal (f n ω) ∂P) atTop
-      (𝓝 (∫⁻ ω, ENNReal.ofReal (F ω) ∂P)) := by
-    apply lintegral_tendsto_of_tendsto_of_monotone
-      (fun n => (hf n).aemeasurable.ennreal_ofReal)
-    · exact Eventually.of_forall fun ω n m hnm => ENNReal.ofReal_le_ofReal (hmono ω hnm)
-    · exact Eventually.of_forall fun ω => ENNReal.continuous_ofReal.continuousAt.tendsto.comp (hlim ω)
-  have hle : (∫⁻ ω, ENNReal.ofReal (F ω) ∂P) ≤ ENNReal.ofReal C := by
-    apply le_of_tendsto hl
-    apply Eventually.of_forall
-    intro n
-    rw [← ofReal_integral_eq_lintegral_ofReal (hf n) (Eventually.of_forall (hn n))]
-    exact ENNReal.ofReal_le_ofReal (hbound n)
-  have hFi : Integrable F P := by
-    refine ⟨hF, ?_⟩
-    rw [hasFiniteIntegral_iff_enorm]
-    have he : (fun ω => ‖F ω‖ₑ) = fun ω => ENNReal.ofReal (F ω) := by
-      funext ω
-      simp [Real.enorm_eq_ofReal_abs, abs_of_nonneg (hFn ω)]
-    rw [he]
-    exact hle.trans_lt ENNReal.ofReal_lt_top
-  exact ⟨hFi, integral_tendsto_of_tendsto_of_monotone hf hFi
-    (Eventually.of_forall hmono) (Eventually.of_forall hlim)⟩
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.DiskHalfPlane
+
+namespace OAI
+
+/-! Angular Power Dynkin for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma powerLogGenerator_lyapunov (κ b : ℝ) (hκ : 0 < κ)
+    (q : ℂ × ℝ) (x : ℝ) :
+    powerLogGenerator κ (codimension κ) (greenExponent κ - b) q x =
+      powerLogDensity κ (codimension κ) (greenExponent κ - b) q x *
+        logClockRate κ q x * (κ * b / 8) *
+          (1 - (greenExponent κ - b) * ((logAngleSq κ q x)⁻¹ - 1)) := by
+  unfold powerLogGenerator
+  rw [← power_physical_generator κ _ _ hκ.le]
+  exact green_weighted_power_generator κ b hκ q x
+
+lemma powerLogDensity_div_angle (κ p α : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    powerLogDensity κ p α q x / logAngleSq κ q x =
+      powerLogDensity κ p (α - 2) q x := by
+  rw [powerLogDensity_radius_angle κ p α 0, powerLogDensity_radius_angle κ p (α - 2) 0,
+    mul_div_assoc]
+  congr 1
+  rw [show (α - 2) / 2 = α / 2 - 1 by ring,
+    Real.rpow_sub (logAngleSq_pos κ q x), Real.rpow_one]
+
+lemma powerLogGenerator_two_powers (κ b : ℝ) (hκ : 0 < κ)
+    (q : ℂ × ℝ) (x : ℝ) :
+    powerLogGenerator κ (codimension κ) (greenExponent κ - b) q x =
+      logClockRate κ q x * (κ * b / 8) *
+        ((1 + (greenExponent κ - b)) *
+            powerLogDensity κ (codimension κ) (greenExponent κ - b) q x -
+          (greenExponent κ - b) *
+            powerLogDensity κ (codimension κ) (greenExponent κ - b - 2) q x) := by
+  rw [powerLogGenerator_lyapunov κ b hκ,
+    ← powerLogDensity_div_angle κ (codimension κ) (greenExponent κ - b)]
+  ring
+
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ}
+  (a : AngularLocalization B hm κ z u ε)
+
+ 
+
+theorem mean_power
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 ≤ κ) (p α : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ α) :
+    Integrable (fun ω => powerLogDensity κ p α (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
+    Integrable (fun ω => ∫ t in (0 : ℝ)..(a.τ ω : ℝ),
+      powerLogGenerator κ p α (a.solution.Q t ω) (B t.toNNReal ω)) P ∧
+      (∫ ω, powerLogDensity κ p α (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) =
+        powerLogDensity κ p α (initialLogCoordinates z) 0 +
+        ∫ ω, (∫ t in (0 : ℝ)..(a.τ ω : ℝ),
+          powerLogGenerator κ p α (a.solution.Q t ω) (B t.toNNReal ω)) ∂P := by
+  let K := Real.exp (p * (u - Real.log 2 - Real.log z.im))
+  let C := K * (|(-4 * α + κ * α * (α + 1) / 2)| + |(4 * p - κ * α / 2)|) /
+    Real.exp (2 * a.lo)
+  apply a.solution.stopped_power_dynkin_bounded hB hBm hBc hB0 hκ p α a.τ a.stopping
+    a.T a.bounded (fun ω t ht => (a.inside ω t ht).2.2) K C (by dsimp [C, K]; positivity)
+  · intro ω t ht
+    rw [abs_of_pos (powerLogDensity_pos _ _ _ _ _)]
+    exact powerLogDensity_le_radius κ p α (Real.log z.im) u hp ha _ _ (a.inside ω t ht).1
+  · intro ω t ht
+    exact powerLogGenerator_le_radius_height κ p α (Real.log z.im) u a.lo hp ha _ _
+      (a.inside ω t ht).1 (a.inside ω t ht).2.2.1
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace AngularLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u v ε δ : ℝ}
+
+ 
+
+lemma synchronize (a : AngularLocalization B hm κ z u ε)
+    (b : AngularLocalization B hm κ z v δ) (ω : Ω)
+    (t : ℝ≥0) (ha : t ≤ a.τ ω) (hb : t ≤ b.τ ω) :
+    a.solution.Q t ω = b.solution.Q t ω := by
+  let lo := min a.lo b.lo
+  let hi := max a.hi b.hi
+  have hle : lo ≤ hi := (min_le_left _ _).trans (a.le.trans (le_max_left _ _))
+  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (s : ℝ) := clippedLogLoewnerField lo hi hle (Real.sqrt κ * B s.toNNReal ω)
+  have hd (c : AngularLocalization B hm κ z u ε) (hc : t ≤ c.τ ω)
+      (hlo : lo ≤ c.lo) (hhi : c.hi ≤ hi) (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
+      HasDerivWithinAt (fun s => c.solution.Q s ω) (V s (c.solution.Q s ω)) (Icc 0 (t : ℝ)) s := by
+    have hsc : s.toNNReal ≤ c.τ ω := (Real.toNNReal_le_iff_le_coe.mpr hs.2).trans hc
+    have hcbar := (c.inside ω s.toNNReal hsc).2.2
+    rw [Real.coe_toNNReal _ hs.1] at hcbar
+    have hder := (c.solution.derivative ω s hs.1).mono
+      (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+    rwa [clippedLogLoewnerField_eq_physical _ _ _ _ _ hcbar,
+      ← clippedLogLoewnerField_eq_physical lo hi hle _ _ ⟨hlo.trans hcbar.1, hcbar.2.trans hhi⟩] at hder
+  have hda := hd a ha (min_le_left _ _) (le_max_left _ _)
+  have hdb (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
+      HasDerivWithinAt (fun s => b.solution.Q s ω) (V s (b.solution.Q s ω)) (Icc 0 (t : ℝ)) s := by
+    have hsb : s.toNNReal ≤ b.τ ω := (Real.toNNReal_le_iff_le_coe.mpr hs.2).trans hb
+    have hcbar := (b.inside ω s.toNNReal hsb).2.2
+    rw [Real.coe_toNNReal _ hs.1] at hcbar
+    have hder := (b.solution.derivative ω s hs.1).mono
+      (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+    rwa [clippedLogLoewnerField_eq_physical _ _ _ _ _ hcbar,
+      ← clippedLogLoewnerField_eq_physical lo hi hle _ _
+        ⟨(min_le_right _ _).trans hcbar.1, hcbar.2.trans (le_max_right _ _)⟩] at hder
+  have he : EqOn (fun s => a.solution.Q s ω) (fun s => b.solution.Q s ω) (Icc 0 (t : ℝ)) := by
+    apply ODE_solution_unique (fun s => hlip (Real.sqrt κ * B s.toNNReal ω))
+      (fun s hs => (hda s hs).continuousWithinAt)
+      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hda s ⟨hs.1, hs.2.le⟩))
+      (fun s hs => (hdb s hs).continuousWithinAt)
+      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hdb s ⟨hs.1, hs.2.le⟩))
+    exact (a.solution.initial ω).trans (b.solution.initial ω).symm
+  exact he ⟨t.coe_nonneg, le_rfl⟩
+
+lemma stopping_le (a : AngularLocalization B hm κ z u ε)
+    (b : AngularLocalization B hm κ z v δ) (huv : u ≤ v) (hδε : δ ≤ ε)
+    (ω : Ω) : a.τ ω ≤ b.τ ω := by
+  by_contra hn
+  have hlt := lt_of_not_ge hn
+  have he := a.synchronize b ω (b.τ ω) hlt.le le_rfl
+  have hh := a.before ω _ hlt
+  rw [he] at hh
+  rcases b.boundary ω with hc | hθ
+  · linarith [hh.1]
+  · linarith [hh.2]
+
+lemma hit_time_unique (a : AngularLocalization B hm κ z u ε)
+    (b : AngularLocalization B hm κ z u δ) (ω : Ω)
+    (ha : logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω) = u)
+    (hb : logRadiusClock (Real.log z.im) (b.solution.Q (b.τ ω) ω) = u) :
+    a.τ ω = b.τ ω := by
+  apply le_antisymm
+  · by_contra hn
+    have hlt := lt_of_not_ge hn
+    have he := a.synchronize b ω (b.τ ω) hlt.le le_rfl
+    have hh := (a.before ω _ hlt).1
+    rw [he, hb] at hh
+    exact lt_irrefl _ hh
+  · by_contra hn
+    have hlt := lt_of_not_ge hn
+    have he := b.synchronize a ω (a.τ ω) hlt.le le_rfl
+    have hh := (b.before ω _ hlt).1
+    rw [he, ha] at hh
+    exact lt_irrefl _ hh
+
+lemma hit_finer (a : AngularLocalization B hm κ z u ε)
+    (b : AngularLocalization B hm κ z u δ) (hδε : δ ≤ ε) (ω : Ω)
+    (ha : logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω) = u) :
+    b.τ ω = a.τ ω ∧ logRadiusClock (Real.log z.im) (b.solution.Q (b.τ ω) ω) = u := by
+  have hab := a.stopping_le b le_rfl hδε ω
+  have he : b.τ ω = a.τ ω := by
+    apply le_antisymm _ hab
+    by_contra hn
+    have hlt := lt_of_not_ge hn
+    have hq := a.synchronize b ω (a.τ ω) le_rfl hlt.le
+    have hh := (b.before ω _ hlt).1
+    rw [← hq, ha] at hh
+    exact lt_irrefl _ hh
+  refine ⟨he, ?_⟩
+  rw [he, ← a.synchronize b ω (a.τ ω) le_rfl hab]
+  exact ha
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma continuous_logClockRate (κ : ℝ) :
+    Continuous (fun p : (ℂ × ℝ) × ℝ => logClockRate κ p.1 p.2) := by
+  have hN : Continuous (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) := by
+    unfold greenLogNormSq
+    fun_prop
+  unfold logClockRate
+  exact (continuous_const.mul (Real.continuous_exp.comp (by fun_prop))).div (hN.pow 2)
+    (fun p => pow_ne_zero 2 (greenLogNormSq_pos κ p.1 p.2).ne')
+
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ}
+  (a : AngularLocalization B hm κ z u ε)
+
+lemma radius_clock_derivative (ω : Ω) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
+    HasDerivWithinAt (fun t => logRadiusClock (Real.log z.im) (a.solution.Q t ω))
+      (logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)) (Icc (0 : ℝ) (a.τ ω : ℝ)) t := by
+  have hd := (a.solution.derivative ω t ht.1).mono
+    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+  have hb := (a.inside ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)).2.2
+  rw [Real.coe_toNNReal t ht.1] at hb
+  rw [clippedLogLoewnerField_eq_physical _ _ _ _ _ hb] at hd
+  exact log_ode_clock_derivative (Real.log z.im) hd
+
+noncomputable def clock : FiniteIncreasingClock Ω where
+  S t ω := logRadiusClock (Real.log z.im) (a.solution.Q t ω)
+  T := a.τ
+  zero ω := by rw [a.solution.initial]; simp [logRadiusClock, initialLogCoordinates]
+  continuous ω := (continuous_logRadiusClock _).comp (a.solution.lipschitz ω).continuous
+  strict ω := by
+    apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc (0 : ℝ) (a.τ ω : ℝ))
+      ((continuous_logRadiusClock _).comp (a.solution.lipschitz ω).continuous).continuousOn
+    · intro t ht
+      exact (a.radius_clock_derivative ω t (interior_subset ht)).mono interior_subset
+    · intro t ht
+      exact logClockRate_pos κ (a.solution.Q t ω) (B t.toNNReal ω)
+
+lemma clock_measurable : Measurable (Function.uncurry a.clock.S) :=
+  (continuous_logRadiusClock _).measurable.comp a.solution.measurable
+
+lemma inverse_measurable : Measurable (Function.uncurry a.clock.inverse) :=
+  a.clock.measurable_inverse a.clock_measurable (stopping_time_measurable_nnreal a.stopping)
+
+lemma clock_upper (ω : Ω) : a.clock.S (a.τ ω) ω ≤ u := (a.inside ω _ le_rfl).1
+
+noncomputable def radiusValue (p α s : ℝ) (ω : Ω) : ℝ :=
+  powerLogDensity κ p α (a.solution.Q (a.clock.inverse s ω) ω)
+    (B (a.clock.inverse s ω).toNNReal ω)
+
+lemma radiusValue_continuous
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω) :
+    Continuous (fun s => a.radiusValue p α s ω) := by
+  exact (powerLogDensity_contDiff κ p α).continuous.comp
+    (((a.solution.lipschitz ω).continuous.comp (a.clock.continuous_inverse ω)).prodMk
+      ((hBc ω).comp (continuous_real_toNNReal.comp (a.clock.continuous_inverse ω))))
+
+lemma radiusValue_measurable
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (p α : ℝ) :
+    Measurable (Function.uncurry (a.radiusValue p α)) := by
+  exact (powerLogDensity_contDiff κ p α).continuous.measurable.comp
+    ((a.solution.measurable.comp (a.inverse_measurable.prodMk measurable_snd)).prodMk
+      (hBm.comp ((continuous_real_toNNReal.measurable.comp a.inverse_measurable).prodMk measurable_snd)))
+
+lemma radiusValue_at_clock (p α : ℝ) (ω : Ω) (t : ℝ)
+    (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
+    a.radiusValue p α (a.clock.S t ω) ω =
+      powerLogDensity κ p α (a.solution.Q t ω) (B t.toNNReal ω) := by
+  unfold radiusValue
+  rw [a.clock.inverse_clock ω t ht]
+
+lemma radius_integral_substitution
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω) :
+    (∫ t in (0 : ℝ)..(a.τ ω : ℝ),
+      powerLogDensity κ p α (a.solution.Q t ω) (B t.toNNReal ω) *
+        logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)) =
+      ∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), a.radiusValue p α s ω := by
+  have hh := intervalIntegral.integral_comp_mul_deriv''
+    (a.clock.continuous ω).continuousOn
+    (f' := fun t => logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω))
+    (a := (0 : ℝ)) (b := (a.τ ω : ℝ)) (g := fun s => a.radiusValue p α s ω)
+    (fun t ht => by
+      have ht' : t ∈ Ioo (0 : ℝ) (a.τ ω : ℝ) := by
+        simpa only [min_eq_left (a.τ ω).coe_nonneg, max_eq_right (a.τ ω).coe_nonneg] using ht
+      exact ((a.radius_clock_derivative ω t ⟨ht'.1.le, ht'.2.le⟩).hasDerivAt
+        (Icc_mem_nhds ht'.1 ht'.2)).hasDerivWithinAt)
+    ((continuous_logClockRate κ).comp
+      ((a.solution.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal))).continuousOn
+    (a.radiusValue_continuous hBc p α ω).continuousOn
+  rw [a.clock.zero ω] at hh
+  rw [← hh]
+  apply intervalIntegral.integral_congr
+  intro t ht
+  have ht' : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ) := by
+    simpa only [uIcc_of_le (a.τ ω).coe_nonneg] using ht
+  dsimp only [Function.comp_def]
+  rw [a.radiusValue_at_clock p α ω t ht']
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma powerLogDensity_le_radius_angle
+    (κ p α η₀ u ε : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (q : ℂ × ℝ) (x : ℝ)
+    (hs : logRadiusClock η₀ q ≤ u) (ha : ε ≤ logAngleSq κ q x) :
+    powerLogDensity κ p α q x ≤
+      Real.exp (p * (u - Real.log 2 - η₀)) * max 1 (ε ^ (α / 2)) := by
+  rw [powerLogDensity_radius_angle κ p α η₀ q x]
+  have hpow : (logAngleSq κ q x) ^ (α / 2) ≤ max 1 (ε ^ (α / 2)) := by
+    by_cases hp' : 0 ≤ α / 2
+    · have hh : (logAngleSq κ q x) ^ (α / 2) ≤ 1 := by
+        simpa only [Real.one_rpow] using
+          (Real.rpow_le_rpow (logAngleSq_pos κ q x).le (logAngleSq_le_one κ q x) hp')
+      exact hh.trans (le_max_left _ _)
+    · exact (Real.rpow_le_rpow_of_nonpos hε ha (le_of_not_ge hp')).trans (le_max_right _ _)
+  exact mul_le_mul (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (by linarith) hp))
+    hpow (Real.rpow_nonneg (logAngleSq_pos κ q x).le _) (Real.exp_pos _).le
+
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε v δ : ℝ}
+  (a : AngularLocalization B hm κ z u ε)
+
+noncomputable def radiusKernel (p α s : ℝ) (ω : Ω) : ℝ :=
+  if s ≤ a.clock.S (a.τ ω) ω then a.radiusValue p α s ω else 0
+
+lemma radiusKernel_nonneg (p α s : ℝ) (ω : Ω) : 0 ≤ a.radiusKernel p α s ω := by
+  unfold radiusKernel
+  split_ifs
+  · exact (powerLogDensity_pos _ _ _ _ _).le
+  · rfl
+
+lemma radiusKernel_le (p α : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (s : ℝ) (ω : Ω) :
+    a.radiusKernel p α s ω ≤
+      Real.exp (p * (u - Real.log 2 - Real.log z.im)) * max 1 (ε ^ (α / 2)) := by
+  have ht := a.clock.inverse_mem s ω
+  have htn := Real.toNNReal_le_iff_le_coe.mpr ht.2
+  have hh := a.inside ω (a.clock.inverse s ω).toNNReal htn
+  rw [Real.coe_toNNReal _ ht.1] at hh
+  unfold radiusKernel
+  split_ifs
+  · exact powerLogDensity_le_radius_angle κ p α (Real.log z.im) u ε hp hε _ _ hh.1 hh.2.1
+  · exact mul_nonneg (Real.exp_pos _).le ((by norm_num : (0 : ℝ) ≤ 1).trans (le_max_left _ _))
+
+lemma radiusKernel_measurable
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (p α : ℝ) :
+    Measurable (Function.uncurry (a.radiusKernel p α)) := by
+  have ht : Measurable (fun q : ℝ × Ω => a.clock.S (a.τ q.2) q.2) :=
+    a.clock_measurable.comp (((measurable_coe_nnreal_real.comp
+      (stopping_time_measurable_nnreal a.stopping)).comp measurable_snd).prodMk measurable_snd)
+  exact Measurable.ite (measurableSet_le measurable_fst ht) (a.radiusValue_measurable hBm p α) measurable_const
+
+lemma radiusKernel_integral (p α : ℝ) (ω : Ω) :
+    (∫ s in (0 : ℝ)..u, a.radiusKernel p α s ω) =
+      ∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), a.radiusValue p α s ω := by
+  change (∫ s in (0 : ℝ)..u,
+    Set.indicator {s | s ≤ a.clock.S (a.τ ω) ω} (fun s => a.radiusValue p α s ω) s) = _
+  exact intervalIntegral.integral_indicator ⟨a.clock.terminal_nonneg ω, a.clock_upper ω⟩
+
+lemma synchronize_real (b : AngularLocalization B hm κ z v δ)
+    (ω : Ω) (t : ℝ) (h0 : 0 ≤ t) (ha : t ≤ (a.τ ω : ℝ)) (hb : t ≤ (b.τ ω : ℝ)) :
+    a.solution.Q t ω = b.solution.Q t ω := by
+  simpa only [Real.coe_toNNReal t h0] using a.synchronize b ω t.toNNReal
+    (Real.toNNReal_le_iff_le_coe.mpr ha) (Real.toNNReal_le_iff_le_coe.mpr hb)
+
+lemma terminal_clock_le (b : AngularLocalization B hm κ z v δ)
+    (huv : u ≤ v) (hδε : δ ≤ ε) (ω : Ω) :
+    a.clock.S (a.τ ω) ω ≤ b.clock.S (b.τ ω) ω := by
+  have ht := a.stopping_le b huv hδε ω
+  have he := a.synchronize b ω (a.τ ω) le_rfl ht
+  change logRadiusClock _ _ ≤ logRadiusClock _ _
+  rw [he]
+  exact (b.clock.strict ω).monotoneOn ⟨(a.τ ω).coe_nonneg, by exact_mod_cast ht⟩
+    ⟨(b.τ ω).coe_nonneg, le_rfl⟩ (by exact_mod_cast ht)
+
+lemma inverse_synchronize (b : AngularLocalization B hm κ z v δ)
+    (huv : u ≤ v) (hδε : δ ≤ ε) (ω : Ω) (s : ℝ)
+    (hs0 : 0 ≤ s) (hs : s ≤ a.clock.S (a.τ ω) ω) :
+    a.clock.inverse s ω = b.clock.inverse s ω := by
+  let t := a.clock.inverse s ω
+  have htm := a.clock.inverse_mem s ω
+  have hab : (a.τ ω : ℝ) ≤ (b.τ ω : ℝ) := by exact_mod_cast a.stopping_le b huv hδε ω
+  have he : b.clock.S t ω = s := by
+    change logRadiusClock _ (b.solution.Q t ω) = s
+    rw [← a.synchronize_real b ω t htm.1 htm.2 (htm.2.trans hab)]
+    change a.clock.S (a.clock.inverse s ω) ω = s
+    rw [a.clock.clock_inverse, max_eq_left hs0]
+    exact min_eq_left hs
+  exact (b.clock.inverse_clock ω t ⟨htm.1, htm.2.trans hab⟩).symm.trans (by rw [he])
+
+lemma radiusKernel_mono (b : AngularLocalization B hm κ z v δ)
+    (huv : u ≤ v) (hδε : δ ≤ ε) (p α s : ℝ) (hs0 : 0 ≤ s) (ω : Ω) :
+    a.radiusKernel p α s ω ≤ b.radiusKernel p α s ω := by
+  by_cases hs : s ≤ a.clock.S (a.τ ω) ω
+  · have hsb := hs.trans (a.terminal_clock_le b huv hδε ω)
+    rw [radiusKernel, ite_eq_left hs, radiusKernel, ite_eq_left hsb]
+    have he := a.inverse_synchronize b huv hδε ω s hs0 hs
+    have htm := a.clock.inverse_mem s ω
+    have hab : (a.τ ω : ℝ) ≤ (b.τ ω : ℝ) := by exact_mod_cast a.stopping_le b huv hδε ω
+    unfold radiusValue
+    rw [← he, a.synchronize_real b ω _ htm.1 htm.2 (htm.2.trans hab)]
+  · rw [radiusKernel, ite_eq_right hs]
+    exact b.radiusKernel_nonneg p α s ω
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ}
+  (a : AngularLocalization B hm κ z u ε)
+
+lemma physical_power_continuous
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω) :
+    Continuous (fun t : ℝ =>
+      powerLogDensity κ p α (a.solution.Q t ω) (B t.toNNReal ω) *
+        logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)) := by
+  have hq := (a.solution.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal)
+  exact ((powerLogDensity_contDiff κ p α).continuous.comp hq).mul
+    ((continuous_logClockRate κ).comp hq)
+
+lemma generator_radial_integral
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (b : ℝ) (ω : Ω) :
+    (∫ t in (0 : ℝ)..(a.τ ω : ℝ),
+      powerLogGenerator κ (codimension κ) (greenExponent κ - b)
+        (a.solution.Q t ω) (B t.toNNReal ω)) =
+      (κ * b / 8) * ((1 + (greenExponent κ - b)) *
+          (∫ s in (0 : ℝ)..u, a.radiusKernel (codimension κ) (greenExponent κ - b) s ω) -
+        (greenExponent κ - b) *
+          (∫ s in (0 : ℝ)..u, a.radiusKernel (codimension κ) (greenExponent κ - b - 2) s ω)) := by
+  let f (α : ℝ) (t : ℝ) :=
+    powerLogDensity κ (codimension κ) α (a.solution.Q t ω) (B t.toNNReal ω) *
+      logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω)
+  have hfi (α : ℝ) : IntervalIntegrable (f α) volume (0 : ℝ) (a.τ ω : ℝ) :=
+    (a.physical_power_continuous hBc (codimension κ) α ω).intervalIntegrable _ _
+  have he (t : ℝ) : powerLogGenerator κ (codimension κ) (greenExponent κ - b)
+      (a.solution.Q t ω) (B t.toNNReal ω) =
+        (κ * b / 8) * ((1 + (greenExponent κ - b)) * f (greenExponent κ - b) t -
+          (greenExponent κ - b) * f (greenExponent κ - b - 2) t) := by
+    rw [powerLogGenerator_two_powers κ b hκ]
+    dsimp only [f]
+    ring
+  simp_rw [he]
+  rw [intervalIntegral.integral_const_mul,
+    intervalIntegral.integral_sub ((hfi _).const_mul _) ((hfi _).const_mul _),
+    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul]
+  rw [a.radiusKernel_integral, a.radiusKernel_integral,
+    ← a.radius_integral_substitution hBc, ← a.radius_integral_substitution hBc]
+
+lemma radiusKernel_cumulative_integrable
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (p α : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (hu : 0 ≤ u) :
+    Integrable (fun ω => ∫ s in (0 : ℝ)..u, a.radiusKernel p α s ω) P := by
+  let C := Real.exp (p * (u - Real.log 2 - Real.log z.im)) * max 1 (ε ^ (α / 2))
+  have hC : 0 ≤ C := mul_nonneg (Real.exp_pos _).le
+    ((by norm_num : (0 : ℝ) ≤ 1).trans (le_max_left _ _))
+  have hh := bounded_stopped_cumulative_integrable (P := P) (a.radiusKernel p α)
+    (a.radiusKernel_measurable hBm p α) (fun _ => u.toNNReal) (fun _ => u.toNNReal)
+    measurable_const (fun _ => le_rfl) u.toNNReal (fun _ => le_rfl) C hC
+    (fun ω s _ _ => by
+      rw [abs_of_nonneg (a.radiusKernel_nonneg p α s ω)]
+      exact a.radiusKernel_le p α hp hε s ω)
+  simpa only [Real.coe_toNNReal u hu] using hh.1
+
+lemma radiusKernel_fubini
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (p α : ℝ) (hp : 0 ≤ p) (hε : 0 < ε) (hu : 0 ≤ u) :
+    (∫ ω, (∫ s in (0 : ℝ)..u, a.radiusKernel p α s ω) ∂P) =
+      ∫ s in (0 : ℝ)..u, ∫ ω, a.radiusKernel p α s ω ∂P := by
+  have hi : Integrable (Function.uncurry (a.radiusKernel p α)) ((volume.restrict (Ioc 0 u)).prod P) :=
+    Integrable.of_bound (a.radiusKernel_measurable hBm p α).aestronglyMeasurable
+      (Real.exp (p * (u - Real.log 2 - Real.log z.im)) * max 1 (ε ^ (α / 2)))
+      (Eventually.of_forall fun q => by
+        rw [Real.norm_eq_abs]
+        change |a.radiusKernel p α q.1 q.2| ≤ _
+        rw [abs_of_nonneg (a.radiusKernel_nonneg p α q.1 q.2)]
+        exact a.radiusKernel_le p α hp hε q.1 q.2)
+  simp_rw [intervalIntegral.integral_of_le hu]
+  exact (integral_integral_swap hi).symm
+
+ 
+
+theorem power_balance
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (b : ℝ) (hb : b ≤ greenExponent κ) (hε : 0 < ε) (hu : 0 ≤ u) :
+    (∫ ω, powerLogDensity κ (codimension κ) (greenExponent κ - b)
+      (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) +
+      (κ * b / 8) * (greenExponent κ - b) *
+        (∫ ω, (∫ s in (0 : ℝ)..u,
+          a.radiusKernel (codimension κ) (greenExponent κ - b - 2) s ω) ∂P) =
+      powerLogDensity κ (codimension κ) (greenExponent κ - b) (initialLogCoordinates z) 0 +
+        (κ * b / 8) * (1 + (greenExponent κ - b)) *
+          (∫ ω, (∫ s in (0 : ℝ)..u,
+            a.radiusKernel (codimension κ) (greenExponent κ - b) s ω) ∂P) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨_, _, he⟩ := a.mean_power hB hBm hBc hB0 hκ.le (codimension κ)
+    (greenExponent κ - b) (codimension_pos κ hκ8).le (sub_nonneg.mpr hb)
+  simp_rw [a.generator_radial_integral hBc hκ b] at he
+  rw [integral_const_mul,
+    integral_sub ((a.radiusKernel_cumulative_integrable hBm _ _ (codimension_pos κ hκ8).le hε hu).const_mul _)
+      ((a.radiusKernel_cumulative_integrable hBm _ _ (codimension_pos κ hκ8).le hε hu).const_mul _),
+    integral_const_mul, integral_const_mul] at he
+  linarith
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u v ε δ : ℝ}
+
+lemma radiusKernel_eq_of_live (a : AngularLocalization B hm κ z u ε)
+    (b : AngularLocalization B hm κ z v δ) (huv : u ≤ v) (hδε : δ ≤ ε)
+    (p α s : ℝ) (hs0 : 0 ≤ s) (ω : Ω) (hs : s ≤ a.clock.S (a.τ ω) ω) :
+    a.radiusKernel p α s ω = b.radiusKernel p α s ω := by
+  have hsb := hs.trans (a.terminal_clock_le b huv hδε ω)
+  rw [radiusKernel, ite_eq_left hs, radiusKernel, ite_eq_left hsb]
+  have he := a.inverse_synchronize b huv hδε ω s hs0 hs
+  have htm := a.clock.inverse_mem s ω
+  have hab : (a.τ ω : ℝ) ≤ (b.τ ω : ℝ) := by exact_mod_cast a.stopping_le b huv hδε ω
+  unfold radiusValue
+  rw [← he, a.synchronize_real b ω _ htm.1 htm.2 (htm.2.trans hab)]
+
+lemma terminal_clock_min (a : AngularLocalization B hm κ z u ε)
+    (b : AngularLocalization B hm κ z v ε) (huv : u ≤ v) (ω : Ω) :
+    a.clock.S (a.τ ω) ω = min u (b.clock.S (b.τ ω) ω) := by
+  have ht := a.stopping_le b huv le_rfl ω
+  apply le_antisymm
+  · exact le_min (a.clock_upper ω) (a.terminal_clock_le b huv le_rfl ω)
+  · rcases a.boundary ω with hc | ha
+    · exact (min_le_left _ _).trans_eq hc.symm
+    · have he : a.τ ω = b.τ ω := by
+        apply le_antisymm ht
+        by_contra hn
+        have hh := (b.before ω _ (lt_of_not_ge hn)).2
+        rw [← a.synchronize b ω (a.τ ω) le_rfl ht, ha] at hh
+        exact lt_irrefl _ hh
+      have hq := a.synchronize b ω (a.τ ω) le_rfl ht
+      change min u (logRadiusClock _ _) ≤ logRadiusClock _ _
+      rw [hq, he]
+      exact min_le_right _ _
+
+lemma radiusKernel_eq_below (a : AngularLocalization B hm κ z u ε)
+    (b : AngularLocalization B hm κ z v ε) (huv : u ≤ v)
+    (p α s : ℝ) (hs0 : 0 ≤ s) (hsu : s ≤ u) (ω : Ω) :
+    a.radiusKernel p α s ω = b.radiusKernel p α s ω := by
+  by_cases hs : s ≤ a.clock.S (a.τ ω) ω
+  · exact a.radiusKernel_eq_of_live b huv le_rfl p α s hs0 ω hs
+  · have hsb : ¬ s ≤ b.clock.S (b.τ ω) ω := by
+      intro h
+      exact hs (by rw [a.terminal_clock_min b huv ω]; exact le_min hsu h)
+    rw [radiusKernel, ite_eq_right hs, radiusKernel, ite_eq_right hsb]
+
+lemma radiusKernel_uniform_bound (a : AngularLocalization B hm κ z u ε)
+    (p α : ℝ) (hp : 0 ≤ p) (hα : 0 ≤ α) (s : ℝ) (ω : Ω) :
+    a.radiusKernel p α s ω ≤ Real.exp (p * (u - Real.log 2 - Real.log z.im)) := by
+  unfold radiusKernel
+  split_ifs
+  · have ht := a.clock.inverse_mem s ω
+    have hh := a.inside ω (a.clock.inverse s ω).toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)
+    rw [Real.coe_toNNReal _ ht.1] at hh
+    exact powerLogDensity_le_radius κ p α (Real.log z.im) u hp hα _ _ hh.1
+  · exact (Real.exp_pos _).le
+
+end AngularLocalization
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+ 
+
+noncomputable def limitRadiusKernel (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (p α s : ℝ) (ω : Ω) : ℝ :=
+  if h : ∃ n, max s 0 ≤ (a n).clock.S ((a n).τ ω) ω then
+    (a (Nat.find h)).radiusKernel p α (max s 0) ω else 0
+
+lemma radiusKernel_eventually_limit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (p α s : ℝ) (ω : Ω) :
+    ∀ᶠ n in atTop, (a n).radiusKernel p α (max s 0) ω = limitRadiusKernel a p α s ω := by
+  unfold limitRadiusKernel
+  split_ifs with hh
+  · filter_upwards [eventually_ge_atTop (Nat.find hh)] with n hn
+    exact ((a (Nat.find hh)).radiusKernel_eq_of_live (a n) le_rfl (hε hn)
+      p α _ (le_max_right _ _) ω (Nat.find_spec hh)).symm
+  · apply Eventually.of_forall
+    intro n
+    have hn : ¬ max s 0 ≤ (a n).clock.S ((a n).τ ω) ω := fun h => hh ⟨n, h⟩
+    rw [AngularLocalization.radiusKernel, ite_eq_right hn]
+
+lemma limitRadiusKernel_nonneg
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (p α s : ℝ) (ω : Ω) :
+    0 ≤ limitRadiusKernel a p α s ω := by
+  unfold limitRadiusKernel
+  split_ifs
+  · exact AngularLocalization.radiusKernel_nonneg _ _ _ _ _
+  · rfl
+
+lemma limitRadiusKernel_measurable
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (p α : ℝ) :
+    Measurable (Function.uncurry (limitRadiusKernel a p α)) := by
+  apply measurable_of_tendsto_metrizable
+    (fun n => ((a n).radiusKernel_measurable hBm p α).comp
+      ((measurable_fst.max measurable_const).prodMk measurable_snd))
+  apply tendsto_pi_nhds.mpr
+  intro q
+  apply tendsto_const_nhds.congr'
+  filter_upwards [radiusKernel_eventually_limit a hε p α q.1 q.2] with n hn
+  exact hn.symm
+
+lemma limitRadiusKernel_uniform_bound
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (p α : ℝ) (hp : 0 ≤ p) (hα : 0 ≤ α) (s : ℝ) (ω : Ω) :
+    limitRadiusKernel a p α s ω ≤ Real.exp (p * (u - Real.log 2 - Real.log z.im)) := by
+  unfold limitRadiusKernel
+  split_ifs
+  · exact AngularLocalization.radiusKernel_uniform_bound _ _ _ hp hα _ _
+  · exact (Real.exp_pos _).le
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ}
+
+def hit (a : AngularLocalization B hm κ z u ε) : Set Ω :=
+  {ω | logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω) = u}
+
+lemma measurable_hit (a : AngularLocalization B hm κ z u ε) :
+    MeasurableSet[a.stopping.measurableSpace] a.hit :=
+  measurableSet_eq_fun a.clock_stopped_measurable measurable_const
+
+noncomputable def hitTime (a : AngularLocalization B hm κ z u ε) (ω : Ω) : WithTop ℝ≥0 :=
+  if ω ∈ a.hit then (a.τ ω : WithTop ℝ≥0) else ⊤
+
+lemma hitTime_stopping (a : AngularLocalization B hm κ z u ε) :
+    IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) a.hitTime := by
+  intro t
+  have he : {ω | a.hitTime ω ≤ (t : WithTop ℝ≥0)} = a.hit ∩ {ω | (a.τ ω : WithTop ℝ≥0) ≤ t} := by
+    ext ω
+    simp only [hitTime, mem_ofPred_eq, mem_inter_iff]
+    split_ifs with h
+    · simp only [h, true_and]
+    · simp only [h, false_and, iff_false]
+      exact WithTop.not_top_le_coe t
+  rw [he]
+  exact a.measurable_hit.2 t
+
+end AngularLocalization
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+ 
+
+noncomputable def localizedRadiusStop
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) : WithTop ℝ≥0 :=
+  ⨅ n, (a n).hitTime ω
+
+lemma localizedRadiusStop_eq_hit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) (n : ℕ)
+    (hn : ω ∈ (a n).hit) : localizedRadiusStop a ω = ((a n).τ ω : WithTop ℝ≥0) := by
+  apply le_antisymm
+  · exact (iInf_le _ n).trans_eq (by rw [AngularLocalization.hitTime, ite_eq_left hn])
+  · apply le_iInf
+    intro m
+    by_cases h : ω ∈ (a m).hit
+    · have he := (a m).hit_time_unique (a n) ω h hn
+      rw [AngularLocalization.hitTime, ite_eq_left h, he]
+    · rw [AngularLocalization.hitTime, ite_eq_right h]
+      exact le_top
+
+lemma localizedRadiusStop_eq_top
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω)
+    (hn : ∀ n, ω ∉ (a n).hit) : localizedRadiusStop a ω = ⊤ := by
+  simp [localizedRadiusStop, AngularLocalization.hitTime, hn]
+
+lemma localizedRadiusStop_le_iff
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω) (t : ℝ≥0) :
+    localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0) ↔ ∃ n, ω ∈ (a n).hit ∧ (a n).τ ω ≤ t := by
+  constructor
+  · intro h
+    by_cases he : ∃ n, ω ∈ (a n).hit
+    · obtain ⟨n, hn⟩ := he
+      rw [localizedRadiusStop_eq_hit a ω n hn, WithTop.coe_le_coe] at h
+      exact ⟨n, hn, h⟩
+    · rw [localizedRadiusStop_eq_top a ω (not_exists.mp he)] at h
+      exact False.elim (WithTop.coe_ne_top (a := t) (top_le_iff.mp h))
+  · rintro ⟨n, hn, ht⟩
+    rw [localizedRadiusStop_eq_hit a ω n hn, WithTop.coe_le_coe]
+    exact ht
+
+lemma localizedRadiusStop_stopping
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) :
+    IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (localizedRadiusStop a) := by
+  intro t
+  have he : {ω | localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0)} =
+      ⋃ n, ((a n).hit ∩ {ω | ((a n).τ ω : WithTop ℝ≥0) ≤ t}) := by
+    ext ω
+    simp only [mem_ofPred_eq, mem_iUnion, mem_inter_iff, WithTop.coe_le_coe]
+    exact localizedRadiusStop_le_iff a ω t
+  rw [he]
+  exact MeasurableSet.iUnion fun n => (a n).measurable_hit.2 t
+
+lemma localizedRadiusStop_history_hit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (n : ℕ) (A : Set Ω)
+    (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
+    MeasurableSet[(a n).stopping.measurableSpace] (A ∩ (a n).hit) := by
+  refine ⟨hA.1.inter (a n).measurable_hit.1, ?_⟩
+  intro t
+  have he : (A ∩ (a n).hit) ∩ {ω | ((a n).τ ω : WithTop ℝ≥0) ≤ t} =
+      (A ∩ {ω | localizedRadiusStop a ω ≤ (t : WithTop ℝ≥0)}) ∩
+        ((a n).hit ∩ {ω | ((a n).τ ω : WithTop ℝ≥0) ≤ t}) := by
+    ext ω
+    simp only [mem_inter_iff, mem_ofPred_eq]
+    constructor
+    · rintro ⟨⟨hωA, hωn⟩, hωt⟩
+      exact ⟨⟨hωA, (localizedRadiusStop_eq_hit a ω n hωn) ▸ hωt⟩, hωn, hωt⟩
+    · rintro ⟨⟨hωA, _⟩, hωn, hωt⟩
+      exact ⟨⟨hωA, hωn⟩, hωt⟩
+  rw [he]
+  exact (hA.2 t).inter ((a n).measurable_hit.2 t)
+
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma integrable_monotone_limit_of_integral_le
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {f : ℕ → Ω → ℝ} {F : Ω → ℝ}
+    (hf : ∀ n, Integrable (f n) P)
+    (hF : AEStronglyMeasurable F P)
+    (hn : ∀ n ω, 0 ≤ f n ω) (hFn : ∀ ω, 0 ≤ F ω)
+    (hmono : ∀ ω, Monotone (fun n => f n ω))
+    (hlim : ∀ ω, Tendsto (fun n => f n ω) atTop (𝓝 (F ω)))
+    (C : ℝ) (hbound : ∀ n, (∫ ω, f n ω ∂P) ≤ C) :
+    Integrable F P ∧
+      Tendsto (fun n => ∫ ω, f n ω ∂P) atTop (𝓝 (∫ ω, F ω ∂P)) := by
+  have hl : Tendsto (fun n => ∫⁻ ω, ENNReal.ofReal (f n ω) ∂P) atTop
+      (𝓝 (∫⁻ ω, ENNReal.ofReal (F ω) ∂P)) := by
+    apply lintegral_tendsto_of_tendsto_of_monotone
+      (fun n => (hf n).aemeasurable.ennreal_ofReal)
+    · exact Eventually.of_forall fun ω n m hnm => ENNReal.ofReal_le_ofReal (hmono ω hnm)
+    · exact Eventually.of_forall fun ω => ENNReal.continuous_ofReal.continuousAt.tendsto.comp (hlim ω)
+  have hle : (∫⁻ ω, ENNReal.ofReal (F ω) ∂P) ≤ ENNReal.ofReal C := by
+    apply le_of_tendsto hl
+    apply Eventually.of_forall
+    intro n
+    rw [← ofReal_integral_eq_lintegral_ofReal (hf n) (Eventually.of_forall (hn n))]
+    exact ENNReal.ofReal_le_ofReal (hbound n)
+  have hFi : Integrable F P := by
+    refine ⟨hF, ?_⟩
+    rw [hasFiniteIntegral_iff_enorm]
+    have he : (fun ω => ‖F ω‖ₑ) = fun ω => ENNReal.ofReal (F ω) := by
+      funext ω
+      simp [Real.enorm_eq_ofReal_abs, abs_of_nonneg (hFn ω)]
+    rw [he]
+    exact hle.trans_lt ENNReal.ofReal_lt_top
+  exact ⟨hFi, integral_tendsto_of_tendsto_of_monotone hf hFi
+    (Eventually.of_forall hmono) (Eventually.of_forall hlim)⟩
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/BoundaryDensity.lean b/lean/OAI/Probability/SLE/BoundaryDensity.lean
index dfb0d8e46ae35e9ee274096643a2946f88c4dec4..2517492f1f400ba80dce2db84c730cb283bf5558 100644
--- a/lean/OAI/Probability/SLE/BoundaryDensity.lean
+++ b/lean/OAI/Probability/SLE/BoundaryDensity.lean
@@ -1,743 +1,743 @@
-import Mathlib
-import OAI.Probability.SLE.LoewnerReflection
-
-namespace OAI
-
-/-! Boundary Density for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-noncomputable def smoothPositiveGap (a x : ℝ) : ℝ :=
-  a + (x-a)*(1-Real.smoothTransition ((2*a-x)/a))
-
-lemma smoothPositiveGap_pos {a : ℝ} (ha : 0 < a) (x : ℝ) :
-    0 < smoothPositiveGap a x := by
-  by_cases hx : x ≤ a
-  · have he : 1 ≤ (2*a-x)/a := (le_div_iff₀ ha).mpr (by linarith)
-    simp only [smoothPositiveGap,Real.smoothTransition.one_of_one_le he,sub_self,mul_zero,add_zero]
-    exact ha
-  · have ht := Real.smoothTransition.le_one ((2*a-x)/a)
-    have hm : 0 ≤ (x-a)*(1-Real.smoothTransition ((2*a-x)/a)) := mul_nonneg (by linarith) (by linarith)
-    unfold smoothPositiveGap
-    linarith
-
-lemma smoothPositiveGap_eq {a x : ℝ} (ha : 0 < a) (hx : 2*a ≤ x) :
-    smoothPositiveGap a x = x := by
-  have he : (2*a-x)/a ≤ 0 := div_nonpos_of_nonpos_of_nonneg (by linarith) ha.le
-  simp only [smoothPositiveGap,Real.smoothTransition.zero_of_nonpos he,sub_zero,mul_one]
-  ring
-
-lemma smoothPositiveGap_contDiff (a : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (smoothPositiveGap a) := by
-  unfold smoothPositiveGap
-  fun_prop
-
-noncomputable def boundaryDensity (κ α β : ℝ) (q : ℂ) (b : ℝ) : ℝ :=
-  Real.exp (-β*α*q.im - β*Real.log (q.re-Real.sqrt κ*b))
-
-noncomputable def smoothBoundaryDensity (κ α β a : ℝ) (q : ℂ) (b : ℝ) : ℝ :=
-  Real.exp (-β*α*q.im - β*Real.log (smoothPositiveGap a (q.re-Real.sqrt κ*b)))
-
-lemma smoothBoundaryDensity_contDiff (κ α β : ℝ) {a : ℝ} (ha : 0 < a) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => smoothBoundaryDensity κ α β a p.1 p.2) := by
-  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => p.1.re) :=
-    Complex.reCLM.contDiff.comp contDiff_fst
-  have him : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => p.1.im) :=
-    Complex.imCLM.contDiff.comp contDiff_fst
-  have hg : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => smoothPositiveGap a (p.1.re-Real.sqrt κ*p.2)) :=
-    (smoothPositiveGap_contDiff a).comp (hre.sub (contDiff_const.mul contDiff_snd))
-  exact ((contDiff_const.mul him).sub (contDiff_const.mul
-    (hg.log (fun p => (smoothPositiveGap_pos ha _).ne')))).exp
-
-lemma smoothBoundaryDensity_eq {κ α β a : ℝ} {q : ℂ} {b : ℝ}
-    (ha : 0 < a) (hx : 2*a ≤ q.re-Real.sqrt κ*b) :
-    smoothBoundaryDensity κ α β a q b = boundaryDensity κ α β q b := by
-  simp only [smoothBoundaryDensity,boundaryDensity,smoothPositiveGap_eq ha hx]
-
-lemma smoothBoundaryDensity_eventuallyEq (κ α β : ℝ) {a : ℝ} (ha : 0 < a)
-    (p : ℂ × ℝ) (hp : 2*a < p.1.re-Real.sqrt κ*p.2) :
-    (fun q : ℂ × ℝ => smoothBoundaryDensity κ α β a q.1 q.2) =ᶠ[𝓝 p]
-      (fun q : ℂ × ℝ => boundaryDensity κ α β q.1 q.2) := by
-  have he : ∀ᶠ q : ℂ × ℝ in 𝓝 p, 2*a < q.1.re-Real.sqrt κ*q.2 :=
-    ((Complex.continuous_re.comp continuous_fst).sub
-      (continuous_const.mul continuous_snd)).continuousAt.eventually (lt_mem_nhds hp)
-  filter_upwards [he] with q hq
-  simp only [smoothBoundaryDensity,boundaryDensity,smoothPositiveGap_eq ha hq.le]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma boundaryDensity_contDiffAt (κ α β : ℝ) (p : ℂ × ℝ)
-    (hp : p.1.re-Real.sqrt κ*p.2 ≠ 0) :
-    ContDiffAt ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => boundaryDensity κ α β r.1 r.2) p := by
-  have hre : ContDiffAt ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => r.1.re) p :=
-    (Complex.reCLM.contDiff.comp contDiff_fst).contDiffAt
-  have him : ContDiffAt ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => r.1.im) p :=
-    (Complex.imCLM.contDiff.comp contDiff_fst).contDiffAt
-  exact ((contDiffAt_const.mul him).sub (contDiffAt_const.mul
-    ((hre.sub (contDiffAt_const.mul contDiffAt_snd)).log hp))).exp
-
-lemma boundaryDensity_spatial_derivative (κ α β : ℝ) (q : ℂ) (b : ℝ)
-    (hp : q.re-Real.sqrt κ*b ≠ 0) :
-    HasDerivAt (boundaryDensity κ α β q)
-      (boundaryDensity κ α β q b*(β*Real.sqrt κ/(q.re-Real.sqrt κ*b))) b := by
-  have hX := (hasDerivAt_const b q.re).sub ((hasDerivAt_id b).const_mul (Real.sqrt κ))
-  have hd := ((hasDerivAt_const b (-β*α*q.im)).sub ((hX.log hp).const_mul β)).exp
-  convert hd using 1 <;> try rfl
-  dsimp only [boundaryDensity,Pi.sub_apply,Pi.mul_apply,Pi.div_apply,Function.comp_def,id_eq]
-  ring
-
-lemma boundaryDensity_spatial_second (κ α β : ℝ) (hκ : 0 ≤ κ) (q : ℂ) (b : ℝ)
-    (hp : q.re-Real.sqrt κ*b ≠ 0) :
-    iteratedDeriv 2 (boundaryDensity κ α β q) b =
-      boundaryDensity κ α β q b*κ*β*(β+1)/(q.re-Real.sqrt κ*b)^2 := by
-  have hn : ∀ᶠ y : ℝ in 𝓝 b, q.re-Real.sqrt κ*y ≠ 0 :=
-    (continuous_const.sub (continuous_const.mul continuous_id)).continuousAt.eventually_ne hp
-  have hd : deriv (boundaryDensity κ α β q) =ᶠ[𝓝 b]
-      (fun y => boundaryDensity κ α β q y*(β*Real.sqrt κ/(q.re-Real.sqrt κ*y))) := by
-    filter_upwards [hn] with y hy
-    exact (boundaryDensity_spatial_derivative κ α β q y hy).deriv
-  have hX := (hasDerivAt_const b q.re).sub ((hasDerivAt_id b).const_mul (Real.sqrt κ))
-  have hh := (boundaryDensity_spatial_derivative κ α β q b hp).mul
-    ((hasDerivAt_const b (β*Real.sqrt κ)).div hX hp)
-  dsimp only [Pi.mul_apply,Pi.div_apply,Pi.sub_apply,id_eq] at hh
-  change HasDerivAt (fun y => boundaryDensity κ α β q y*(β*Real.sqrt κ/(q.re-Real.sqrt κ*y))) _ b at hh
-  rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one,hd.deriv_eq,hh.deriv]
-  have hs := Real.sq_sqrt hκ
-  generalize boundaryDensity κ α β q b = f
-  generalize q.re-Real.sqrt κ*b = X
-  generalize Real.sqrt κ = r at *
-  rw [← hs]
-  field_simp
-  ring
-
-lemma boundaryDensity_directional (κ α β : ℝ) (q v : ℂ) (b : ℝ)
-    (hp : q.re-Real.sqrt κ*b ≠ 0) :
-    fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b) (v,0) =
-      boundaryDensity κ α β q b * (-β*α*v.im - β*v.re/(q.re-Real.sqrt κ*b)) := by
-  have hr : HasDerivAt (fun t : ℝ => (q+t • v).re) v.re 0 := by
-    simpa only [Complex.add_re,Complex.smul_re,smul_eq_mul,id_eq,one_mul] using
-      ((hasDerivAt_id (0:ℝ)).mul_const v.re).const_add q.re
-  have hi : HasDerivAt (fun t : ℝ => (q+t • v).im) v.im 0 := by
-    simpa only [Complex.add_im,Complex.smul_im,smul_eq_mul,id_eq,one_mul] using
-      ((hasDerivAt_id (0:ℝ)).mul_const v.im).const_add q.im
-  have hd := ((hi.const_mul (-β*α)).sub
-    (((hr.sub_const (Real.sqrt κ*b)).log (by simpa only [zero_smul,add_zero] using hp)).const_mul β)).exp
-  have hline : HasDerivAt (fun t : ℝ => boundaryDensity κ α β (q+t • v) b)
-      (boundaryDensity κ α β q b * (-β*α*v.im-β*v.re/(q.re-Real.sqrt κ*b))) 0 := by
-    convert hd using 1 <;> try rfl
-    simp only [zero_smul,add_zero,boundaryDensity,Pi.sub_apply]
-    ring
-  have haff : HasDerivAt (fun t : ℝ => (q+t • v,b)) (v,0) 0 := by
-    simpa only [id_eq,one_smul] using (((hasDerivAt_id (0:ℝ)).smul_const v).const_add q).prodMk (hasDerivAt_const 0 b)
-  have hfd : HasFDerivAt (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2)
-      (fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b)) (q+(0:ℝ) • v,b) := by
-    simpa only [zero_smul,add_zero] using ((boundaryDensity_contDiffAt κ α β (q,b) hp).differentiableAt (by simp)).hasFDerivAt
-  have hh := hfd.comp_hasDerivAt 0 haff
-  exact hh.unique hline
-
-lemma boundaryDensity_generator (κ α β : ℝ) (hκ : 0 ≤ κ) (q : ℂ) (b : ℝ)
-    (hp : q.re-Real.sqrt κ*b ≠ 0) :
-    fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b)
-      (((2/(q.re-Real.sqrt κ*b):ℝ):ℂ) + (((q.re-Real.sqrt κ*b)⁻¹^2:ℝ):ℂ)*I,0) +
-      (1/2:ℝ)*iteratedDeriv 2 (boundaryDensity κ α β q) b =
-      boundaryDensity κ α β q b * β * (-α-2+κ*(β+1)/2)/(q.re-Real.sqrt κ*b)^2 := by
-  rw [boundaryDensity_directional κ α β q _ b hp,boundaryDensity_spatial_second κ α β hκ q b hp]
-  simp only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,Complex.I_im,mul_one,
-    Complex.I_re,mul_zero,add_zero,zero_add,Complex.add_re,Complex.mul_re,sub_zero]
-  field_simp
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma exists_compact_boundary_field (κ α β a R : ℝ) (ha : 0 < a) (hR : 0 ≤ R) :
-    ∃ φ : ℂ × ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
-      (∀ q b, ‖(q,b)‖ ≤ R → 2*a < q.re-Real.sqrt κ*b → φ (q,b) = boundaryDensity κ α β q b) ∧
-      (∀ q b, ‖(q,b)‖ ≤ R → 2*a < q.re-Real.sqrt κ*b →
-        fderiv ℝ φ (q,b) = fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b)) ∧
-      (∀ q b, ‖(q,b)‖ ≤ R → 2*a < q.re-Real.sqrt κ*b →
-        iteratedDeriv 2 (fun x => φ (q,x)) b = iteratedDeriv 2 (boundaryDensity κ α β q) b) := by
-  let cut : ContDiffBump (0 : ℂ × ℝ) := ⟨R+1,R+2,by linarith,by linarith⟩
-  let φ (p : ℂ × ℝ) := cut p*smoothBoundaryDensity κ α β a p.1 p.2
-  have hnear (p : ℂ × ℝ) (hball : ‖p‖ ≤ R) (hp : 2*a < p.1.re-Real.sqrt κ*p.2) :
-      φ =ᶠ[𝓝 p] (fun z : ℂ × ℝ => boundaryDensity κ α β z.1 z.2) := by
-    have hb : p ∈ Metric.ball (0:ℂ × ℝ) cut.rIn := by
-      simpa only [Metric.mem_ball,dist_zero_right] using (show ‖p‖ < R+1 by linarith)
-    filter_upwards [cut.eventuallyEq_one_of_mem_ball hb,smoothBoundaryDensity_eventuallyEq κ α β ha p hp] with z hz he
-    simp only [φ,hz,Pi.one_apply,one_mul,he]
-  refine ⟨φ,cut.contDiff.mul (smoothBoundaryDensity_contDiff κ α β ha),cut.hasCompactSupport.mul_right,
-    (fun q b h h' => (hnear (q,b) h h').eq_of_nhds),?_,?_⟩
-  · intro q b h h'
-    exact (hnear (q,b) h h').fderiv_eq
-  · intro q b h h'
-    exact ((hnear (q,b) h h').comp_tendsto ((continuous_const.prodMk continuous_id).tendsto b)).iteratedDeriv_eq 2
-
-lemma compact_boundary_drift_zero (κ α β a : ℝ) (hκ : 0 < κ) (ha : 0 < a)
-    (hβ : -α-2+κ*(β+1)/2 = 0) (φ : ℂ × ℝ → ℝ) (q : ℂ) (b : ℝ)
-    (hD : fderiv ℝ φ (q,b) = fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b))
-    (h2 : iteratedDeriv 2 (fun x => φ (q,x)) b = iteratedDeriv 2 (boundaryDensity κ α β q) b)
-    (hx : 2*a < q.re-Real.sqrt κ*b) :
-    fderiv ℝ φ (q,b) (boundaryClockField a (Real.sqrt κ*b) q,0) +
-      (1/2:ℝ)*iteratedDeriv 2 (fun x => φ (q,x)) b = 0 := by
-  have hgap : a ≤ q.re-Real.sqrt κ*b := by linarith
-  have he : boundaryClockField a (Real.sqrt κ*b) q =
-      ((2/(q.re-Real.sqrt κ*b):ℝ):ℂ) + (((q.re-Real.sqrt κ*b)⁻¹^2:ℝ):ℂ)*I := by
-    simp only [boundaryClockField,positiveBoundaryField_eq hgap,max_eq_right hgap]
-  rw [hD,h2,he,boundaryDensity_generator κ α β hκ.le q b (by linarith),hβ,mul_zero,zero_div]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-theorem BoundaryClockSolution.stopped_boundary_density
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ a : ℝ} {q₀ : ℂ}
-    (c : BoundaryClockSolution B hm κ a q₀)
-    (hκ : 0 < κ) (ha : 0 < a) (α β : ℝ)
-    (hβ : -α-2+κ*(β+1)/2 = 0)
-    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hinside : ∀ ω t, t ≤ τ ω →
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ 2*a < (c.Q t ω).re-Real.sqrt κ*B t ω) :
-    Integrable (fun ω => boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
-      (∫ ω, boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
-        boundaryDensity κ α β q₀ 0 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_boundary_field κ α β a R ha hR
-  let L₀ : ℝ≥0 := ⟨2*a⁻¹+a⁻¹^2,by positivity⟩
-  have hL₀ := boundaryClockField_bound ha
-  let V (t : ℝ) (ω : Ω) :=
-    boundaryClockField a (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
-  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
-    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
-      measurable_snd)).const_mul _
-  have hpair : Measurable (fun p : ℝ × Ω =>
-      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
-  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
-    change Measurable ((fun p : ℝ × ℂ => boundaryClockField a p.1 p.2) ∘
-      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
-    exact (continuous_boundaryClockField ha).measurable.comp hpair
-  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
-    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
-  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
-    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
-    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
-  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
-  let drift (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
-  have hzero (ω : Ω) (u : ℝ) (hu : 0 ≤ u) (hut : u < (τ ω : ℝ)) :
-      brownianFieldDrift B F drift u ω = 0 := by
-    have ht : u.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr hut.le
-    obtain ⟨hball, hgap⟩ := hinside ω u.toNNReal ht
-    change fderiv ℝ φ (c.Q (u.toNNReal : ℝ) ω, B u.toNNReal ω)
-      (V (u.toNNReal : ℝ) ω, 0) + _ = 0
-    dsimp only [V]
-    rw [Real.toNNReal_coe]
-    exact compact_boundary_drift_zero κ α β a hκ ha hβ φ _ _
-      (hφD _ _ hball hgap) (hφ2 _ _ hball hgap) hgap
-  have hopt := brownian_time_field_optional hB hm hBm hBc K F drift hfield τ hτ T hb hzero
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
-  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
-    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
-  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
-    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
-  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
-    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
-    exact Eventually.of_forall fun ω => by
-      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
-        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
-  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
-      boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω) :=
-    hφeq _ _ (hinside ω (τ ω) le_rfl).1 (hinside ω (τ ω) le_rfl).2
-  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
-  calc
-    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
-      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
-    _ = ∫ ω, F 0 ω (B 0 ω) ∂P := hopt
-    _ = ∫ _, boundaryDensity κ α β q₀ 0 ∂P := by
-      apply integral_congr_ae
-      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
-      change φ (c.Q 0 ω, B 0 ω) = _
-      have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
-        simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
-      rw [hφeq _ _ hh (hinside ω 0 zero_le).2, c.initial ω, hB0]
-    _ = boundaryDensity κ α β q₀ 0 := by simp
-
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma continuous_hittingBtwn_mem {Ω : Type*} (H : ℝ≥0 → Ω → ℝ)
-    (hHc : ∀ ω, Continuous (fun t => H t ω)) (T : ℝ≥0) (ω : Ω)
-    (he : ∃ j ∈ Icc 0 T, H j ω ∈ ({0} : Set ℝ)) :
-    H (hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω) ω = 0 := by
-  have hc : IsClosed (Icc 0 T ∩ {t : ℝ≥0 | H t ω ∈ ({0} : Set ℝ)}) :=
-    isClosed_Icc.inter (isClosed_singleton.preimage (hHc ω))
-  have hn : (Icc 0 T ∩ {t : ℝ≥0 | H t ω ∈ ({0} : Set ℝ)}).Nonempty := by
-    obtain ⟨j, hj, hh⟩ := he
-    exact ⟨j, hj, hh⟩
-  have hh := hc.csInf_mem hn (bddBelow_Icc.inter_of_left)
-  simpa only [hittingBtwn_def, ite_eq_left he, mem_inter_iff, mem_ofPred_eq, mem_singleton_iff] using hh.2
-
-lemma continuous_zero_iff_dense
-    (H : ℝ≥0 → ℝ) (hHc : Continuous H) (hHn : ∀ t, 0 ≤ H t) (t : ℝ≥0) :
-    (∃ s ≤ t, H s = 0) ↔
-      ∀ n : ℕ, ∃ k : ℕ,
-        H (TopologicalSpace.denseSeq (Iic t) k) < 1 / ((n : ℝ) + 1) := by
-  let : Nonempty (Iic t) := ⟨⟨0, show (0 : ℝ≥0) ≤ t from zero_le⟩⟩
-  constructor
-  · rintro ⟨s, hs, he⟩ n
-    have hpos : (0 : ℝ) < 1 / ((n : ℝ) + 1) := by positivity
-    exact (TopologicalSpace.denseRange_denseSeq (Iic t)).exists_mem_open
-      (isOpen_lt (hHc.comp continuous_subtype_val) continuous_const)
-      ⟨⟨s, hs⟩, by simpa only [mem_ofPred_eq, Function.comp_apply, he] using hpos⟩
-  · intro he
-    have hcc : IsCompact (Iic t) := by
-      have heq : Iic t = Icc 0 t := by ext s; simp
-      rw [heq]
-      exact isCompact_Icc
-    obtain ⟨s, hs, hmin⟩ := hcc.exists_isMinOn
-      ⟨0, show (0 : ℝ≥0) ≤ t from zero_le⟩ hHc.continuousOn
-    refine ⟨s, hs, le_antisymm ?_ (hHn s)⟩
-    apply ge_of_tendsto (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
-    exact Eventually.of_forall fun n => by
-      obtain ⟨k, hk⟩ := he n
-      exact (hmin (TopologicalSpace.denseSeq (Iic t) k).property).trans hk.le
-
-theorem continuous_nonnegative_first_zero_stopping
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
-    (H : ℝ≥0 → Ω → ℝ) (hHa : Adapted f H)
-    (hHc : ∀ ω, Continuous (fun t => H t ω))
-    (hHn : ∀ t ω, 0 ≤ H t ω) (T : ℝ≥0) :
-    IsStoppingTime f (fun ω => ((hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω : ℝ≥0) : WithTop ℝ≥0)) ∧
-    (∀ ω, hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω ≤ T) ∧
-    (∀ ω t, t < hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω → 0 < H t ω) ∧
-    (∀ ω, hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω < T → H (hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω) ω = 0) := by
-  classical
-  have hb (ω : Ω) := hittingBtwn_le (u := H) (s := ({0} : Set ℝ)) (n := 0) (m := T) ω
-  have hmem (ω : Ω) (h : hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω < T) :
-      H (hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω) ω = 0 := by
-    apply continuous_hittingBtwn_mem H hHc T ω
-    by_contra he
-    simp only [hittingBtwn_def, ite_eq_right he] at h
-    exact lt_irrefl _ h
-  refine ⟨?_, hb, ?_, hmem⟩
-  · intro t
-    by_cases hT : T ≤ t
-    · have he : {ω : Ω | ((hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} = univ := by
-        ext ω
-        simp only [mem_ofPred_eq, mem_univ, iff_true, WithTop.coe_le_coe]
-        exact (hb ω).trans hT
-      rw [he]
-      exact MeasurableSet.univ
-    · have ht : t < T := lt_of_not_ge hT
-      let : Nonempty (Iic t) := ⟨⟨0, show (0 : ℝ≥0) ≤ t from zero_le⟩⟩
-      have he (ω : Ω) : hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω ≤ t ↔ ∃ s ≤ t, H s ω = 0 := by
-        constructor
-        · intro h
-          exact ⟨_, h, hmem ω (h.trans_lt ht)⟩
-        · rintro ⟨s, hs, h0⟩
-          exact (hittingBtwn_le_of_mem (zero_le) (hs.trans ht.le) (by simpa using h0)).trans hs
-      have hset : {ω : Ω | ((hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
-          ⋂ n : ℕ, ⋃ k : ℕ, {ω : Ω | H (TopologicalSpace.denseSeq (Iic t) k) ω < 1 / ((n : ℝ) + 1)} := by
-        ext ω
-        simp only [mem_ofPred_eq, WithTop.coe_le_coe, mem_iInter, mem_iUnion]
-        exact (he ω).trans (continuous_zero_iff_dense (fun s => H s ω) (hHc ω) (fun s => hHn s ω) t)
-      rw [hset]
-      apply MeasurableSet.iInter
-      intro n
-      apply MeasurableSet.iUnion
-      intro k
-      have hm := (hHa (TopologicalSpace.denseSeq (Iic t) k)).mono
-        (f.mono (TopologicalSpace.denseSeq (Iic t) k).property) le_rfl
-      exact measurableSet_lt hm measurable_const
-  · intro ω t ht
-    have hn := notMem_of_lt_hittingBtwn ht (zero_le)
-    exact lt_of_le_of_ne (hHn t ω) (by simpa only [mem_singleton_iff, ne_eq, eq_comm] using hn)
-
-end SLEExactGauge
-end
-
- 
-section
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def logRadiusClock (η₀ : ℝ) (q : ℂ × ℝ) : ℝ :=
-  η₀ - q.1.im + q.2
-
-noncomputable def logAngleSq (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) : ℝ :=
-  Real.exp (2 * q.1.im) / greenLogNormSq κ q b
-
-noncomputable def logClockRate (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) : ℝ :=
-  4 * Real.exp (2 * q.1.im) / greenLogNormSq κ q b ^ 2
-
-lemma logAngleSq_pos (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    0 < logAngleSq κ q b :=
-  div_pos (Real.exp_pos _) (greenLogNormSq_pos κ q b)
-
-lemma logAngleSq_le_one (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    logAngleSq κ q b ≤ 1 := by
-  apply (div_le_one (greenLogNormSq_pos κ q b)).mpr
-  exact le_add_of_nonneg_left (sq_nonneg _)
-
-lemma continuous_logAngleSq (κ : ℝ) :
-    Continuous (fun p : (ℂ × ℝ) × ℝ => logAngleSq κ p.1 p.2) := by
-  have hi : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
-    Complex.continuous_im.comp (continuous_fst.fst)
-  have hr : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
-    Complex.continuous_re.comp (continuous_fst.fst)
-  exact (Real.continuous_exp.comp (continuous_const.mul hi)).div
-    (((hr.sub (continuous_const.mul continuous_snd)).pow 2).add
-      (Real.continuous_exp.comp (continuous_const.mul hi)))
-    (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
-
-lemma continuous_logRadiusClock (η₀ : ℝ) : Continuous (logRadiusClock η₀) :=
-  (continuous_const.sub (Complex.continuous_im.comp continuous_fst)).add continuous_snd
-
-lemma greenLogDensity_radius_angle (κ η₀ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    greenLogDensity κ q b =
-      Real.exp (codimension κ * (logRadiusClock η₀ q - Real.log 2 - η₀)) *
-        (logAngleSq κ q b) ^ (greenExponent κ / 2) := by
-  rw [Real.rpow_def_of_pos (logAngleSq_pos κ q b), ← Real.exp_add]
-  unfold logAngleSq
-  rw [Real.log_div (Real.exp_ne_zero _) (greenLogNormSq_pos κ q b).ne', Real.log_exp]
-  unfold greenLogDensity logRadiusClock
-  congr 1
-  ring
-
-lemma codimension_pos (κ : ℝ) (hκ8 : κ < 8) : 0 < codimension κ := by
-  dsimp [codimension, dimension]
-  linarith
-
-lemma greenExponent_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < greenExponent κ := by
-  unfold greenExponent
-  have : 1 < 8 / κ := (lt_div_iff₀ hκ).mpr (by simpa using hκ8)
-  linarith
-
-lemma greenLogDensity_le_at_angular_exit
-    (κ η₀ u ε : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (q : ℂ × ℝ) (b : ℝ)
-    (hclock : logRadiusClock η₀ q ≤ u) (hangle : logAngleSq κ q b ≤ ε) :
-    greenLogDensity κ q b ≤
-      Real.exp (codimension κ * (u - Real.log 2 - η₀)) * ε ^ (greenExponent κ / 2) := by
-  rw [greenLogDensity_radius_angle κ η₀ q b]
-  apply mul_le_mul
-  · exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (by linarith)
-      (codimension_pos κ hκ8).le)
-  · exact Real.rpow_le_rpow (logAngleSq_pos κ q b).le hangle
-      (div_nonneg (greenExponent_pos κ hκ hκ8).le (by norm_num))
-  · exact Real.rpow_nonneg (logAngleSq_pos κ q b).le _
-  · exact (Real.exp_pos _).le
-
-lemma logClockRate_pos (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    0 < logClockRate κ q b := by
-  unfold logClockRate
-  exact div_pos (mul_pos (by norm_num) (Real.exp_pos _))
-    (sq_pos_of_pos (greenLogNormSq_pos κ q b))
-
-lemma logClockRate_angle (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    logClockRate κ q b = 4 * logAngleSq κ q b ^ 2 / Real.exp (2 * q.1.im) := by
-  unfold logClockRate logAngleSq
-  field_simp [(Real.exp_pos (2 * q.1.im)).ne', (greenLogNormSq_pos κ q b).ne']
-
-lemma log_ode_im_derivative {Q : ℝ → ℂ × ℝ} {D : Set ℝ} {t κ b : ℝ}
-    (hQ : HasDerivWithinAt Q (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)) D t) :
-    HasDerivWithinAt (fun s => (Q s).1.im) (-2 / greenLogNormSq κ (Q t) b) D t := by
-  have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hQ.fst
-  simpa only [Function.comp_def, Complex.imCLM_apply,
-    (physicalLogLoewnerField_components κ (Q t) b).2.1] using hi
-
-lemma log_ode_clock_derivative {Q : ℝ → ℂ × ℝ} {D : Set ℝ} {t κ b : ℝ}
-    (η₀ : ℝ)
-    (hQ : HasDerivWithinAt Q (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)) D t) :
-    HasDerivWithinAt (fun s => logRadiusClock η₀ (Q s))
-      (logClockRate κ (Q t) b) D t := by
-  have hs : HasDerivWithinAt (fun s => (Q s).2)
-      (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)).2 D t := hQ.snd
-  have hd : HasDerivWithinAt (fun s => η₀ - (Q s).1.im + (Q s).2)
-      (-(-2 / greenLogNormSq κ (Q t) b) +
-        (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)).2) D t :=
-    ((log_ode_im_derivative hQ).const_sub η₀).add hs
-  have he : -(-2 / greenLogNormSq κ (Q t) b) +
-      (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)).2 = logClockRate κ (Q t) b := by
-    rw [(physicalLogLoewnerField_components κ (Q t) b).2.2]
-    unfold logClockRate
-    have hN := (greenLogNormSq_pos κ (Q t) b).ne'
-    field_simp
-    unfold greenLogNormSq
-    ring
-  simpa only [he, logRadiusClock, Pi.add_apply] using hd
-
-lemma log_height_clock_derivative_nonneg
-    (κ ε : ℝ) (hε : 0 < ε) (q : ℂ × ℝ) (b : ℝ)
-    (ha : ε ≤ logAngleSq κ q b) :
-    0 ≤ -2 / greenLogNormSq κ q b + logClockRate κ q b / (2 * ε) := by
-  have hN := greenLogNormSq_pos κ q b
-  have h1 : 1 ≤ logAngleSq κ q b / ε := (le_div_iff₀ hε).mpr (by simpa using ha)
-  have h2 := mul_le_mul_of_nonneg_left h1 (show 0 ≤ 2 / greenLogNormSq κ q b by positivity)
-  have he : logClockRate κ q b / (2 * ε) =
-      (2 / greenLogNormSq κ q b) * (logAngleSq κ q b / ε) := by
-    unfold logClockRate logAngleSq
-    field_simp
-    ring
-  rw [he, neg_div]
-  simpa only [mul_one, sub_eq_add_neg, add_comm] using sub_nonneg.mpr h2
-
-lemma log_clock_rate_lower (κ ε η₀ : ℝ) (hε : 0 < ε) (q : ℂ × ℝ) (b : ℝ)
-    (ha : ε ≤ logAngleSq κ q b) (hi : q.1.im ≤ η₀) :
-    4 * ε ^ 2 / Real.exp (2 * η₀) ≤ logClockRate κ q b := by
-  rw [logClockRate_angle]
-  have hs : ε ^ 2 ≤ logAngleSq κ q b ^ 2 := sq_le_sq₀ hε.le (logAngleSq_pos κ q b).le |>.mpr ha
-  have he : Real.exp (2 * q.1.im) ≤ Real.exp (2 * η₀) :=
-    Real.exp_le_exp.mpr (by linarith)
-  exact div_le_div₀ (by positivity) (by nlinarith) (Real.exp_pos _) he
-
- 
-
-theorem log_loewner_height_clock_bounds
-    {Q : ℝ → ℂ × ℝ} {B : ℝ → ℝ} {κ T ε : ℝ}
-    (hT : 0 ≤ T) (hε : 0 < ε) (hQ0 : (Q 0).2 = 0)
-    (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q
-      (physicalLogLoewnerField (Real.sqrt κ * B t) (Q t)) (Icc 0 T) t)
-    (hangle : ∀ t ∈ Icc 0 T, ε ≤ logAngleSq κ (Q t) (B t)) :
-    (Q T).1.im ≤ (Q 0).1.im ∧
-    (Q 0).1.im - logRadiusClock (Q 0).1.im (Q T) / (2 * ε) ≤ (Q T).1.im ∧
-    (4 * ε ^ 2 / Real.exp (2 * (Q 0).1.im)) * T ≤ logRadiusClock (Q 0).1.im (Q T) := by
-  let η₀ := (Q 0).1.im
-  let c := 4 * ε ^ 2 / Real.exp (2 * η₀)
-  have h0 : (0 : ℝ) ∈ Icc 0 T := ⟨le_rfl, hT⟩
-  have hTT : T ∈ Icc 0 T := ⟨hT, le_rfl⟩
-  have hQc : ContinuousOn Q (Icc 0 T) := fun t ht => (hQ t ht).continuousWithinAt
-  have hi : AntitoneOn (fun t => (Q t).1.im) (Icc 0 T) := by
-    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
-      (Complex.continuous_im.comp_continuousOn (continuous_fst.comp_continuousOn hQc))
-      (fun t ht => (log_ode_im_derivative (hQ t (interior_subset ht))).mono interior_subset)
-    intro t ht
-    exact div_nonpos_of_nonpos_of_nonneg (by norm_num)
-      (greenLogNormSq_pos κ (Q t) (B t)).le
-  have hib (t : ℝ) (ht : t ∈ Icc 0 T) : (Q t).1.im ≤ η₀ := hi h0 ht ht.1
-  have hS0 : logRadiusClock η₀ (Q 0) = 0 := by simp [logRadiusClock, η₀, hQ0]
-  have hSc : ContinuousOn (fun t => logRadiusClock η₀ (Q t)) (Icc 0 T) :=
-    (continuous_logRadiusClock η₀).comp_continuousOn hQc
-  have hlo : MonotoneOn (fun t => (Q t).1.im +
-      logRadiusClock η₀ (Q t) / (2 * ε)) (Icc 0 T) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
-      ((Complex.continuous_im.comp_continuousOn (continuous_fst.comp_continuousOn hQc)).add
-        (hSc.div_const _))
-      (fun t ht => ((log_ode_im_derivative (hQ t (interior_subset ht))).add
-        ((log_ode_clock_derivative η₀ (hQ t (interior_subset ht))).div_const _)).mono interior_subset)
-    intro t ht
-    exact log_height_clock_derivative_nonneg κ ε hε (Q t) (B t) (hangle t (interior_subset ht))
-  have hrate : MonotoneOn (fun t => logRadiusClock η₀ (Q t) - c * t) (Icc 0 T) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
-      (hSc.sub (continuous_const.mul continuous_id).continuousOn)
-      (fun t ht => ((log_ode_clock_derivative η₀ (hQ t (interior_subset ht))).sub
-        ((hasDerivWithinAt_id t (Icc 0 T)).const_mul c)).mono interior_subset)
-    intro t ht
-    have hb := log_clock_rate_lower κ ε η₀ hε (Q t) (B t)
-      (hangle t (interior_subset ht)) (hib t (interior_subset ht))
-    change 0 ≤ logClockRate κ (Q t) (B t) - c * 1
-    simpa only [mul_one] using sub_nonneg.mpr hb
-  refine ⟨hib T hTT, ?_, ?_⟩
-  · have hh := hlo h0 hTT hT
-    dsimp only at hh
-    rw [hS0, zero_div, add_zero] at hh
-    linarith
-  · have hh := hrate h0 hTT hT
-    dsimp only at hh
-    rw [hS0, mul_zero, sub_zero] at hh
-    linarith
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-structure ClippedLogSolution {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (κ lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) where
-  L : ℝ≥0
-  Q : ℝ → Ω → ℂ × ℝ
-  measurable : Measurable (fun p : ℝ × Ω => Q p.1 p.2)
-  lipschitz : ∀ ω, LipschitzWith L (fun t => Q t ω)
-  initial : ∀ ω, Q 0 ω = q₀
-  derivative : ∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω)
-    (clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (Q t ω))
-    (Ici 0) t
-  adapted : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-    (fun t : ℝ≥0 => Q t)
-
-lemma exists_clippedLogSolution
-    {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) :
-    Nonempty (ClippedLogSolution B hm κ lo hi hle q₀) := by
-  let U (t : ℝ≥0) (ω : Ω) := Real.sqrt κ * B t ω
-  have hUm : Measurable (fun p : ℝ≥0 × Ω => U p.1 p.2) := hBm.const_mul _
-  have hUc (ω : Ω) : Continuous (fun t => U t ω) := (hBc ω).const_mul _
-  obtain ⟨L, Q, hQm, hQl, hQ0, hQd, hQaU⟩ :=
-    clipped_log_loewner_global_adapted_solution U hUm hUc lo hi hle q₀
-  have hBa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) B :=
-    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
-  refine ⟨⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩⟩
-  intro t
-  apply (hQaU t).mono _ le_rfl
-  change (⨆ j ≤ t, MeasurableSpace.comap (U j) (borel ℝ)) ≤ _
-  refine iSup₂_le fun j hj => ?_
-  exact ((hBa.measurable_le hj).const_mul (Real.sqrt κ)).comap_le
-
-lemma compact_green_drift_zero
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (lo hi : ℝ) (hle : lo ≤ hi)
-    (φ : (ℂ × ℝ) × ℝ → ℝ) (q : ℂ × ℝ) (x : ℝ)
-    (hφD : fderiv ℝ φ (q, x) =
-      fderiv ℝ (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (q, x))
-    (hφ2 : iteratedDeriv 2 (fun y => φ (q, y)) x =
-      iteratedDeriv 2 (greenLogDensity κ q) x)
-    (hq : q.1.im ∈ Icc lo hi) :
-    fderiv ℝ φ (q, x)
-        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * x) q, 0) +
-      (1 / 2 : ℝ) * iteratedDeriv 2 (fun y => φ (q, y)) x = 0 := by
-  rw [hφD, hφ2, greenLogDensity_pair_directional,
-    clippedLogLoewnerField_eq_physical lo hi hle _ _ hq]
-  exact green_physical_generator_zero κ hκ hκ8 q x
-
- 
-
-theorem ClippedLogSolution.stopped_green
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hinside : ∀ ω t, t ≤ τ ω →
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
-    Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
-      (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
-        greenLogDensity κ q₀ 0 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
-  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) :=
-    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
-  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
-    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
-      measurable_snd)).const_mul _
-  have hpair : Measurable (fun p : ℝ × Ω =>
-      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
-  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
-    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
-  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
-    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
-  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
-    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
-    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
-  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
-  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
-  have hzero (ω : Ω) (u : ℝ) (hu : 0 ≤ u) (hut : u < (τ ω : ℝ)) :
-      brownianFieldDrift B F a u ω = 0 := by
-    have ht : u.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr hut.le
-    obtain ⟨hball, hlo, hhi⟩ := hinside ω u.toNNReal ht
-    change fderiv ℝ φ (c.Q (u.toNNReal : ℝ) ω, B u.toNNReal ω)
-      (V (u.toNNReal : ℝ) ω, 0) + _ = 0
-    dsimp only [V]
-    rw [Real.toNNReal_coe]
-    exact compact_green_drift_zero κ hκ hκ8 lo hi hle φ _ _
-      (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
-  have hopt := brownian_time_field_optional hB hm hBm hBc K F a hfield τ hτ T hb hzero
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
-  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
-    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
-  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
-    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
-  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
-    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
-    exact Eventually.of_forall fun ω => by
-      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
-        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
-  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
-      greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) :=
-    hφeq _ _ (hinside ω (τ ω) le_rfl).1
-  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
-  calc
-    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
-      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
-    _ = ∫ ω, F 0 ω (B 0 ω) ∂P := hopt
-    _ = ∫ _, greenLogDensity κ q₀ 0 ∂P := by
-      apply integral_congr_ae
-      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
-      change φ (c.Q 0 ω, B 0 ω) = _
-      have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
-        simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
-      rw [hφeq _ _ hh, c.initial ω, hB0]
-    _ = greenLogDensity κ q₀ 0 := by simp
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.LoewnerReflection
+
+namespace OAI
+
+/-! Boundary Density for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+noncomputable def smoothPositiveGap (a x : ℝ) : ℝ :=
+  a + (x-a)*(1-Real.smoothTransition ((2*a-x)/a))
+
+lemma smoothPositiveGap_pos {a : ℝ} (ha : 0 < a) (x : ℝ) :
+    0 < smoothPositiveGap a x := by
+  by_cases hx : x ≤ a
+  · have he : 1 ≤ (2*a-x)/a := (le_div_iff₀ ha).mpr (by linarith)
+    simp only [smoothPositiveGap,Real.smoothTransition.one_of_one_le he,sub_self,mul_zero,add_zero]
+    exact ha
+  · have ht := Real.smoothTransition.le_one ((2*a-x)/a)
+    have hm : 0 ≤ (x-a)*(1-Real.smoothTransition ((2*a-x)/a)) := mul_nonneg (by linarith) (by linarith)
+    unfold smoothPositiveGap
+    linarith
+
+lemma smoothPositiveGap_eq {a x : ℝ} (ha : 0 < a) (hx : 2*a ≤ x) :
+    smoothPositiveGap a x = x := by
+  have he : (2*a-x)/a ≤ 0 := div_nonpos_of_nonpos_of_nonneg (by linarith) ha.le
+  simp only [smoothPositiveGap,Real.smoothTransition.zero_of_nonpos he,sub_zero,mul_one]
+  ring
+
+lemma smoothPositiveGap_contDiff (a : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (smoothPositiveGap a) := by
+  unfold smoothPositiveGap
+  fun_prop
+
+noncomputable def boundaryDensity (κ α β : ℝ) (q : ℂ) (b : ℝ) : ℝ :=
+  Real.exp (-β*α*q.im - β*Real.log (q.re-Real.sqrt κ*b))
+
+noncomputable def smoothBoundaryDensity (κ α β a : ℝ) (q : ℂ) (b : ℝ) : ℝ :=
+  Real.exp (-β*α*q.im - β*Real.log (smoothPositiveGap a (q.re-Real.sqrt κ*b)))
+
+lemma smoothBoundaryDensity_contDiff (κ α β : ℝ) {a : ℝ} (ha : 0 < a) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => smoothBoundaryDensity κ α β a p.1 p.2) := by
+  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => p.1.re) :=
+    Complex.reCLM.contDiff.comp contDiff_fst
+  have him : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => p.1.im) :=
+    Complex.imCLM.contDiff.comp contDiff_fst
+  have hg : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℂ × ℝ => smoothPositiveGap a (p.1.re-Real.sqrt κ*p.2)) :=
+    (smoothPositiveGap_contDiff a).comp (hre.sub (contDiff_const.mul contDiff_snd))
+  exact ((contDiff_const.mul him).sub (contDiff_const.mul
+    (hg.log (fun p => (smoothPositiveGap_pos ha _).ne')))).exp
+
+lemma smoothBoundaryDensity_eq {κ α β a : ℝ} {q : ℂ} {b : ℝ}
+    (ha : 0 < a) (hx : 2*a ≤ q.re-Real.sqrt κ*b) :
+    smoothBoundaryDensity κ α β a q b = boundaryDensity κ α β q b := by
+  simp only [smoothBoundaryDensity,boundaryDensity,smoothPositiveGap_eq ha hx]
+
+lemma smoothBoundaryDensity_eventuallyEq (κ α β : ℝ) {a : ℝ} (ha : 0 < a)
+    (p : ℂ × ℝ) (hp : 2*a < p.1.re-Real.sqrt κ*p.2) :
+    (fun q : ℂ × ℝ => smoothBoundaryDensity κ α β a q.1 q.2) =ᶠ[𝓝 p]
+      (fun q : ℂ × ℝ => boundaryDensity κ α β q.1 q.2) := by
+  have he : ∀ᶠ q : ℂ × ℝ in 𝓝 p, 2*a < q.1.re-Real.sqrt κ*q.2 :=
+    ((Complex.continuous_re.comp continuous_fst).sub
+      (continuous_const.mul continuous_snd)).continuousAt.eventually (lt_mem_nhds hp)
+  filter_upwards [he] with q hq
+  simp only [smoothBoundaryDensity,boundaryDensity,smoothPositiveGap_eq ha hq.le]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma boundaryDensity_contDiffAt (κ α β : ℝ) (p : ℂ × ℝ)
+    (hp : p.1.re-Real.sqrt κ*p.2 ≠ 0) :
+    ContDiffAt ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => boundaryDensity κ α β r.1 r.2) p := by
+  have hre : ContDiffAt ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => r.1.re) p :=
+    (Complex.reCLM.contDiff.comp contDiff_fst).contDiffAt
+  have him : ContDiffAt ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => r.1.im) p :=
+    (Complex.imCLM.contDiff.comp contDiff_fst).contDiffAt
+  exact ((contDiffAt_const.mul him).sub (contDiffAt_const.mul
+    ((hre.sub (contDiffAt_const.mul contDiffAt_snd)).log hp))).exp
+
+lemma boundaryDensity_spatial_derivative (κ α β : ℝ) (q : ℂ) (b : ℝ)
+    (hp : q.re-Real.sqrt κ*b ≠ 0) :
+    HasDerivAt (boundaryDensity κ α β q)
+      (boundaryDensity κ α β q b*(β*Real.sqrt κ/(q.re-Real.sqrt κ*b))) b := by
+  have hX := (hasDerivAt_const b q.re).sub ((hasDerivAt_id b).const_mul (Real.sqrt κ))
+  have hd := ((hasDerivAt_const b (-β*α*q.im)).sub ((hX.log hp).const_mul β)).exp
+  convert hd using 1 <;> try rfl
+  dsimp only [boundaryDensity,Pi.sub_apply,Pi.mul_apply,Pi.div_apply,Function.comp_def,id_eq]
+  ring
+
+lemma boundaryDensity_spatial_second (κ α β : ℝ) (hκ : 0 ≤ κ) (q : ℂ) (b : ℝ)
+    (hp : q.re-Real.sqrt κ*b ≠ 0) :
+    iteratedDeriv 2 (boundaryDensity κ α β q) b =
+      boundaryDensity κ α β q b*κ*β*(β+1)/(q.re-Real.sqrt κ*b)^2 := by
+  have hn : ∀ᶠ y : ℝ in 𝓝 b, q.re-Real.sqrt κ*y ≠ 0 :=
+    (continuous_const.sub (continuous_const.mul continuous_id)).continuousAt.eventually_ne hp
+  have hd : deriv (boundaryDensity κ α β q) =ᶠ[𝓝 b]
+      (fun y => boundaryDensity κ α β q y*(β*Real.sqrt κ/(q.re-Real.sqrt κ*y))) := by
+    filter_upwards [hn] with y hy
+    exact (boundaryDensity_spatial_derivative κ α β q y hy).deriv
+  have hX := (hasDerivAt_const b q.re).sub ((hasDerivAt_id b).const_mul (Real.sqrt κ))
+  have hh := (boundaryDensity_spatial_derivative κ α β q b hp).mul
+    ((hasDerivAt_const b (β*Real.sqrt κ)).div hX hp)
+  dsimp only [Pi.mul_apply,Pi.div_apply,Pi.sub_apply,id_eq] at hh
+  change HasDerivAt (fun y => boundaryDensity κ α β q y*(β*Real.sqrt κ/(q.re-Real.sqrt κ*y))) _ b at hh
+  rw [show (2:ℕ)=1+1 from rfl,iteratedDeriv_succ,iteratedDeriv_one,hd.deriv_eq,hh.deriv]
+  have hs := Real.sq_sqrt hκ
+  generalize boundaryDensity κ α β q b = f
+  generalize q.re-Real.sqrt κ*b = X
+  generalize Real.sqrt κ = r at *
+  rw [← hs]
+  field_simp
+  ring
+
+lemma boundaryDensity_directional (κ α β : ℝ) (q v : ℂ) (b : ℝ)
+    (hp : q.re-Real.sqrt κ*b ≠ 0) :
+    fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b) (v,0) =
+      boundaryDensity κ α β q b * (-β*α*v.im - β*v.re/(q.re-Real.sqrt κ*b)) := by
+  have hr : HasDerivAt (fun t : ℝ => (q+t • v).re) v.re 0 := by
+    simpa only [Complex.add_re,Complex.smul_re,smul_eq_mul,id_eq,one_mul] using
+      ((hasDerivAt_id (0:ℝ)).mul_const v.re).const_add q.re
+  have hi : HasDerivAt (fun t : ℝ => (q+t • v).im) v.im 0 := by
+    simpa only [Complex.add_im,Complex.smul_im,smul_eq_mul,id_eq,one_mul] using
+      ((hasDerivAt_id (0:ℝ)).mul_const v.im).const_add q.im
+  have hd := ((hi.const_mul (-β*α)).sub
+    (((hr.sub_const (Real.sqrt κ*b)).log (by simpa only [zero_smul,add_zero] using hp)).const_mul β)).exp
+  have hline : HasDerivAt (fun t : ℝ => boundaryDensity κ α β (q+t • v) b)
+      (boundaryDensity κ α β q b * (-β*α*v.im-β*v.re/(q.re-Real.sqrt κ*b))) 0 := by
+    convert hd using 1 <;> try rfl
+    simp only [zero_smul,add_zero,boundaryDensity,Pi.sub_apply]
+    ring
+  have haff : HasDerivAt (fun t : ℝ => (q+t • v,b)) (v,0) 0 := by
+    simpa only [id_eq,one_smul] using (((hasDerivAt_id (0:ℝ)).smul_const v).const_add q).prodMk (hasDerivAt_const 0 b)
+  have hfd : HasFDerivAt (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2)
+      (fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b)) (q+(0:ℝ) • v,b) := by
+    simpa only [zero_smul,add_zero] using ((boundaryDensity_contDiffAt κ α β (q,b) hp).differentiableAt (by simp)).hasFDerivAt
+  have hh := hfd.comp_hasDerivAt 0 haff
+  exact hh.unique hline
+
+lemma boundaryDensity_generator (κ α β : ℝ) (hκ : 0 ≤ κ) (q : ℂ) (b : ℝ)
+    (hp : q.re-Real.sqrt κ*b ≠ 0) :
+    fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b)
+      (((2/(q.re-Real.sqrt κ*b):ℝ):ℂ) + (((q.re-Real.sqrt κ*b)⁻¹^2:ℝ):ℂ)*I,0) +
+      (1/2:ℝ)*iteratedDeriv 2 (boundaryDensity κ α β q) b =
+      boundaryDensity κ α β q b * β * (-α-2+κ*(β+1)/2)/(q.re-Real.sqrt κ*b)^2 := by
+  rw [boundaryDensity_directional κ α β q _ b hp,boundaryDensity_spatial_second κ α β hκ q b hp]
+  simp only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,Complex.I_im,mul_one,
+    Complex.I_re,mul_zero,add_zero,zero_add,Complex.add_re,Complex.mul_re,sub_zero]
+  field_simp
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma exists_compact_boundary_field (κ α β a R : ℝ) (ha : 0 < a) (hR : 0 ≤ R) :
+    ∃ φ : ℂ × ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
+      (∀ q b, ‖(q,b)‖ ≤ R → 2*a < q.re-Real.sqrt κ*b → φ (q,b) = boundaryDensity κ α β q b) ∧
+      (∀ q b, ‖(q,b)‖ ≤ R → 2*a < q.re-Real.sqrt κ*b →
+        fderiv ℝ φ (q,b) = fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b)) ∧
+      (∀ q b, ‖(q,b)‖ ≤ R → 2*a < q.re-Real.sqrt κ*b →
+        iteratedDeriv 2 (fun x => φ (q,x)) b = iteratedDeriv 2 (boundaryDensity κ α β q) b) := by
+  let cut : ContDiffBump (0 : ℂ × ℝ) := ⟨R+1,R+2,by linarith,by linarith⟩
+  let φ (p : ℂ × ℝ) := cut p*smoothBoundaryDensity κ α β a p.1 p.2
+  have hnear (p : ℂ × ℝ) (hball : ‖p‖ ≤ R) (hp : 2*a < p.1.re-Real.sqrt κ*p.2) :
+      φ =ᶠ[𝓝 p] (fun z : ℂ × ℝ => boundaryDensity κ α β z.1 z.2) := by
+    have hb : p ∈ Metric.ball (0:ℂ × ℝ) cut.rIn := by
+      simpa only [Metric.mem_ball,dist_zero_right] using (show ‖p‖ < R+1 by linarith)
+    filter_upwards [cut.eventuallyEq_one_of_mem_ball hb,smoothBoundaryDensity_eventuallyEq κ α β ha p hp] with z hz he
+    simp only [φ,hz,Pi.one_apply,one_mul,he]
+  refine ⟨φ,cut.contDiff.mul (smoothBoundaryDensity_contDiff κ α β ha),cut.hasCompactSupport.mul_right,
+    (fun q b h h' => (hnear (q,b) h h').eq_of_nhds),?_,?_⟩
+  · intro q b h h'
+    exact (hnear (q,b) h h').fderiv_eq
+  · intro q b h h'
+    exact ((hnear (q,b) h h').comp_tendsto ((continuous_const.prodMk continuous_id).tendsto b)).iteratedDeriv_eq 2
+
+lemma compact_boundary_drift_zero (κ α β a : ℝ) (hκ : 0 < κ) (ha : 0 < a)
+    (hβ : -α-2+κ*(β+1)/2 = 0) (φ : ℂ × ℝ → ℝ) (q : ℂ) (b : ℝ)
+    (hD : fderiv ℝ φ (q,b) = fderiv ℝ (fun p : ℂ × ℝ => boundaryDensity κ α β p.1 p.2) (q,b))
+    (h2 : iteratedDeriv 2 (fun x => φ (q,x)) b = iteratedDeriv 2 (boundaryDensity κ α β q) b)
+    (hx : 2*a < q.re-Real.sqrt κ*b) :
+    fderiv ℝ φ (q,b) (boundaryClockField a (Real.sqrt κ*b) q,0) +
+      (1/2:ℝ)*iteratedDeriv 2 (fun x => φ (q,x)) b = 0 := by
+  have hgap : a ≤ q.re-Real.sqrt κ*b := by linarith
+  have he : boundaryClockField a (Real.sqrt κ*b) q =
+      ((2/(q.re-Real.sqrt κ*b):ℝ):ℂ) + (((q.re-Real.sqrt κ*b)⁻¹^2:ℝ):ℂ)*I := by
+    simp only [boundaryClockField,positiveBoundaryField_eq hgap,max_eq_right hgap]
+  rw [hD,h2,he,boundaryDensity_generator κ α β hκ.le q b (by linarith),hβ,mul_zero,zero_div]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+theorem BoundaryClockSolution.stopped_boundary_density
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ a : ℝ} {q₀ : ℂ}
+    (c : BoundaryClockSolution B hm κ a q₀)
+    (hκ : 0 < κ) (ha : 0 < a) (α β : ℝ)
+    (hβ : -α-2+κ*(β+1)/2 = 0)
+    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hinside : ∀ ω t, t ≤ τ ω →
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ 2*a < (c.Q t ω).re-Real.sqrt κ*B t ω) :
+    Integrable (fun ω => boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
+      (∫ ω, boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
+        boundaryDensity κ α β q₀ 0 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_boundary_field κ α β a R ha hR
+  let L₀ : ℝ≥0 := ⟨2*a⁻¹+a⁻¹^2,by positivity⟩
+  have hL₀ := boundaryClockField_bound ha
+  let V (t : ℝ) (ω : Ω) :=
+    boundaryClockField a (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
+  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
+    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
+      measurable_snd)).const_mul _
+  have hpair : Measurable (fun p : ℝ × Ω =>
+      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
+  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
+    change Measurable ((fun p : ℝ × ℂ => boundaryClockField a p.1 p.2) ∘
+      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
+    exact (continuous_boundaryClockField ha).measurable.comp hpair
+  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
+    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
+  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
+    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
+    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
+  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
+  let drift (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
+  have hzero (ω : Ω) (u : ℝ) (hu : 0 ≤ u) (hut : u < (τ ω : ℝ)) :
+      brownianFieldDrift B F drift u ω = 0 := by
+    have ht : u.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr hut.le
+    obtain ⟨hball, hgap⟩ := hinside ω u.toNNReal ht
+    change fderiv ℝ φ (c.Q (u.toNNReal : ℝ) ω, B u.toNNReal ω)
+      (V (u.toNNReal : ℝ) ω, 0) + _ = 0
+    dsimp only [V]
+    rw [Real.toNNReal_coe]
+    exact compact_boundary_drift_zero κ α β a hκ ha hβ φ _ _
+      (hφD _ _ hball hgap) (hφ2 _ _ hball hgap) hgap
+  have hopt := brownian_time_field_optional hB hm hBm hBc K F drift hfield τ hτ T hb hzero
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
+  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
+    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
+  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
+    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
+  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
+    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
+    exact Eventually.of_forall fun ω => by
+      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
+        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
+  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
+      boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω) :=
+    hφeq _ _ (hinside ω (τ ω) le_rfl).1 (hinside ω (τ ω) le_rfl).2
+  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
+  calc
+    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
+      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
+    _ = ∫ ω, F 0 ω (B 0 ω) ∂P := hopt
+    _ = ∫ _, boundaryDensity κ α β q₀ 0 ∂P := by
+      apply integral_congr_ae
+      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
+      change φ (c.Q 0 ω, B 0 ω) = _
+      have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
+        simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
+      rw [hφeq _ _ hh (hinside ω 0 zero_le).2, c.initial ω, hB0]
+    _ = boundaryDensity κ α β q₀ 0 := by simp
+
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma continuous_hittingBtwn_mem {Ω : Type*} (H : ℝ≥0 → Ω → ℝ)
+    (hHc : ∀ ω, Continuous (fun t => H t ω)) (T : ℝ≥0) (ω : Ω)
+    (he : ∃ j ∈ Icc 0 T, H j ω ∈ ({0} : Set ℝ)) :
+    H (hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω) ω = 0 := by
+  have hc : IsClosed (Icc 0 T ∩ {t : ℝ≥0 | H t ω ∈ ({0} : Set ℝ)}) :=
+    isClosed_Icc.inter (isClosed_singleton.preimage (hHc ω))
+  have hn : (Icc 0 T ∩ {t : ℝ≥0 | H t ω ∈ ({0} : Set ℝ)}).Nonempty := by
+    obtain ⟨j, hj, hh⟩ := he
+    exact ⟨j, hj, hh⟩
+  have hh := hc.csInf_mem hn (bddBelow_Icc.inter_of_left)
+  simpa only [hittingBtwn_def, ite_eq_left he, mem_inter_iff, mem_ofPred_eq, mem_singleton_iff] using hh.2
+
+lemma continuous_zero_iff_dense
+    (H : ℝ≥0 → ℝ) (hHc : Continuous H) (hHn : ∀ t, 0 ≤ H t) (t : ℝ≥0) :
+    (∃ s ≤ t, H s = 0) ↔
+      ∀ n : ℕ, ∃ k : ℕ,
+        H (TopologicalSpace.denseSeq (Iic t) k) < 1 / ((n : ℝ) + 1) := by
+  let : Nonempty (Iic t) := ⟨⟨0, show (0 : ℝ≥0) ≤ t from zero_le⟩⟩
+  constructor
+  · rintro ⟨s, hs, he⟩ n
+    have hpos : (0 : ℝ) < 1 / ((n : ℝ) + 1) := by positivity
+    exact (TopologicalSpace.denseRange_denseSeq (Iic t)).exists_mem_open
+      (isOpen_lt (hHc.comp continuous_subtype_val) continuous_const)
+      ⟨⟨s, hs⟩, by simpa only [mem_ofPred_eq, Function.comp_apply, he] using hpos⟩
+  · intro he
+    have hcc : IsCompact (Iic t) := by
+      have heq : Iic t = Icc 0 t := by ext s; simp
+      rw [heq]
+      exact isCompact_Icc
+    obtain ⟨s, hs, hmin⟩ := hcc.exists_isMinOn
+      ⟨0, show (0 : ℝ≥0) ≤ t from zero_le⟩ hHc.continuousOn
+    refine ⟨s, hs, le_antisymm ?_ (hHn s)⟩
+    apply ge_of_tendsto (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
+    exact Eventually.of_forall fun n => by
+      obtain ⟨k, hk⟩ := he n
+      exact (hmin (TopologicalSpace.denseSeq (Iic t) k).property).trans hk.le
+
+theorem continuous_nonnegative_first_zero_stopping
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
+    (H : ℝ≥0 → Ω → ℝ) (hHa : Adapted f H)
+    (hHc : ∀ ω, Continuous (fun t => H t ω))
+    (hHn : ∀ t ω, 0 ≤ H t ω) (T : ℝ≥0) :
+    IsStoppingTime f (fun ω => ((hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω : ℝ≥0) : WithTop ℝ≥0)) ∧
+    (∀ ω, hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω ≤ T) ∧
+    (∀ ω t, t < hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω → 0 < H t ω) ∧
+    (∀ ω, hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω < T → H (hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω) ω = 0) := by
+  classical
+  have hb (ω : Ω) := hittingBtwn_le (u := H) (s := ({0} : Set ℝ)) (n := 0) (m := T) ω
+  have hmem (ω : Ω) (h : hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω < T) :
+      H (hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω) ω = 0 := by
+    apply continuous_hittingBtwn_mem H hHc T ω
+    by_contra he
+    simp only [hittingBtwn_def, ite_eq_right he] at h
+    exact lt_irrefl _ h
+  refine ⟨?_, hb, ?_, hmem⟩
+  · intro t
+    by_cases hT : T ≤ t
+    · have he : {ω : Ω | ((hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} = univ := by
+        ext ω
+        simp only [mem_ofPred_eq, mem_univ, iff_true, WithTop.coe_le_coe]
+        exact (hb ω).trans hT
+      rw [he]
+      exact MeasurableSet.univ
+    · have ht : t < T := lt_of_not_ge hT
+      let : Nonempty (Iic t) := ⟨⟨0, show (0 : ℝ≥0) ≤ t from zero_le⟩⟩
+      have he (ω : Ω) : hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω ≤ t ↔ ∃ s ≤ t, H s ω = 0 := by
+        constructor
+        · intro h
+          exact ⟨_, h, hmem ω (h.trans_lt ht)⟩
+        · rintro ⟨s, hs, h0⟩
+          exact (hittingBtwn_le_of_mem (zero_le) (hs.trans ht.le) (by simpa using h0)).trans hs
+      have hset : {ω : Ω | ((hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
+          ⋂ n : ℕ, ⋃ k : ℕ, {ω : Ω | H (TopologicalSpace.denseSeq (Iic t) k) ω < 1 / ((n : ℝ) + 1)} := by
+        ext ω
+        simp only [mem_ofPred_eq, WithTop.coe_le_coe, mem_iInter, mem_iUnion]
+        exact (he ω).trans (continuous_zero_iff_dense (fun s => H s ω) (hHc ω) (fun s => hHn s ω) t)
+      rw [hset]
+      apply MeasurableSet.iInter
+      intro n
+      apply MeasurableSet.iUnion
+      intro k
+      have hm := (hHa (TopologicalSpace.denseSeq (Iic t) k)).mono
+        (f.mono (TopologicalSpace.denseSeq (Iic t) k).property) le_rfl
+      exact measurableSet_lt hm measurable_const
+  · intro ω t ht
+    have hn := notMem_of_lt_hittingBtwn ht (zero_le)
+    exact lt_of_le_of_ne (hHn t ω) (by simpa only [mem_singleton_iff, ne_eq, eq_comm] using hn)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def logRadiusClock (η₀ : ℝ) (q : ℂ × ℝ) : ℝ :=
+  η₀ - q.1.im + q.2
+
+noncomputable def logAngleSq (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) : ℝ :=
+  Real.exp (2 * q.1.im) / greenLogNormSq κ q b
+
+noncomputable def logClockRate (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) : ℝ :=
+  4 * Real.exp (2 * q.1.im) / greenLogNormSq κ q b ^ 2
+
+lemma logAngleSq_pos (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    0 < logAngleSq κ q b :=
+  div_pos (Real.exp_pos _) (greenLogNormSq_pos κ q b)
+
+lemma logAngleSq_le_one (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    logAngleSq κ q b ≤ 1 := by
+  apply (div_le_one (greenLogNormSq_pos κ q b)).mpr
+  exact le_add_of_nonneg_left (sq_nonneg _)
+
+lemma continuous_logAngleSq (κ : ℝ) :
+    Continuous (fun p : (ℂ × ℝ) × ℝ => logAngleSq κ p.1 p.2) := by
+  have hi : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
+    Complex.continuous_im.comp (continuous_fst.fst)
+  have hr : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
+    Complex.continuous_re.comp (continuous_fst.fst)
+  exact (Real.continuous_exp.comp (continuous_const.mul hi)).div
+    (((hr.sub (continuous_const.mul continuous_snd)).pow 2).add
+      (Real.continuous_exp.comp (continuous_const.mul hi)))
+    (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
+
+lemma continuous_logRadiusClock (η₀ : ℝ) : Continuous (logRadiusClock η₀) :=
+  (continuous_const.sub (Complex.continuous_im.comp continuous_fst)).add continuous_snd
+
+lemma greenLogDensity_radius_angle (κ η₀ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    greenLogDensity κ q b =
+      Real.exp (codimension κ * (logRadiusClock η₀ q - Real.log 2 - η₀)) *
+        (logAngleSq κ q b) ^ (greenExponent κ / 2) := by
+  rw [Real.rpow_def_of_pos (logAngleSq_pos κ q b), ← Real.exp_add]
+  unfold logAngleSq
+  rw [Real.log_div (Real.exp_ne_zero _) (greenLogNormSq_pos κ q b).ne', Real.log_exp]
+  unfold greenLogDensity logRadiusClock
+  congr 1
+  ring
+
+lemma codimension_pos (κ : ℝ) (hκ8 : κ < 8) : 0 < codimension κ := by
+  dsimp [codimension, dimension]
+  linarith
+
+lemma greenExponent_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < greenExponent κ := by
+  unfold greenExponent
+  have : 1 < 8 / κ := (lt_div_iff₀ hκ).mpr (by simpa using hκ8)
+  linarith
+
+lemma greenLogDensity_le_at_angular_exit
+    (κ η₀ u ε : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (q : ℂ × ℝ) (b : ℝ)
+    (hclock : logRadiusClock η₀ q ≤ u) (hangle : logAngleSq κ q b ≤ ε) :
+    greenLogDensity κ q b ≤
+      Real.exp (codimension κ * (u - Real.log 2 - η₀)) * ε ^ (greenExponent κ / 2) := by
+  rw [greenLogDensity_radius_angle κ η₀ q b]
+  apply mul_le_mul
+  · exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (by linarith)
+      (codimension_pos κ hκ8).le)
+  · exact Real.rpow_le_rpow (logAngleSq_pos κ q b).le hangle
+      (div_nonneg (greenExponent_pos κ hκ hκ8).le (by norm_num))
+  · exact Real.rpow_nonneg (logAngleSq_pos κ q b).le _
+  · exact (Real.exp_pos _).le
+
+lemma logClockRate_pos (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    0 < logClockRate κ q b := by
+  unfold logClockRate
+  exact div_pos (mul_pos (by norm_num) (Real.exp_pos _))
+    (sq_pos_of_pos (greenLogNormSq_pos κ q b))
+
+lemma logClockRate_angle (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    logClockRate κ q b = 4 * logAngleSq κ q b ^ 2 / Real.exp (2 * q.1.im) := by
+  unfold logClockRate logAngleSq
+  field_simp [(Real.exp_pos (2 * q.1.im)).ne', (greenLogNormSq_pos κ q b).ne']
+
+lemma log_ode_im_derivative {Q : ℝ → ℂ × ℝ} {D : Set ℝ} {t κ b : ℝ}
+    (hQ : HasDerivWithinAt Q (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)) D t) :
+    HasDerivWithinAt (fun s => (Q s).1.im) (-2 / greenLogNormSq κ (Q t) b) D t := by
+  have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hQ.fst
+  simpa only [Function.comp_def, Complex.imCLM_apply,
+    (physicalLogLoewnerField_components κ (Q t) b).2.1] using hi
+
+lemma log_ode_clock_derivative {Q : ℝ → ℂ × ℝ} {D : Set ℝ} {t κ b : ℝ}
+    (η₀ : ℝ)
+    (hQ : HasDerivWithinAt Q (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)) D t) :
+    HasDerivWithinAt (fun s => logRadiusClock η₀ (Q s))
+      (logClockRate κ (Q t) b) D t := by
+  have hs : HasDerivWithinAt (fun s => (Q s).2)
+      (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)).2 D t := hQ.snd
+  have hd : HasDerivWithinAt (fun s => η₀ - (Q s).1.im + (Q s).2)
+      (-(-2 / greenLogNormSq κ (Q t) b) +
+        (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)).2) D t :=
+    ((log_ode_im_derivative hQ).const_sub η₀).add hs
+  have he : -(-2 / greenLogNormSq κ (Q t) b) +
+      (physicalLogLoewnerField (Real.sqrt κ * b) (Q t)).2 = logClockRate κ (Q t) b := by
+    rw [(physicalLogLoewnerField_components κ (Q t) b).2.2]
+    unfold logClockRate
+    have hN := (greenLogNormSq_pos κ (Q t) b).ne'
+    field_simp
+    unfold greenLogNormSq
+    ring
+  simpa only [he, logRadiusClock, Pi.add_apply] using hd
+
+lemma log_height_clock_derivative_nonneg
+    (κ ε : ℝ) (hε : 0 < ε) (q : ℂ × ℝ) (b : ℝ)
+    (ha : ε ≤ logAngleSq κ q b) :
+    0 ≤ -2 / greenLogNormSq κ q b + logClockRate κ q b / (2 * ε) := by
+  have hN := greenLogNormSq_pos κ q b
+  have h1 : 1 ≤ logAngleSq κ q b / ε := (le_div_iff₀ hε).mpr (by simpa using ha)
+  have h2 := mul_le_mul_of_nonneg_left h1 (show 0 ≤ 2 / greenLogNormSq κ q b by positivity)
+  have he : logClockRate κ q b / (2 * ε) =
+      (2 / greenLogNormSq κ q b) * (logAngleSq κ q b / ε) := by
+    unfold logClockRate logAngleSq
+    field_simp
+    ring
+  rw [he, neg_div]
+  simpa only [mul_one, sub_eq_add_neg, add_comm] using sub_nonneg.mpr h2
+
+lemma log_clock_rate_lower (κ ε η₀ : ℝ) (hε : 0 < ε) (q : ℂ × ℝ) (b : ℝ)
+    (ha : ε ≤ logAngleSq κ q b) (hi : q.1.im ≤ η₀) :
+    4 * ε ^ 2 / Real.exp (2 * η₀) ≤ logClockRate κ q b := by
+  rw [logClockRate_angle]
+  have hs : ε ^ 2 ≤ logAngleSq κ q b ^ 2 := sq_le_sq₀ hε.le (logAngleSq_pos κ q b).le |>.mpr ha
+  have he : Real.exp (2 * q.1.im) ≤ Real.exp (2 * η₀) :=
+    Real.exp_le_exp.mpr (by linarith)
+  exact div_le_div₀ (by positivity) (by nlinarith) (Real.exp_pos _) he
+
+ 
+
+theorem log_loewner_height_clock_bounds
+    {Q : ℝ → ℂ × ℝ} {B : ℝ → ℝ} {κ T ε : ℝ}
+    (hT : 0 ≤ T) (hε : 0 < ε) (hQ0 : (Q 0).2 = 0)
+    (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q
+      (physicalLogLoewnerField (Real.sqrt κ * B t) (Q t)) (Icc 0 T) t)
+    (hangle : ∀ t ∈ Icc 0 T, ε ≤ logAngleSq κ (Q t) (B t)) :
+    (Q T).1.im ≤ (Q 0).1.im ∧
+    (Q 0).1.im - logRadiusClock (Q 0).1.im (Q T) / (2 * ε) ≤ (Q T).1.im ∧
+    (4 * ε ^ 2 / Real.exp (2 * (Q 0).1.im)) * T ≤ logRadiusClock (Q 0).1.im (Q T) := by
+  let η₀ := (Q 0).1.im
+  let c := 4 * ε ^ 2 / Real.exp (2 * η₀)
+  have h0 : (0 : ℝ) ∈ Icc 0 T := ⟨le_rfl, hT⟩
+  have hTT : T ∈ Icc 0 T := ⟨hT, le_rfl⟩
+  have hQc : ContinuousOn Q (Icc 0 T) := fun t ht => (hQ t ht).continuousWithinAt
+  have hi : AntitoneOn (fun t => (Q t).1.im) (Icc 0 T) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
+      (Complex.continuous_im.comp_continuousOn (continuous_fst.comp_continuousOn hQc))
+      (fun t ht => (log_ode_im_derivative (hQ t (interior_subset ht))).mono interior_subset)
+    intro t ht
+    exact div_nonpos_of_nonpos_of_nonneg (by norm_num)
+      (greenLogNormSq_pos κ (Q t) (B t)).le
+  have hib (t : ℝ) (ht : t ∈ Icc 0 T) : (Q t).1.im ≤ η₀ := hi h0 ht ht.1
+  have hS0 : logRadiusClock η₀ (Q 0) = 0 := by simp [logRadiusClock, η₀, hQ0]
+  have hSc : ContinuousOn (fun t => logRadiusClock η₀ (Q t)) (Icc 0 T) :=
+    (continuous_logRadiusClock η₀).comp_continuousOn hQc
+  have hlo : MonotoneOn (fun t => (Q t).1.im +
+      logRadiusClock η₀ (Q t) / (2 * ε)) (Icc 0 T) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
+      ((Complex.continuous_im.comp_continuousOn (continuous_fst.comp_continuousOn hQc)).add
+        (hSc.div_const _))
+      (fun t ht => ((log_ode_im_derivative (hQ t (interior_subset ht))).add
+        ((log_ode_clock_derivative η₀ (hQ t (interior_subset ht))).div_const _)).mono interior_subset)
+    intro t ht
+    exact log_height_clock_derivative_nonneg κ ε hε (Q t) (B t) (hangle t (interior_subset ht))
+  have hrate : MonotoneOn (fun t => logRadiusClock η₀ (Q t) - c * t) (Icc 0 T) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
+      (hSc.sub (continuous_const.mul continuous_id).continuousOn)
+      (fun t ht => ((log_ode_clock_derivative η₀ (hQ t (interior_subset ht))).sub
+        ((hasDerivWithinAt_id t (Icc 0 T)).const_mul c)).mono interior_subset)
+    intro t ht
+    have hb := log_clock_rate_lower κ ε η₀ hε (Q t) (B t)
+      (hangle t (interior_subset ht)) (hib t (interior_subset ht))
+    change 0 ≤ logClockRate κ (Q t) (B t) - c * 1
+    simpa only [mul_one] using sub_nonneg.mpr hb
+  refine ⟨hib T hTT, ?_, ?_⟩
+  · have hh := hlo h0 hTT hT
+    dsimp only at hh
+    rw [hS0, zero_div, add_zero] at hh
+    linarith
+  · have hh := hrate h0 hTT hT
+    dsimp only at hh
+    rw [hS0, mul_zero, sub_zero] at hh
+    linarith
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+structure ClippedLogSolution {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (κ lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) where
+  L : ℝ≥0
+  Q : ℝ → Ω → ℂ × ℝ
+  measurable : Measurable (fun p : ℝ × Ω => Q p.1 p.2)
+  lipschitz : ∀ ω, LipschitzWith L (fun t => Q t ω)
+  initial : ∀ ω, Q 0 ω = q₀
+  derivative : ∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω)
+    (clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (Q t ω))
+    (Ici 0) t
+  adapted : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+    (fun t : ℝ≥0 => Q t)
+
+lemma exists_clippedLogSolution
+    {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) :
+    Nonempty (ClippedLogSolution B hm κ lo hi hle q₀) := by
+  let U (t : ℝ≥0) (ω : Ω) := Real.sqrt κ * B t ω
+  have hUm : Measurable (fun p : ℝ≥0 × Ω => U p.1 p.2) := hBm.const_mul _
+  have hUc (ω : Ω) : Continuous (fun t => U t ω) := (hBc ω).const_mul _
+  obtain ⟨L, Q, hQm, hQl, hQ0, hQd, hQaU⟩ :=
+    clipped_log_loewner_global_adapted_solution U hUm hUc lo hi hle q₀
+  have hBa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) B :=
+    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
+  refine ⟨⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩⟩
+  intro t
+  apply (hQaU t).mono _ le_rfl
+  change (⨆ j ≤ t, MeasurableSpace.comap (U j) (borel ℝ)) ≤ _
+  refine iSup₂_le fun j hj => ?_
+  exact ((hBa.measurable_le hj).const_mul (Real.sqrt κ)).comap_le
+
+lemma compact_green_drift_zero
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (lo hi : ℝ) (hle : lo ≤ hi)
+    (φ : (ℂ × ℝ) × ℝ → ℝ) (q : ℂ × ℝ) (x : ℝ)
+    (hφD : fderiv ℝ φ (q, x) =
+      fderiv ℝ (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (q, x))
+    (hφ2 : iteratedDeriv 2 (fun y => φ (q, y)) x =
+      iteratedDeriv 2 (greenLogDensity κ q) x)
+    (hq : q.1.im ∈ Icc lo hi) :
+    fderiv ℝ φ (q, x)
+        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * x) q, 0) +
+      (1 / 2 : ℝ) * iteratedDeriv 2 (fun y => φ (q, y)) x = 0 := by
+  rw [hφD, hφ2, greenLogDensity_pair_directional,
+    clippedLogLoewnerField_eq_physical lo hi hle _ _ hq]
+  exact green_physical_generator_zero κ hκ hκ8 q x
+
+ 
+
+theorem ClippedLogSolution.stopped_green
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hinside : ∀ ω t, t ≤ τ ω →
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
+    Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
+      (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
+        greenLogDensity κ q₀ 0 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
+  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) :=
+    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
+  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
+    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
+      measurable_snd)).const_mul _
+  have hpair : Measurable (fun p : ℝ × Ω =>
+      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
+  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
+    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
+  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
+    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
+  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
+    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
+    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
+  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
+  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
+  have hzero (ω : Ω) (u : ℝ) (hu : 0 ≤ u) (hut : u < (τ ω : ℝ)) :
+      brownianFieldDrift B F a u ω = 0 := by
+    have ht : u.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr hut.le
+    obtain ⟨hball, hlo, hhi⟩ := hinside ω u.toNNReal ht
+    change fderiv ℝ φ (c.Q (u.toNNReal : ℝ) ω, B u.toNNReal ω)
+      (V (u.toNNReal : ℝ) ω, 0) + _ = 0
+    dsimp only [V]
+    rw [Real.toNNReal_coe]
+    exact compact_green_drift_zero κ hκ hκ8 lo hi hle φ _ _
+      (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
+  have hopt := brownian_time_field_optional hB hm hBm hBc K F a hfield τ hτ T hb hzero
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
+  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
+    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
+  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
+    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
+  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
+    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
+    exact Eventually.of_forall fun ω => by
+      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
+        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
+  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
+      greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) :=
+    hφeq _ _ (hinside ω (τ ω) le_rfl).1
+  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
+  calc
+    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
+      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
+    _ = ∫ ω, F 0 ω (B 0 ω) ∂P := hopt
+    _ = ∫ _, greenLogDensity κ q₀ 0 ∂P := by
+      apply integral_congr_ae
+      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
+      change φ (c.Q 0 ω, B 0 ω) = _
+      have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
+        simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
+      rw [hφeq _ _ hh, c.initial ω, hB0]
+    _ = greenLogDensity κ q₀ 0 := by simp
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/BrownianGenerators.lean b/lean/OAI/Probability/SLE/BrownianGenerators.lean
index 6e1af05458c2111ea23b829ae1175005c8092532..77e0dedf9ea71c4ea93bcc0e4e601d99ac034033 100644
--- a/lean/OAI/Probability/SLE/BrownianGenerators.lean
+++ b/lean/OAI/Probability/SLE/BrownianGenerators.lean
@@ -1,683 +1,683 @@
-import Mathlib
-
-namespace OAI
-
-/-! Brownian Generators for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-structure SmoothTimeBrownianField {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (K : ℝ≥0)
-    (F a : ℝ≥0 → Ω → ℝ → ℝ) : Prop where
-  smooth : ∀ t ω, ContDiff ℝ 3 (F t ω)
-  adapted : ∀ t j, j ≤ 2 → Measurable[
-    (Filtration.natural B (fun t => (hm t).stronglyMeasurable) t).prod (borel ℝ)]
-    (fun q : Ω × ℝ => iteratedDeriv j (F t q.1) q.2)
-  joint : ∀ j, j ≤ 2 → Measurable
-    (fun q : (ℝ≥0 × Ω) × ℝ => iteratedDeriv j (F q.1.1 q.1.2) q.2)
-  drift_joint : Measurable (fun q : (ℝ≥0 × Ω) × ℝ => a q.1.1 q.1.2 q.2)
-  bound : ∀ t ω x j, j ≤ 3 → |iteratedDeriv j (F t ω) x| ≤ K
-  drift_bound : ∀ t ω x, |a t ω x| ≤ K
-  drift_lipschitz : ∀ t ω x y, |a t ω x - a t ω y| ≤ (K : ℝ) * |x - y|
-  second_lipschitz : ∀ s t ω x y,
-    |iteratedDeriv 2 (F t ω) x - iteratedDeriv 2 (F s ω) y| ≤
-      (K : ℝ) * (|(t : ℝ) - s| + |x - y|)
-  time_increment : ∀ s t : ℝ≥0, s ≤ t → ∀ ω x,
-    F t ω x - F s ω x = ∫ u in (s : ℝ)..(t : ℝ), a u.toNNReal ω x
-
-noncomputable def brownianFieldDrift {Ω : Type*}
-    (B : ℝ≥0 → Ω → ℝ) (F a : ℝ≥0 → Ω → ℝ → ℝ) (u : ℝ) (ω : Ω) : ℝ :=
-  a u.toNNReal ω (B u.toNNReal ω) +
-    (1 / 2 : ℝ) * iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)
-
-lemma brownian_increment_indep_natural
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (s t : ℝ≥0) :
-    Indep (Filtration.natural B (fun t => (hm t).stronglyMeasurable) s)
-      ((borel ℝ).comap (fun ω => B (s + t) ω - B s ω)) P := by
-  have hi := ((hB.indepFun_shift s).comp (measurable_pi_apply t) measurable_id).symm
-  rw [Filtration.natural_eq_comap B _ s]
-  exact (IndepFun_iff_Indep _ _ _).mp hi
-
-lemma brownian_increment_indep_adapted
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (s t : ℝ≥0) {f : Ω → ℝ}
-    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f) :
-    IndepFun f (fun ω => B (s + t) ω - B s ω) P := by
-  rw [IndepFun_iff_Indep]
-  exact indep_of_indep_of_le (brownian_increment_indep_natural hB hm s t) hf.comap_le le_rfl
-
-lemma gaussian_square_memLp_two {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
-    MemLp (fun ω => X ω ^ 2) 2 P := by
-  have h4 : MemLp X 4 P := hX.hasGaussianLaw.memLp (by norm_num)
-  apply (memLp_two_iff_integrable_sq (hX.aemeasurable.pow_const 2).aestronglyMeasurable).mpr
-  have hint := h4.integrable_norm_pow (p := 4) (by decide)
-  convert hint using 1
-  funext ω
-  rw [Real.norm_eq_abs, show (4 : ℕ) = 2 * 2 from rfl, pow_mul, sq_abs]
-
-lemma gaussian_centered_square_memLp_two {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
-    MemLp (fun ω => X ω ^ 2 - (δ : ℝ)) 2 P := by
-  let : IsProbabilityMeasure P := hX.isProbabilityMeasure
-  exact (gaussian_square_memLp_two hX).sub (memLp_const _)
-
-lemma gaussian_square_integral (δ : ℝ≥0) :
-    (∫ x : ℝ, x ^ 2 ∂gaussianReal 0 δ) = δ := by
-  have h := variance_fun_id_gaussianReal (μ := 0) (v := δ)
-  rw [variance_eq_integral (X := fun x : ℝ => x) (by fun_prop)] at h
-  simpa using h
-
-lemma gaussian_centered_square_integral {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
-    (∫ ω, X ω ^ 2 - (δ : ℝ) ∂P) = 0 := by
-  have he := hX.integral_comp (f := fun x : ℝ => x ^ 2 - (δ : ℝ)) (by fun_prop)
-  simp only [Function.comp_apply] at he
-  rw [he]
-  have h2 : Integrable (fun x : ℝ => x ^ 2) (gaussianReal 0 δ) :=
-    (memLp_id_gaussianReal (μ := 0) (v := δ) 2).integrable_sq
-  rw [integral_sub h2 (integrable_const _), gaussian_square_integral]
-  simp
-
-lemma real_second_taylor_bound {f : ℝ → ℝ} (hf : ContDiff ℝ 3 f)
-    (K : ℝ≥0) (hK : ∀ z, |iteratedDeriv 3 f z| ≤ K) (x y : ℝ) :
-    |f (x + y) - f x - deriv f x * y - (1 / 2 : ℝ) * iteratedDeriv 2 f x * y ^ 2| ≤
-      (K : ℝ) * |y| ^ 3 := by
-  have ht := map_add_eq_sum_add_integral_iteratedFDeriv (n := 2) (x := x) (y := y)
-    (fun (_t : ℝ) (_ht : _ ∈ Icc (0 : ℝ) 1) => hf.contDiffAt)
-  simp only [iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod, Finset.prod_const,
-    Finset.card_univ, Fintype.card_fin, smul_eq_mul] at ht
-  norm_num [Finset.sum_range_succ, iteratedDeriv_zero, iteratedDeriv_one] at ht
-  have hi : |∫ t : ℝ in 0..1, (1 - t) ^ 2 * (y ^ 3 * iteratedDeriv 3 f (x + t * y))| ≤
-      (K : ℝ) * |y| ^ 3 := by
-    have h := intervalIntegral.norm_integral_le_of_norm_le_const (C := (K : ℝ) * |y| ^ 3)
-      (a := (0 : ℝ)) (b := 1)
-      (f := fun t : ℝ => (1 - t) ^ 2 * (y ^ 3 * iteratedDeriv 3 f (x + t * y))) ?_
-    · simpa only [Real.norm_eq_abs, sub_zero, abs_one, mul_one] using h
-    · intro t ht
-      have ht' : t ∈ Ioc (0 : ℝ) 1 := by simpa using ht
-      have ht1 : |1 - t| ≤ 1 := by rw [abs_of_nonneg (by linarith [ht'.2])]; linarith [ht'.1]
-      simp only [Real.norm_eq_abs, abs_mul, abs_pow]
-      calc
-        |1 - t| ^ 2 * (|y| ^ 3 * |iteratedDeriv 3 f (x + t * y)|) ≤
-            1 ^ 2 * (|y| ^ 3 * (K : ℝ)) := by gcongr; exact hK _
-        _ = (K : ℝ) * |y| ^ 3 := by ring
-  have he : f (x + y) - f x - deriv f x * y - (1 / 2 : ℝ) * iteratedDeriv 2 f x * y ^ 2 =
-      (1 / 2 : ℝ) * ∫ t : ℝ in 0..1, (1 - t) ^ 2 * (y ^ 3 * iteratedDeriv 3 f (x + t * y)) := by
-    rw [ht]
-    ring
-  rw [he, abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2)]
-  nlinarith [show 0 ≤ (K : ℝ) * |y| ^ 3 by positivity]
-
-noncomputable def standardGaussianAbsMoment (m : ℕ) : ℝ :=
-  ∫ x : ℝ, |x| ^ m ∂gaussianReal 0 1
-
-lemma standardGaussianAbsMoment_nonneg (m : ℕ) : 0 ≤ standardGaussianAbsMoment m :=
-  integral_nonneg fun x => pow_nonneg (abs_nonneg x) m
-
-lemma gaussian_abs_moment_integrable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P)
-    (m : ℕ) (hm : m ≠ 0) : Integrable (fun ω => |X ω| ^ m) P := by
-  have hm' : (m : ℝ≥0∞) ≠ ∞ := ENNReal.natCast_ne_top m
-  have hp : MemLp X m P := hX.hasGaussianLaw.memLp hm'
-  simpa only [Real.norm_eq_abs] using hp.integrable_norm_pow hm
-
-lemma gaussian_abs_moment {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P)
-    (m : ℕ) : (∫ ω, |X ω| ^ m ∂P) = (Real.sqrt δ) ^ m * standardGaussianAbsMoment m := by
-  have he := hX.integral_comp (f := fun x : ℝ => |x| ^ m) (by fun_prop)
-  simp only [Function.comp_apply] at he
-  rw [he]
-  have hm : (gaussianReal 0 1).map (fun x : ℝ => Real.sqrt δ * x) = gaussianReal 0 δ := by
-    rw [gaussianReal_map_const_mul]
-    congr 1
-    · simp
-    · ext
-      simp [Real.sq_sqrt δ.coe_nonneg]
-  rw [← hm, integral_map (by fun_prop) (by fun_prop)]
-  simp_rw [abs_mul, mul_pow, abs_of_nonneg (Real.sqrt_nonneg _)]
-  rw [integral_const_mul]
-  rfl
-
-lemma gaussian_abs_cube {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
-    (∫ ω, |X ω| ^ 3 ∂P) = (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 3 := by
-  rw [gaussian_abs_moment hX]
-  congr 1
-  rw [pow_succ, Real.sq_sqrt δ.coe_nonneg]
-
-lemma SmoothTimeBrownianField.eval_measurable
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (j : ℕ) (hj : j ≤ 2)
-    {V : Ω → ℝ} (hV : Measurable V) :
-    Measurable (fun ω => iteratedDeriv j (F t ω) (V ω)) :=
-  (hF.joint j hj).comp ((measurable_const.prodMk measurable_id).prodMk hV)
-
-lemma SmoothTimeBrownianField.eval_adapted
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (j : ℕ) (hj : j ≤ 2) :
-    Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) t]
-      (fun ω => iteratedDeriv j (F t ω) (B t ω)) := by
-  have hBt := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable) t).measurable
-  exact (hF.adapted t j hj).comp (measurable_id.prodMk hBt)
-
-lemma brownian_increment_orthogonal
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (s δ : ℝ≥0) {f : Ω → ℝ}
-    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f) :
-    (∫ ω, f ω * (B (s + δ) ω - B s ω) ∂P) = 0 := by
-  have hif := brownian_increment_indep_adapted hB hm s δ hf
-  have hfm : Measurable f := hf.mono
-    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s) le_rfl
-  have hlaw := (hB.shift s).hasLaw_eval δ
-  rw [hif.integral_fun_mul_eq_mul_integral hfm.aestronglyMeasurable
-    ((hm _).sub (hm _)).aestronglyMeasurable, hlaw.integral_eq]
-  simp
-
-lemma brownian_centered_square_orthogonal
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (s δ : ℝ≥0) {f : Ω → ℝ}
-    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f) :
-    (∫ ω, f ω * ((B (s + δ) ω - B s ω) ^ 2 - (δ : ℝ)) ∂P) = 0 := by
-  have hif := (brownian_increment_indep_adapted hB hm s δ hf).comp
-    measurable_id (by fun_prop : Measurable (fun x : ℝ => x ^ 2 - (δ : ℝ)))
-  have hfm : Measurable f := hf.mono
-    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s) le_rfl
-  have hlaw := (hB.shift s).hasLaw_eval δ
-  have he := hif.integral_fun_mul_eq_mul_integral
-    hfm.aestronglyMeasurable (((hm _).sub (hm _)).pow_const 2 |>.sub measurable_const).aestronglyMeasurable
-  simp only [Function.comp_apply, id_eq] at he
-  rw [he, gaussian_centered_square_integral hlaw, mul_zero]
-
-lemma brownian_set_integral_increment_zero
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (s δ : ℝ≥0) {f : Ω → ℝ} {E : Set Ω}
-    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f)
-    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
-    (∫ ω in E, f ω * (B (s + δ) ω - B s ω) ∂P) = 0 := by
-  have hEm : MeasurableSet E :=
-    (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s _ hE
-  have h := brownian_increment_orthogonal hB hm s δ (hf.indicator hE)
-  rw [← integral_indicator hEm]
-  convert h using 1
-  congr 1
-  funext ω
-  by_cases hω : ω ∈ E <;> simp [hω]
-
-lemma brownian_set_integral_square
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (s δ C : ℝ≥0) {f : Ω → ℝ} {E : Set Ω}
-    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f)
-    (hbound : ∀ ω, |f ω| ≤ C)
-    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
-    (∫ ω in E, f ω * (B (s + δ) ω - B s ω) ^ 2 ∂P) =
-      (δ : ℝ) * ∫ ω in E, f ω ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hEm : MeasurableSet E :=
-    (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s _ hE
-  have hfm := hf.mono ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s) le_rfl
-  have hfi : Integrable f (P.restrict E) :=
-    Integrable.of_bound hfm.aestronglyMeasurable _ (Eventually.of_forall fun ω => by
-      simpa only [Real.norm_eq_abs] using hbound ω)
-  have hi2 : Integrable (fun ω => (B (s + δ) ω - B s ω) ^ 2) (P.restrict E) :=
-    ((hB.shift s).hasLaw_eval δ).hasGaussianLaw.memLp_two.integrable_sq.restrict
-  have hprod := hi2.bdd_mul hfm.aestronglyMeasurable
-    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hbound ω)
-  have h := brownian_centered_square_orthogonal hB hm s δ (hf.indicator hE)
-  have hz : (∫ ω in E, f ω * ((B (s + δ) ω - B s ω) ^ 2 - (δ : ℝ)) ∂P) = 0 := by
-    rw [← integral_indicator hEm]
-    convert h using 1
-    congr 1
-    funext ω
-    by_cases hω : ω ∈ E <;> simp [hω]
-  simp_rw [mul_sub] at hz
-  rw [integral_sub hprod (hfi.mul_const _), integral_mul_const] at hz
-  linarith
-
-lemma smooth_field_taylor_set_error
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (s δ : ℝ≥0) {E : Set Ω}
-    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
-    |(∫ ω in E, F s ω (B (s + δ) ω) - F s ω (B s ω) ∂P) -
-      (1 / 2 : ℝ) * (δ : ℝ) * ∫ ω in E, iteratedDeriv 2 (F s ω) (B s ω) ∂P| ≤
-      (K : ℝ) * (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 3 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let X (ω : Ω) := B (s + δ) ω - B s ω
-  let c₁ (ω : Ω) := iteratedDeriv 1 (F s ω) (B s ω)
-  let c₂ (ω : Ω) := iteratedDeriv 2 (F s ω) (B s ω)
-  let T (ω : Ω) := F s ω (B (s + δ) ω) - F s ω (B s ω)
-  let R (ω : Ω) := T ω - c₁ ω * X ω - (1 / 2 : ℝ) * (c₂ ω * X ω ^ 2)
-  have hXlaw : HasLaw X (gaussianReal 0 δ) P := (hB.shift s).hasLaw_eval δ
-  have hXm : Measurable X := (hm _).sub (hm _)
-  have hc₁m : Measurable c₁ := hF.eval_measurable s 1 (by decide) (hm s)
-  have hc₂m : Measurable c₂ := hF.eval_measurable s 2 (by decide) (hm s)
-  have hc₁b (ω) : |c₁ ω| ≤ K := hF.bound s ω _ 1 (by decide)
-  have hc₂b (ω) : |c₂ ω| ≤ K := hF.bound s ω _ 2 (by decide)
-  have hI (t : ℝ≥0) : Integrable (fun ω => F s ω (B t ω)) (P.restrict E) := by
-    have hm' : Measurable (fun ω => F s ω (B t ω)) := by
-      simpa only [iteratedDeriv_zero] using hF.eval_measurable s 0 (by decide) (hm t)
-    refine Integrable.of_bound hm'.aestronglyMeasurable K (Eventually.of_forall fun ω => ?_)
-    simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using hF.bound s ω (B t ω) 0 (by decide)
-  have hTi : Integrable T (P.restrict E) := (hI _).sub (hI _)
-  have hlin : Integrable (fun ω => c₁ ω * X ω) (P.restrict E) :=
-    hXlaw.hasGaussianLaw.integrable.restrict.bdd_mul hc₁m.aestronglyMeasurable
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hc₁b ω)
-  have hquad : Integrable (fun ω => c₂ ω * X ω ^ 2) (P.restrict E) :=
-    hXlaw.hasGaussianLaw.memLp_two.integrable_sq.restrict.bdd_mul hc₂m.aestronglyMeasurable
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hc₂b ω)
-  have hRi : Integrable R (P.restrict E) := (hTi.sub hlin).sub (hquad.const_mul _)
-  have hlin0 : (∫ ω in E, c₁ ω * X ω ∂P) = 0 :=
-    brownian_set_integral_increment_zero hB hm s δ (hF.eval_adapted s 1 (by decide)) hE
-  have hquad0 : (∫ ω in E, c₂ ω * X ω ^ 2 ∂P) = (δ : ℝ) * ∫ ω in E, c₂ ω ∂P :=
-    brownian_set_integral_square hB hm s δ K (hF.eval_adapted s 2 (by decide)) hc₂b hE
-  have hRint : (∫ ω in E, R ω ∂P) = (∫ ω in E, T ω ∂P) -
-      (1 / 2 : ℝ) * (δ : ℝ) * ∫ ω in E, c₂ ω ∂P := by
-    dsimp only [R]
-    have hsubi : Integrable (fun ω => T ω - c₁ ω * X ω) (P.restrict E) := hTi.sub hlin
-    rw [integral_sub hsubi (hquad.const_mul (1 / 2 : ℝ)), integral_sub hTi hlin,
-      hlin0, sub_zero, integral_const_mul, hquad0]
-    ring
-  have hRb (ω) : |R ω| ≤ (K : ℝ) * |X ω| ^ 3 := by
-    have h := real_second_taylor_bound (hF.smooth s ω) K
-      (fun z => hF.bound s ω z 3 (by decide)) (B s ω) (X ω)
-    have he : B s ω + X ω = B (s + δ) ω := by dsimp [X]; ring
-    rw [he] at h
-    simpa only [R, T, c₁, c₂, iteratedDeriv_one, mul_assoc] using h
-  change |(∫ ω in E, T ω ∂P) - (1 / 2 : ℝ) * (δ : ℝ) * ∫ ω in E, c₂ ω ∂P| ≤ _
-  rw [← hRint]
-  have hcube := gaussian_abs_moment_integrable hXlaw 3 (by decide)
-  calc
-    |∫ ω in E, R ω ∂P| ≤ ∫ ω in E, |R ω| ∂P := by
-      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := P.restrict E) R
-    _ ≤ ∫ ω in E, (K : ℝ) * |X ω| ^ 3 ∂P :=
-      integral_mono hRi.abs (hcube.restrict.const_mul _) hRb
-    _ ≤ ∫ ω, (K : ℝ) * |X ω| ^ 3 ∂P :=
-      integral_mono_measure Measure.restrict_le_self
-        (Eventually.of_forall fun _ => by positivity) (hcube.const_mul _)
-    _ = (K : ℝ) * (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 3 := by
-      rw [integral_const_mul, gaussian_abs_cube hXlaw]
-      ring
-
-lemma brownian_expected_abs_increment_le
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (r t δ : ℝ≥0) (hgap : |(t : ℝ) - r| ≤ δ) (E : Set Ω) :
-    (∫ ω in E, |B t ω - B r ω| ∂P) ≤ Real.sqrt δ * standardGaussianAbsMoment 1 := by
-  have hlaw : HasLaw (fun ω => B t ω - B r ω)
-      (gaussianReal 0 (nndist (t : ℝ) (r : ℝ))) P := hB.hasLaw_sub t r
-  have hint : Integrable (fun ω => |B t ω - B r ω|) P := hlaw.hasGaussianLaw.integrable.abs
-  calc
-    (∫ ω in E, |B t ω - B r ω| ∂P) ≤ ∫ ω, |B t ω - B r ω| ∂P :=
-      integral_mono_measure Measure.restrict_le_self
-        (Eventually.of_forall fun _ => abs_nonneg _) hint
-    _ = Real.sqrt (nndist (t : ℝ) (r : ℝ)) * standardGaussianAbsMoment 1 := by
-      simpa only [pow_one, Pi.sub_apply] using gaussian_abs_moment hlaw 1
-    _ ≤ Real.sqrt δ * standardGaussianAbsMoment 1 := by
-      apply mul_le_mul_of_nonneg_right _ (standardGaussianAbsMoment_nonneg 1)
-      apply Real.sqrt_le_sqrt
-      simpa only [coe_nndist, Real.dist_eq] using hgap
-
-lemma bounded_interval_integral_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    (a b : ℝ) (hab : a ≤ b) (G : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) (C : ℝ)
-    (hbound : ∀ u ω, |G u ω| ≤ C) :
-    Integrable (fun ω => ∫ u in a..b, G u ω) P := by
-  have hI : Integrable (Function.uncurry G) ((volume.restrict (Ioc a b)).prod P) :=
-    Integrable.of_bound hG.aestronglyMeasurable C
-      (Eventually.of_forall fun q => by simpa only [Real.norm_eq_abs, Function.uncurry] using hbound q.1 q.2)
-  simpa only [intervalIntegral.integral_of_le hab, Function.uncurry] using hI.integral_prod_right
-
-lemma bounded_interval_integral_expectation_bound
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    (a b : ℝ) (hab : a ≤ b) (G : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) (C D : ℝ)
-    (hbound : ∀ u ω, |G u ω| ≤ C)
-    (hmean : ∀ u ∈ Ioc a b, |∫ ω, G u ω ∂P| ≤ D) :
-    |∫ ω, (∫ u in a..b, G u ω) ∂P| ≤ D * (b - a) := by
-  have hI : Integrable (Function.uncurry G) ((volume.restrict (Ioc a b)).prod P) :=
-    Integrable.of_bound hG.aestronglyMeasurable C
-      (Eventually.of_forall fun q => by simpa only [Real.norm_eq_abs, Function.uncurry] using hbound q.1 q.2)
-  simp_rw [intervalIntegral.integral_of_le hab]
-  rw [← integral_integral_swap hI]
-  have h := intervalIntegral.norm_integral_le_of_norm_le_const (a := a) (b := b)
-    (f := fun u => ∫ ω, G u ω ∂P) (C := D) (fun u hu => by
-      simpa only [Real.norm_eq_abs] using hmean u (by simpa only [uIoc_of_le hab] using hu))
-  simpa only [Real.norm_eq_abs, intervalIntegral.integral_of_le hab,
-    abs_of_nonneg (sub_nonneg.mpr hab)] using h
-
-lemma bounded_time_section_intervalIntegrable
-    {Ω : Type*} [MeasurableSpace Ω] (a b : ℝ) (hab : a ≤ b)
-    (G : ℝ → Ω → ℝ) (hG : Measurable (Function.uncurry G)) (C : ℝ)
-    (hbound : ∀ u ω, |G u ω| ≤ C) (ω : Ω) :
-    IntervalIntegrable (fun u => G u ω) volume a b := by
-  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab]
-  apply Integrable.of_bound ((hG.comp (measurable_id.prodMk measurable_const)).aestronglyMeasurable) C
-  exact Eventually.of_forall fun u => by simpa only [Real.norm_eq_abs, Function.comp_apply, Function.uncurry, id_eq] using hbound u ω
-
-lemma smooth_field_time_set_error
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) (E : Set Ω) :
-    |(∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) -
-      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), a u.toNNReal ω (B u.toNNReal ω)) ∂P| ≤
-      (K : ℝ) * (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let C (u : ℝ) (ω : Ω) := a u.toNNReal ω (B (s + δ) ω)
-  let D (u : ℝ) (ω : Ω) := a u.toNNReal ω (B u.toNNReal ω)
-  let G (u : ℝ) (ω : Ω) := C u ω - D u ω
-  have htime : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
-  have hC : Measurable (Function.uncurry C) :=
-    hF.drift_joint.comp (htime.prodMk ((hm (s + δ)).comp measurable_snd))
-  have hD : Measurable (Function.uncurry D) :=
-    hF.drift_joint.comp (htime.prodMk (hBm.comp htime))
-  have hG : Measurable (Function.uncurry G) := hC.sub hD
-  have hCb (u ω) : |C u ω| ≤ K := hF.drift_bound _ _ _
-  have hDb (u ω) : |D u ω| ≤ K := hF.drift_bound _ _ _
-  have hGb (u ω) : |G u ω| ≤ 2 * (K : ℝ) := by
-    exact (abs_sub _ _).trans (by linarith [hCb u ω, hDb u ω])
-  have hle : (s : ℝ) ≤ (s + δ : ℝ≥0) := by
-    simp only [NNReal.coe_add]; exact le_add_of_nonneg_right δ.coe_nonneg
-  have hCI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle C hC K hCb
-  have hDI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle D hD K hDb
-  have he : (∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) -
-      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P) =
-      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), G u ω) ∂P := by
-    simp_rw [hF.time_increment s (s + δ) (le_add_of_nonneg_right (zero_le))]
-    change (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), C u ω) ∂P) - _ = _
-    rw [← integral_sub hCI hDI]
-    apply integral_congr_ae
-    exact Eventually.of_forall fun ω =>
-      (intervalIntegral.integral_sub
-        (bounded_time_section_intervalIntegrable _ _ hle C hC K hCb ω)
-        (bounded_time_section_intervalIntegrable _ _ hle D hD K hDb ω)).symm
-  change |(∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) -
-    (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P)| ≤ _
-  rw [he]
-  have hmean (u : ℝ) (hu : u ∈ Ioc (s : ℝ) (s + δ : ℝ≥0)) :
-      |∫ ω in E, G u ω ∂P| ≤ (K : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1 := by
-    have hgi : Integrable (G u) (P.restrict E) :=
-      Integrable.of_bound (hG.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable
-        (2 * (K : ℝ)) (Eventually.of_forall fun ω => by
-          simpa only [Real.norm_eq_abs] using hGb u ω)
-    have hxi : Integrable (fun ω => |B (s + δ) ω - B u.toNNReal ω|) (P.restrict E) :=
-      ((hB.hasLaw_sub (s + δ) u.toNNReal).hasGaussianLaw.integrable.abs).restrict
-    have hgap : |((s + δ : ℝ≥0) : ℝ) - (u.toNNReal : ℝ)| ≤ δ := by
-      rw [Real.coe_toNNReal _ (le_trans s.coe_nonneg hu.1.le),
-        abs_of_nonneg (sub_nonneg.mpr hu.2), NNReal.coe_add]
-      linarith [hu.1]
-    calc
-      |∫ ω in E, G u ω ∂P| ≤ ∫ ω in E, |G u ω| ∂P := by
-        simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := P.restrict E) (G u)
-      _ ≤ ∫ ω in E, (K : ℝ) * |B (s + δ) ω - B u.toNNReal ω| ∂P :=
-        integral_mono hgi.abs (hxi.const_mul _) (fun ω => hF.drift_lipschitz _ _ _ _)
-      _ = (K : ℝ) * ∫ ω in E, |B (s + δ) ω - B u.toNNReal ω| ∂P := integral_const_mul _ _
-      _ ≤ (K : ℝ) * (Real.sqrt δ * standardGaussianAbsMoment 1) :=
-        mul_le_mul_of_nonneg_left (brownian_expected_abs_increment_le hB _ _ _ hgap E) K.coe_nonneg
-      _ = (K : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1 := by ring
-  have hb := bounded_interval_integral_expectation_bound (P := P.restrict E) _ _ hle G hG
-    (2 * (K : ℝ)) ((K : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1) hGb hmean
-  simpa only [NNReal.coe_add, add_sub_cancel_left, mul_right_comm, mul_assoc] using hb
-
-lemma smooth_field_second_mean_error
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) (E : Set Ω)
-    (u : ℝ) (hu : u ∈ Ioc (s : ℝ) (s + δ : ℝ≥0)) :
-    |∫ ω in E, iteratedDeriv 2 (F s ω) (B s ω) -
-      iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω) ∂P| ≤
-      (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let C (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F s ω) (B s ω)
-  let D (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)
-  let G (u : ℝ) (ω : Ω) := C u ω - D u ω
-  have htime : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
-  have hC : Measurable (Function.uncurry C) :=
-    (hF.eval_measurable s 2 (by decide) (hm s)).comp measurable_snd
-  have hD : Measurable (Function.uncurry D) := by
-    change Measurable (fun q : ℝ × Ω =>
-      iteratedDeriv 2 (F q.1.toNNReal q.2) (B q.1.toNNReal q.2))
-    simpa only [Function.comp_def] using
-      (hF.joint 2 (by decide)).comp (htime.prodMk (hBm.comp htime))
-  have hG : Measurable (Function.uncurry G) := hC.sub hD
-  have hCb (u ω) : |C u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
-  have hDb (u ω) : |D u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
-  have hGb (u ω) : |G u ω| ≤ 2 * (K : ℝ) :=
-    (abs_sub _ _).trans (by linarith [hCb u ω, hDb u ω])
-  have hgi : Integrable (G u) (P.restrict E) :=
-    Integrable.of_bound (hG.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable
-      (2 * (K : ℝ)) (Eventually.of_forall fun ω => by
-        simpa only [Real.norm_eq_abs] using hGb u ω)
-  have hxi : Integrable (fun ω => |B u.toNNReal ω - B s ω|) (P.restrict E) :=
-    ((hB.hasLaw_sub u.toNNReal s).hasGaussianLaw.integrable.abs).restrict
-  have hgap : |(u.toNNReal : ℝ) - (s : ℝ)| ≤ δ := by
-    rw [Real.coe_toNNReal _ (le_trans s.coe_nonneg hu.1.le), abs_of_nonneg (sub_nonneg.mpr hu.1.le)]
-    have hh := hu.2
-    simp only [NNReal.coe_add] at hh
-    linarith
-  have hpoint (ω) : |G u ω| ≤ (K : ℝ) * ((δ : ℝ) + |B u.toNNReal ω - B s ω|) := by
-    calc
-      |G u ω| = |D u ω - C u ω| := abs_sub_comm _ _
-      _ ≤ (K : ℝ) * (|(u.toNNReal : ℝ) - (s : ℝ)| + |B u.toNNReal ω - B s ω|) :=
-        hF.second_lipschitz s u.toNNReal ω _ _
-      _ ≤ (K : ℝ) * ((δ : ℝ) + |B u.toNNReal ω - B s ω|) :=
-        mul_le_mul_of_nonneg_left (add_le_add hgap le_rfl) K.coe_nonneg
-  have hconst : (∫ _ω in E, (δ : ℝ) ∂P) ≤ δ := by
-    calc
-      (∫ _ω in E, (δ : ℝ) ∂P) ≤ ∫ _ω, (δ : ℝ) ∂P :=
-        integral_mono_measure Measure.restrict_le_self
-          (Eventually.of_forall fun _ => δ.coe_nonneg) (integrable_const _)
-      _ = δ := by simp
-  calc
-    |∫ ω in E, G u ω ∂P| ≤ ∫ ω in E, |G u ω| ∂P := by
-      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := P.restrict E) (G u)
-    _ ≤ ∫ ω in E, (K : ℝ) * ((δ : ℝ) + |B u.toNNReal ω - B s ω|) ∂P :=
-      integral_mono hgi.abs (((integrable_const _).add hxi).const_mul _) hpoint
-    _ = (K : ℝ) * ((∫ _ω in E, (δ : ℝ) ∂P) +
-        ∫ ω in E, |B u.toNNReal ω - B s ω| ∂P) := by
-      rw [integral_const_mul, integral_add (integrable_const _) hxi]
-    _ ≤ (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) :=
-      mul_le_mul_of_nonneg_left
-        (add_le_add hconst (brownian_expected_abs_increment_le hB _ _ _ hgap E)) K.coe_nonneg
-
-lemma smooth_field_second_set_error
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) (E : Set Ω) :
-    |(δ : ℝ) * (∫ ω in E, iteratedDeriv 2 (F s ω) (B s ω) ∂P) -
-      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0),
-        iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)) ∂P| ≤
-      (K : ℝ) * (δ : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let C (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F s ω) (B s ω)
-  let D (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)
-  let G (u : ℝ) (ω : Ω) := C u ω - D u ω
-  have htime : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
-  have hC : Measurable (Function.uncurry C) :=
-    (hF.eval_measurable s 2 (by decide) (hm s)).comp measurable_snd
-  have hD : Measurable (Function.uncurry D) := by
-    change Measurable (fun q : ℝ × Ω =>
-      iteratedDeriv 2 (F q.1.toNNReal q.2) (B q.1.toNNReal q.2))
-    simpa only [Function.comp_def] using
-      (hF.joint 2 (by decide)).comp (htime.prodMk (hBm.comp htime))
-  have hG : Measurable (Function.uncurry G) := hC.sub hD
-  have hCb (u ω) : |C u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
-  have hDb (u ω) : |D u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
-  have hGb (u ω) : |G u ω| ≤ 2 * (K : ℝ) :=
-    (abs_sub _ _).trans (by linarith [hCb u ω, hDb u ω])
-  have hle : (s : ℝ) ≤ (s + δ : ℝ≥0) := by
-    simp only [NNReal.coe_add]; exact le_add_of_nonneg_right δ.coe_nonneg
-  have hCI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle C hC K hCb
-  have hDI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle D hD K hDb
-  have he : (δ : ℝ) * (∫ ω in E, C 0 ω ∂P) -
-      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P) =
-      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), G u ω) ∂P := by
-    have hconst : (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), C u ω) ∂P) =
-        (δ : ℝ) * ∫ ω in E, C 0 ω ∂P := by
-      simp only [C, intervalIntegral.integral_const, NNReal.coe_add,
-        add_sub_cancel_left, smul_eq_mul, integral_const_mul]
-    rw [← hconst, ← integral_sub hCI hDI]
-    apply integral_congr_ae
-    exact Eventually.of_forall fun ω =>
-      (intervalIntegral.integral_sub
-        (bounded_time_section_intervalIntegrable _ _ hle C hC K hCb ω)
-        (bounded_time_section_intervalIntegrable _ _ hle D hD K hDb ω)).symm
-  change |(δ : ℝ) * (∫ ω in E, C 0 ω ∂P) -
-    (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P)| ≤ _
-  rw [he]
-  have hmean (u : ℝ) (hu : u ∈ Ioc (s : ℝ) (s + δ : ℝ≥0)) :
-      |∫ ω in E, G u ω ∂P| ≤ (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) :=
-    smooth_field_second_mean_error hB hm hBm K hF s δ E u hu
-  have hb := bounded_interval_integral_expectation_bound (P := P.restrict E) _ _ hle G hG
-    (2 * (K : ℝ)) ((K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1)) hGb hmean
-  calc
-    _ ≤ (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) *
-        (((s + δ : ℝ≥0) : ℝ) - s) := hb
-    _ = _ := by simp only [NNReal.coe_add]; ring
-
-lemma SmoothTimeBrownianField.eval_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (j : ℕ) (hj : j ≤ 2)
-    {V : Ω → ℝ} (hV : Measurable V) :
-    Integrable (fun ω => iteratedDeriv j (F t ω) (V ω)) P := by
-  apply Integrable.of_bound (hF.eval_measurable t j hj hV).aestronglyMeasurable K
-  exact Eventually.of_forall fun ω => by
-    simpa only [Real.norm_eq_abs] using hF.bound t ω (V ω) j (by omega)
-
-lemma SmoothTimeBrownianField.deriv_real_joint
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (j : ℕ) (hj : j ≤ 2) :
-    Measurable (fun q : ℝ × Ω => iteratedDeriv j (F q.1.toNNReal q.2) (B q.1.toNNReal q.2)) := by
-  have ht : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
-  simpa only [Function.comp_def] using (hF.joint j hj).comp (ht.prodMk (hBm.comp ht))
-
-lemma SmoothTimeBrownianField.drift_real_joint
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
-    Measurable (fun q : ℝ × Ω => a q.1.toNNReal q.2 (B q.1.toNNReal q.2)) := by
-  have ht : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
-  exact hF.drift_joint.comp (ht.prodMk (hBm.comp ht))
-
-lemma bounded_interval_integral_expectation_add
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    (a b c : ℝ) (hab : a ≤ b) (G H : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) (hH : Measurable (Function.uncurry H))
-    (C D : ℝ) (hGb : ∀ u ω, |G u ω| ≤ C) (hHb : ∀ u ω, |H u ω| ≤ D) :
-    (∫ ω, (∫ u in a..b, G u ω + c * H u ω) ∂P) =
-      (∫ ω, (∫ u in a..b, G u ω) ∂P) + c * (∫ ω, (∫ u in a..b, H u ω) ∂P) := by
-  have hGI := bounded_interval_integral_integrable a b hab G hG C hGb (P := P)
-  have hHI := bounded_interval_integral_integrable a b hab H hH D hHb (P := P)
-  calc
-    _ = ∫ ω, ((∫ u in a..b, G u ω) + c * (∫ u in a..b, H u ω)) ∂P := by
-      apply integral_congr_ae
-      filter_upwards [] with ω
-      rw [intervalIntegral.integral_add
-        (bounded_time_section_intervalIntegrable a b hab G hG C hGb ω)
-        ((bounded_time_section_intervalIntegrable a b hab H hH D hHb ω).const_mul c),
-        intervalIntegral.integral_const_mul]
-    _ = _ := by rw [integral_add hGI (hHI.const_mul c), integral_const_mul]
-
-noncomputable def dynkinRate (δ : ℝ≥0) : ℝ :=
-  Real.sqrt δ * standardGaussianAbsMoment 3 + Real.sqrt δ * standardGaussianAbsMoment 1 +
-    (1 / 2 : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1)
-
-lemma dynkinRate_tendsto_zero : Tendsto dynkinRate (𝓝 0) (𝓝 0) := by
-  have hc : Continuous dynkinRate := by unfold dynkinRate; fun_prop
-  simpa [ContinuousAt, dynkinRate] using hc.continuousAt (x := 0)
-
-lemma abs_dynkin_algebra_bound (T A S I J eT eA eS δ : ℝ)
-    (hT : |T - (1 / 2 : ℝ) * δ * S| ≤ eT)
-    (hA : |A - I| ≤ eA) (hS : |δ * S - J| ≤ eS) :
-    |(T + A) - (I + (1 / 2 : ℝ) * J)| ≤ eT + eA + (1 / 2 : ℝ) * eS := by
-  have he : (T + A) - (I + (1 / 2 : ℝ) * J) =
-      (T - (1 / 2 : ℝ) * δ * S) + (A - I) + (1 / 2 : ℝ) * (δ * S - J) := by ring
-  rw [he]
-  calc
-    _ ≤ |T - (1 / 2 : ℝ) * δ * S| + |A - I| + |(1 / 2 : ℝ) * (δ * S - J)| :=
-      (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
-    _ = |T - (1 / 2 : ℝ) * δ * S| + |A - I| + (1 / 2 : ℝ) * |δ * S - J| := by
-      rw [abs_mul]; norm_num
-    _ ≤ _ := add_le_add (add_le_add hT hA) (mul_le_mul_of_nonneg_left hS (by norm_num))
-
-lemma smooth_field_one_step_error
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) {E : Set Ω}
-    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
-    |(∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B s ω) ∂P) -
-      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P)| ≤
-      (K : ℝ) * (δ : ℝ) * dynkinRate δ := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hI (v t : ℝ≥0) : Integrable (fun ω => F v ω (B t ω)) (P.restrict E) := by
-    simpa only [iteratedDeriv_zero] using hF.eval_integrable v 0 (by decide) (hm t) (P := P.restrict E)
-  have hsplit : (∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B s ω) ∂P) =
-      (∫ ω in E, F s ω (B (s + δ) ω) - F s ω (B s ω) ∂P) +
-      (∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) := by
-    rw [integral_sub (hI _ _) (hI _ _), integral_sub (hI _ _) (hI _ _),
-      integral_sub (hI _ _) (hI _ _)]
-    ring
-  have hle : (s : ℝ) ≤ (s + δ : ℝ≥0) := by
-    simp only [NNReal.coe_add]; exact le_add_of_nonneg_right δ.coe_nonneg
-  have hsplitD : (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P) =
-      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), a u.toNNReal ω (B u.toNNReal ω)) ∂P) +
-      (1 / 2 : ℝ) * (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0),
-        iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)) ∂P) := by
-    apply bounded_interval_integral_expectation_add _ _ _ hle _ _
-      (hF.drift_real_joint hBm) (hF.deriv_real_joint hBm 2 (by decide)) (K : ℝ) (K : ℝ)
-    · exact fun u ω => hF.drift_bound _ _ _
-    · exact fun u ω => hF.bound _ _ _ 2 (by decide)
-  rw [hsplit, hsplitD]
-  have h := abs_dynkin_algebra_bound _ _ _ _ _ _ _ _ _
-    (smooth_field_taylor_set_error hB hm K hF s δ hE)
-    (smooth_field_time_set_error hB hm hBm K hF s δ E)
-    (smooth_field_second_set_error hB hm hBm K hF s δ E)
-  convert h using 1
-  unfold dynkinRate
-  ring
-
- 
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Brownian Generators for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+structure SmoothTimeBrownianField {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (K : ℝ≥0)
+    (F a : ℝ≥0 → Ω → ℝ → ℝ) : Prop where
+  smooth : ∀ t ω, ContDiff ℝ 3 (F t ω)
+  adapted : ∀ t j, j ≤ 2 → Measurable[
+    (Filtration.natural B (fun t => (hm t).stronglyMeasurable) t).prod (borel ℝ)]
+    (fun q : Ω × ℝ => iteratedDeriv j (F t q.1) q.2)
+  joint : ∀ j, j ≤ 2 → Measurable
+    (fun q : (ℝ≥0 × Ω) × ℝ => iteratedDeriv j (F q.1.1 q.1.2) q.2)
+  drift_joint : Measurable (fun q : (ℝ≥0 × Ω) × ℝ => a q.1.1 q.1.2 q.2)
+  bound : ∀ t ω x j, j ≤ 3 → |iteratedDeriv j (F t ω) x| ≤ K
+  drift_bound : ∀ t ω x, |a t ω x| ≤ K
+  drift_lipschitz : ∀ t ω x y, |a t ω x - a t ω y| ≤ (K : ℝ) * |x - y|
+  second_lipschitz : ∀ s t ω x y,
+    |iteratedDeriv 2 (F t ω) x - iteratedDeriv 2 (F s ω) y| ≤
+      (K : ℝ) * (|(t : ℝ) - s| + |x - y|)
+  time_increment : ∀ s t : ℝ≥0, s ≤ t → ∀ ω x,
+    F t ω x - F s ω x = ∫ u in (s : ℝ)..(t : ℝ), a u.toNNReal ω x
+
+noncomputable def brownianFieldDrift {Ω : Type*}
+    (B : ℝ≥0 → Ω → ℝ) (F a : ℝ≥0 → Ω → ℝ → ℝ) (u : ℝ) (ω : Ω) : ℝ :=
+  a u.toNNReal ω (B u.toNNReal ω) +
+    (1 / 2 : ℝ) * iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)
+
+lemma brownian_increment_indep_natural
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (s t : ℝ≥0) :
+    Indep (Filtration.natural B (fun t => (hm t).stronglyMeasurable) s)
+      ((borel ℝ).comap (fun ω => B (s + t) ω - B s ω)) P := by
+  have hi := ((hB.indepFun_shift s).comp (measurable_pi_apply t) measurable_id).symm
+  rw [Filtration.natural_eq_comap B _ s]
+  exact (IndepFun_iff_Indep _ _ _).mp hi
+
+lemma brownian_increment_indep_adapted
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (s t : ℝ≥0) {f : Ω → ℝ}
+    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f) :
+    IndepFun f (fun ω => B (s + t) ω - B s ω) P := by
+  rw [IndepFun_iff_Indep]
+  exact indep_of_indep_of_le (brownian_increment_indep_natural hB hm s t) hf.comap_le le_rfl
+
+lemma gaussian_square_memLp_two {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
+    MemLp (fun ω => X ω ^ 2) 2 P := by
+  have h4 : MemLp X 4 P := hX.hasGaussianLaw.memLp (by norm_num)
+  apply (memLp_two_iff_integrable_sq (hX.aemeasurable.pow_const 2).aestronglyMeasurable).mpr
+  have hint := h4.integrable_norm_pow (p := 4) (by decide)
+  convert hint using 1
+  funext ω
+  rw [Real.norm_eq_abs, show (4 : ℕ) = 2 * 2 from rfl, pow_mul, sq_abs]
+
+lemma gaussian_centered_square_memLp_two {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
+    MemLp (fun ω => X ω ^ 2 - (δ : ℝ)) 2 P := by
+  let : IsProbabilityMeasure P := hX.isProbabilityMeasure
+  exact (gaussian_square_memLp_two hX).sub (memLp_const _)
+
+lemma gaussian_square_integral (δ : ℝ≥0) :
+    (∫ x : ℝ, x ^ 2 ∂gaussianReal 0 δ) = δ := by
+  have h := variance_fun_id_gaussianReal (μ := 0) (v := δ)
+  rw [variance_eq_integral (X := fun x : ℝ => x) (by fun_prop)] at h
+  simpa using h
+
+lemma gaussian_centered_square_integral {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
+    (∫ ω, X ω ^ 2 - (δ : ℝ) ∂P) = 0 := by
+  have he := hX.integral_comp (f := fun x : ℝ => x ^ 2 - (δ : ℝ)) (by fun_prop)
+  simp only [Function.comp_apply] at he
+  rw [he]
+  have h2 : Integrable (fun x : ℝ => x ^ 2) (gaussianReal 0 δ) :=
+    (memLp_id_gaussianReal (μ := 0) (v := δ) 2).integrable_sq
+  rw [integral_sub h2 (integrable_const _), gaussian_square_integral]
+  simp
+
+lemma real_second_taylor_bound {f : ℝ → ℝ} (hf : ContDiff ℝ 3 f)
+    (K : ℝ≥0) (hK : ∀ z, |iteratedDeriv 3 f z| ≤ K) (x y : ℝ) :
+    |f (x + y) - f x - deriv f x * y - (1 / 2 : ℝ) * iteratedDeriv 2 f x * y ^ 2| ≤
+      (K : ℝ) * |y| ^ 3 := by
+  have ht := map_add_eq_sum_add_integral_iteratedFDeriv (n := 2) (x := x) (y := y)
+    (fun (_t : ℝ) (_ht : _ ∈ Icc (0 : ℝ) 1) => hf.contDiffAt)
+  simp only [iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod, Finset.prod_const,
+    Finset.card_univ, Fintype.card_fin, smul_eq_mul] at ht
+  norm_num [Finset.sum_range_succ, iteratedDeriv_zero, iteratedDeriv_one] at ht
+  have hi : |∫ t : ℝ in 0..1, (1 - t) ^ 2 * (y ^ 3 * iteratedDeriv 3 f (x + t * y))| ≤
+      (K : ℝ) * |y| ^ 3 := by
+    have h := intervalIntegral.norm_integral_le_of_norm_le_const (C := (K : ℝ) * |y| ^ 3)
+      (a := (0 : ℝ)) (b := 1)
+      (f := fun t : ℝ => (1 - t) ^ 2 * (y ^ 3 * iteratedDeriv 3 f (x + t * y))) ?_
+    · simpa only [Real.norm_eq_abs, sub_zero, abs_one, mul_one] using h
+    · intro t ht
+      have ht' : t ∈ Ioc (0 : ℝ) 1 := by simpa using ht
+      have ht1 : |1 - t| ≤ 1 := by rw [abs_of_nonneg (by linarith [ht'.2])]; linarith [ht'.1]
+      simp only [Real.norm_eq_abs, abs_mul, abs_pow]
+      calc
+        |1 - t| ^ 2 * (|y| ^ 3 * |iteratedDeriv 3 f (x + t * y)|) ≤
+            1 ^ 2 * (|y| ^ 3 * (K : ℝ)) := by gcongr; exact hK _
+        _ = (K : ℝ) * |y| ^ 3 := by ring
+  have he : f (x + y) - f x - deriv f x * y - (1 / 2 : ℝ) * iteratedDeriv 2 f x * y ^ 2 =
+      (1 / 2 : ℝ) * ∫ t : ℝ in 0..1, (1 - t) ^ 2 * (y ^ 3 * iteratedDeriv 3 f (x + t * y)) := by
+    rw [ht]
+    ring
+  rw [he, abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2)]
+  nlinarith [show 0 ≤ (K : ℝ) * |y| ^ 3 by positivity]
+
+noncomputable def standardGaussianAbsMoment (m : ℕ) : ℝ :=
+  ∫ x : ℝ, |x| ^ m ∂gaussianReal 0 1
+
+lemma standardGaussianAbsMoment_nonneg (m : ℕ) : 0 ≤ standardGaussianAbsMoment m :=
+  integral_nonneg fun x => pow_nonneg (abs_nonneg x) m
+
+lemma gaussian_abs_moment_integrable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P)
+    (m : ℕ) (hm : m ≠ 0) : Integrable (fun ω => |X ω| ^ m) P := by
+  have hm' : (m : ℝ≥0∞) ≠ ∞ := ENNReal.natCast_ne_top m
+  have hp : MemLp X m P := hX.hasGaussianLaw.memLp hm'
+  simpa only [Real.norm_eq_abs] using hp.integrable_norm_pow hm
+
+lemma gaussian_abs_moment {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P)
+    (m : ℕ) : (∫ ω, |X ω| ^ m ∂P) = (Real.sqrt δ) ^ m * standardGaussianAbsMoment m := by
+  have he := hX.integral_comp (f := fun x : ℝ => |x| ^ m) (by fun_prop)
+  simp only [Function.comp_apply] at he
+  rw [he]
+  have hm : (gaussianReal 0 1).map (fun x : ℝ => Real.sqrt δ * x) = gaussianReal 0 δ := by
+    rw [gaussianReal_map_const_mul]
+    congr 1
+    · simp
+    · ext
+      simp [Real.sq_sqrt δ.coe_nonneg]
+  rw [← hm, integral_map (by fun_prop) (by fun_prop)]
+  simp_rw [abs_mul, mul_pow, abs_of_nonneg (Real.sqrt_nonneg _)]
+  rw [integral_const_mul]
+  rfl
+
+lemma gaussian_abs_cube {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {X : Ω → ℝ} {δ : ℝ≥0} (hX : HasLaw X (gaussianReal 0 δ) P) :
+    (∫ ω, |X ω| ^ 3 ∂P) = (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 3 := by
+  rw [gaussian_abs_moment hX]
+  congr 1
+  rw [pow_succ, Real.sq_sqrt δ.coe_nonneg]
+
+lemma SmoothTimeBrownianField.eval_measurable
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (j : ℕ) (hj : j ≤ 2)
+    {V : Ω → ℝ} (hV : Measurable V) :
+    Measurable (fun ω => iteratedDeriv j (F t ω) (V ω)) :=
+  (hF.joint j hj).comp ((measurable_const.prodMk measurable_id).prodMk hV)
+
+lemma SmoothTimeBrownianField.eval_adapted
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (j : ℕ) (hj : j ≤ 2) :
+    Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) t]
+      (fun ω => iteratedDeriv j (F t ω) (B t ω)) := by
+  have hBt := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable) t).measurable
+  exact (hF.adapted t j hj).comp (measurable_id.prodMk hBt)
+
+lemma brownian_increment_orthogonal
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (s δ : ℝ≥0) {f : Ω → ℝ}
+    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f) :
+    (∫ ω, f ω * (B (s + δ) ω - B s ω) ∂P) = 0 := by
+  have hif := brownian_increment_indep_adapted hB hm s δ hf
+  have hfm : Measurable f := hf.mono
+    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s) le_rfl
+  have hlaw := (hB.shift s).hasLaw_eval δ
+  rw [hif.integral_fun_mul_eq_mul_integral hfm.aestronglyMeasurable
+    ((hm _).sub (hm _)).aestronglyMeasurable, hlaw.integral_eq]
+  simp
+
+lemma brownian_centered_square_orthogonal
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (s δ : ℝ≥0) {f : Ω → ℝ}
+    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f) :
+    (∫ ω, f ω * ((B (s + δ) ω - B s ω) ^ 2 - (δ : ℝ)) ∂P) = 0 := by
+  have hif := (brownian_increment_indep_adapted hB hm s δ hf).comp
+    measurable_id (by fun_prop : Measurable (fun x : ℝ => x ^ 2 - (δ : ℝ)))
+  have hfm : Measurable f := hf.mono
+    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s) le_rfl
+  have hlaw := (hB.shift s).hasLaw_eval δ
+  have he := hif.integral_fun_mul_eq_mul_integral
+    hfm.aestronglyMeasurable (((hm _).sub (hm _)).pow_const 2 |>.sub measurable_const).aestronglyMeasurable
+  simp only [Function.comp_apply, id_eq] at he
+  rw [he, gaussian_centered_square_integral hlaw, mul_zero]
+
+lemma brownian_set_integral_increment_zero
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (s δ : ℝ≥0) {f : Ω → ℝ} {E : Set Ω}
+    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f)
+    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
+    (∫ ω in E, f ω * (B (s + δ) ω - B s ω) ∂P) = 0 := by
+  have hEm : MeasurableSet E :=
+    (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s _ hE
+  have h := brownian_increment_orthogonal hB hm s δ (hf.indicator hE)
+  rw [← integral_indicator hEm]
+  convert h using 1
+  congr 1
+  funext ω
+  by_cases hω : ω ∈ E <;> simp [hω]
+
+lemma brownian_set_integral_square
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (s δ C : ℝ≥0) {f : Ω → ℝ} {E : Set Ω}
+    (hf : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] f)
+    (hbound : ∀ ω, |f ω| ≤ C)
+    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
+    (∫ ω in E, f ω * (B (s + δ) ω - B s ω) ^ 2 ∂P) =
+      (δ : ℝ) * ∫ ω in E, f ω ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hEm : MeasurableSet E :=
+    (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s _ hE
+  have hfm := hf.mono ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s) le_rfl
+  have hfi : Integrable f (P.restrict E) :=
+    Integrable.of_bound hfm.aestronglyMeasurable _ (Eventually.of_forall fun ω => by
+      simpa only [Real.norm_eq_abs] using hbound ω)
+  have hi2 : Integrable (fun ω => (B (s + δ) ω - B s ω) ^ 2) (P.restrict E) :=
+    ((hB.shift s).hasLaw_eval δ).hasGaussianLaw.memLp_two.integrable_sq.restrict
+  have hprod := hi2.bdd_mul hfm.aestronglyMeasurable
+    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hbound ω)
+  have h := brownian_centered_square_orthogonal hB hm s δ (hf.indicator hE)
+  have hz : (∫ ω in E, f ω * ((B (s + δ) ω - B s ω) ^ 2 - (δ : ℝ)) ∂P) = 0 := by
+    rw [← integral_indicator hEm]
+    convert h using 1
+    congr 1
+    funext ω
+    by_cases hω : ω ∈ E <;> simp [hω]
+  simp_rw [mul_sub] at hz
+  rw [integral_sub hprod (hfi.mul_const _), integral_mul_const] at hz
+  linarith
+
+lemma smooth_field_taylor_set_error
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (s δ : ℝ≥0) {E : Set Ω}
+    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
+    |(∫ ω in E, F s ω (B (s + δ) ω) - F s ω (B s ω) ∂P) -
+      (1 / 2 : ℝ) * (δ : ℝ) * ∫ ω in E, iteratedDeriv 2 (F s ω) (B s ω) ∂P| ≤
+      (K : ℝ) * (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 3 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let X (ω : Ω) := B (s + δ) ω - B s ω
+  let c₁ (ω : Ω) := iteratedDeriv 1 (F s ω) (B s ω)
+  let c₂ (ω : Ω) := iteratedDeriv 2 (F s ω) (B s ω)
+  let T (ω : Ω) := F s ω (B (s + δ) ω) - F s ω (B s ω)
+  let R (ω : Ω) := T ω - c₁ ω * X ω - (1 / 2 : ℝ) * (c₂ ω * X ω ^ 2)
+  have hXlaw : HasLaw X (gaussianReal 0 δ) P := (hB.shift s).hasLaw_eval δ
+  have hXm : Measurable X := (hm _).sub (hm _)
+  have hc₁m : Measurable c₁ := hF.eval_measurable s 1 (by decide) (hm s)
+  have hc₂m : Measurable c₂ := hF.eval_measurable s 2 (by decide) (hm s)
+  have hc₁b (ω) : |c₁ ω| ≤ K := hF.bound s ω _ 1 (by decide)
+  have hc₂b (ω) : |c₂ ω| ≤ K := hF.bound s ω _ 2 (by decide)
+  have hI (t : ℝ≥0) : Integrable (fun ω => F s ω (B t ω)) (P.restrict E) := by
+    have hm' : Measurable (fun ω => F s ω (B t ω)) := by
+      simpa only [iteratedDeriv_zero] using hF.eval_measurable s 0 (by decide) (hm t)
+    refine Integrable.of_bound hm'.aestronglyMeasurable K (Eventually.of_forall fun ω => ?_)
+    simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using hF.bound s ω (B t ω) 0 (by decide)
+  have hTi : Integrable T (P.restrict E) := (hI _).sub (hI _)
+  have hlin : Integrable (fun ω => c₁ ω * X ω) (P.restrict E) :=
+    hXlaw.hasGaussianLaw.integrable.restrict.bdd_mul hc₁m.aestronglyMeasurable
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hc₁b ω)
+  have hquad : Integrable (fun ω => c₂ ω * X ω ^ 2) (P.restrict E) :=
+    hXlaw.hasGaussianLaw.memLp_two.integrable_sq.restrict.bdd_mul hc₂m.aestronglyMeasurable
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hc₂b ω)
+  have hRi : Integrable R (P.restrict E) := (hTi.sub hlin).sub (hquad.const_mul _)
+  have hlin0 : (∫ ω in E, c₁ ω * X ω ∂P) = 0 :=
+    brownian_set_integral_increment_zero hB hm s δ (hF.eval_adapted s 1 (by decide)) hE
+  have hquad0 : (∫ ω in E, c₂ ω * X ω ^ 2 ∂P) = (δ : ℝ) * ∫ ω in E, c₂ ω ∂P :=
+    brownian_set_integral_square hB hm s δ K (hF.eval_adapted s 2 (by decide)) hc₂b hE
+  have hRint : (∫ ω in E, R ω ∂P) = (∫ ω in E, T ω ∂P) -
+      (1 / 2 : ℝ) * (δ : ℝ) * ∫ ω in E, c₂ ω ∂P := by
+    dsimp only [R]
+    have hsubi : Integrable (fun ω => T ω - c₁ ω * X ω) (P.restrict E) := hTi.sub hlin
+    rw [integral_sub hsubi (hquad.const_mul (1 / 2 : ℝ)), integral_sub hTi hlin,
+      hlin0, sub_zero, integral_const_mul, hquad0]
+    ring
+  have hRb (ω) : |R ω| ≤ (K : ℝ) * |X ω| ^ 3 := by
+    have h := real_second_taylor_bound (hF.smooth s ω) K
+      (fun z => hF.bound s ω z 3 (by decide)) (B s ω) (X ω)
+    have he : B s ω + X ω = B (s + δ) ω := by dsimp [X]; ring
+    rw [he] at h
+    simpa only [R, T, c₁, c₂, iteratedDeriv_one, mul_assoc] using h
+  change |(∫ ω in E, T ω ∂P) - (1 / 2 : ℝ) * (δ : ℝ) * ∫ ω in E, c₂ ω ∂P| ≤ _
+  rw [← hRint]
+  have hcube := gaussian_abs_moment_integrable hXlaw 3 (by decide)
+  calc
+    |∫ ω in E, R ω ∂P| ≤ ∫ ω in E, |R ω| ∂P := by
+      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := P.restrict E) R
+    _ ≤ ∫ ω in E, (K : ℝ) * |X ω| ^ 3 ∂P :=
+      integral_mono hRi.abs (hcube.restrict.const_mul _) hRb
+    _ ≤ ∫ ω, (K : ℝ) * |X ω| ^ 3 ∂P :=
+      integral_mono_measure Measure.restrict_le_self
+        (Eventually.of_forall fun _ => by positivity) (hcube.const_mul _)
+    _ = (K : ℝ) * (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 3 := by
+      rw [integral_const_mul, gaussian_abs_cube hXlaw]
+      ring
+
+lemma brownian_expected_abs_increment_le
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (r t δ : ℝ≥0) (hgap : |(t : ℝ) - r| ≤ δ) (E : Set Ω) :
+    (∫ ω in E, |B t ω - B r ω| ∂P) ≤ Real.sqrt δ * standardGaussianAbsMoment 1 := by
+  have hlaw : HasLaw (fun ω => B t ω - B r ω)
+      (gaussianReal 0 (nndist (t : ℝ) (r : ℝ))) P := hB.hasLaw_sub t r
+  have hint : Integrable (fun ω => |B t ω - B r ω|) P := hlaw.hasGaussianLaw.integrable.abs
+  calc
+    (∫ ω in E, |B t ω - B r ω| ∂P) ≤ ∫ ω, |B t ω - B r ω| ∂P :=
+      integral_mono_measure Measure.restrict_le_self
+        (Eventually.of_forall fun _ => abs_nonneg _) hint
+    _ = Real.sqrt (nndist (t : ℝ) (r : ℝ)) * standardGaussianAbsMoment 1 := by
+      simpa only [pow_one, Pi.sub_apply] using gaussian_abs_moment hlaw 1
+    _ ≤ Real.sqrt δ * standardGaussianAbsMoment 1 := by
+      apply mul_le_mul_of_nonneg_right _ (standardGaussianAbsMoment_nonneg 1)
+      apply Real.sqrt_le_sqrt
+      simpa only [coe_nndist, Real.dist_eq] using hgap
+
+lemma bounded_interval_integral_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    (a b : ℝ) (hab : a ≤ b) (G : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) (C : ℝ)
+    (hbound : ∀ u ω, |G u ω| ≤ C) :
+    Integrable (fun ω => ∫ u in a..b, G u ω) P := by
+  have hI : Integrable (Function.uncurry G) ((volume.restrict (Ioc a b)).prod P) :=
+    Integrable.of_bound hG.aestronglyMeasurable C
+      (Eventually.of_forall fun q => by simpa only [Real.norm_eq_abs, Function.uncurry] using hbound q.1 q.2)
+  simpa only [intervalIntegral.integral_of_le hab, Function.uncurry] using hI.integral_prod_right
+
+lemma bounded_interval_integral_expectation_bound
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    (a b : ℝ) (hab : a ≤ b) (G : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) (C D : ℝ)
+    (hbound : ∀ u ω, |G u ω| ≤ C)
+    (hmean : ∀ u ∈ Ioc a b, |∫ ω, G u ω ∂P| ≤ D) :
+    |∫ ω, (∫ u in a..b, G u ω) ∂P| ≤ D * (b - a) := by
+  have hI : Integrable (Function.uncurry G) ((volume.restrict (Ioc a b)).prod P) :=
+    Integrable.of_bound hG.aestronglyMeasurable C
+      (Eventually.of_forall fun q => by simpa only [Real.norm_eq_abs, Function.uncurry] using hbound q.1 q.2)
+  simp_rw [intervalIntegral.integral_of_le hab]
+  rw [← integral_integral_swap hI]
+  have h := intervalIntegral.norm_integral_le_of_norm_le_const (a := a) (b := b)
+    (f := fun u => ∫ ω, G u ω ∂P) (C := D) (fun u hu => by
+      simpa only [Real.norm_eq_abs] using hmean u (by simpa only [uIoc_of_le hab] using hu))
+  simpa only [Real.norm_eq_abs, intervalIntegral.integral_of_le hab,
+    abs_of_nonneg (sub_nonneg.mpr hab)] using h
+
+lemma bounded_time_section_intervalIntegrable
+    {Ω : Type*} [MeasurableSpace Ω] (a b : ℝ) (hab : a ≤ b)
+    (G : ℝ → Ω → ℝ) (hG : Measurable (Function.uncurry G)) (C : ℝ)
+    (hbound : ∀ u ω, |G u ω| ≤ C) (ω : Ω) :
+    IntervalIntegrable (fun u => G u ω) volume a b := by
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab]
+  apply Integrable.of_bound ((hG.comp (measurable_id.prodMk measurable_const)).aestronglyMeasurable) C
+  exact Eventually.of_forall fun u => by simpa only [Real.norm_eq_abs, Function.comp_apply, Function.uncurry, id_eq] using hbound u ω
+
+lemma smooth_field_time_set_error
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) (E : Set Ω) :
+    |(∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) -
+      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), a u.toNNReal ω (B u.toNNReal ω)) ∂P| ≤
+      (K : ℝ) * (δ : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let C (u : ℝ) (ω : Ω) := a u.toNNReal ω (B (s + δ) ω)
+  let D (u : ℝ) (ω : Ω) := a u.toNNReal ω (B u.toNNReal ω)
+  let G (u : ℝ) (ω : Ω) := C u ω - D u ω
+  have htime : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
+  have hC : Measurable (Function.uncurry C) :=
+    hF.drift_joint.comp (htime.prodMk ((hm (s + δ)).comp measurable_snd))
+  have hD : Measurable (Function.uncurry D) :=
+    hF.drift_joint.comp (htime.prodMk (hBm.comp htime))
+  have hG : Measurable (Function.uncurry G) := hC.sub hD
+  have hCb (u ω) : |C u ω| ≤ K := hF.drift_bound _ _ _
+  have hDb (u ω) : |D u ω| ≤ K := hF.drift_bound _ _ _
+  have hGb (u ω) : |G u ω| ≤ 2 * (K : ℝ) := by
+    exact (abs_sub _ _).trans (by linarith [hCb u ω, hDb u ω])
+  have hle : (s : ℝ) ≤ (s + δ : ℝ≥0) := by
+    simp only [NNReal.coe_add]; exact le_add_of_nonneg_right δ.coe_nonneg
+  have hCI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle C hC K hCb
+  have hDI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle D hD K hDb
+  have he : (∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) -
+      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P) =
+      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), G u ω) ∂P := by
+    simp_rw [hF.time_increment s (s + δ) (le_add_of_nonneg_right (zero_le))]
+    change (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), C u ω) ∂P) - _ = _
+    rw [← integral_sub hCI hDI]
+    apply integral_congr_ae
+    exact Eventually.of_forall fun ω =>
+      (intervalIntegral.integral_sub
+        (bounded_time_section_intervalIntegrable _ _ hle C hC K hCb ω)
+        (bounded_time_section_intervalIntegrable _ _ hle D hD K hDb ω)).symm
+  change |(∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) -
+    (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P)| ≤ _
+  rw [he]
+  have hmean (u : ℝ) (hu : u ∈ Ioc (s : ℝ) (s + δ : ℝ≥0)) :
+      |∫ ω in E, G u ω ∂P| ≤ (K : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1 := by
+    have hgi : Integrable (G u) (P.restrict E) :=
+      Integrable.of_bound (hG.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable
+        (2 * (K : ℝ)) (Eventually.of_forall fun ω => by
+          simpa only [Real.norm_eq_abs] using hGb u ω)
+    have hxi : Integrable (fun ω => |B (s + δ) ω - B u.toNNReal ω|) (P.restrict E) :=
+      ((hB.hasLaw_sub (s + δ) u.toNNReal).hasGaussianLaw.integrable.abs).restrict
+    have hgap : |((s + δ : ℝ≥0) : ℝ) - (u.toNNReal : ℝ)| ≤ δ := by
+      rw [Real.coe_toNNReal _ (le_trans s.coe_nonneg hu.1.le),
+        abs_of_nonneg (sub_nonneg.mpr hu.2), NNReal.coe_add]
+      linarith [hu.1]
+    calc
+      |∫ ω in E, G u ω ∂P| ≤ ∫ ω in E, |G u ω| ∂P := by
+        simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := P.restrict E) (G u)
+      _ ≤ ∫ ω in E, (K : ℝ) * |B (s + δ) ω - B u.toNNReal ω| ∂P :=
+        integral_mono hgi.abs (hxi.const_mul _) (fun ω => hF.drift_lipschitz _ _ _ _)
+      _ = (K : ℝ) * ∫ ω in E, |B (s + δ) ω - B u.toNNReal ω| ∂P := integral_const_mul _ _
+      _ ≤ (K : ℝ) * (Real.sqrt δ * standardGaussianAbsMoment 1) :=
+        mul_le_mul_of_nonneg_left (brownian_expected_abs_increment_le hB _ _ _ hgap E) K.coe_nonneg
+      _ = (K : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1 := by ring
+  have hb := bounded_interval_integral_expectation_bound (P := P.restrict E) _ _ hle G hG
+    (2 * (K : ℝ)) ((K : ℝ) * Real.sqrt δ * standardGaussianAbsMoment 1) hGb hmean
+  simpa only [NNReal.coe_add, add_sub_cancel_left, mul_right_comm, mul_assoc] using hb
+
+lemma smooth_field_second_mean_error
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) (E : Set Ω)
+    (u : ℝ) (hu : u ∈ Ioc (s : ℝ) (s + δ : ℝ≥0)) :
+    |∫ ω in E, iteratedDeriv 2 (F s ω) (B s ω) -
+      iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω) ∂P| ≤
+      (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let C (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F s ω) (B s ω)
+  let D (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)
+  let G (u : ℝ) (ω : Ω) := C u ω - D u ω
+  have htime : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
+  have hC : Measurable (Function.uncurry C) :=
+    (hF.eval_measurable s 2 (by decide) (hm s)).comp measurable_snd
+  have hD : Measurable (Function.uncurry D) := by
+    change Measurable (fun q : ℝ × Ω =>
+      iteratedDeriv 2 (F q.1.toNNReal q.2) (B q.1.toNNReal q.2))
+    simpa only [Function.comp_def] using
+      (hF.joint 2 (by decide)).comp (htime.prodMk (hBm.comp htime))
+  have hG : Measurable (Function.uncurry G) := hC.sub hD
+  have hCb (u ω) : |C u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
+  have hDb (u ω) : |D u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
+  have hGb (u ω) : |G u ω| ≤ 2 * (K : ℝ) :=
+    (abs_sub _ _).trans (by linarith [hCb u ω, hDb u ω])
+  have hgi : Integrable (G u) (P.restrict E) :=
+    Integrable.of_bound (hG.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable
+      (2 * (K : ℝ)) (Eventually.of_forall fun ω => by
+        simpa only [Real.norm_eq_abs] using hGb u ω)
+  have hxi : Integrable (fun ω => |B u.toNNReal ω - B s ω|) (P.restrict E) :=
+    ((hB.hasLaw_sub u.toNNReal s).hasGaussianLaw.integrable.abs).restrict
+  have hgap : |(u.toNNReal : ℝ) - (s : ℝ)| ≤ δ := by
+    rw [Real.coe_toNNReal _ (le_trans s.coe_nonneg hu.1.le), abs_of_nonneg (sub_nonneg.mpr hu.1.le)]
+    have hh := hu.2
+    simp only [NNReal.coe_add] at hh
+    linarith
+  have hpoint (ω) : |G u ω| ≤ (K : ℝ) * ((δ : ℝ) + |B u.toNNReal ω - B s ω|) := by
+    calc
+      |G u ω| = |D u ω - C u ω| := abs_sub_comm _ _
+      _ ≤ (K : ℝ) * (|(u.toNNReal : ℝ) - (s : ℝ)| + |B u.toNNReal ω - B s ω|) :=
+        hF.second_lipschitz s u.toNNReal ω _ _
+      _ ≤ (K : ℝ) * ((δ : ℝ) + |B u.toNNReal ω - B s ω|) :=
+        mul_le_mul_of_nonneg_left (add_le_add hgap le_rfl) K.coe_nonneg
+  have hconst : (∫ _ω in E, (δ : ℝ) ∂P) ≤ δ := by
+    calc
+      (∫ _ω in E, (δ : ℝ) ∂P) ≤ ∫ _ω, (δ : ℝ) ∂P :=
+        integral_mono_measure Measure.restrict_le_self
+          (Eventually.of_forall fun _ => δ.coe_nonneg) (integrable_const _)
+      _ = δ := by simp
+  calc
+    |∫ ω in E, G u ω ∂P| ≤ ∫ ω in E, |G u ω| ∂P := by
+      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := P.restrict E) (G u)
+    _ ≤ ∫ ω in E, (K : ℝ) * ((δ : ℝ) + |B u.toNNReal ω - B s ω|) ∂P :=
+      integral_mono hgi.abs (((integrable_const _).add hxi).const_mul _) hpoint
+    _ = (K : ℝ) * ((∫ _ω in E, (δ : ℝ) ∂P) +
+        ∫ ω in E, |B u.toNNReal ω - B s ω| ∂P) := by
+      rw [integral_const_mul, integral_add (integrable_const _) hxi]
+    _ ≤ (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) :=
+      mul_le_mul_of_nonneg_left
+        (add_le_add hconst (brownian_expected_abs_increment_le hB _ _ _ hgap E)) K.coe_nonneg
+
+lemma smooth_field_second_set_error
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) (E : Set Ω) :
+    |(δ : ℝ) * (∫ ω in E, iteratedDeriv 2 (F s ω) (B s ω) ∂P) -
+      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0),
+        iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)) ∂P| ≤
+      (K : ℝ) * (δ : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let C (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F s ω) (B s ω)
+  let D (u : ℝ) (ω : Ω) := iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)
+  let G (u : ℝ) (ω : Ω) := C u ω - D u ω
+  have htime : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
+  have hC : Measurable (Function.uncurry C) :=
+    (hF.eval_measurable s 2 (by decide) (hm s)).comp measurable_snd
+  have hD : Measurable (Function.uncurry D) := by
+    change Measurable (fun q : ℝ × Ω =>
+      iteratedDeriv 2 (F q.1.toNNReal q.2) (B q.1.toNNReal q.2))
+    simpa only [Function.comp_def] using
+      (hF.joint 2 (by decide)).comp (htime.prodMk (hBm.comp htime))
+  have hG : Measurable (Function.uncurry G) := hC.sub hD
+  have hCb (u ω) : |C u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
+  have hDb (u ω) : |D u ω| ≤ K := hF.bound _ _ _ 2 (by decide)
+  have hGb (u ω) : |G u ω| ≤ 2 * (K : ℝ) :=
+    (abs_sub _ _).trans (by linarith [hCb u ω, hDb u ω])
+  have hle : (s : ℝ) ≤ (s + δ : ℝ≥0) := by
+    simp only [NNReal.coe_add]; exact le_add_of_nonneg_right δ.coe_nonneg
+  have hCI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle C hC K hCb
+  have hDI := bounded_interval_integral_integrable (P := P.restrict E) _ _ hle D hD K hDb
+  have he : (δ : ℝ) * (∫ ω in E, C 0 ω ∂P) -
+      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P) =
+      ∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), G u ω) ∂P := by
+    have hconst : (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), C u ω) ∂P) =
+        (δ : ℝ) * ∫ ω in E, C 0 ω ∂P := by
+      simp only [C, intervalIntegral.integral_const, NNReal.coe_add,
+        add_sub_cancel_left, smul_eq_mul, integral_const_mul]
+    rw [← hconst, ← integral_sub hCI hDI]
+    apply integral_congr_ae
+    exact Eventually.of_forall fun ω =>
+      (intervalIntegral.integral_sub
+        (bounded_time_section_intervalIntegrable _ _ hle C hC K hCb ω)
+        (bounded_time_section_intervalIntegrable _ _ hle D hD K hDb ω)).symm
+  change |(δ : ℝ) * (∫ ω in E, C 0 ω ∂P) -
+    (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), D u ω) ∂P)| ≤ _
+  rw [he]
+  have hmean (u : ℝ) (hu : u ∈ Ioc (s : ℝ) (s + δ : ℝ≥0)) :
+      |∫ ω in E, G u ω ∂P| ≤ (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) :=
+    smooth_field_second_mean_error hB hm hBm K hF s δ E u hu
+  have hb := bounded_interval_integral_expectation_bound (P := P.restrict E) _ _ hle G hG
+    (2 * (K : ℝ)) ((K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1)) hGb hmean
+  calc
+    _ ≤ (K : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1) *
+        (((s + δ : ℝ≥0) : ℝ) - s) := hb
+    _ = _ := by simp only [NNReal.coe_add]; ring
+
+lemma SmoothTimeBrownianField.eval_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (j : ℕ) (hj : j ≤ 2)
+    {V : Ω → ℝ} (hV : Measurable V) :
+    Integrable (fun ω => iteratedDeriv j (F t ω) (V ω)) P := by
+  apply Integrable.of_bound (hF.eval_measurable t j hj hV).aestronglyMeasurable K
+  exact Eventually.of_forall fun ω => by
+    simpa only [Real.norm_eq_abs] using hF.bound t ω (V ω) j (by omega)
+
+lemma SmoothTimeBrownianField.deriv_real_joint
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (j : ℕ) (hj : j ≤ 2) :
+    Measurable (fun q : ℝ × Ω => iteratedDeriv j (F q.1.toNNReal q.2) (B q.1.toNNReal q.2)) := by
+  have ht : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
+  simpa only [Function.comp_def] using (hF.joint j hj).comp (ht.prodMk (hBm.comp ht))
+
+lemma SmoothTimeBrownianField.drift_real_joint
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
+    Measurable (fun q : ℝ × Ω => a q.1.toNNReal q.2 (B q.1.toNNReal q.2)) := by
+  have ht : Measurable (fun q : ℝ × Ω => (q.1.toNNReal, q.2)) := by fun_prop
+  exact hF.drift_joint.comp (ht.prodMk (hBm.comp ht))
+
+lemma bounded_interval_integral_expectation_add
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    (a b c : ℝ) (hab : a ≤ b) (G H : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) (hH : Measurable (Function.uncurry H))
+    (C D : ℝ) (hGb : ∀ u ω, |G u ω| ≤ C) (hHb : ∀ u ω, |H u ω| ≤ D) :
+    (∫ ω, (∫ u in a..b, G u ω + c * H u ω) ∂P) =
+      (∫ ω, (∫ u in a..b, G u ω) ∂P) + c * (∫ ω, (∫ u in a..b, H u ω) ∂P) := by
+  have hGI := bounded_interval_integral_integrable a b hab G hG C hGb (P := P)
+  have hHI := bounded_interval_integral_integrable a b hab H hH D hHb (P := P)
+  calc
+    _ = ∫ ω, ((∫ u in a..b, G u ω) + c * (∫ u in a..b, H u ω)) ∂P := by
+      apply integral_congr_ae
+      filter_upwards [] with ω
+      rw [intervalIntegral.integral_add
+        (bounded_time_section_intervalIntegrable a b hab G hG C hGb ω)
+        ((bounded_time_section_intervalIntegrable a b hab H hH D hHb ω).const_mul c),
+        intervalIntegral.integral_const_mul]
+    _ = _ := by rw [integral_add hGI (hHI.const_mul c), integral_const_mul]
+
+noncomputable def dynkinRate (δ : ℝ≥0) : ℝ :=
+  Real.sqrt δ * standardGaussianAbsMoment 3 + Real.sqrt δ * standardGaussianAbsMoment 1 +
+    (1 / 2 : ℝ) * ((δ : ℝ) + Real.sqrt δ * standardGaussianAbsMoment 1)
+
+lemma dynkinRate_tendsto_zero : Tendsto dynkinRate (𝓝 0) (𝓝 0) := by
+  have hc : Continuous dynkinRate := by unfold dynkinRate; fun_prop
+  simpa [ContinuousAt, dynkinRate] using hc.continuousAt (x := 0)
+
+lemma abs_dynkin_algebra_bound (T A S I J eT eA eS δ : ℝ)
+    (hT : |T - (1 / 2 : ℝ) * δ * S| ≤ eT)
+    (hA : |A - I| ≤ eA) (hS : |δ * S - J| ≤ eS) :
+    |(T + A) - (I + (1 / 2 : ℝ) * J)| ≤ eT + eA + (1 / 2 : ℝ) * eS := by
+  have he : (T + A) - (I + (1 / 2 : ℝ) * J) =
+      (T - (1 / 2 : ℝ) * δ * S) + (A - I) + (1 / 2 : ℝ) * (δ * S - J) := by ring
+  rw [he]
+  calc
+    _ ≤ |T - (1 / 2 : ℝ) * δ * S| + |A - I| + |(1 / 2 : ℝ) * (δ * S - J)| :=
+      (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
+    _ = |T - (1 / 2 : ℝ) * δ * S| + |A - I| + (1 / 2 : ℝ) * |δ * S - J| := by
+      rw [abs_mul]; norm_num
+    _ ≤ _ := add_le_add (add_le_add hT hA) (mul_le_mul_of_nonneg_left hS (by norm_num))
+
+lemma smooth_field_one_step_error
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (K : ℝ≥0) {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (s δ : ℝ≥0) {E : Set Ω}
+    (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
+    |(∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B s ω) ∂P) -
+      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P)| ≤
+      (K : ℝ) * (δ : ℝ) * dynkinRate δ := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hI (v t : ℝ≥0) : Integrable (fun ω => F v ω (B t ω)) (P.restrict E) := by
+    simpa only [iteratedDeriv_zero] using hF.eval_integrable v 0 (by decide) (hm t) (P := P.restrict E)
+  have hsplit : (∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B s ω) ∂P) =
+      (∫ ω in E, F s ω (B (s + δ) ω) - F s ω (B s ω) ∂P) +
+      (∫ ω in E, F (s + δ) ω (B (s + δ) ω) - F s ω (B (s + δ) ω) ∂P) := by
+    rw [integral_sub (hI _ _) (hI _ _), integral_sub (hI _ _) (hI _ _),
+      integral_sub (hI _ _) (hI _ _)]
+    ring
+  have hle : (s : ℝ) ≤ (s + δ : ℝ≥0) := by
+    simp only [NNReal.coe_add]; exact le_add_of_nonneg_right δ.coe_nonneg
+  have hsplitD : (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P) =
+      (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0), a u.toNNReal ω (B u.toNNReal ω)) ∂P) +
+      (1 / 2 : ℝ) * (∫ ω in E, (∫ u in (s : ℝ)..(s + δ : ℝ≥0),
+        iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω)) ∂P) := by
+    apply bounded_interval_integral_expectation_add _ _ _ hle _ _
+      (hF.drift_real_joint hBm) (hF.deriv_real_joint hBm 2 (by decide)) (K : ℝ) (K : ℝ)
+    · exact fun u ω => hF.drift_bound _ _ _
+    · exact fun u ω => hF.bound _ _ _ 2 (by decide)
+  rw [hsplit, hsplitD]
+  have h := abs_dynkin_algebra_bound _ _ _ _ _ _ _ _ _
+    (smooth_field_taylor_set_error hB hm K hF s δ hE)
+    (smooth_field_time_set_error hB hm hBm K hF s δ E)
+    (smooth_field_second_set_error hB hm hBm K hF s δ E)
+  convert h using 1
+  unfold dynkinRate
+  ring
+
+ 
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/BrownianStrongMarkov.lean b/lean/OAI/Probability/SLE/BrownianStrongMarkov.lean
index fac0aefca270755d90e7b82ebef49461f37236dc..67a4f06cd9837779deb81d9066f1337e2a860bff 100644
--- a/lean/OAI/Probability/SLE/BrownianStrongMarkov.lean
+++ b/lean/OAI/Probability/SLE/BrownianStrongMarkov.lean
@@ -1,798 +1,798 @@
-import Mathlib
-import OAI.Probability.SLE.NearRealContraction
-
-namespace OAI
-
-/-! Brownian Strong Markov for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-noncomputable def shiftedFDD {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (I : Finset ℝ≥0) (s : ℝ≥0) (ω : Ω) : I → ℝ :=
-  fun i => B (s + i) ω - B s ω
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-  (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-
-include hm in
-lemma shiftedFDD_measurable (I : Finset ℝ≥0) (s : ℝ≥0) :
-    Measurable (shiftedFDD B I s) :=
-  Measurable.of_eval (fun i => (hm (s + i)).sub (hm s))
-
-include hB hm in
-omit [IsProbabilityMeasure P] in
-lemma shiftedFDD_indep (I : Finset ℝ≥0) (s : ℝ≥0) :
-    Indep (Filtration.natural B (fun t => (hm t).stronglyMeasurable) s)
-      (MeasurableSpace.comap (shiftedFDD B I s) inferInstance) P := by
-  have hi := ((hB.indepFun_shift s).comp
-    (Measurable.of_eval (fun i : I => measurable_pi_apply (i : ℝ≥0))) measurable_id).symm
-  rw [Filtration.natural_eq_comap B _ s]
-  exact (IndepFun_iff_Indep _ _ _).mp hi
-
-include hB hm in
-omit [IsProbabilityMeasure P] in
-lemma shiftedFDD_setIntegral (I : Finset ℝ≥0) (s : ℝ≥0)
-    (φ : BoundedContinuousFunction (I → ℝ) ℝ) {A : Set Ω}
-    (hA : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] A) :
-    (∫ ω in A, φ (shiftedFDD B I s ω) ∂P) =
-      P.real A * ∫ x, φ x ∂BrownianReal.projectiveFamily I := by
-  rw [(shiftedFDD_indep hB hm I s).setIntegral_eq_mul
-    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s)
-    (shiftedFDD_measurable hm I s).aemeasurable hA φ.continuous.aestronglyMeasurable]
-  congr 1
-  exact ((hB.shift s).hasLaw I).integral_comp φ.continuous.aestronglyMeasurable
-
-include hm in
-lemma shiftedFDD_grid_measurable {τ : Ω → ℝ≥0} (hτm : Measurable τ)
-    (I : Finset ℝ≥0) (n : ℕ) :
-    Measurable (fun ω => shiftedFDD B I (stopGrid n (τ ω)) ω) := by
-  apply Measurable.of_eval
-  intro i
-  exact measurable_discrete_eval
-    (fun k ω => B (stopGridTime n k + i) ω - B (stopGridTime n k) ω)
-    (fun k => (hm _).sub (hm _)) (stopGridIndex_measurable hτm n)
-
-include hB hm in
-lemma shiftedFDD_grid_setIntegral {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A)
-    (I : Finset ℝ≥0) (φ : BoundedContinuousFunction (I → ℝ) ℝ) (n : ℕ) :
-    (∫ ω in A, φ (shiftedFDD B I (stopGrid n (τ ω)) ω) ∂P) =
-      P.real A * ∫ x, φ x ∂BrownianReal.projectiveFamily I := by
-  apply integral_stopping_grid_glue hτ hA n _
-    (Integrable.of_bound (φ.continuous.measurable.comp
-      (shiftedFDD_grid_measurable hm (stopping_time_measurable_nnreal hτ) I n)).aestronglyMeasurable
-      ‖φ‖ (Eventually.of_forall (fun ω => φ.norm_coe_le_norm _)))
-  intro k
-  have hk := stopping_grid_piece_measurable hτ hA n k
-  have hk' := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le _ _ hk
-  rw [setIntegral_congr_fun hk' (g := fun ω => φ (shiftedFDD B I (stopGridTime n k) ω))
-    (fun ω hω => by
-      have he : stopGridIndex n (τ ω) = k := hω.2
-      simp only [Function.comp_def, stopGrid, he])]
-  exact shiftedFDD_setIntegral hB hm I _ φ hk
-
-omit mΩ in
-lemma shiftedFDD_grid_tendsto
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (I : Finset ℝ≥0) (τ : Ω → ℝ≥0) (ω : Ω) :
-    Tendsto (fun n => shiftedFDD B I (stopGrid n (τ ω)) ω) atTop
-      (𝓝 (shiftedFDD B I (τ ω) ω)) := by
-  apply tendsto_pi_nhds.mpr
-  intro i
-  exact ((hc ω).continuousAt.tendsto.comp ((stopGrid_tendsto (τ ω)).add_const (i : ℝ≥0))).sub
-    ((hc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω)))
-
-include hB hm in
- 
-
-theorem brownian_strongMarkov_integral
-    (hc : ∀ ω, Continuous (fun t => B t ω)) {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A)
-    (I : Finset ℝ≥0) (φ : BoundedContinuousFunction (I → ℝ) ℝ) :
-    (∫ ω in A, φ (shiftedFDD B I (τ ω) ω) ∂P) =
-      P.real A * ∫ x, φ x ∂BrownianReal.projectiveFamily I := by
-  have hlim := tendsto_integral_of_dominated_convergence (μ := P.restrict A)
-    (F := fun n ω => φ (shiftedFDD B I (stopGrid n (τ ω)) ω))
-    (f := fun ω => φ (shiftedFDD B I (τ ω) ω)) (fun _ => ‖φ‖)
-    (fun n => (φ.continuous.measurable.comp
-      (shiftedFDD_grid_measurable hm (stopping_time_measurable_nnreal hτ) I n)).aestronglyMeasurable)
-    (integrable_const _)
-    (fun n => Eventually.of_forall (fun ω => φ.norm_coe_le_norm _))
-    (Eventually.of_forall (fun ω => φ.continuous.continuousAt.tendsto.comp
-      (shiftedFDD_grid_tendsto hc I τ ω)))
-  simp only [shiftedFDD_grid_setIntegral hB hm hτ hA I φ] at hlim
-  exact tendsto_nhds_unique hlim tendsto_const_nhds
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def brownianRestart {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : ℝ := B (τ ω + t) ω - B (τ ω) ω
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-  (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-  (hc : ∀ ω, Continuous (fun t => B t ω))
-
-include hm hc in
-lemma brownianRestart_measurable {τ : Ω → ℝ≥0} (hτm : Measurable τ) :
-    Measurable (fun ω t => brownianRestart B τ t ω) := by
-  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable hc hm
-  apply Measurable.of_eval
-  intro t
-  exact (hBm.comp ((hτm.add_const t).prodMk measurable_id)).sub
-    (hBm.comp (hτm.prodMk measurable_id))
-
-include hB hm hc in
-lemma brownian_restart_fdd_restrict {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) (I : Finset ℝ≥0) :
-    (P.restrict A).map (fun ω => shiftedFDD B I (τ ω) ω) =
-      P A • BrownianReal.projectiveFamily I := by
-  have hXm : Measurable (fun ω => shiftedFDD B I (τ ω) ω) :=
-    (Finset.measurable_restrict I).comp
-      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))
-  let : IsFiniteMeasure (P A • BrownianReal.projectiveFamily I) :=
-    Measure.smul_finite _ (measure_ne_top P A)
-  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
-  intro φ
-  rw [integral_map hXm.aemeasurable φ.continuous.aestronglyMeasurable,
-    integral_smul_measure]
-  exact brownian_strongMarkov_integral hB hm hc hτ hA I φ
-
-include hB hm hc in
- 
-
-theorem brownian_restart_path_law {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) :
-    (P.restrict A).map (fun ω t => brownianRestart B τ t ω) =
-      P A • P.map (fun ω t => B t ω) := by
-  have hXm := brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)
-  have hBm : Measurable (fun ω t => B t ω) := Measurable.of_eval hm
-  let (I : Finset ℝ≥0) : IsFiniteMeasure (P A • BrownianReal.projectiveFamily I) :=
-    Measure.smul_finite _ (measure_ne_top P A)
-  have hL : IsProjectiveLimit ((P.restrict A).map (fun ω t => brownianRestart B τ t ω))
-      (fun I => P A • BrownianReal.projectiveFamily I) := by
-    intro I
-    rw [Measure.map_map (Finset.measurable_restrict I) hXm]
-    exact brownian_restart_fdd_restrict hB hm hc hτ hA I
-  have hR : IsProjectiveLimit (P A • P.map (fun ω t => B t ω))
-      (fun I => P A • BrownianReal.projectiveFamily I) := by
-    intro I
-    rw [Measure.map_smul _ (Finset.measurable_restrict I).aemeasurable,
-      Measure.map_map (Finset.measurable_restrict I) hBm]
-    exact congrArg (fun ν => P A • ν) (hB.hasLaw I).map_eq
-  exact hL.unique hR
-
-include hB hm hc in
-lemma brownian_restart_path_map {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) :
-    P.map (fun ω t => brownianRestart B τ t ω) = P.map (fun ω t => B t ω) := by
-  simpa only [Measure.restrict_univ, measure_univ, one_smul] using
-    brownian_restart_path_law hB hm hc hτ (A := univ) MeasurableSet.univ
-
-include hB hm hc in
-lemma brownian_restart_event_factorizes {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A)
-    {S : Set (ℝ≥0 → ℝ)} (hS : MeasurableSet S) :
-    P (A ∩ {ω | (fun t => brownianRestart B τ t ω) ∈ S}) =
-      P A * P {ω | (fun t => B t ω) ∈ S} := by
-  have hXm := brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)
-  have he := congrArg (fun ν : Measure (ℝ≥0 → ℝ) => ν S)
-    (brownian_restart_path_law hB hm hc hτ hA)
-  rw [Measure.map_apply hXm hS, Measure.restrict_apply (hXm hS),
-    Measure.smul_apply, Measure.map_apply (Measurable.of_eval hm) hS] at he
-  simpa only [Set.preimage, Set.inter_comm, smul_eq_mul] using he
-
-include hB hm hc in
- 
-
-theorem brownian_restart_independent {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) :
-    Indep hτ.measurableSpace
-      (MeasurableSpace.comap (fun ω t => brownianRestart B τ t ω) inferInstance) P := by
-  rw [Indep_iff]
-  intro A C hA hC
-  rcases hC with ⟨S, hS, rfl⟩
-  have he := congrArg (fun ν : Measure (ℝ≥0 → ℝ) => ν S)
-    (brownian_restart_path_map hB hm hc hτ)
-  rw [Measure.map_apply (brownianRestart_measurable hm hc
-    (stopping_time_measurable_nnreal hτ)) hS,
-    Measure.map_apply (Measurable.of_eval hm) hS] at he
-  change P (A ∩ {ω | (fun t => brownianRestart B τ t ω) ∈ S}) = _
-  rw [brownian_restart_event_factorizes hB hm hc hτ hA hS]
-  rw [he]
-  rfl
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-section
-variable {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
-  (U : ℝ≥0 → Ω → ℝ) (hUa : Adapted ℱ U)
-  (hUc : ∀ ω, Continuous (fun t => U t ω))
-include hUa hUc
-
-lemma measurable_reverse_driver (T : ℝ≥0) :
-    @Measurable (ℝ × Ω) ℝ ((borel ℝ).prod (ℱ T)) (borel ℝ)
-      (fun p => U (min ((T : ℝ) - p.1).toNNReal T) p.2) := by
-  have hsa : StronglyAdapted ℱ U := fun t => (hUa t).stronglyMeasurable
-  have hp := (hsa.isStronglyProgressive_of_continuous hUc T).measurable
-  let : MeasurableSpace Ω := ℱ T
-  have ht : Measurable (fun s : ℝ => (⟨min ((T : ℝ) - s).toNNReal T,
-      by change min ((T : ℝ) - s).toNNReal T ≤ T; exact min_le_right _ _⟩ : Iic T)) :=
-    ((continuous_real_toNNReal.comp (continuous_const.sub continuous_id)).min continuous_const).measurable.subtype_mk
-  exact hp.comp ((ht.comp measurable_fst).prodMk measurable_snd)
-
- 
-
-theorem inverse_vertical_measurable
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (h : ∀ ω, IsCapacityTwoWitness (fun t => U t ω) (γ ω) (G ω) (F ω))
-    (T : ℝ≥0) {a : ℝ} (ha : 0 < a) :
-    @Measurable Ω ℂ (ℱ T) (borel ℂ)
-      (fun ω => F ω T ((U T ω : ℂ) + (a : ℂ) * I)) := by
-  let V (s : ℝ) (ω : Ω) := U (min ((T : ℝ) - s).toNNReal T) ω
-  have hVm := measurable_reverse_driver ℱ U hUa hUc T
-  have hVc (ω : Ω) : Continuous (fun s : ℝ => V s ω) :=
-    (hUc ω).comp ((continuous_real_toNNReal.comp (continuous_const.sub continuous_id)).min continuous_const)
-  let : MeasurableSpace Ω := ℱ T
-  have hwm : Measurable (fun ω => (U T ω : ℂ) + (a : ℂ) * I) :=
-    (Complex.continuous_ofReal.measurable.comp (hUa T)).add_const _
-  obtain ⟨Q, hQm, hQ0, hQd⟩ := exists_measurable_reverse_flow V hVm hVc
-    (fun ω => (U T ω : ℂ) + (a : ℂ) * I) hwm ha
-  have he (ω : Ω) : Q (T : ℝ) ω = F ω T ((U T ω : ℂ) + (a : ℂ) * I) := by
-    apply (h ω).reverse_flow_identifies T _ ha (by simp) (fun s => Q s ω) (hQ0 ω)
-    intro s hs
-    have hsT : ((T : ℝ) - s).toNNReal ≤ T :=
-      Real.toNNReal_le_iff_le_coe.mpr (by linarith [hs.1])
-    simpa only [V, min_eq_left hsT] using
-      (hQd ω s hs.1).mono (show Icc (0 : ℝ) T ⊆ Ici 0 from Icc_subset_Ici_self)
-  have hefun : (fun ω => Q (T : ℝ) ω) = (fun ω => F ω T ((U T ω : ℂ) + (a : ℂ) * I)) := funext he
-  rw [← hefun]
-  exact hQm.comp (measurable_prodMk_left (x := (T : ℝ)))
-
- 
-
-theorem capacity_trace_adapted
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (h : ∀ ω, IsCapacityTwoWitness (fun t => U t ω) (γ ω) (G ω) (F ω)) :
-    Adapted ℱ (fun t ω => γ ω t) := by
-  intro T
-  have hn (n : ℕ) : @Measurable Ω ℂ (ℱ T) (borel ℂ)
-      (fun ω => F ω T ((U T ω : ℂ) + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * I)) :=
-    inverse_vertical_measurable ℱ U hUa hUc γ G F h T (by positivity)
-  let : MeasurableSpace Ω := ℱ T
-  apply measurable_of_tendsto_metrizable
-    (f := fun n ω => F ω T ((U T ω : ℂ) + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * I))
-  · exact hn
-  · apply tendsto_pi_nhds.mpr
-    intro ω
-    have hy : Tendsto (fun n : ℕ => 1 / ((n : ℝ) + 1)) atTop (𝓝[>] (0 : ℝ)) :=
-      tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat,
-        Eventually.of_forall (fun n => by change (0 : ℝ) < 1 / ((n : ℝ) + 1); positivity)⟩
-    exact ((h ω).2.2.2.2.2.2.2.2.2.2.2.2 T).comp hy
-
-end
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-def IsCapacityTwoTrace (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) : Prop :=
-  ∃ G F : ℝ≥0 → ℂ → ℂ, IsCapacityTwoWitness U γ G F
-
- 
-
-def IsOrdinaryChordalSLE {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (P : Measure Ω) : Prop :=
-  (∀ t, Measurable (fun ω => γ ω t)) ∧
-  ∃ B : ℝ≥0 → Ω → ℝ, ProbabilityTheory.IsBrownianReal B P ∧
-    ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * B t ω) (γ ω)
-
- 
-
-structure RegularSLEPresentation {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) where
-  samples : Set Ω
-  measurable_samples : MeasurableSet samples
-  full_samples : ∀ᵐ ω ∂P, ω ∈ samples
-  B : ℝ≥0 → samples → ℝ
-  measurable_B : ∀ t, Measurable (B t)
-  continuous_B : ∀ ω, Continuous (fun t => B t ω)
-  zero_B : ∀ ω, B 0 ω = 0
-  brownian : IsBrownianReal B (P.comap (Subtype.val : samples → Ω))
-  G : samples → ℝ≥0 → ℂ → ℂ
-  F : samples → ℝ≥0 → ℂ → ℂ
-  witness : ∀ ω : samples, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-    (γ ω) (G ω) (F ω)
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-
-lemma measurePreserving_val (a : RegularSLEPresentation P κ γ) :
-    MeasurePreserving (Subtype.val : a.samples → Ω) (P.comap Subtype.val) P := by
-  refine ⟨measurable_subtype_coe, ?_⟩
-  rw [map_comap_subtype_coe a.measurable_samples,
-    Measure.restrict_eq_self_of_ae_mem a.full_samples]
-
-instance probability [IsProbabilityMeasure P] (a : RegularSLEPresentation P κ γ) :
-    IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) :=
-  (MeasurableEmbedding.subtype_coe a.measurable_samples).isProbabilityMeasure_comap
-    (by simpa only [Subtype.range_coe] using a.full_samples)
-
-def filtration (a : RegularSLEPresentation P κ γ) :
-    Filtration ℝ≥0 (inferInstance : MeasurableSpace a.samples) :=
-  Filtration.natural a.B (fun t => (a.measurable_B t).stronglyMeasurable)
-
-lemma trace_adapted (a : RegularSLEPresentation P κ γ) :
-    Adapted a.filtration (fun t ω => γ ω t) := by
-  apply capacity_trace_adapted a.filtration (fun t ω => Real.sqrt κ * a.B t ω)
-    ?_ ?_ (γ ∘ Subtype.val) a.G a.F a.witness
-  · intro t
-    exact measurable_const.mul
-      ((Filtration.stronglyAdapted_natural (fun t => (a.measurable_B t).stronglyMeasurable)) t).measurable
-  · intro ω
-    exact continuous_const.mul (a.continuous_B ω)
-
-lemma trace_continuous (a : RegularSLEPresentation P κ γ) (ω : a.samples) :
-    Continuous (γ ω) := (a.witness ω).1
-
-end RegularSLEPresentation
-
- 
-
-theorem exists_regular_sle_presentation
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) :
-    Nonempty (RegularSLEPresentation P κ γ) := by
-  classical
-  obtain ⟨B, hB, htrace⟩ := hSLE.2
-  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
-  have htraceC : ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * C t ω) (γ ω) := by
-    filter_upwards [htrace, hCB] with ω hω heq
-    simpa only [heq] using hω
-  let N := toMeasurable P {ω | ¬ IsCapacityTwoTrace (fun t => Real.sqrt κ * C t ω) (γ ω)}
-  let S := Nᶜ
-  have hSm : MeasurableSet S := (measurableSet_toMeasurable _ _).compl
-  have hNa : P N = 0 := by
-    rw [show N = toMeasurable P {ω | ¬ IsCapacityTwoTrace
-      (fun t => Real.sqrt κ * C t ω) (γ ω)} from rfl, measure_toMeasurable]
-    exact ae_iff.mp htraceC
-  have hSa : ∀ᵐ ω ∂P, ω ∈ S := by
-    apply ae_iff.mpr
-    simpa only [S, mem_compl_iff, not_not, ofPred_mem_eq] using hNa
-  have hw (ω : S) : IsCapacityTwoTrace (fun t => Real.sqrt κ * C t ω) (γ ω) := by
-    by_contra hh
-    exact ω.property (subset_toMeasurable P _ hh)
-  choose G F hGF using hw
-  have hval : MeasurePreserving (Subtype.val : S → Ω) (P.comap Subtype.val) P := by
-    refine ⟨measurable_subtype_coe, ?_⟩
-    rw [map_comap_subtype_coe hSm, Measure.restrict_eq_self_of_ae_mem hSa]
-  refine ⟨{
-    samples := S
-    measurable_samples := hSm
-    full_samples := hSa
-    B := fun t ω => C t ω
-    measurable_B := fun t => (hCm.comp (measurable_prodMk_left (x := t))).comp measurable_subtype_coe
-    continuous_B := fun ω => hCc ω
-    zero_B := fun ω => hC0 ω
-    brownian := ?_
-    G := G
-    F := F
-    witness := hGF }⟩
-  exact ⟨⟨fun I => (hC.hasLaw I).comp hval.hasLaw⟩, Eventually.of_forall (fun ω => hCc ω)⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem ordinary_positive_real_center_approach
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (x r : ℝ) (hx : 0 < x) (hr : 0 < r) :
-    P {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤
-      ENNReal.ofReal ((r/(((8-κ)/32)*x))^(3*(8-κ)/(4*κ))) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hδ : 0 < (8-κ)/32 := by positivity
-  have hδ2 : (8-κ)/32 ≤ 1/2 := by linarith
-  have hβ : 0 < 3*(8-κ)/(4*κ) := by positivity
-  by_cases hrx : r < ((8-κ)/32)*x
-  · apply (measure_mono_ae (show {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤ᵐ[P]
-      {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r ((8-κ)/32)} from ?_)).trans
-      (boundary_comparison_failure_sle_exponent hB κ x r hκ hκ8 hx hr)
-    filter_upwards [hw,hB.cont,hB.eval_zero_ae_eq_zero] with ω hω hc h0
-    rintro ⟨t,ht⟩
-    by_contra hn
-    have hh := hω.all_time_real_center_avoidance (hc.const_mul _) (by simp only [h0,mul_zero])
-      hx hrx hδ2 hn t
-    exact (not_lt_of_ge hh) (by simpa only [dist_eq_norm] using ht)
-  · have hq : 1 ≤ r/(((8-κ)/32)*x) :=
-      (le_div_iff₀ (mul_pos hδ hx)).mpr (by simpa only [one_mul] using le_of_not_gt hrx)
-    have hh := ENNReal.ofReal_le_ofReal (Real.one_le_rpow hq hβ.le)
-    rw [ENNReal.ofReal_one] at hh
-    exact prob_le_one.trans hh
-
- 
-
-theorem sle_positive_real_center_approach
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (x r : ℝ) (hx : 0 < x) (hr : 0 < r) :
-    P {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤
-      ENNReal.ofReal ((r/(((8-κ)/32)*x))^(3*(8-κ)/(4*κ))) := by
-  classical
-  obtain ⟨B,hB,htrace⟩ := hSLE.2
-  have hchoice (ω : Ω) : ∃ G F : ℝ≥0 → ℂ → ℂ,
-      IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω) →
-        IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) G F := by
-    by_cases h : IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω)
-    · obtain ⟨G,F,hw⟩ := h
-      exact ⟨G,F,fun _ => hw⟩
-    · exact ⟨fun _ _ => 0,fun _ _ => 0,fun hh => False.elim (h hh)⟩
-  choose G F hw using hchoice
-  exact ordinary_positive_real_center_approach hB κ hκ hκ8 γ G F
-    (htrace.mono fun ω hω => hw ω hω) x r hx hr
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Complex ComplexConjugate
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma IsOrdinaryChordalSLE.mirror
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (h : IsOrdinaryChordalSLE κ γ P) :
-    IsOrdinaryChordalSLE κ (fun ω => loewnerMirror ∘ γ ω) P := by
-  obtain ⟨hm,B,hB,hw⟩ := h
-  refine ⟨fun t => loewnerMirror.continuous.measurable.comp (hm t),-B,hB.neg,?_⟩
-  filter_upwards [hw] with ω hω
-  obtain ⟨G,F,hGF⟩ := hω
-  refine ⟨fun t z => loewnerMirror (G t (loewnerMirror z)),
-    fun t z => loewnerMirror (F t (loewnerMirror z)),?_⟩
-  simpa only [Pi.neg_apply,mul_neg] using hGF.mirror
-
- 
-theorem sle_real_center_approach
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (x r : ℝ) (hx : x ≠ 0) (hr : 0 < r) :
-    P {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤
-      ENNReal.ofReal ((r/(((8-κ)/32)*|x|))^(3*(8-κ)/(4*κ))) := by
-  rcases lt_or_gt_of_ne hx with hx | hx
-  · have hh := sle_positive_real_center_approach κ hκ hκ8
-      (fun ω => loewnerMirror ∘ γ ω) hSLE.mirror (-x) r (neg_pos.mpr hx) hr
-    have he : {ω | ∃ t : ℝ≥0, dist ((loewnerMirror ∘ γ ω) t) ((-x:ℝ):ℂ) < r} =
-        {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} := by
-      ext ω
-      have hrx : ((-x:ℝ):ℂ) = loewnerMirror (x:ℂ) := by simp
-      simp only [Function.comp_apply,hrx,loewnerMirror_dist]
-    rw [he] at hh
-    simpa only [abs_of_neg hx] using hh
-  · simpa only [abs_of_pos hx] using
-      sle_positive_real_center_approach κ hκ hκ8 γ hSLE x r hx hr
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma derivative_close_of_cauchy {f : ℂ → ℂ} {U : Set ℂ} {b : ℂ} {r : ℝ}
-    (hr : 0 < r) (hU : IsOpen U) (hd : DifferentiableOn ℂ f U)
-    (hinc : closedBall b r ⊆ U)
-    (hbound : ∀ w ∈ closedBall b r, ‖deriv f w‖ ≤ 2 * ‖deriv f b‖) :
-    ∀ w ∈ closedBall b (r / 8), ‖deriv f w - deriv f b‖ ≤ ‖deriv f b‖ / 2 := by
-  have hdr := hd.deriv hU
-  have hsmall : closedBall b (r / 8) ⊆ closedBall b r := closedBall_subset_closedBall (by linarith)
-  have hsecond (w : ℂ) (hw : w ∈ closedBall b (r / 8)) :
-      ‖deriv (deriv f) w‖ ≤ 4 * ‖deriv f b‖ / r := by
-    have hs : closedBall w (r / 2) ⊆ closedBall b r := by
-      intro z hz
-      apply mem_closedBall.mpr
-      have hw' := mem_closedBall.mp hw
-      have hz' := mem_closedBall.mp hz
-      exact (dist_triangle z w b).trans (by linarith)
-    have hh := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le (by positivity : 0 < r / 2)
-      (hdr.diffContOnCl_ball (hs.trans hinc))
-      (fun z hz => hbound z (hs (sphere_subset_closedBall hz)))
-    convert hh using 1
-    ring
-  intro w hw
-  have hh := (convex_closedBall b (r / 8)).norm_image_sub_le_of_norm_deriv_le
-    (fun z hz => hdr.differentiableAt (hU.mem_nhds (hinc (hsmall hz)))) hsecond
-    (mem_closedBall_self (by positivity : 0 ≤ r / 8)) hw
-  have hn : ‖w - b‖ ≤ r / 8 := mem_closedBall_iff_norm.mp hw
-  calc
-    ‖deriv f w - deriv f b‖ ≤ (4 * ‖deriv f b‖ / r) * ‖w - b‖ := hh
-    _ ≤ (4 * ‖deriv f b‖ / r) * (r / 8) := mul_le_mul_of_nonneg_left hn (by positivity)
-    _ = ‖deriv f b‖ / 2 := by field_simp; ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma disk_image_of_deriv_close {f f' : ℂ → ℂ} {b a : ℂ} {r : ℝ}
-    (hr : 0 ≤ r) (ha : a ≠ 0)
-    (hd : ∀ z ∈ closedBall b r, HasDerivAt f (f' z) z)
-    (hb : ∀ z ∈ closedBall b r, ‖f' z / a - 1‖ ≤ 1 / 2) :
-    closedBall (f b) (‖a‖ * r / 2) ⊆ f '' closedBall b r := by
-  let g : ℂ → ℂ := fun z => f z / a
-  have hl : LipschitzOnWith (1 / 2 : ℝ≥0) (fun z => g z - z) (closedBall b r) := by
-    apply (convex_closedBall b r).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
-      (f' := fun z => f' z / a - 1)
-    · intro z hz
-      exact (((hd z hz).div_const a).sub (hasDerivAt_id z)).hasDerivWithinAt
-    · intro z hz
-      exact_mod_cast hb z hz
-  have hl' : ApproximatesLinearOn g (ContinuousLinearMap.id ℂ ℂ) (closedBall b r) (1 / 2) := by
-    apply LipschitzOnWith.approximatesLinearOn
-    exact hl
-  have hs := hl'.surjOn_closedBall_of_nonlinearRightInverse
-    (ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse hr (Subset.refl _)
-  have hs' : SurjOn g (closedBall b r) (closedBall (g b) (r / 2)) := by
-    norm_num [ContinuousLinearEquiv.toNonlinearRightInverse] at hs
-    convert hs using 1
-    ring_nf
-  intro y hy
-  have hy' : y / a ∈ closedBall (g b) (r / 2) := by
-    rw [mem_closedBall_iff_norm] at hy ⊢
-    change ‖y / a - f b / a‖ ≤ r / 2
-    rw [← sub_div, norm_div]
-    apply (div_le_iff₀ (norm_pos_iff.mpr ha)).mpr
-    nlinarith
-  obtain ⟨z, hz, he⟩ := hs' hy'
-  exact ⟨z, hz, (div_left_inj' ha).mp he⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma bloch_disk {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) :
-    ∃ b ∈ ball (0 : ℂ) 1,
-      closedBall (f b) (‖deriv f 0‖ / 64) ⊆ f '' ball 0 1 := by
-  have hzero : (0 : ℂ) ∈ closedBall 0 (1 / 2 : ℝ) := mem_closedBall_self (by norm_num)
-  have hinc : closedBall (0 : ℂ) (1 / 2 : ℝ) ⊆ ball 0 1 :=
-    closedBall_subset_ball (by norm_num)
-  by_cases h0 : deriv f 0 = 0
-  · refine ⟨0, mem_ball_self (by norm_num), ?_⟩
-    simp only [h0, norm_zero, zero_div, closedBall_zero, singleton_subset_iff]
-    exact mem_image_of_mem f (mem_ball_self (by norm_num))
-  have hd0 : 0 < ‖deriv f 0‖ := norm_pos_iff.mpr h0
-  let Φ : ℂ → ℝ := fun z => (1 / 2 - ‖z‖) * ‖deriv f z‖
-  have hc : ContinuousOn Φ (closedBall 0 (1 / 2 : ℝ)) :=
-    (continuousOn_const.sub continuous_norm.continuousOn).mul
-      ((hd.deriv isOpen_ball).continuousOn.mono hinc).norm
-  obtain ⟨b, hb, hmax⟩ := (isCompact_closedBall (0 : ℂ) (1 / 2 : ℝ)).exists_isMaxOn
-    ⟨0, hzero⟩ hc
-  have hM : ‖deriv f 0‖ / 2 ≤ (1 / 2 - ‖b‖) * ‖deriv f b‖ := by
-    have hh := hmax hzero
-    simpa [Φ, norm_zero, mul_comm, div_eq_mul_inv] using hh
-  have hb' : ‖b‖ ≤ 1 / 2 := mem_closedBall_zero_iff.mp hb
-  have hab : 0 < ‖deriv f b‖ := by
-    by_contra hn
-    have hh : ‖deriv f b‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
-    rw [hh, mul_zero] at hM
-    linarith
-  have hmargin : 0 < 1 / 2 - ‖b‖ := by
-    by_contra hn
-    have hh := mul_nonpos_of_nonpos_of_nonneg (le_of_not_gt hn) hab.le
-    linarith
-  let r : ℝ := (1 / 2 - ‖b‖) / 2
-  have hr : 0 < r := div_pos hmargin (by norm_num)
-  have hsub : closedBall b r ⊆ closedBall 0 (1 / 2 : ℝ) := by
-    intro w hw
-    rw [mem_closedBall_zero_iff]
-    have hw' : ‖w - b‖ ≤ r := mem_closedBall_iff_norm.mp hw
-    have hnorm : ‖w‖ ≤ ‖w - b‖ + ‖b‖ := by simpa [norm_sub_rev, add_comm] using norm_le_insert b w
-    dsimp [r] at hw'
-    linarith
-  have hbound (w : ℂ) (hw : w ∈ closedBall b r) : ‖deriv f w‖ ≤ 2 * ‖deriv f b‖ := by
-    have hh := hmax (hsub hw)
-    change (1 / 2 - ‖w‖) * ‖deriv f w‖ ≤ (1 / 2 - ‖b‖) * ‖deriv f b‖ at hh
-    have hw' : ‖w - b‖ ≤ r := mem_closedBall_iff_norm.mp hw
-    have hnorm : ‖w‖ ≤ ‖w - b‖ + ‖b‖ := by simpa [norm_sub_rev, add_comm] using norm_le_insert b w
-    have hm : r ≤ 1 / 2 - ‖w‖ := by dsimp [r] at *; linarith
-    have he : 1 / 2 - ‖b‖ = 2 * r := by dsimp [r]; ring
-    have hprod := mul_le_mul_of_nonneg_right hm (norm_nonneg (deriv f w))
-    rw [he] at hh
-    nlinarith
-  have hclose := derivative_close_of_cauchy hr isOpen_ball hd (hsub.trans hinc) hbound
-  have hsmall : closedBall b (r / 8) ⊆ closedBall b r := closedBall_subset_closedBall (by linarith)
-  have himage := disk_image_of_deriv_close (f := f) (f' := deriv f) (a := deriv f b)
-    (by positivity : 0 ≤ r / 8) (norm_pos_iff.mp hab)
-    (fun w hw => (hd.differentiableAt (isOpen_ball.mem_nhds (hinc (hsub (hsmall hw))))).hasDerivAt)
-    (by
-      intro w hw
-      rw [← div_self (norm_pos_iff.mp hab), ← sub_div, norm_div]
-      exact (div_le_iff₀ hab).mpr (by simpa [div_eq_mul_inv, mul_comm] using hclose w hw))
-  refine ⟨b, hinc hb, ?_⟩
-  have hlower : ‖deriv f 0‖ / 64 ≤ ‖deriv f b‖ * (r / 8) / 2 := by
-    dsimp [r]
-    nlinarith
-  exact (closedBall_subset_closedBall hlower).trans
-    (himage.trans (image_mono (hsmall.trans (hsub.trans hinc))))
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma hasDerivAt_log_lift {g h : ℂ → ℂ} {U : Set ℂ} (hU : IsOpen U)
-    (hg : DifferentiableOn ℂ g U) (hh : ContinuousOn h U)
-    (he : EqOn (Complex.exp ∘ h) g U) {z : ℂ} (hz : z ∈ U) :
-    HasDerivAt h (deriv g z / g z) z := by
-  have hgz : g z ≠ 0 := by rw [← he hz]; exact Complex.exp_ne_zero _
-  have hdz := (hg.differentiableAt (hU.mem_nhds hz)).hasDerivAt
-  have hcx := hh.continuousAt (hU.mem_nhds hz)
-  have hdisc : ∀ᶠ w in 𝓝 z, ‖h w - h z‖ < Real.pi := by
-    have hc : ContinuousAt (fun w => ‖h w - h z‖) z := (hcx.sub continuousAt_const).norm
-    exact hc (Iio_mem_nhds (by simpa using Real.pi_pos))
-  have hlocal : (fun w => Complex.log (g w / g z) + h z) =ᶠ[𝓝 z] h := by
-    filter_upwards [hU.mem_nhds hz, hdisc] with w hw hw'
-    have him : |(h w - h z).im| < Real.pi := (Complex.abs_im_le_norm _).trans_lt hw'
-    rw [← he hw, ← he hz]
-    change Complex.log (Complex.exp (h w) / Complex.exp (h z)) + h z = h w
-    rw [← Complex.exp_sub, Complex.log_exp (by linarith [(abs_lt.mp him).1])
-      (le_of_lt (abs_lt.mp him).2)]
-    ring
-  have hlog := (hdz.div_const (g z)).clog (by simp [hgz])
-  have hf : HasDerivAt (fun w => Complex.log (g w / g z) + h z) (deriv g z / g z) z := by
-    simpa [hgz] using hlog.add_const (h z)
-  exact hf.congr_of_eventuallyEq hlocal.symm
-
-lemma differentiableOn_log_lift {g h : ℂ → ℂ} {U : Set ℂ} (hU : IsOpen U)
-    (hg : DifferentiableOn ℂ g U) (hh : ContinuousOn h U)
-    (he : EqOn (Complex.exp ∘ h) g U) : DifferentiableOn ℂ h U :=
-  fun _ hz => (hasDerivAt_log_lift hU hg hh he hz).differentiableAt.differentiableWithinAt
-
-lemma exists_holomorphic_log_on_ball {g : ℂ → ℂ} {b : ℂ} {r : ℝ}
-    (hr : 0 < r) (hg : DifferentiableOn ℂ g (ball b r))
-    (h0 : ∀ z ∈ ball b r, g z ≠ 0) :
-    ∃ h : ℂ → ℂ, DifferentiableOn ℂ h (ball b r) ∧
-      EqOn (Complex.exp ∘ h) g (ball b r) ∧
-      ∀ z ∈ ball b r, deriv h z = deriv g z / g z := by
-  let := (convex_ball b r).contractibleSpace ⟨b, mem_ball_self hr⟩
-  obtain ⟨h, hh, he⟩ := Complex.exists_continuousOn_eqOn_exp_comp
-    (by change SimplyConnectedSpace (ball b r); infer_instance) isOpen_ball hg.continuousOn
-    (by rintro ⟨z, hz, he⟩; exact h0 z hz he)
-  exact ⟨h, differentiableOn_log_lift isOpen_ball hg hh he, he,
-    fun z hz => (hasDerivAt_log_lift isOpen_ball hg hh he hz).deriv⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma univalent_omitted_value_bound {f : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (a : ℂ) (ha : a ∉ f '' ball 0 1) :
-    ‖deriv f 0‖ ≤ (128 * Real.pi) * ‖f 0 - a‖ := by
-  have hnon (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) : f z - a ≠ 0 := by
-    intro he
-    exact ha ⟨z, hz, sub_eq_zero.mp he⟩
-  have hz0 : (0 : ℂ) ∈ ball 0 1 := mem_ball_self (by norm_num)
-  obtain ⟨h, hhd, he, hhder⟩ := exists_holomorphic_log_on_ball (by norm_num : (0 : ℝ) < 1)
-    (hd.sub_const a) hnon
-  obtain ⟨b, hb, hball⟩ := bloch_disk hhd
-  have hbound : ‖deriv h 0‖ < 128 * Real.pi := by
-    by_contra hn
-    have hn' : 128 * Real.pi ≤ ‖deriv h 0‖ := le_of_not_gt hn
-    have hw : h b + 2 * Real.pi * Complex.I ∈ closedBall (h b) (‖deriv h 0‖ / 64) := by
-      rw [mem_closedBall_iff_norm, add_sub_cancel_left, norm_mul, norm_mul,
-        norm_ofNat, Complex.norm_real, Real.norm_eq_abs, abs_of_pos Real.pi_pos, Complex.norm_I, mul_one]
-      linarith
-    obtain ⟨z, hz, hzh⟩ := hball hw
-    have hfe : f z - a = f b - a := by
-      calc
-        f z - a = Complex.exp (h z) := (he hz).symm
-        _ = Complex.exp (h b) := by rw [hzh]; exact Complex.exp_periodic (h b)
-        _ = f b - a := he hb
-    have hzb : z = b := hi hz hb (sub_left_inj.mp hfe)
-    rw [hzb] at hzh
-    have him := congrArg Complex.im hzh
-    norm_num [Complex.add_im, Complex.mul_im, Complex.mul_re] at him
-  have hd0 := (hd.differentiableAt (isOpen_ball.mem_nhds hz0)).hasDerivAt
-  have hdsub : deriv (fun z => f z - a) 0 = deriv f 0 := (hd0.sub_const a).deriv
-  rw [hhder 0 hz0, hdsub, norm_div] at hbound
-  exact ((div_lt_iff₀ (norm_pos_iff.mpr (hnon 0 hz0))).mp hbound).le
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.NearRealContraction
+
+namespace OAI
+
+/-! Brownian Strong Markov for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+noncomputable def shiftedFDD {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (I : Finset ℝ≥0) (s : ℝ≥0) (ω : Ω) : I → ℝ :=
+  fun i => B (s + i) ω - B s ω
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+  (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+
+include hm in
+lemma shiftedFDD_measurable (I : Finset ℝ≥0) (s : ℝ≥0) :
+    Measurable (shiftedFDD B I s) :=
+  Measurable.of_eval (fun i => (hm (s + i)).sub (hm s))
+
+include hB hm in
+omit [IsProbabilityMeasure P] in
+lemma shiftedFDD_indep (I : Finset ℝ≥0) (s : ℝ≥0) :
+    Indep (Filtration.natural B (fun t => (hm t).stronglyMeasurable) s)
+      (MeasurableSpace.comap (shiftedFDD B I s) inferInstance) P := by
+  have hi := ((hB.indepFun_shift s).comp
+    (Measurable.of_eval (fun i : I => measurable_pi_apply (i : ℝ≥0))) measurable_id).symm
+  rw [Filtration.natural_eq_comap B _ s]
+  exact (IndepFun_iff_Indep _ _ _).mp hi
+
+include hB hm in
+omit [IsProbabilityMeasure P] in
+lemma shiftedFDD_setIntegral (I : Finset ℝ≥0) (s : ℝ≥0)
+    (φ : BoundedContinuousFunction (I → ℝ) ℝ) {A : Set Ω}
+    (hA : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] A) :
+    (∫ ω in A, φ (shiftedFDD B I s ω) ∂P) =
+      P.real A * ∫ x, φ x ∂BrownianReal.projectiveFamily I := by
+  rw [(shiftedFDD_indep hB hm I s).setIntegral_eq_mul
+    ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le s)
+    (shiftedFDD_measurable hm I s).aemeasurable hA φ.continuous.aestronglyMeasurable]
+  congr 1
+  exact ((hB.shift s).hasLaw I).integral_comp φ.continuous.aestronglyMeasurable
+
+include hm in
+lemma shiftedFDD_grid_measurable {τ : Ω → ℝ≥0} (hτm : Measurable τ)
+    (I : Finset ℝ≥0) (n : ℕ) :
+    Measurable (fun ω => shiftedFDD B I (stopGrid n (τ ω)) ω) := by
+  apply Measurable.of_eval
+  intro i
+  exact measurable_discrete_eval
+    (fun k ω => B (stopGridTime n k + i) ω - B (stopGridTime n k) ω)
+    (fun k => (hm _).sub (hm _)) (stopGridIndex_measurable hτm n)
+
+include hB hm in
+lemma shiftedFDD_grid_setIntegral {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A)
+    (I : Finset ℝ≥0) (φ : BoundedContinuousFunction (I → ℝ) ℝ) (n : ℕ) :
+    (∫ ω in A, φ (shiftedFDD B I (stopGrid n (τ ω)) ω) ∂P) =
+      P.real A * ∫ x, φ x ∂BrownianReal.projectiveFamily I := by
+  apply integral_stopping_grid_glue hτ hA n _
+    (Integrable.of_bound (φ.continuous.measurable.comp
+      (shiftedFDD_grid_measurable hm (stopping_time_measurable_nnreal hτ) I n)).aestronglyMeasurable
+      ‖φ‖ (Eventually.of_forall (fun ω => φ.norm_coe_le_norm _)))
+  intro k
+  have hk := stopping_grid_piece_measurable hτ hA n k
+  have hk' := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le _ _ hk
+  rw [setIntegral_congr_fun hk' (g := fun ω => φ (shiftedFDD B I (stopGridTime n k) ω))
+    (fun ω hω => by
+      have he : stopGridIndex n (τ ω) = k := hω.2
+      simp only [Function.comp_def, stopGrid, he])]
+  exact shiftedFDD_setIntegral hB hm I _ φ hk
+
+omit mΩ in
+lemma shiftedFDD_grid_tendsto
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (I : Finset ℝ≥0) (τ : Ω → ℝ≥0) (ω : Ω) :
+    Tendsto (fun n => shiftedFDD B I (stopGrid n (τ ω)) ω) atTop
+      (𝓝 (shiftedFDD B I (τ ω) ω)) := by
+  apply tendsto_pi_nhds.mpr
+  intro i
+  exact ((hc ω).continuousAt.tendsto.comp ((stopGrid_tendsto (τ ω)).add_const (i : ℝ≥0))).sub
+    ((hc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω)))
+
+include hB hm in
+ 
+
+theorem brownian_strongMarkov_integral
+    (hc : ∀ ω, Continuous (fun t => B t ω)) {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A)
+    (I : Finset ℝ≥0) (φ : BoundedContinuousFunction (I → ℝ) ℝ) :
+    (∫ ω in A, φ (shiftedFDD B I (τ ω) ω) ∂P) =
+      P.real A * ∫ x, φ x ∂BrownianReal.projectiveFamily I := by
+  have hlim := tendsto_integral_of_dominated_convergence (μ := P.restrict A)
+    (F := fun n ω => φ (shiftedFDD B I (stopGrid n (τ ω)) ω))
+    (f := fun ω => φ (shiftedFDD B I (τ ω) ω)) (fun _ => ‖φ‖)
+    (fun n => (φ.continuous.measurable.comp
+      (shiftedFDD_grid_measurable hm (stopping_time_measurable_nnreal hτ) I n)).aestronglyMeasurable)
+    (integrable_const _)
+    (fun n => Eventually.of_forall (fun ω => φ.norm_coe_le_norm _))
+    (Eventually.of_forall (fun ω => φ.continuous.continuousAt.tendsto.comp
+      (shiftedFDD_grid_tendsto hc I τ ω)))
+  simp only [shiftedFDD_grid_setIntegral hB hm hτ hA I φ] at hlim
+  exact tendsto_nhds_unique hlim tendsto_const_nhds
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def brownianRestart {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : ℝ := B (τ ω + t) ω - B (τ ω) ω
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+  (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+  (hc : ∀ ω, Continuous (fun t => B t ω))
+
+include hm hc in
+lemma brownianRestart_measurable {τ : Ω → ℝ≥0} (hτm : Measurable τ) :
+    Measurable (fun ω t => brownianRestart B τ t ω) := by
+  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable hc hm
+  apply Measurable.of_eval
+  intro t
+  exact (hBm.comp ((hτm.add_const t).prodMk measurable_id)).sub
+    (hBm.comp (hτm.prodMk measurable_id))
+
+include hB hm hc in
+lemma brownian_restart_fdd_restrict {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) (I : Finset ℝ≥0) :
+    (P.restrict A).map (fun ω => shiftedFDD B I (τ ω) ω) =
+      P A • BrownianReal.projectiveFamily I := by
+  have hXm : Measurable (fun ω => shiftedFDD B I (τ ω) ω) :=
+    (Finset.measurable_restrict I).comp
+      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))
+  let : IsFiniteMeasure (P A • BrownianReal.projectiveFamily I) :=
+    Measure.smul_finite _ (measure_ne_top P A)
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro φ
+  rw [integral_map hXm.aemeasurable φ.continuous.aestronglyMeasurable,
+    integral_smul_measure]
+  exact brownian_strongMarkov_integral hB hm hc hτ hA I φ
+
+include hB hm hc in
+ 
+
+theorem brownian_restart_path_law {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) :
+    (P.restrict A).map (fun ω t => brownianRestart B τ t ω) =
+      P A • P.map (fun ω t => B t ω) := by
+  have hXm := brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)
+  have hBm : Measurable (fun ω t => B t ω) := Measurable.of_eval hm
+  let (I : Finset ℝ≥0) : IsFiniteMeasure (P A • BrownianReal.projectiveFamily I) :=
+    Measure.smul_finite _ (measure_ne_top P A)
+  have hL : IsProjectiveLimit ((P.restrict A).map (fun ω t => brownianRestart B τ t ω))
+      (fun I => P A • BrownianReal.projectiveFamily I) := by
+    intro I
+    rw [Measure.map_map (Finset.measurable_restrict I) hXm]
+    exact brownian_restart_fdd_restrict hB hm hc hτ hA I
+  have hR : IsProjectiveLimit (P A • P.map (fun ω t => B t ω))
+      (fun I => P A • BrownianReal.projectiveFamily I) := by
+    intro I
+    rw [Measure.map_smul _ (Finset.measurable_restrict I).aemeasurable,
+      Measure.map_map (Finset.measurable_restrict I) hBm]
+    exact congrArg (fun ν => P A • ν) (hB.hasLaw I).map_eq
+  exact hL.unique hR
+
+include hB hm hc in
+lemma brownian_restart_path_map {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) :
+    P.map (fun ω t => brownianRestart B τ t ω) = P.map (fun ω t => B t ω) := by
+  simpa only [Measure.restrict_univ, measure_univ, one_smul] using
+    brownian_restart_path_law hB hm hc hτ (A := univ) MeasurableSet.univ
+
+include hB hm hc in
+lemma brownian_restart_event_factorizes {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A)
+    {S : Set (ℝ≥0 → ℝ)} (hS : MeasurableSet S) :
+    P (A ∩ {ω | (fun t => brownianRestart B τ t ω) ∈ S}) =
+      P A * P {ω | (fun t => B t ω) ∈ S} := by
+  have hXm := brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)
+  have he := congrArg (fun ν : Measure (ℝ≥0 → ℝ) => ν S)
+    (brownian_restart_path_law hB hm hc hτ hA)
+  rw [Measure.map_apply hXm hS, Measure.restrict_apply (hXm hS),
+    Measure.smul_apply, Measure.map_apply (Measurable.of_eval hm) hS] at he
+  simpa only [Set.preimage, Set.inter_comm, smul_eq_mul] using he
+
+include hB hm hc in
+ 
+
+theorem brownian_restart_independent {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) :
+    Indep hτ.measurableSpace
+      (MeasurableSpace.comap (fun ω t => brownianRestart B τ t ω) inferInstance) P := by
+  rw [Indep_iff]
+  intro A C hA hC
+  rcases hC with ⟨S, hS, rfl⟩
+  have he := congrArg (fun ν : Measure (ℝ≥0 → ℝ) => ν S)
+    (brownian_restart_path_map hB hm hc hτ)
+  rw [Measure.map_apply (brownianRestart_measurable hm hc
+    (stopping_time_measurable_nnreal hτ)) hS,
+    Measure.map_apply (Measurable.of_eval hm) hS] at he
+  change P (A ∩ {ω | (fun t => brownianRestart B τ t ω) ∈ S}) = _
+  rw [brownian_restart_event_factorizes hB hm hc hτ hA hS]
+  rw [he]
+  rfl
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+section
+variable {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
+  (U : ℝ≥0 → Ω → ℝ) (hUa : Adapted ℱ U)
+  (hUc : ∀ ω, Continuous (fun t => U t ω))
+include hUa hUc
+
+lemma measurable_reverse_driver (T : ℝ≥0) :
+    @Measurable (ℝ × Ω) ℝ ((borel ℝ).prod (ℱ T)) (borel ℝ)
+      (fun p => U (min ((T : ℝ) - p.1).toNNReal T) p.2) := by
+  have hsa : StronglyAdapted ℱ U := fun t => (hUa t).stronglyMeasurable
+  have hp := (hsa.isStronglyProgressive_of_continuous hUc T).measurable
+  let : MeasurableSpace Ω := ℱ T
+  have ht : Measurable (fun s : ℝ => (⟨min ((T : ℝ) - s).toNNReal T,
+      by change min ((T : ℝ) - s).toNNReal T ≤ T; exact min_le_right _ _⟩ : Iic T)) :=
+    ((continuous_real_toNNReal.comp (continuous_const.sub continuous_id)).min continuous_const).measurable.subtype_mk
+  exact hp.comp ((ht.comp measurable_fst).prodMk measurable_snd)
+
+ 
+
+theorem inverse_vertical_measurable
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (h : ∀ ω, IsCapacityTwoWitness (fun t => U t ω) (γ ω) (G ω) (F ω))
+    (T : ℝ≥0) {a : ℝ} (ha : 0 < a) :
+    @Measurable Ω ℂ (ℱ T) (borel ℂ)
+      (fun ω => F ω T ((U T ω : ℂ) + (a : ℂ) * I)) := by
+  let V (s : ℝ) (ω : Ω) := U (min ((T : ℝ) - s).toNNReal T) ω
+  have hVm := measurable_reverse_driver ℱ U hUa hUc T
+  have hVc (ω : Ω) : Continuous (fun s : ℝ => V s ω) :=
+    (hUc ω).comp ((continuous_real_toNNReal.comp (continuous_const.sub continuous_id)).min continuous_const)
+  let : MeasurableSpace Ω := ℱ T
+  have hwm : Measurable (fun ω => (U T ω : ℂ) + (a : ℂ) * I) :=
+    (Complex.continuous_ofReal.measurable.comp (hUa T)).add_const _
+  obtain ⟨Q, hQm, hQ0, hQd⟩ := exists_measurable_reverse_flow V hVm hVc
+    (fun ω => (U T ω : ℂ) + (a : ℂ) * I) hwm ha
+  have he (ω : Ω) : Q (T : ℝ) ω = F ω T ((U T ω : ℂ) + (a : ℂ) * I) := by
+    apply (h ω).reverse_flow_identifies T _ ha (by simp) (fun s => Q s ω) (hQ0 ω)
+    intro s hs
+    have hsT : ((T : ℝ) - s).toNNReal ≤ T :=
+      Real.toNNReal_le_iff_le_coe.mpr (by linarith [hs.1])
+    simpa only [V, min_eq_left hsT] using
+      (hQd ω s hs.1).mono (show Icc (0 : ℝ) T ⊆ Ici 0 from Icc_subset_Ici_self)
+  have hefun : (fun ω => Q (T : ℝ) ω) = (fun ω => F ω T ((U T ω : ℂ) + (a : ℂ) * I)) := funext he
+  rw [← hefun]
+  exact hQm.comp (measurable_prodMk_left (x := (T : ℝ)))
+
+ 
+
+theorem capacity_trace_adapted
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (h : ∀ ω, IsCapacityTwoWitness (fun t => U t ω) (γ ω) (G ω) (F ω)) :
+    Adapted ℱ (fun t ω => γ ω t) := by
+  intro T
+  have hn (n : ℕ) : @Measurable Ω ℂ (ℱ T) (borel ℂ)
+      (fun ω => F ω T ((U T ω : ℂ) + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * I)) :=
+    inverse_vertical_measurable ℱ U hUa hUc γ G F h T (by positivity)
+  let : MeasurableSpace Ω := ℱ T
+  apply measurable_of_tendsto_metrizable
+    (f := fun n ω => F ω T ((U T ω : ℂ) + ((1 / ((n : ℝ) + 1) : ℝ) : ℂ) * I))
+  · exact hn
+  · apply tendsto_pi_nhds.mpr
+    intro ω
+    have hy : Tendsto (fun n : ℕ => 1 / ((n : ℝ) + 1)) atTop (𝓝[>] (0 : ℝ)) :=
+      tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat,
+        Eventually.of_forall (fun n => by change (0 : ℝ) < 1 / ((n : ℝ) + 1); positivity)⟩
+    exact ((h ω).2.2.2.2.2.2.2.2.2.2.2.2 T).comp hy
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+def IsCapacityTwoTrace (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) : Prop :=
+  ∃ G F : ℝ≥0 → ℂ → ℂ, IsCapacityTwoWitness U γ G F
+
+ 
+
+def IsOrdinaryChordalSLE {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (P : Measure Ω) : Prop :=
+  (∀ t, Measurable (fun ω => γ ω t)) ∧
+  ∃ B : ℝ≥0 → Ω → ℝ, ProbabilityTheory.IsBrownianReal B P ∧
+    ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * B t ω) (γ ω)
+
+ 
+
+structure RegularSLEPresentation {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) where
+  samples : Set Ω
+  measurable_samples : MeasurableSet samples
+  full_samples : ∀ᵐ ω ∂P, ω ∈ samples
+  B : ℝ≥0 → samples → ℝ
+  measurable_B : ∀ t, Measurable (B t)
+  continuous_B : ∀ ω, Continuous (fun t => B t ω)
+  zero_B : ∀ ω, B 0 ω = 0
+  brownian : IsBrownianReal B (P.comap (Subtype.val : samples → Ω))
+  G : samples → ℝ≥0 → ℂ → ℂ
+  F : samples → ℝ≥0 → ℂ → ℂ
+  witness : ∀ ω : samples, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+    (γ ω) (G ω) (F ω)
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+
+lemma measurePreserving_val (a : RegularSLEPresentation P κ γ) :
+    MeasurePreserving (Subtype.val : a.samples → Ω) (P.comap Subtype.val) P := by
+  refine ⟨measurable_subtype_coe, ?_⟩
+  rw [map_comap_subtype_coe a.measurable_samples,
+    Measure.restrict_eq_self_of_ae_mem a.full_samples]
+
+instance probability [IsProbabilityMeasure P] (a : RegularSLEPresentation P κ γ) :
+    IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) :=
+  (MeasurableEmbedding.subtype_coe a.measurable_samples).isProbabilityMeasure_comap
+    (by simpa only [Subtype.range_coe] using a.full_samples)
+
+def filtration (a : RegularSLEPresentation P κ γ) :
+    Filtration ℝ≥0 (inferInstance : MeasurableSpace a.samples) :=
+  Filtration.natural a.B (fun t => (a.measurable_B t).stronglyMeasurable)
+
+lemma trace_adapted (a : RegularSLEPresentation P κ γ) :
+    Adapted a.filtration (fun t ω => γ ω t) := by
+  apply capacity_trace_adapted a.filtration (fun t ω => Real.sqrt κ * a.B t ω)
+    ?_ ?_ (γ ∘ Subtype.val) a.G a.F a.witness
+  · intro t
+    exact measurable_const.mul
+      ((Filtration.stronglyAdapted_natural (fun t => (a.measurable_B t).stronglyMeasurable)) t).measurable
+  · intro ω
+    exact continuous_const.mul (a.continuous_B ω)
+
+lemma trace_continuous (a : RegularSLEPresentation P κ γ) (ω : a.samples) :
+    Continuous (γ ω) := (a.witness ω).1
+
+end RegularSLEPresentation
+
+ 
+
+theorem exists_regular_sle_presentation
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) :
+    Nonempty (RegularSLEPresentation P κ γ) := by
+  classical
+  obtain ⟨B, hB, htrace⟩ := hSLE.2
+  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
+  have htraceC : ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * C t ω) (γ ω) := by
+    filter_upwards [htrace, hCB] with ω hω heq
+    simpa only [heq] using hω
+  let N := toMeasurable P {ω | ¬ IsCapacityTwoTrace (fun t => Real.sqrt κ * C t ω) (γ ω)}
+  let S := Nᶜ
+  have hSm : MeasurableSet S := (measurableSet_toMeasurable _ _).compl
+  have hNa : P N = 0 := by
+    rw [show N = toMeasurable P {ω | ¬ IsCapacityTwoTrace
+      (fun t => Real.sqrt κ * C t ω) (γ ω)} from rfl, measure_toMeasurable]
+    exact ae_iff.mp htraceC
+  have hSa : ∀ᵐ ω ∂P, ω ∈ S := by
+    apply ae_iff.mpr
+    simpa only [S, mem_compl_iff, not_not, ofPred_mem_eq] using hNa
+  have hw (ω : S) : IsCapacityTwoTrace (fun t => Real.sqrt κ * C t ω) (γ ω) := by
+    by_contra hh
+    exact ω.property (subset_toMeasurable P _ hh)
+  choose G F hGF using hw
+  have hval : MeasurePreserving (Subtype.val : S → Ω) (P.comap Subtype.val) P := by
+    refine ⟨measurable_subtype_coe, ?_⟩
+    rw [map_comap_subtype_coe hSm, Measure.restrict_eq_self_of_ae_mem hSa]
+  refine ⟨{
+    samples := S
+    measurable_samples := hSm
+    full_samples := hSa
+    B := fun t ω => C t ω
+    measurable_B := fun t => (hCm.comp (measurable_prodMk_left (x := t))).comp measurable_subtype_coe
+    continuous_B := fun ω => hCc ω
+    zero_B := fun ω => hC0 ω
+    brownian := ?_
+    G := G
+    F := F
+    witness := hGF }⟩
+  exact ⟨⟨fun I => (hC.hasLaw I).comp hval.hasLaw⟩, Eventually.of_forall (fun ω => hCc ω)⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem ordinary_positive_real_center_approach
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (x r : ℝ) (hx : 0 < x) (hr : 0 < r) :
+    P {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤
+      ENNReal.ofReal ((r/(((8-κ)/32)*x))^(3*(8-κ)/(4*κ))) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hδ : 0 < (8-κ)/32 := by positivity
+  have hδ2 : (8-κ)/32 ≤ 1/2 := by linarith
+  have hβ : 0 < 3*(8-κ)/(4*κ) := by positivity
+  by_cases hrx : r < ((8-κ)/32)*x
+  · apply (measure_mono_ae (show {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤ᵐ[P]
+      {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r ((8-κ)/32)} from ?_)).trans
+      (boundary_comparison_failure_sle_exponent hB κ x r hκ hκ8 hx hr)
+    filter_upwards [hw,hB.cont,hB.eval_zero_ae_eq_zero] with ω hω hc h0
+    rintro ⟨t,ht⟩
+    by_contra hn
+    have hh := hω.all_time_real_center_avoidance (hc.const_mul _) (by simp only [h0,mul_zero])
+      hx hrx hδ2 hn t
+    exact (not_lt_of_ge hh) (by simpa only [dist_eq_norm] using ht)
+  · have hq : 1 ≤ r/(((8-κ)/32)*x) :=
+      (le_div_iff₀ (mul_pos hδ hx)).mpr (by simpa only [one_mul] using le_of_not_gt hrx)
+    have hh := ENNReal.ofReal_le_ofReal (Real.one_le_rpow hq hβ.le)
+    rw [ENNReal.ofReal_one] at hh
+    exact prob_le_one.trans hh
+
+ 
+
+theorem sle_positive_real_center_approach
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (x r : ℝ) (hx : 0 < x) (hr : 0 < r) :
+    P {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤
+      ENNReal.ofReal ((r/(((8-κ)/32)*x))^(3*(8-κ)/(4*κ))) := by
+  classical
+  obtain ⟨B,hB,htrace⟩ := hSLE.2
+  have hchoice (ω : Ω) : ∃ G F : ℝ≥0 → ℂ → ℂ,
+      IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω) →
+        IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) G F := by
+    by_cases h : IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω)
+    · obtain ⟨G,F,hw⟩ := h
+      exact ⟨G,F,fun _ => hw⟩
+    · exact ⟨fun _ _ => 0,fun _ _ => 0,fun hh => False.elim (h hh)⟩
+  choose G F hw using hchoice
+  exact ordinary_positive_real_center_approach hB κ hκ hκ8 γ G F
+    (htrace.mono fun ω hω => hw ω hω) x r hx hr
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Complex ComplexConjugate
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma IsOrdinaryChordalSLE.mirror
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (h : IsOrdinaryChordalSLE κ γ P) :
+    IsOrdinaryChordalSLE κ (fun ω => loewnerMirror ∘ γ ω) P := by
+  obtain ⟨hm,B,hB,hw⟩ := h
+  refine ⟨fun t => loewnerMirror.continuous.measurable.comp (hm t),-B,hB.neg,?_⟩
+  filter_upwards [hw] with ω hω
+  obtain ⟨G,F,hGF⟩ := hω
+  refine ⟨fun t z => loewnerMirror (G t (loewnerMirror z)),
+    fun t z => loewnerMirror (F t (loewnerMirror z)),?_⟩
+  simpa only [Pi.neg_apply,mul_neg] using hGF.mirror
+
+ 
+theorem sle_real_center_approach
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (x r : ℝ) (hx : x ≠ 0) (hr : 0 < r) :
+    P {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} ≤
+      ENNReal.ofReal ((r/(((8-κ)/32)*|x|))^(3*(8-κ)/(4*κ))) := by
+  rcases lt_or_gt_of_ne hx with hx | hx
+  · have hh := sle_positive_real_center_approach κ hκ hκ8
+      (fun ω => loewnerMirror ∘ γ ω) hSLE.mirror (-x) r (neg_pos.mpr hx) hr
+    have he : {ω | ∃ t : ℝ≥0, dist ((loewnerMirror ∘ γ ω) t) ((-x:ℝ):ℂ) < r} =
+        {ω | ∃ t : ℝ≥0, dist (γ ω t) (x:ℂ) < r} := by
+      ext ω
+      have hrx : ((-x:ℝ):ℂ) = loewnerMirror (x:ℂ) := by simp
+      simp only [Function.comp_apply,hrx,loewnerMirror_dist]
+    rw [he] at hh
+    simpa only [abs_of_neg hx] using hh
+  · simpa only [abs_of_pos hx] using
+      sle_positive_real_center_approach κ hκ hκ8 γ hSLE x r hx hr
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma derivative_close_of_cauchy {f : ℂ → ℂ} {U : Set ℂ} {b : ℂ} {r : ℝ}
+    (hr : 0 < r) (hU : IsOpen U) (hd : DifferentiableOn ℂ f U)
+    (hinc : closedBall b r ⊆ U)
+    (hbound : ∀ w ∈ closedBall b r, ‖deriv f w‖ ≤ 2 * ‖deriv f b‖) :
+    ∀ w ∈ closedBall b (r / 8), ‖deriv f w - deriv f b‖ ≤ ‖deriv f b‖ / 2 := by
+  have hdr := hd.deriv hU
+  have hsmall : closedBall b (r / 8) ⊆ closedBall b r := closedBall_subset_closedBall (by linarith)
+  have hsecond (w : ℂ) (hw : w ∈ closedBall b (r / 8)) :
+      ‖deriv (deriv f) w‖ ≤ 4 * ‖deriv f b‖ / r := by
+    have hs : closedBall w (r / 2) ⊆ closedBall b r := by
+      intro z hz
+      apply mem_closedBall.mpr
+      have hw' := mem_closedBall.mp hw
+      have hz' := mem_closedBall.mp hz
+      exact (dist_triangle z w b).trans (by linarith)
+    have hh := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le (by positivity : 0 < r / 2)
+      (hdr.diffContOnCl_ball (hs.trans hinc))
+      (fun z hz => hbound z (hs (sphere_subset_closedBall hz)))
+    convert hh using 1
+    ring
+  intro w hw
+  have hh := (convex_closedBall b (r / 8)).norm_image_sub_le_of_norm_deriv_le
+    (fun z hz => hdr.differentiableAt (hU.mem_nhds (hinc (hsmall hz)))) hsecond
+    (mem_closedBall_self (by positivity : 0 ≤ r / 8)) hw
+  have hn : ‖w - b‖ ≤ r / 8 := mem_closedBall_iff_norm.mp hw
+  calc
+    ‖deriv f w - deriv f b‖ ≤ (4 * ‖deriv f b‖ / r) * ‖w - b‖ := hh
+    _ ≤ (4 * ‖deriv f b‖ / r) * (r / 8) := mul_le_mul_of_nonneg_left hn (by positivity)
+    _ = ‖deriv f b‖ / 2 := by field_simp; ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma disk_image_of_deriv_close {f f' : ℂ → ℂ} {b a : ℂ} {r : ℝ}
+    (hr : 0 ≤ r) (ha : a ≠ 0)
+    (hd : ∀ z ∈ closedBall b r, HasDerivAt f (f' z) z)
+    (hb : ∀ z ∈ closedBall b r, ‖f' z / a - 1‖ ≤ 1 / 2) :
+    closedBall (f b) (‖a‖ * r / 2) ⊆ f '' closedBall b r := by
+  let g : ℂ → ℂ := fun z => f z / a
+  have hl : LipschitzOnWith (1 / 2 : ℝ≥0) (fun z => g z - z) (closedBall b r) := by
+    apply (convex_closedBall b r).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
+      (f' := fun z => f' z / a - 1)
+    · intro z hz
+      exact (((hd z hz).div_const a).sub (hasDerivAt_id z)).hasDerivWithinAt
+    · intro z hz
+      exact_mod_cast hb z hz
+  have hl' : ApproximatesLinearOn g (ContinuousLinearMap.id ℂ ℂ) (closedBall b r) (1 / 2) := by
+    apply LipschitzOnWith.approximatesLinearOn
+    exact hl
+  have hs := hl'.surjOn_closedBall_of_nonlinearRightInverse
+    (ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse hr (Subset.refl _)
+  have hs' : SurjOn g (closedBall b r) (closedBall (g b) (r / 2)) := by
+    norm_num [ContinuousLinearEquiv.toNonlinearRightInverse] at hs
+    convert hs using 1
+    ring_nf
+  intro y hy
+  have hy' : y / a ∈ closedBall (g b) (r / 2) := by
+    rw [mem_closedBall_iff_norm] at hy ⊢
+    change ‖y / a - f b / a‖ ≤ r / 2
+    rw [← sub_div, norm_div]
+    apply (div_le_iff₀ (norm_pos_iff.mpr ha)).mpr
+    nlinarith
+  obtain ⟨z, hz, he⟩ := hs' hy'
+  exact ⟨z, hz, (div_left_inj' ha).mp he⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma bloch_disk {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1)) :
+    ∃ b ∈ ball (0 : ℂ) 1,
+      closedBall (f b) (‖deriv f 0‖ / 64) ⊆ f '' ball 0 1 := by
+  have hzero : (0 : ℂ) ∈ closedBall 0 (1 / 2 : ℝ) := mem_closedBall_self (by norm_num)
+  have hinc : closedBall (0 : ℂ) (1 / 2 : ℝ) ⊆ ball 0 1 :=
+    closedBall_subset_ball (by norm_num)
+  by_cases h0 : deriv f 0 = 0
+  · refine ⟨0, mem_ball_self (by norm_num), ?_⟩
+    simp only [h0, norm_zero, zero_div, closedBall_zero, singleton_subset_iff]
+    exact mem_image_of_mem f (mem_ball_self (by norm_num))
+  have hd0 : 0 < ‖deriv f 0‖ := norm_pos_iff.mpr h0
+  let Φ : ℂ → ℝ := fun z => (1 / 2 - ‖z‖) * ‖deriv f z‖
+  have hc : ContinuousOn Φ (closedBall 0 (1 / 2 : ℝ)) :=
+    (continuousOn_const.sub continuous_norm.continuousOn).mul
+      ((hd.deriv isOpen_ball).continuousOn.mono hinc).norm
+  obtain ⟨b, hb, hmax⟩ := (isCompact_closedBall (0 : ℂ) (1 / 2 : ℝ)).exists_isMaxOn
+    ⟨0, hzero⟩ hc
+  have hM : ‖deriv f 0‖ / 2 ≤ (1 / 2 - ‖b‖) * ‖deriv f b‖ := by
+    have hh := hmax hzero
+    simpa [Φ, norm_zero, mul_comm, div_eq_mul_inv] using hh
+  have hb' : ‖b‖ ≤ 1 / 2 := mem_closedBall_zero_iff.mp hb
+  have hab : 0 < ‖deriv f b‖ := by
+    by_contra hn
+    have hh : ‖deriv f b‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
+    rw [hh, mul_zero] at hM
+    linarith
+  have hmargin : 0 < 1 / 2 - ‖b‖ := by
+    by_contra hn
+    have hh := mul_nonpos_of_nonpos_of_nonneg (le_of_not_gt hn) hab.le
+    linarith
+  let r : ℝ := (1 / 2 - ‖b‖) / 2
+  have hr : 0 < r := div_pos hmargin (by norm_num)
+  have hsub : closedBall b r ⊆ closedBall 0 (1 / 2 : ℝ) := by
+    intro w hw
+    rw [mem_closedBall_zero_iff]
+    have hw' : ‖w - b‖ ≤ r := mem_closedBall_iff_norm.mp hw
+    have hnorm : ‖w‖ ≤ ‖w - b‖ + ‖b‖ := by simpa [norm_sub_rev, add_comm] using norm_le_insert b w
+    dsimp [r] at hw'
+    linarith
+  have hbound (w : ℂ) (hw : w ∈ closedBall b r) : ‖deriv f w‖ ≤ 2 * ‖deriv f b‖ := by
+    have hh := hmax (hsub hw)
+    change (1 / 2 - ‖w‖) * ‖deriv f w‖ ≤ (1 / 2 - ‖b‖) * ‖deriv f b‖ at hh
+    have hw' : ‖w - b‖ ≤ r := mem_closedBall_iff_norm.mp hw
+    have hnorm : ‖w‖ ≤ ‖w - b‖ + ‖b‖ := by simpa [norm_sub_rev, add_comm] using norm_le_insert b w
+    have hm : r ≤ 1 / 2 - ‖w‖ := by dsimp [r] at *; linarith
+    have he : 1 / 2 - ‖b‖ = 2 * r := by dsimp [r]; ring
+    have hprod := mul_le_mul_of_nonneg_right hm (norm_nonneg (deriv f w))
+    rw [he] at hh
+    nlinarith
+  have hclose := derivative_close_of_cauchy hr isOpen_ball hd (hsub.trans hinc) hbound
+  have hsmall : closedBall b (r / 8) ⊆ closedBall b r := closedBall_subset_closedBall (by linarith)
+  have himage := disk_image_of_deriv_close (f := f) (f' := deriv f) (a := deriv f b)
+    (by positivity : 0 ≤ r / 8) (norm_pos_iff.mp hab)
+    (fun w hw => (hd.differentiableAt (isOpen_ball.mem_nhds (hinc (hsub (hsmall hw))))).hasDerivAt)
+    (by
+      intro w hw
+      rw [← div_self (norm_pos_iff.mp hab), ← sub_div, norm_div]
+      exact (div_le_iff₀ hab).mpr (by simpa [div_eq_mul_inv, mul_comm] using hclose w hw))
+  refine ⟨b, hinc hb, ?_⟩
+  have hlower : ‖deriv f 0‖ / 64 ≤ ‖deriv f b‖ * (r / 8) / 2 := by
+    dsimp [r]
+    nlinarith
+  exact (closedBall_subset_closedBall hlower).trans
+    (himage.trans (image_mono (hsmall.trans (hsub.trans hinc))))
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma hasDerivAt_log_lift {g h : ℂ → ℂ} {U : Set ℂ} (hU : IsOpen U)
+    (hg : DifferentiableOn ℂ g U) (hh : ContinuousOn h U)
+    (he : EqOn (Complex.exp ∘ h) g U) {z : ℂ} (hz : z ∈ U) :
+    HasDerivAt h (deriv g z / g z) z := by
+  have hgz : g z ≠ 0 := by rw [← he hz]; exact Complex.exp_ne_zero _
+  have hdz := (hg.differentiableAt (hU.mem_nhds hz)).hasDerivAt
+  have hcx := hh.continuousAt (hU.mem_nhds hz)
+  have hdisc : ∀ᶠ w in 𝓝 z, ‖h w - h z‖ < Real.pi := by
+    have hc : ContinuousAt (fun w => ‖h w - h z‖) z := (hcx.sub continuousAt_const).norm
+    exact hc (Iio_mem_nhds (by simpa using Real.pi_pos))
+  have hlocal : (fun w => Complex.log (g w / g z) + h z) =ᶠ[𝓝 z] h := by
+    filter_upwards [hU.mem_nhds hz, hdisc] with w hw hw'
+    have him : |(h w - h z).im| < Real.pi := (Complex.abs_im_le_norm _).trans_lt hw'
+    rw [← he hw, ← he hz]
+    change Complex.log (Complex.exp (h w) / Complex.exp (h z)) + h z = h w
+    rw [← Complex.exp_sub, Complex.log_exp (by linarith [(abs_lt.mp him).1])
+      (le_of_lt (abs_lt.mp him).2)]
+    ring
+  have hlog := (hdz.div_const (g z)).clog (by simp [hgz])
+  have hf : HasDerivAt (fun w => Complex.log (g w / g z) + h z) (deriv g z / g z) z := by
+    simpa [hgz] using hlog.add_const (h z)
+  exact hf.congr_of_eventuallyEq hlocal.symm
+
+lemma differentiableOn_log_lift {g h : ℂ → ℂ} {U : Set ℂ} (hU : IsOpen U)
+    (hg : DifferentiableOn ℂ g U) (hh : ContinuousOn h U)
+    (he : EqOn (Complex.exp ∘ h) g U) : DifferentiableOn ℂ h U :=
+  fun _ hz => (hasDerivAt_log_lift hU hg hh he hz).differentiableAt.differentiableWithinAt
+
+lemma exists_holomorphic_log_on_ball {g : ℂ → ℂ} {b : ℂ} {r : ℝ}
+    (hr : 0 < r) (hg : DifferentiableOn ℂ g (ball b r))
+    (h0 : ∀ z ∈ ball b r, g z ≠ 0) :
+    ∃ h : ℂ → ℂ, DifferentiableOn ℂ h (ball b r) ∧
+      EqOn (Complex.exp ∘ h) g (ball b r) ∧
+      ∀ z ∈ ball b r, deriv h z = deriv g z / g z := by
+  let := (convex_ball b r).contractibleSpace ⟨b, mem_ball_self hr⟩
+  obtain ⟨h, hh, he⟩ := Complex.exists_continuousOn_eqOn_exp_comp
+    (by change SimplyConnectedSpace (ball b r); infer_instance) isOpen_ball hg.continuousOn
+    (by rintro ⟨z, hz, he⟩; exact h0 z hz he)
+  exact ⟨h, differentiableOn_log_lift isOpen_ball hg hh he, he,
+    fun z hz => (hasDerivAt_log_lift isOpen_ball hg hh he hz).deriv⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma univalent_omitted_value_bound {f : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (a : ℂ) (ha : a ∉ f '' ball 0 1) :
+    ‖deriv f 0‖ ≤ (128 * Real.pi) * ‖f 0 - a‖ := by
+  have hnon (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) : f z - a ≠ 0 := by
+    intro he
+    exact ha ⟨z, hz, sub_eq_zero.mp he⟩
+  have hz0 : (0 : ℂ) ∈ ball 0 1 := mem_ball_self (by norm_num)
+  obtain ⟨h, hhd, he, hhder⟩ := exists_holomorphic_log_on_ball (by norm_num : (0 : ℝ) < 1)
+    (hd.sub_const a) hnon
+  obtain ⟨b, hb, hball⟩ := bloch_disk hhd
+  have hbound : ‖deriv h 0‖ < 128 * Real.pi := by
+    by_contra hn
+    have hn' : 128 * Real.pi ≤ ‖deriv h 0‖ := le_of_not_gt hn
+    have hw : h b + 2 * Real.pi * Complex.I ∈ closedBall (h b) (‖deriv h 0‖ / 64) := by
+      rw [mem_closedBall_iff_norm, add_sub_cancel_left, norm_mul, norm_mul,
+        norm_ofNat, Complex.norm_real, Real.norm_eq_abs, abs_of_pos Real.pi_pos, Complex.norm_I, mul_one]
+      linarith
+    obtain ⟨z, hz, hzh⟩ := hball hw
+    have hfe : f z - a = f b - a := by
+      calc
+        f z - a = Complex.exp (h z) := (he hz).symm
+        _ = Complex.exp (h b) := by rw [hzh]; exact Complex.exp_periodic (h b)
+        _ = f b - a := he hb
+    have hzb : z = b := hi hz hb (sub_left_inj.mp hfe)
+    rw [hzb] at hzh
+    have him := congrArg Complex.im hzh
+    norm_num [Complex.add_im, Complex.mul_im, Complex.mul_re] at him
+  have hd0 := (hd.differentiableAt (isOpen_ball.mem_nhds hz0)).hasDerivAt
+  have hdsub : deriv (fun z => f z - a) 0 = deriv f 0 := (hd0.sub_const a).deriv
+  rw [hhder 0 hz0, hdsub, norm_div] at hbound
+  exact ((div_lt_iff₀ (norm_pos_iff.mpr (hnon 0 hz0))).mp hbound).le
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/ChartMeasurability.lean b/lean/OAI/Probability/SLE/ChartMeasurability.lean
index 338c485ddc4908a7f758ac48f1fd22f318d71013..3fbbcd5093c9de118e64be996dde99dafbada005 100644
--- a/lean/OAI/Probability/SLE/ChartMeasurability.lean
+++ b/lean/OAI/Probability/SLE/ChartMeasurability.lean
@@ -1,689 +1,689 @@
-import Mathlib
-import OAI.Probability.SLE.DiskHalfPlane
-import OAI.Probability.SLE.StoppedMeasures
-
-namespace OAI
-
-/-! Chart Measurability for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_killedChart_stopped_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)) (z : ℂ)
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) :
-    Measurable[hτ.measurableSpace]
-      (fun ω => if z ∈ survivingDomain (γ ω) (τ ω) then G ω (τ ω) z else 0) := by
-  exact measurable_stoppedValue
-    (source_killedChart_progressive hm hc κ γ G F hw z) hτ
-
-end
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory Complex TopologicalSpace Classical
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma source_measurable_inverse_chart
-    {Ω : Type*} [MeasurableSpace Ω]
-    (S : Ω → Set ℂ) (G F K : Ω → ℂ → ℂ) (H : Set ℂ)
-    (hS : ∀ ω, IsOpen (S ω)) (hH : IsOpen H)
-    (hG : ∀ ω, ContinuousOn (G ω) (S ω))
-    (hF : ∀ ω, ContinuousOn (F ω) H)
-    (hFH : ∀ ω, MapsTo (F ω) H (S ω))
-    (hGF : ∀ ω z, z ∈ H → G ω (F ω z) = z)
-    (hFG : ∀ ω z, z ∈ S ω → F ω (G ω z) = z)
-    (hK : ∀ z, Measurable (fun ω => K ω z))
-    (hKeq : ∀ ω z, K ω z = if z ∈ S ω then G ω z else 0)
-    (hGH : ∀ ω z, z ∈ S ω → 0 < (G ω z).im)
-    (w : Ω → ℂ) (hwm : Measurable w) (hw : ∀ ω, w ω ∈ H) :
-    Measurable (fun ω => F ω (w ω)) := by
-  classical
-  let d := denseSeq ℂ
-  let v (k : ℕ) (ω : Ω) (n : ℕ) : Prop :=
-    0 < (K ω (d n)).im ∧ dist (K ω (d n)) (w ω) < 1/((k:ℝ)+1)
-  have hv (k : ℕ) (ω : Ω) : ∃ n, v k ω n := by
-    have hpos : 0 < 1/((k:ℝ)+1) := by positivity
-    have hx : F ω (w ω) ∈ S ω := hFH ω (hw ω)
-    have hct := (hG ω).continuousAt ((hS ω).mem_nhds hx)
-    have hn : {z | dist (G ω z) (w ω) < 1/((k:ℝ)+1)} ∈ 𝓝 (F ω (w ω)) := by
-      change G ω ⁻¹' Metric.ball (w ω) (1/((k:ℝ)+1)) ∈ 𝓝 (F ω (w ω))
-      apply hct.preimage_mem_nhds
-      rw [hGF ω _ (hw ω)]
-      exact Metric.ball_mem_nhds (w ω) hpos
-    obtain ⟨n,hn⟩ := (denseRange_denseSeq ℂ).mem_nhds (inter_mem ((hS ω).mem_nhds hx) hn)
-    refine ⟨n,?_⟩
-    dsimp [v,d]
-    rw [hKeq,ite_eq_left hn.1]
-    exact ⟨hGH ω _ hn.1,hn.2⟩
-  have hvm (k n : ℕ) : MeasurableSet {ω | v k ω n} :=
-    (measurableSet_lt measurable_const (Complex.measurable_im.comp (hK _))).inter
-      (measurableSet_lt ((hK _).dist hwm) measurable_const)
-  let j (k : ℕ) (ω : Ω) : ℕ := Nat.find (hv k ω)
-  have hj (k : ℕ) : Measurable (j k) := measurable_find (hv k) (hvm k)
-  have hjv (k : ℕ) (ω : Ω) : v k ω (j k ω) := Nat.find_spec (hv k ω)
-  have hjS (k : ℕ) (ω : Ω) : d (j k ω) ∈ S ω := by
-    by_contra hn
-    have hh := (hjv k ω).1
-    rw [hKeq,ite_eq_right hn] at hh
-    norm_num at hh
-  have hlim (ω : Ω) : Tendsto (fun k => K ω (d (j k ω))) atTop (𝓝 (w ω)) := by
-    apply Metric.tendsto_atTop.mpr
-    intro ε hε
-    obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) ε hε
-    refine ⟨N,fun k hk => (hjv k ω).2.trans ?_⟩
-    simpa only [Real.dist_eq,sub_zero,abs_of_pos (by positivity : 0 < 1/((k:ℝ)+1))]
-      using hN k hk
-  have hflim (ω : Ω) : Tendsto (fun k => d (j k ω)) atTop (𝓝 (F ω (w ω))) := by
-    have hh := ((hF ω).continuousAt (hH.mem_nhds (hw ω))).tendsto.comp (hlim ω)
-    convert hh using 1
-    ext k
-    simp only [Function.comp_apply,hKeq,ite_eq_left (hjS k ω),hFG ω _ (hjS k ω)]
-  exact measurable_of_tendsto_metrizable (fun k => (measurable_of_countable d).comp (hj k)) (tendsto_pi_nhds.mpr hflim)
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical
-open scoped Topology NNReal
-namespace SLEExactGauge
-lemma source_stopped_inverseChart_measurable
-    {Ω : Type*} [m : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω))
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) (w : Ω → ℂ)
-    (hwm : Measurable[hτ.measurableSpace] w) (hwH : ∀ ω, w ω ∈ upperHalfPlane) :
-    Measurable[hτ.measurableSpace] (fun ω => F ω (τ ω) (w ω)) := by
-  apply @source_measurable_inverse_chart Ω hτ.measurableSpace
-    (fun ω => survivingDomain (γ ω) (τ ω)) (fun ω => G ω (τ ω))
-    (fun ω => F ω (τ ω))
-    (fun ω z => if z ∈ survivingDomain (γ ω) (τ ω) then G ω (τ ω) z else 0)
-    upperHalfPlane
-    (fun ω => isOpen_survivingDomain (hw ω).1 _) isOpen_upperHalfPlane
-    (fun ω => ((hw ω).2.2.2.1 _).continuousOn)
-    (fun ω => ((hw ω).2.2.2.2.1 _).continuousOn)
-    (fun ω => (hw ω).2.2.2.2.2.2.1 _)
-    (fun ω => (hw ω).2.2.2.2.2.2.2.2.1 _)
-    (fun ω => (hw ω).2.2.2.2.2.2.2.1 _)
-    (fun z => source_killedChart_stopped_measurable hm hc κ γ G F hw z τ hτ)
-    (fun _ _ => rfl)
-    (fun ω z hz => (hw ω).imag_positive _ z hz) w hwm hwH
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
-open scoped Topology NNReal
-namespace SLEExactGauge
-noncomputable section
-local instance : MeasurableSpace C(upperHalfPlane,ℂ) := borel _
-local instance : BorelSpace C(upperHalfPlane,ℂ) := ⟨rfl⟩
-
-lemma source_halfplane_cmap_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (f : Ω → C(upperHalfPlane,ℂ))
-    (hf : ∀ z, Measurable (fun ω => f ω z)) : Measurable f := by
-  let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-  let : PolishSpace upperHalfPlane := isOpen_upperHalfPlane.polishSpace
-  let : Nonempty upperHalfPlane := ⟨⟨Complex.I,by simp [upperHalfPlane]⟩⟩
-  have he : MeasurableEmbedding
-      (fun F : C(upperHalfPlane,ℂ) => fun n => F (denseSeq upperHalfPlane n)) := by
-    apply Continuous.measurableEmbedding
-    · exact continuous_pi fun n => continuous_eval_const _
-    · intro F G heq
-      apply DFunLike.coe_injective
-      exact (denseRange_denseSeq upperHalfPlane).equalizer F.continuous G.continuous heq
-  exact he.measurable_comp_iff.mp (Measurable.of_eval (fun _ => hf _))
-
-def sourceStoppedInversePath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (τ : Ω → ℝ≥0) (ω : Ω) : C(upperHalfPlane,ℂ) :=
-  ⟨fun z => F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ)),
-    ContinuousOn.comp_continuous ((hw ω).2.2.2.2.1 _).continuousOn
-      (continuous_subtype_val.add continuous_const)
-      (fun z => by simpa [upperHalfPlane] using z.2)⟩
-
-lemma source_stopped_inversePath_measurable
-    {Ω : Type*} [m : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω))
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) :
-    Measurable[hτ.measurableSpace] (sourceStoppedInversePath B κ γ G F hw τ) := by
-  apply @source_halfplane_cmap_measurable Ω hτ.measurableSpace
-  intro z
-  have hB := measurable_stoppedValue
-    ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc) hτ
-  apply source_stopped_inverseChart_measurable B hm hc κ γ G F hw τ hτ
-    (fun ω => z+(Real.sqrt κ * B (τ ω) ω : ℝ))
-  · exact measurable_const.add (Complex.measurable_ofReal.comp (measurable_const.mul hB))
-  · intro ω
-    simpa [upperHalfPlane] using z.2
-end
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory Metric Bornology
-open scoped Topology NNReal
-namespace SLEExactGauge
- 
-
-def square (R : ℝ) : Set ℂ := {z | |z.re| ≤ R ∧ |z.im| ≤ R}
-
-def upperExterior (R : ℝ) : Set ℂ := upperHalfPlane ∩ (square R)ᶜ
-
-def lowerExterior (R : ℝ) : Set ℂ := {z | z.im ≤ 0} ∩ (square R)ᶜ
-
-lemma square_isCompact (R : ℝ) : IsCompact (square R) := by
-  have hc : IsClosed (square R) :=
-    (isClosed_le Complex.continuous_re.abs continuous_const).inter
-      (isClosed_le Complex.continuous_im.abs continuous_const)
-  apply isCompact_of_isClosed_isBounded hc
-  apply isBounded_iff_forall_norm_le.mpr
-  refine ⟨2*R,fun z hz => ?_⟩
-  exact (Complex.norm_le_abs_re_add_abs_im z).trans (by linarith [hz.1,hz.2])
-
-lemma bounded_subset_square {s : Set ℂ} (hs : Bornology.IsBounded s) :
-    ∃ R > 0, s ⊆ square R := by
-  obtain ⟨C,hC⟩ := hs.exists_norm_le
-  refine ⟨max C 1,lt_of_lt_of_le zero_lt_one (le_max_right _ _),?_⟩
-  intro z hz
-  exact ⟨(Complex.abs_re_le_norm z).trans ((hC z hz).trans (le_max_left _ _)),
-    (Complex.abs_im_le_norm z).trans ((hC z hz).trans (le_max_left _ _))⟩
-
-lemma upperExterior_eq (R : ℝ) (hR : 0 ≤ R) : upperExterior R =
-    ({z : ℂ | z.re < -R} ∩ {z | 0 < z.im}) ∪ {z | R < z.im} ∪
-    ({z : ℂ | R < z.re} ∩ {z | 0 < z.im}) := by
-  ext z
-  simp only [upperExterior,upperHalfPlane,square,mem_inter_iff,mem_ofPred_eq,
-    mem_compl_iff,mem_union,abs_le]
-  constructor
-  · rintro ⟨hi,hs⟩
-    by_cases hl : z.re < -R
-    · exact Or.inl (Or.inl ⟨hl,hi⟩)
-    by_cases hr : R < z.re
-    · exact Or.inr ⟨hr,hi⟩
-    exact Or.inl (Or.inr (by by_contra hn; exact hs ⟨⟨by linarith,by linarith⟩,
-      ⟨by linarith,by linarith⟩⟩))
-  · rintro ((⟨hl,hi⟩ | hi) | ⟨hr,hi⟩)
-    · exact ⟨hi,fun hs => (not_lt_of_ge hs.1.1) hl⟩
-    · exact ⟨by linarith,fun hs => (not_lt_of_ge hs.2.2) hi⟩
-    · exact ⟨hi,fun hs => (not_lt_of_ge hs.1.2) hr⟩
-
-lemma upperExterior_preconnected (R : ℝ) (hR : 0 ≤ R) : IsPreconnected (upperExterior R) := by
-  rw [upperExterior_eq R hR]
-  have hL := ((convex_halfSpace_re_lt (-R)).inter (convex_halfSpace_im_gt 0)).isPreconnected
-  have hU := (convex_halfSpace_im_gt R).isPreconnected
-  have hV := ((convex_halfSpace_re_gt R).inter (convex_halfSpace_im_gt 0)).isPreconnected
-  have hLU : (({z : ℂ | z.re < -R} ∩ {z | 0 < z.im}) ∩ {z | R < z.im}).Nonempty := by
-    refine ⟨⟨-R-1,R+1⟩,?_,?_⟩ <;> simp only [mem_inter_iff,mem_ofPred_eq] <;> (try constructor) <;> linarith
-  have hUV : ((({z : ℂ | z.re < -R} ∩ {z | 0 < z.im}) ∪ {z | R < z.im}) ∩
-      ({z : ℂ | R < z.re} ∩ {z | 0 < z.im})).Nonempty := by
-    refine ⟨⟨R+1,R+1⟩,Or.inr ?_,?_,?_⟩ <;> simp only [mem_ofPred_eq] <;> linarith
-  exact (hL.union' hLU hU).union' hUV hV
-
-lemma upperExterior_unbounded (R : ℝ) (hR : 0 ≤ R) : ¬ Bornology.IsBounded (upperExterior R) := by
-  intro hb
-  obtain ⟨C,hC⟩ := hb.exists_norm_le
-  let y : ℝ := max C R + 1
-  have hyR : R < y := by dsimp [y]; linarith [le_max_right C R]
-  have hyC : C < y := by dsimp [y]; linarith [le_max_left C R]
-  have hy0 : 0 < y := by linarith
-  have hm : (⟨0,y⟩ : ℂ) ∈ upperExterior R := by
-    exact ⟨hy0,fun h => (not_le_of_gt hyR) ((le_abs_self y).trans h.2)⟩
-  have hb' := hC _ hm
-  have hn := Complex.abs_im_le_norm (⟨0,y⟩ : ℂ)
-  change |y| ≤ ‖(⟨0,y⟩ : ℂ)‖ at hn
-  rw [abs_of_pos hy0] at hn
-  linarith
-
-lemma upperExterior_subset_survivingDomain {γ : ℝ≥0 → ℂ} {T : ℝ≥0} {R : ℝ}
-    (hR : 0 ≤ R) (hγ : γ '' Icc 0 T ⊆ square R) :
-    upperExterior R ⊆ survivingDomain γ T := by
-  intro z hz
-  have hs : upperExterior R ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by
-    intro w hw
-    exact ⟨hw.1,fun ht => hw.2 (hγ ht)⟩
-  refine ⟨hs hz,?_⟩
-  intro hb
-  exact upperExterior_unbounded R hR (hb.subset
-    ((upperExterior_preconnected R hR).subset_connectedComponentIn hz hs))
-
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma source_live_compact_bridge (γ : C(ℝ≥0,ℂ)) (t : ℝ≥0) (z : ℂ)
-    (hz : z ∈ survivingDomain γ t) :
-    ∃ (R δ : ℝ) (K : Set ℂ), 0 < R ∧ 0 < δ ∧ IsCompact K ∧
-      IsPreconnected K ∧ Metric.closedBall z δ ⊆ K ∧
-      (⟨0,R+1⟩ : ℂ) ∈ K ∧ K ⊆ upperHalfPlane ∧
-      MapsTo γ (Icc 0 t) (Metric.ball 0 R \ K) := by
-  obtain ⟨M,hM⟩ := (isCompact_Icc.image γ.continuous).isBounded.exists_norm_le
-  let R := max M 0 + 1
-  have hR : 0 < R := by dsimp [R]; linarith [le_max_right M 0]
-  have hnorm (s : ℝ≥0) (hs : s ∈ Icc 0 t) : ‖γ s‖ < R := by
-    have hh := hM _ ⟨s,hs,rfl⟩
-    dsimp [R]; linarith [le_max_left M 0]
-  have htr : γ '' Icc 0 t ⊆ square R := by
-    rintro _ ⟨s,hs,rfl⟩
-    exact ⟨(abs_re_le_norm _).trans (hnorm s hs).le,(abs_im_le_norm _).trans (hnorm s hs).le⟩
-  let A : Set ℂ := upperHalfPlane \ γ '' Icc 0 t
-  let C := connectedComponentIn A z
-  have hA : IsOpen A := isOpen_upperHalfPlane.sdiff (isCompact_Icc.image γ.continuous).isClosed
-  have hzC : z ∈ C := mem_connectedComponentIn hz.1
-  have hCsub : C ⊆ A := connectedComponentIn_subset _ _
-  have hnot : ¬ C ⊆ square R := fun hh => hz.2 ((square_isCompact R).isBounded.subset hh)
-  obtain ⟨v,hv,hvR⟩ := Set.not_subset.mp hnot
-  have hvE : v ∈ upperExterior R := ⟨(hCsub hv).1,hvR⟩
-  have hEsub : upperExterior R ⊆ A := by
-    intro w hw
-    exact ⟨hw.1,fun ht => hw.2 (htr ht)⟩
-  have hEC : upperExterior R ⊆ C := by
-    have hc := (upperExterior_preconnected R hR.le).subset_connectedComponentIn hvE hEsub
-    rwa [← connectedComponentIn_eq hv] at hc
-  let p : ℂ := ⟨0,R+1⟩
-  have hpE : p ∈ upperExterior R := by
-    refine ⟨by change 0 < R+1; linarith,?_⟩
-    intro hh
-    have hhi : |R+1| ≤ R := hh.2
-    linarith [le_abs_self (R+1)]
-  have hpathc : IsPathConnected C := (hA.connectedComponentIn).isConnected_iff_isPathConnected.mp
-    ⟨⟨z,hzC⟩,isPreconnected_connectedComponentIn⟩
-  let η := (hpathc.joinedIn z hzC p (hEC hpE)).somePath
-  have hη : range η ⊆ A := by
-    rintro _ ⟨s,rfl⟩
-    exact hCsub ((hpathc.joinedIn z hzC p (hEC hpE)).somePath_mem s)
-  obtain ⟨ε,hε,hεsub⟩ := Metric.isOpen_iff.mp hA z hz.1
-  let δ := ε/2
-  have hδ : 0 < δ := by dsimp [δ]; positivity
-  have hball : Metric.closedBall z δ ⊆ A := by
-    apply subset_trans _ hεsub
-    exact Metric.closedBall_subset_ball (by dsimp [δ]; linarith)
-  let K := range η ∪ Metric.closedBall z δ
-  have hKc : IsCompact K := (isCompact_range η.continuous).union (isCompact_closedBall z δ)
-  have hKconn : IsPreconnected K :=
-    (isConnected_range η.continuous).2.union z η.source_mem_range (Metric.mem_closedBall_self hδ.le)
-      (convex_closedBall z δ).isPreconnected
-  have hKsub : K ⊆ A := union_subset hη hball
-  refine ⟨R,δ,K,hR,hδ,hKc,hKconn,subset_union_right,
-    Or.inl η.target_mem_range,fun w hw => (hKsub hw).1,?_⟩
-  intro s hs
-  refine ⟨by simpa only [Metric.mem_ball,dist_zero_right] using hnorm s hs,?_⟩
-  intro hh
-  exact (hKsub hh).2 ⟨s,hs,rfl⟩
-
- 
-
-lemma source_live_of_bridge {γ : ℝ≥0 → ℂ} {t : ℝ≥0} {z : ℂ} {R : ℝ} {K : Set ℂ}
-    (hR : 0 ≤ R) (hK : IsPreconnected K) (hz : z ∈ K)
-    (hp : (⟨0,R+1⟩ : ℂ) ∈ K) (hKH : K ⊆ upperHalfPlane)
-    (hγ : MapsTo γ (Icc 0 t) (Metric.ball 0 R \ K)) :
-    z ∈ survivingDomain γ t := by
-  have hnorm (s : ℝ≥0) (hs : s ∈ Icc 0 t) : ‖γ s‖ < R := by
-    simpa only [Metric.mem_ball,dist_zero_right] using (hγ hs).1
-  have htr : γ '' Icc 0 t ⊆ square R := by
-    rintro _ ⟨s,hs,rfl⟩
-    exact ⟨(abs_re_le_norm _).trans (hnorm s hs).le,(abs_im_le_norm _).trans (hnorm s hs).le⟩
-  have hpE : (⟨0,R+1⟩ : ℂ) ∈ upperExterior R := by
-    refine ⟨by change 0 < R+1; linarith,?_⟩
-    intro hh
-    have hhi : |R+1| ≤ R := hh.2
-    linarith [le_abs_self (R+1)]
-  have hpS := upperExterior_subset_survivingDomain hR htr hpE
-  have hKa : K ⊆ upperHalfPlane \ γ '' Icc 0 t := by
-    intro w hw
-    refine ⟨hKH hw,?_⟩
-    rintro ⟨s,hs,rfl⟩
-    exact (hγ hs).2 hw
-  have hpC := hK.subset_connectedComponentIn hz hKa hp
-  refine ⟨hKa hz,?_⟩
-  rw [connectedComponentIn_eq hpC]
-  exact hpS.2
-
-noncomputable def sourceRescaledPath (p : C(ℝ≥0,ℂ) × ℝ≥0) : C(ℝ≥0,ℂ) :=
-  ⟨fun s => p.1 (p.2*s),p.1.continuous.comp (continuous_const.mul continuous_id)⟩
-
-lemma sourceRescaledPath_continuous : Continuous sourceRescaledPath := by
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  exact continuous_eval.comp
-    ((continuous_fst.fst).prodMk ((continuous_fst.snd).mul continuous_snd))
-
-lemma sourceRescaledPath_mapsTo {p : C(ℝ≥0,ℂ) × ℝ≥0} {A : Set ℂ} :
-    MapsTo (sourceRescaledPath p) (Icc 0 1) A ↔ MapsTo p.1 (Icc 0 p.2) A := by
-  have he : (fun s : ℝ≥0 => p.2*s) '' Icc 0 1 = Icc 0 p.2 := by
-    simpa using image_mul_left_Icc (show 0 ≤ p.2 from bot_le) (show (0 : ℝ≥0) ≤ 1 by norm_num)
-  rw [← he]
-  constructor
-  · intro h v hv
-    rcases hv with ⟨s,hs,rfl⟩
-    exact h hs
-  · intro h s hs
-    exact h ⟨s,hs,rfl⟩
-
-lemma source_survival_open :
-    IsOpen {p : (C(ℝ≥0,ℂ) × ℝ≥0) × ℂ | p.2 ∈ survivingDomain p.1.1 p.1.2} := by
-  apply isOpen_iff_forall_mem_open.mpr
-  rintro ⟨⟨γ,t⟩,z⟩ hz
-  obtain ⟨R,δ,K,hR,hδ,hKc,hKconn,hball,hp,hKH,hγ⟩ := source_live_compact_bridge γ t z hz
-  let W : Set (C(ℝ≥0,ℂ) × ℝ≥0) := {p | MapsTo p.1 (Icc 0 p.2) (Metric.ball 0 R \ K)}
-  have hWo : IsOpen W := by
-    have hh := (ContinuousMap.isOpen_setOfPred_mapsTo (K := Icc (0 : ℝ≥0) 1) (U := Metric.ball (0 : ℂ) R \ K) isCompact_Icc
-      (Metric.isOpen_ball.sdiff hKc.isClosed)).preimage sourceRescaledPath_continuous
-    convert hh using 1
-    ext p
-    exact sourceRescaledPath_mapsTo.symm
-  refine ⟨W ×ˢ Metric.ball z δ,?_,hWo.prod Metric.isOpen_ball,⟨hγ,Metric.mem_ball_self hδ⟩⟩
-  rintro ⟨⟨β,s⟩,w⟩ ⟨hβ,hw⟩
-  exact source_live_of_bridge hR.le hKconn (hball (Metric.ball_subset_closedBall hw)) hp hKH hβ
-
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-abbrev SourceDiskParameter := (C(upperHalfPlane,ℂ) × ℂ) × (ℂ × ℝ)
-
-lemma source_physical_disk_kernel_open :
-  IsOpen {p : SourceDiskParameter × C(ℝ≥0,ℂ) |
-    ∃ t : ℝ≥0, p.1.1.2 ∈ survivingDomain p.2 t ∧
-      ∃ h : p.2 t ∈ upperHalfPlane,
-        p.1.2.2 < ‖p.1.1.1 ⟨p.2 t,h⟩ - p.1.2.1‖} := by
-  let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-  have ht (t : ℝ≥0) :
-      IsOpen {p : SourceDiskParameter × C(ℝ≥0,ℂ) |
-        p.1.1.2 ∈ survivingDomain p.2 t ∧
-          ∃ h : p.2 t ∈ upperHalfPlane,
-            p.1.2.2 < ‖p.1.1.1 ⟨p.2 t,h⟩ - p.1.2.1‖} := by
-    let S : Set (SourceDiskParameter × C(ℝ≥0,ℂ)) := {p | p.2 t ∈ upperHalfPlane}
-    have hS : IsOpen S := isOpen_upperHalfPlane.preimage
-      ((continuous_eval_const t).comp continuous_snd)
-    let Z : S → upperHalfPlane := fun p => ⟨p.1.2 t,p.2⟩
-    have hZ : Continuous Z := Continuous.subtype_mk
-      ((continuous_eval_const t).comp (continuous_snd.comp continuous_subtype_val)) _
-    have hF : Continuous (fun p : S => p.1.1.1.1 (Z p)) :=
-      continuous_eval.comp
-        ((continuous_fst.fst.fst.comp continuous_subtype_val).prodMk hZ)
-    have hLive : IsOpen {p : S | p.1.1.1.2 ∈ survivingDomain p.1.2 t} :=
-      source_survival_open.preimage
-        (((continuous_snd.comp continuous_subtype_val).prodMk continuous_const).prodMk
-          (continuous_fst.fst.snd.comp continuous_subtype_val))
-    have hExit : IsOpen {p : S | p.1.1.2.2 < ‖p.1.1.1.1 (Z p) - p.1.1.2.1‖} :=
-      isOpen_lt (continuous_fst.snd.snd.comp continuous_subtype_val)
-        ((hF.sub (continuous_fst.snd.fst.comp continuous_subtype_val)).norm)
-    have hO := hS.isOpenMap_subtype_val _ (hLive.inter hExit)
-    convert hO using 1
-    ext p
-    constructor
-    · rintro ⟨hl,h,he⟩
-      exact ⟨⟨p,h⟩,⟨hl,he⟩,rfl⟩
-    · rintro ⟨q,⟨hl,he⟩,rfl⟩
-      exact ⟨hl,q.2,he⟩
-  simpa only [ofPred_exists] using isOpen_iUnion ht
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma source_high_vertical_survives {γ : ℝ≥0 → ℂ} {t : ℝ≥0} {R : ℝ}
-    (hR : 0 ≤ R) (hb : ∀ s ∈ Icc 0 t, ‖γ s‖ ≤ R) {y : ℝ} (hy : R < y) :
-    (⟨0,y⟩ : ℂ) ∈ survivingDomain γ t := by
-  let A : Set ℂ := {z | R < z.im}
-  have hA : A ⊆ upperHalfPlane \ γ '' Icc 0 t := by
-    intro z hz
-    refine ⟨lt_of_le_of_lt hR hz, ?_⟩
-    rintro ⟨s,hs,rfl⟩
-    have hh := (le_abs_self (γ s).im).trans ((Complex.abs_im_le_norm _).trans (hb s hs))
-    exact (not_lt_of_ge hh) hz
-  refine ⟨hA hy, ?_⟩
-  intro hbd
-  obtain ⟨C,hC⟩ := (hbd.subset ((convex_halfSpace_im_gt R).isPreconnected.subset_connectedComponentIn hy hA)).exists_norm_le
-  let v : ℝ := max C R + 1
-  have hv : R < v := by dsimp [v]; linarith [le_max_right C R]
-  have hvC : C < v := by dsimp [v]; linarith [le_max_left C R]
-  have hh := hC (⟨0,v⟩ : ℂ) hv
-  have hh' := Complex.abs_im_le_norm (⟨0,v⟩ : ℂ)
-  change |v| ≤ _ at hh'
-  linarith [le_abs_self v]
-
-lemma IsCapacityTwoWitness.source_later_image_unbounded
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) :
-    ¬ Bornology.IsBounded ((fun z => G T z - (U T : ℂ)) '' survivingDomain γ (T+t)) := by
-  intro hbd
-  obtain ⟨C,hC⟩ := hbd.exists_norm_le
-  obtain ⟨R,hR⟩ := (isCompact_Icc.image h.1).isBounded.exists_norm_le
-  have hb : ∀ s ∈ Icc 0 (T+t), ‖γ s‖ ≤ max R 0 := by
-    intro s hs
-    exact (hR _ ⟨s,hs,rfl⟩).trans (le_max_left _ _)
-  have hlim := h.2.2.2.2.2.2.2.2.2.2.2.1 T
-  have hnorm := hlim.norm
-  have he : ∀ᶠ z in cocompact ℂ, z ∈ survivingDomain γ T →
-      ‖z * (G T z - z)‖ < ‖(2 : ℂ)*(T : ℂ)‖ + 1 := by
-    apply eventually_inf_principal.mp
-    exact (tendsto_order.mp hnorm).2 _ (lt_add_one _)
-  obtain ⟨K,hK,hKe⟩ := mem_cocompact.mp he
-  obtain ⟨D,hD⟩ := hK.isBounded.exists_norm_le
-  let A : ℝ := ‖(2 : ℂ)*(T : ℂ)‖ + 1
-  let y : ℝ := max (max (max R 0) D) (max A (C + |U T|)) + 2
-  have hyR : max R 0 < y := by dsimp [y]; linarith [le_max_left (max R 0) D,le_max_left (max (max R 0) D) (max A (C+|U T|))]
-  have hyD : D < y := by dsimp [y]; linarith [le_max_right (max R 0) D,le_max_left (max (max R 0) D) (max A (C+|U T|))]
-  have hyA : A < y := by dsimp [y]; linarith [le_max_left A (C+|U T|),le_max_right (max (max R 0) D) (max A (C+|U T|))]
-  have hyC : C + |U T| + 1 < y := by dsimp [y]; linarith [le_max_right A (C+|U T|),le_max_right (max (max R 0) D) (max A (C+|U T|))]
-  have hy0 : 0 < y := lt_of_le_of_lt (le_max_right R 0) hyR
-  let z : ℂ := ⟨0,y⟩
-  have hnz : ‖z‖ = y := by simp [z,Complex.norm_def,Real.sqrt_mul_self hy0.le]
-  have hz : z ∈ survivingDomain γ (T+t) := source_high_vertical_survives (le_max_right R 0) hb hyR
-  have hzt : z ∈ survivingDomain γ T := survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le)) hz
-  have hzK : z ∈ Kᶜ := by intro h'; have := hD z h'; rw [hnz] at this; linarith
-  have hsmall := hKe hzK hzt
-  have hG : ‖G T z‖ ≤ C + |U T| := by
-    calc
-      ‖G T z‖ = ‖(G T z-(U T : ℂ))+(U T : ℂ)‖ := by rw [sub_add_cancel]
-      _ ≤ ‖G T z-(U T : ℂ)‖+‖(U T : ℂ)‖ := norm_add_le _ _
-      _ ≤ C + |U T| := by gcongr; exact hC _ ⟨z,hz,rfl⟩; simp
-  have htri := norm_sub_norm_le z (G T z)
-  rw [hnz,norm_sub_rev] at htri
-  rw [norm_mul,hnz] at hsmall
-  change y*‖G T z-z‖ < A at hsmall
-  have hpos : 0 < A := by dsimp [A]; positivity
-  nlinarith
-
-lemma IsCapacityTwoWitness.source_surviving_preconnected
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) :
-    IsPreconnected (survivingDomain γ t) := by
-  have he : F t '' upperHalfPlane = survivingDomain γ t := by
-    apply Subset.antisymm
-    · rintro _ ⟨w,hw,rfl⟩
-      exact h.2.2.2.2.2.2.1 t hw
-    · intro z hz
-      exact ⟨G t z,h.2.2.2.2.2.1 t hz,h.2.2.2.2.2.2.2.1 t z hz⟩
-  rw [← he]
-  exact (convex_halfSpace_im_gt 0).isPreconnected.image _ (h.2.2.2.2.1 t).continuousOn
-
-lemma source_stopped_live_transfer
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) (β : ℝ≥0 → ℂ)
-    (hid : ∀ s ∈ Icc 0 t, β s ∈ upperHalfPlane →
-      γ (T+s) = F T (β s + (U T : ℂ)))
-    (z : ℂ) (hz : z ∈ survivingDomain γ (T+t)) :
-    G T z - (U T : ℂ) ∈ survivingDomain β t := by
-  let f : ℂ → ℂ := fun z => G T z - (U T : ℂ)
-  let D := survivingDomain γ (T+t)
-  have hsub : D ⊆ survivingDomain γ T :=
-    survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le))
-  have hmap : f '' D ⊆ upperHalfPlane \ β '' Icc 0 t := by
-    rintro w ⟨v,hv,rfl⟩
-    have hvT := hsub hv
-    have him : f v ∈ upperHalfPlane := by
-      simpa only [f,upperHalfPlane,mem_ofPred_eq,sub_im,ofReal_im,sub_zero]
-        using h.2.2.2.2.2.1 T hvT
-    refine ⟨him,?_⟩
-    rintro ⟨s,hs,he⟩
-    have hβ : β s ∈ upperHalfPlane := he ▸ him
-    have hv' : γ (T+s) = v := by
-      rw [hid s hs hβ,he]
-      simp only [f,sub_add_cancel,h.2.2.2.2.2.2.2.1 T v hvT]
-    exact hv.1.2 ⟨T+s,⟨bot_le,by simpa only [add_comm] using add_le_add_left hs.2 T⟩,hv'⟩
-  have hconn : IsPreconnected (f '' D) :=
-    (h.source_surviving_preconnected (T+t)).image _
-      (((h.2.2.2.1 T).continuousOn.mono hsub).sub continuousOn_const)
-  have hmem : f z ∈ f '' D := ⟨z,hz,rfl⟩
-  refine ⟨hmap hmem,?_⟩
-  intro hb
-  exact h.source_later_image_unbounded T t
-    (hb.subset (hconn.subset_connectedComponentIn hmem hmap))
-
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory Complex TopologicalSpace
-open scoped Topology NNReal
-namespace SLEExactGauge
-noncomputable section
-local instance : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-def SourcePathInteriorDense (β : C(ℝ≥0,ℂ)) : Prop :=
-  ∀ s t : ℝ≥0, s < t → ∃ u ∈ Ioo s t, 0 < (β u).im
-
-lemma source_path_interior_dense_measurable :
-    MeasurableSet {β : C(ℝ≥0,ℂ) | SourcePathInteriorDense β} := by
-  have he : {β : C(ℝ≥0,ℂ) | SourcePathInteriorDense β} =
-      ⋂ n : ℕ, ⋂ m : ℕ, {β | denseSeq ℝ≥0 n < denseSeq ℝ≥0 m →
-        ∃ u ∈ Ioo (denseSeq ℝ≥0 n) (denseSeq ℝ≥0 m), 0 < (β u).im} := by
-    ext β
-    simp only [mem_iInter,mem_ofPred_eq]
-    constructor
-    · exact fun h _ _ => h _ _
-    · intro h s t hst
-      obtain ⟨a,ha⟩ := (denseRange_denseSeq ℝ≥0).exists_mem_open isOpen_Ioo (nonempty_Ioo.mpr hst)
-      obtain ⟨b,hb⟩ := (denseRange_denseSeq ℝ≥0).exists_mem_open isOpen_Ioo (nonempty_Ioo.mpr ha.2)
-      obtain ⟨u,hu,hui⟩ := h a b hb.1
-      exact ⟨u,⟨ha.1.trans hu.1,hu.2.trans hb.2⟩,hui⟩
-  rw [he]
-  apply MeasurableSet.iInter
-  intro n
-  apply MeasurableSet.iInter
-  intro m
-  by_cases hnm : denseSeq ℝ≥0 n < denseSeq ℝ≥0 m
-  · simp only [hnm,true_implies,ofPred_exists]
-    apply IsOpen.measurableSet
-    apply isOpen_iUnion
-    intro u
-    by_cases hu : u ∈ Ioo (denseSeq ℝ≥0 n) (denseSeq ℝ≥0 m)
-    · simpa only [hu,true_and,Function.comp_apply] using isOpen_lt continuous_const (Complex.continuous_im.comp (continuous_eval_const u))
-    · simp [hu]
-  · simp [hnm]
-end
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma IsCapacityTwoWitness.radius_le_of_domain_subset
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (s t : ℝ≥0) (hsub : survivingDomain γ s ⊆ survivingDomain γ t)
-    (z : ℂ) (hz : z ∈ survivingDomain γ s) :
-    fullRadius G s z ≤ fullRadius G t z := by
-  have hγ := h.1
-  have ht : z ∈ survivingDomain γ t := hsub hz
-  have hgs := h.derivative_norm_positive hU s z hz
-  have hgt := h.derivative_norm_positive hU t z ht
-  have hys := h.imag_positive s z hz
-  rcases h with ⟨_,h0,hupper,hG,hF,hGm,hFm,hinv,hinv',hinit,hode,hnorm,htip⟩
-  have hfs := ((hF s).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hGm s hz))).hasDerivAt
-  have hgsd := ((hG s).differentiableAt ((isOpen_survivingDomain hγ s).mem_nhds hz)).hasDerivAt
-  have hgtd := ((hG t).differentiableAt ((isOpen_survivingDomain hγ t).mem_nhds ht)).hasDerivAt
-  have hinvder : deriv (F s) (G s z)*deriv (G s) z = 1 := by
-    have he : (fun w => F s (G s w)) =ᶠ[𝓝 z] (fun w => w) := by
-      filter_upwards [(isOpen_survivingDomain hγ s).mem_nhds hz] with w hw
-      exact hinv s w hw
-    exact ((hfs.comp z hgsd).congr_of_eventuallyEq he.symm).unique (hasDerivAt_id z)
-  have hnp : ‖deriv (F s) (G s z)‖*‖deriv (G s) z‖ = 1 := by
-    simpa only [norm_mul,norm_one] using congrArg norm hinvder
-  let f : ℂ → ℂ := fun w => G t (F s (diskToHalfPlane (G s z) w))
-  have hfd : DifferentiableOn ℂ f (Metric.ball 0 1) :=
-    (hG t).comp ((hF s).comp (diskToHalfPlane_differentiable _) (diskToHalfPlane_maps hys))
-      (fun w hw => hsub (hFm s (diskToHalfPlane_maps hys hw)))
-  have hfm : MapsTo f (Metric.ball 0 1) upperHalfPlane :=
-    fun w hw => hGm t (hsub (hFm s (diskToHalfPlane_maps hys hw)))
-  have hf0 : f 0 = G t z := by simp only [f,diskToHalfPlane_zero,hinv s z hz]
-  have hder : deriv f 0 = deriv (G t) z*deriv (F s) (G s z)*(G s z-star (G s z)) := by
-    have hfs' : HasDerivAt (F s) (deriv (F s) (G s z)) (diskToHalfPlane (G s z) 0) := by
-      simpa only [diskToHalfPlane_zero] using hfs
-    have hgt' : HasDerivAt (G t) (deriv (G t) z) (F s (diskToHalfPlane (G s z) 0)) := by
-      simpa only [diskToHalfPlane_zero,hinv s z hz] using hgtd
-    have hh := hgt'.comp 0 (hfs'.comp 0 (diskToHalfPlane_deriv_zero (G s z)))
-    simpa only [f,Function.comp_def,mul_assoc] using hh.deriv
-  have hn : ‖deriv f 0‖ = ‖deriv (G t) z‖*fullRadius G s z := by
-    rw [hder,norm_mul,norm_mul,norm_sub_conj_of_im_pos hys,fullRadius]
-    have hi : ‖deriv (F s) (G s z)‖ = 1/‖deriv (G s) z‖ := (eq_div_iff hgs.ne').mpr hnp
-    rw [hi]
-    ring
-  have hpos : 0 < ‖deriv f 0‖ := by
-    rw [hn,fullRadius]
-    positivity
-  have hbound := radius_of_halfplane_map (by norm_num : (0:ℝ) < 1) hfd hfm
-    (norm_pos_iff.mp hpos)
-  rw [hf0,hn] at hbound
-  have hy : 0 < (G t z).im := hGm t ht
-  have hh := (le_div_iff₀ (show 0 < ‖deriv (G t) z‖*fullRadius G s z by rwa [← hn])).mp hbound
-  change _ ≤ 2*(G t z).im/‖deriv (G t) z‖
-  exact (le_div_iff₀ hgt).mpr (by nlinarith [hh])
-
-end SLEExactGauge
-namespace SLEExactGauge
- 
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.DiskHalfPlane
+import OAI.Probability.SLE.StoppedMeasures
+
+namespace OAI
+
+/-! Chart Measurability for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_killedChart_stopped_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)) (z : ℂ)
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) :
+    Measurable[hτ.measurableSpace]
+      (fun ω => if z ∈ survivingDomain (γ ω) (τ ω) then G ω (τ ω) z else 0) := by
+  exact measurable_stoppedValue
+    (source_killedChart_progressive hm hc κ γ G F hw z) hτ
+
+end
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory Complex TopologicalSpace Classical
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma source_measurable_inverse_chart
+    {Ω : Type*} [MeasurableSpace Ω]
+    (S : Ω → Set ℂ) (G F K : Ω → ℂ → ℂ) (H : Set ℂ)
+    (hS : ∀ ω, IsOpen (S ω)) (hH : IsOpen H)
+    (hG : ∀ ω, ContinuousOn (G ω) (S ω))
+    (hF : ∀ ω, ContinuousOn (F ω) H)
+    (hFH : ∀ ω, MapsTo (F ω) H (S ω))
+    (hGF : ∀ ω z, z ∈ H → G ω (F ω z) = z)
+    (hFG : ∀ ω z, z ∈ S ω → F ω (G ω z) = z)
+    (hK : ∀ z, Measurable (fun ω => K ω z))
+    (hKeq : ∀ ω z, K ω z = if z ∈ S ω then G ω z else 0)
+    (hGH : ∀ ω z, z ∈ S ω → 0 < (G ω z).im)
+    (w : Ω → ℂ) (hwm : Measurable w) (hw : ∀ ω, w ω ∈ H) :
+    Measurable (fun ω => F ω (w ω)) := by
+  classical
+  let d := denseSeq ℂ
+  let v (k : ℕ) (ω : Ω) (n : ℕ) : Prop :=
+    0 < (K ω (d n)).im ∧ dist (K ω (d n)) (w ω) < 1/((k:ℝ)+1)
+  have hv (k : ℕ) (ω : Ω) : ∃ n, v k ω n := by
+    have hpos : 0 < 1/((k:ℝ)+1) := by positivity
+    have hx : F ω (w ω) ∈ S ω := hFH ω (hw ω)
+    have hct := (hG ω).continuousAt ((hS ω).mem_nhds hx)
+    have hn : {z | dist (G ω z) (w ω) < 1/((k:ℝ)+1)} ∈ 𝓝 (F ω (w ω)) := by
+      change G ω ⁻¹' Metric.ball (w ω) (1/((k:ℝ)+1)) ∈ 𝓝 (F ω (w ω))
+      apply hct.preimage_mem_nhds
+      rw [hGF ω _ (hw ω)]
+      exact Metric.ball_mem_nhds (w ω) hpos
+    obtain ⟨n,hn⟩ := (denseRange_denseSeq ℂ).mem_nhds (inter_mem ((hS ω).mem_nhds hx) hn)
+    refine ⟨n,?_⟩
+    dsimp [v,d]
+    rw [hKeq,ite_eq_left hn.1]
+    exact ⟨hGH ω _ hn.1,hn.2⟩
+  have hvm (k n : ℕ) : MeasurableSet {ω | v k ω n} :=
+    (measurableSet_lt measurable_const (Complex.measurable_im.comp (hK _))).inter
+      (measurableSet_lt ((hK _).dist hwm) measurable_const)
+  let j (k : ℕ) (ω : Ω) : ℕ := Nat.find (hv k ω)
+  have hj (k : ℕ) : Measurable (j k) := measurable_find (hv k) (hvm k)
+  have hjv (k : ℕ) (ω : Ω) : v k ω (j k ω) := Nat.find_spec (hv k ω)
+  have hjS (k : ℕ) (ω : Ω) : d (j k ω) ∈ S ω := by
+    by_contra hn
+    have hh := (hjv k ω).1
+    rw [hKeq,ite_eq_right hn] at hh
+    norm_num at hh
+  have hlim (ω : Ω) : Tendsto (fun k => K ω (d (j k ω))) atTop (𝓝 (w ω)) := by
+    apply Metric.tendsto_atTop.mpr
+    intro ε hε
+    obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) ε hε
+    refine ⟨N,fun k hk => (hjv k ω).2.trans ?_⟩
+    simpa only [Real.dist_eq,sub_zero,abs_of_pos (by positivity : 0 < 1/((k:ℝ)+1))]
+      using hN k hk
+  have hflim (ω : Ω) : Tendsto (fun k => d (j k ω)) atTop (𝓝 (F ω (w ω))) := by
+    have hh := ((hF ω).continuousAt (hH.mem_nhds (hw ω))).tendsto.comp (hlim ω)
+    convert hh using 1
+    ext k
+    simp only [Function.comp_apply,hKeq,ite_eq_left (hjS k ω),hFG ω _ (hjS k ω)]
+  exact measurable_of_tendsto_metrizable (fun k => (measurable_of_countable d).comp (hj k)) (tendsto_pi_nhds.mpr hflim)
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical
+open scoped Topology NNReal
+namespace SLEExactGauge
+lemma source_stopped_inverseChart_measurable
+    {Ω : Type*} [m : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω))
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) (w : Ω → ℂ)
+    (hwm : Measurable[hτ.measurableSpace] w) (hwH : ∀ ω, w ω ∈ upperHalfPlane) :
+    Measurable[hτ.measurableSpace] (fun ω => F ω (τ ω) (w ω)) := by
+  apply @source_measurable_inverse_chart Ω hτ.measurableSpace
+    (fun ω => survivingDomain (γ ω) (τ ω)) (fun ω => G ω (τ ω))
+    (fun ω => F ω (τ ω))
+    (fun ω z => if z ∈ survivingDomain (γ ω) (τ ω) then G ω (τ ω) z else 0)
+    upperHalfPlane
+    (fun ω => isOpen_survivingDomain (hw ω).1 _) isOpen_upperHalfPlane
+    (fun ω => ((hw ω).2.2.2.1 _).continuousOn)
+    (fun ω => ((hw ω).2.2.2.2.1 _).continuousOn)
+    (fun ω => (hw ω).2.2.2.2.2.2.1 _)
+    (fun ω => (hw ω).2.2.2.2.2.2.2.2.1 _)
+    (fun ω => (hw ω).2.2.2.2.2.2.2.1 _)
+    (fun z => source_killedChart_stopped_measurable hm hc κ γ G F hw z τ hτ)
+    (fun _ _ => rfl)
+    (fun ω z hz => (hw ω).imag_positive _ z hz) w hwm hwH
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
+open scoped Topology NNReal
+namespace SLEExactGauge
+noncomputable section
+local instance : MeasurableSpace C(upperHalfPlane,ℂ) := borel _
+local instance : BorelSpace C(upperHalfPlane,ℂ) := ⟨rfl⟩
+
+lemma source_halfplane_cmap_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (f : Ω → C(upperHalfPlane,ℂ))
+    (hf : ∀ z, Measurable (fun ω => f ω z)) : Measurable f := by
+  let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+  let : PolishSpace upperHalfPlane := isOpen_upperHalfPlane.polishSpace
+  let : Nonempty upperHalfPlane := ⟨⟨Complex.I,by simp [upperHalfPlane]⟩⟩
+  have he : MeasurableEmbedding
+      (fun F : C(upperHalfPlane,ℂ) => fun n => F (denseSeq upperHalfPlane n)) := by
+    apply Continuous.measurableEmbedding
+    · exact continuous_pi fun n => continuous_eval_const _
+    · intro F G heq
+      apply DFunLike.coe_injective
+      exact (denseRange_denseSeq upperHalfPlane).equalizer F.continuous G.continuous heq
+  exact he.measurable_comp_iff.mp (Measurable.of_eval (fun _ => hf _))
+
+def sourceStoppedInversePath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (τ : Ω → ℝ≥0) (ω : Ω) : C(upperHalfPlane,ℂ) :=
+  ⟨fun z => F ω (τ ω) (z + (Real.sqrt κ * B (τ ω) ω : ℝ)),
+    ContinuousOn.comp_continuous ((hw ω).2.2.2.2.1 _).continuousOn
+      (continuous_subtype_val.add continuous_const)
+      (fun z => by simpa [upperHalfPlane] using z.2)⟩
+
+lemma source_stopped_inversePath_measurable
+    {Ω : Type*} [m : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω))
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) :
+    Measurable[hτ.measurableSpace] (sourceStoppedInversePath B κ γ G F hw τ) := by
+  apply @source_halfplane_cmap_measurable Ω hτ.measurableSpace
+  intro z
+  have hB := measurable_stoppedValue
+    ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc) hτ
+  apply source_stopped_inverseChart_measurable B hm hc κ γ G F hw τ hτ
+    (fun ω => z+(Real.sqrt κ * B (τ ω) ω : ℝ))
+  · exact measurable_const.add (Complex.measurable_ofReal.comp (measurable_const.mul hB))
+  · intro ω
+    simpa [upperHalfPlane] using z.2
+end
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory Metric Bornology
+open scoped Topology NNReal
+namespace SLEExactGauge
+ 
+
+def square (R : ℝ) : Set ℂ := {z | |z.re| ≤ R ∧ |z.im| ≤ R}
+
+def upperExterior (R : ℝ) : Set ℂ := upperHalfPlane ∩ (square R)ᶜ
+
+def lowerExterior (R : ℝ) : Set ℂ := {z | z.im ≤ 0} ∩ (square R)ᶜ
+
+lemma square_isCompact (R : ℝ) : IsCompact (square R) := by
+  have hc : IsClosed (square R) :=
+    (isClosed_le Complex.continuous_re.abs continuous_const).inter
+      (isClosed_le Complex.continuous_im.abs continuous_const)
+  apply isCompact_of_isClosed_isBounded hc
+  apply isBounded_iff_forall_norm_le.mpr
+  refine ⟨2*R,fun z hz => ?_⟩
+  exact (Complex.norm_le_abs_re_add_abs_im z).trans (by linarith [hz.1,hz.2])
+
+lemma bounded_subset_square {s : Set ℂ} (hs : Bornology.IsBounded s) :
+    ∃ R > 0, s ⊆ square R := by
+  obtain ⟨C,hC⟩ := hs.exists_norm_le
+  refine ⟨max C 1,lt_of_lt_of_le zero_lt_one (le_max_right _ _),?_⟩
+  intro z hz
+  exact ⟨(Complex.abs_re_le_norm z).trans ((hC z hz).trans (le_max_left _ _)),
+    (Complex.abs_im_le_norm z).trans ((hC z hz).trans (le_max_left _ _))⟩
+
+lemma upperExterior_eq (R : ℝ) (hR : 0 ≤ R) : upperExterior R =
+    ({z : ℂ | z.re < -R} ∩ {z | 0 < z.im}) ∪ {z | R < z.im} ∪
+    ({z : ℂ | R < z.re} ∩ {z | 0 < z.im}) := by
+  ext z
+  simp only [upperExterior,upperHalfPlane,square,mem_inter_iff,mem_ofPred_eq,
+    mem_compl_iff,mem_union,abs_le]
+  constructor
+  · rintro ⟨hi,hs⟩
+    by_cases hl : z.re < -R
+    · exact Or.inl (Or.inl ⟨hl,hi⟩)
+    by_cases hr : R < z.re
+    · exact Or.inr ⟨hr,hi⟩
+    exact Or.inl (Or.inr (by by_contra hn; exact hs ⟨⟨by linarith,by linarith⟩,
+      ⟨by linarith,by linarith⟩⟩))
+  · rintro ((⟨hl,hi⟩ | hi) | ⟨hr,hi⟩)
+    · exact ⟨hi,fun hs => (not_lt_of_ge hs.1.1) hl⟩
+    · exact ⟨by linarith,fun hs => (not_lt_of_ge hs.2.2) hi⟩
+    · exact ⟨hi,fun hs => (not_lt_of_ge hs.1.2) hr⟩
+
+lemma upperExterior_preconnected (R : ℝ) (hR : 0 ≤ R) : IsPreconnected (upperExterior R) := by
+  rw [upperExterior_eq R hR]
+  have hL := ((convex_halfSpace_re_lt (-R)).inter (convex_halfSpace_im_gt 0)).isPreconnected
+  have hU := (convex_halfSpace_im_gt R).isPreconnected
+  have hV := ((convex_halfSpace_re_gt R).inter (convex_halfSpace_im_gt 0)).isPreconnected
+  have hLU : (({z : ℂ | z.re < -R} ∩ {z | 0 < z.im}) ∩ {z | R < z.im}).Nonempty := by
+    refine ⟨⟨-R-1,R+1⟩,?_,?_⟩ <;> simp only [mem_inter_iff,mem_ofPred_eq] <;> (try constructor) <;> linarith
+  have hUV : ((({z : ℂ | z.re < -R} ∩ {z | 0 < z.im}) ∪ {z | R < z.im}) ∩
+      ({z : ℂ | R < z.re} ∩ {z | 0 < z.im})).Nonempty := by
+    refine ⟨⟨R+1,R+1⟩,Or.inr ?_,?_,?_⟩ <;> simp only [mem_ofPred_eq] <;> linarith
+  exact (hL.union' hLU hU).union' hUV hV
+
+lemma upperExterior_unbounded (R : ℝ) (hR : 0 ≤ R) : ¬ Bornology.IsBounded (upperExterior R) := by
+  intro hb
+  obtain ⟨C,hC⟩ := hb.exists_norm_le
+  let y : ℝ := max C R + 1
+  have hyR : R < y := by dsimp [y]; linarith [le_max_right C R]
+  have hyC : C < y := by dsimp [y]; linarith [le_max_left C R]
+  have hy0 : 0 < y := by linarith
+  have hm : (⟨0,y⟩ : ℂ) ∈ upperExterior R := by
+    exact ⟨hy0,fun h => (not_le_of_gt hyR) ((le_abs_self y).trans h.2)⟩
+  have hb' := hC _ hm
+  have hn := Complex.abs_im_le_norm (⟨0,y⟩ : ℂ)
+  change |y| ≤ ‖(⟨0,y⟩ : ℂ)‖ at hn
+  rw [abs_of_pos hy0] at hn
+  linarith
+
+lemma upperExterior_subset_survivingDomain {γ : ℝ≥0 → ℂ} {T : ℝ≥0} {R : ℝ}
+    (hR : 0 ≤ R) (hγ : γ '' Icc 0 T ⊆ square R) :
+    upperExterior R ⊆ survivingDomain γ T := by
+  intro z hz
+  have hs : upperExterior R ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by
+    intro w hw
+    exact ⟨hw.1,fun ht => hw.2 (hγ ht)⟩
+  refine ⟨hs hz,?_⟩
+  intro hb
+  exact upperExterior_unbounded R hR (hb.subset
+    ((upperExterior_preconnected R hR).subset_connectedComponentIn hz hs))
+
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma source_live_compact_bridge (γ : C(ℝ≥0,ℂ)) (t : ℝ≥0) (z : ℂ)
+    (hz : z ∈ survivingDomain γ t) :
+    ∃ (R δ : ℝ) (K : Set ℂ), 0 < R ∧ 0 < δ ∧ IsCompact K ∧
+      IsPreconnected K ∧ Metric.closedBall z δ ⊆ K ∧
+      (⟨0,R+1⟩ : ℂ) ∈ K ∧ K ⊆ upperHalfPlane ∧
+      MapsTo γ (Icc 0 t) (Metric.ball 0 R \ K) := by
+  obtain ⟨M,hM⟩ := (isCompact_Icc.image γ.continuous).isBounded.exists_norm_le
+  let R := max M 0 + 1
+  have hR : 0 < R := by dsimp [R]; linarith [le_max_right M 0]
+  have hnorm (s : ℝ≥0) (hs : s ∈ Icc 0 t) : ‖γ s‖ < R := by
+    have hh := hM _ ⟨s,hs,rfl⟩
+    dsimp [R]; linarith [le_max_left M 0]
+  have htr : γ '' Icc 0 t ⊆ square R := by
+    rintro _ ⟨s,hs,rfl⟩
+    exact ⟨(abs_re_le_norm _).trans (hnorm s hs).le,(abs_im_le_norm _).trans (hnorm s hs).le⟩
+  let A : Set ℂ := upperHalfPlane \ γ '' Icc 0 t
+  let C := connectedComponentIn A z
+  have hA : IsOpen A := isOpen_upperHalfPlane.sdiff (isCompact_Icc.image γ.continuous).isClosed
+  have hzC : z ∈ C := mem_connectedComponentIn hz.1
+  have hCsub : C ⊆ A := connectedComponentIn_subset _ _
+  have hnot : ¬ C ⊆ square R := fun hh => hz.2 ((square_isCompact R).isBounded.subset hh)
+  obtain ⟨v,hv,hvR⟩ := Set.not_subset.mp hnot
+  have hvE : v ∈ upperExterior R := ⟨(hCsub hv).1,hvR⟩
+  have hEsub : upperExterior R ⊆ A := by
+    intro w hw
+    exact ⟨hw.1,fun ht => hw.2 (htr ht)⟩
+  have hEC : upperExterior R ⊆ C := by
+    have hc := (upperExterior_preconnected R hR.le).subset_connectedComponentIn hvE hEsub
+    rwa [← connectedComponentIn_eq hv] at hc
+  let p : ℂ := ⟨0,R+1⟩
+  have hpE : p ∈ upperExterior R := by
+    refine ⟨by change 0 < R+1; linarith,?_⟩
+    intro hh
+    have hhi : |R+1| ≤ R := hh.2
+    linarith [le_abs_self (R+1)]
+  have hpathc : IsPathConnected C := (hA.connectedComponentIn).isConnected_iff_isPathConnected.mp
+    ⟨⟨z,hzC⟩,isPreconnected_connectedComponentIn⟩
+  let η := (hpathc.joinedIn z hzC p (hEC hpE)).somePath
+  have hη : range η ⊆ A := by
+    rintro _ ⟨s,rfl⟩
+    exact hCsub ((hpathc.joinedIn z hzC p (hEC hpE)).somePath_mem s)
+  obtain ⟨ε,hε,hεsub⟩ := Metric.isOpen_iff.mp hA z hz.1
+  let δ := ε/2
+  have hδ : 0 < δ := by dsimp [δ]; positivity
+  have hball : Metric.closedBall z δ ⊆ A := by
+    apply subset_trans _ hεsub
+    exact Metric.closedBall_subset_ball (by dsimp [δ]; linarith)
+  let K := range η ∪ Metric.closedBall z δ
+  have hKc : IsCompact K := (isCompact_range η.continuous).union (isCompact_closedBall z δ)
+  have hKconn : IsPreconnected K :=
+    (isConnected_range η.continuous).2.union z η.source_mem_range (Metric.mem_closedBall_self hδ.le)
+      (convex_closedBall z δ).isPreconnected
+  have hKsub : K ⊆ A := union_subset hη hball
+  refine ⟨R,δ,K,hR,hδ,hKc,hKconn,subset_union_right,
+    Or.inl η.target_mem_range,fun w hw => (hKsub hw).1,?_⟩
+  intro s hs
+  refine ⟨by simpa only [Metric.mem_ball,dist_zero_right] using hnorm s hs,?_⟩
+  intro hh
+  exact (hKsub hh).2 ⟨s,hs,rfl⟩
+
+ 
+
+lemma source_live_of_bridge {γ : ℝ≥0 → ℂ} {t : ℝ≥0} {z : ℂ} {R : ℝ} {K : Set ℂ}
+    (hR : 0 ≤ R) (hK : IsPreconnected K) (hz : z ∈ K)
+    (hp : (⟨0,R+1⟩ : ℂ) ∈ K) (hKH : K ⊆ upperHalfPlane)
+    (hγ : MapsTo γ (Icc 0 t) (Metric.ball 0 R \ K)) :
+    z ∈ survivingDomain γ t := by
+  have hnorm (s : ℝ≥0) (hs : s ∈ Icc 0 t) : ‖γ s‖ < R := by
+    simpa only [Metric.mem_ball,dist_zero_right] using (hγ hs).1
+  have htr : γ '' Icc 0 t ⊆ square R := by
+    rintro _ ⟨s,hs,rfl⟩
+    exact ⟨(abs_re_le_norm _).trans (hnorm s hs).le,(abs_im_le_norm _).trans (hnorm s hs).le⟩
+  have hpE : (⟨0,R+1⟩ : ℂ) ∈ upperExterior R := by
+    refine ⟨by change 0 < R+1; linarith,?_⟩
+    intro hh
+    have hhi : |R+1| ≤ R := hh.2
+    linarith [le_abs_self (R+1)]
+  have hpS := upperExterior_subset_survivingDomain hR htr hpE
+  have hKa : K ⊆ upperHalfPlane \ γ '' Icc 0 t := by
+    intro w hw
+    refine ⟨hKH hw,?_⟩
+    rintro ⟨s,hs,rfl⟩
+    exact (hγ hs).2 hw
+  have hpC := hK.subset_connectedComponentIn hz hKa hp
+  refine ⟨hKa hz,?_⟩
+  rw [connectedComponentIn_eq hpC]
+  exact hpS.2
+
+noncomputable def sourceRescaledPath (p : C(ℝ≥0,ℂ) × ℝ≥0) : C(ℝ≥0,ℂ) :=
+  ⟨fun s => p.1 (p.2*s),p.1.continuous.comp (continuous_const.mul continuous_id)⟩
+
+lemma sourceRescaledPath_continuous : Continuous sourceRescaledPath := by
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  exact continuous_eval.comp
+    ((continuous_fst.fst).prodMk ((continuous_fst.snd).mul continuous_snd))
+
+lemma sourceRescaledPath_mapsTo {p : C(ℝ≥0,ℂ) × ℝ≥0} {A : Set ℂ} :
+    MapsTo (sourceRescaledPath p) (Icc 0 1) A ↔ MapsTo p.1 (Icc 0 p.2) A := by
+  have he : (fun s : ℝ≥0 => p.2*s) '' Icc 0 1 = Icc 0 p.2 := by
+    simpa using image_mul_left_Icc (show 0 ≤ p.2 from bot_le) (show (0 : ℝ≥0) ≤ 1 by norm_num)
+  rw [← he]
+  constructor
+  · intro h v hv
+    rcases hv with ⟨s,hs,rfl⟩
+    exact h hs
+  · intro h s hs
+    exact h ⟨s,hs,rfl⟩
+
+lemma source_survival_open :
+    IsOpen {p : (C(ℝ≥0,ℂ) × ℝ≥0) × ℂ | p.2 ∈ survivingDomain p.1.1 p.1.2} := by
+  apply isOpen_iff_forall_mem_open.mpr
+  rintro ⟨⟨γ,t⟩,z⟩ hz
+  obtain ⟨R,δ,K,hR,hδ,hKc,hKconn,hball,hp,hKH,hγ⟩ := source_live_compact_bridge γ t z hz
+  let W : Set (C(ℝ≥0,ℂ) × ℝ≥0) := {p | MapsTo p.1 (Icc 0 p.2) (Metric.ball 0 R \ K)}
+  have hWo : IsOpen W := by
+    have hh := (ContinuousMap.isOpen_setOfPred_mapsTo (K := Icc (0 : ℝ≥0) 1) (U := Metric.ball (0 : ℂ) R \ K) isCompact_Icc
+      (Metric.isOpen_ball.sdiff hKc.isClosed)).preimage sourceRescaledPath_continuous
+    convert hh using 1
+    ext p
+    exact sourceRescaledPath_mapsTo.symm
+  refine ⟨W ×ˢ Metric.ball z δ,?_,hWo.prod Metric.isOpen_ball,⟨hγ,Metric.mem_ball_self hδ⟩⟩
+  rintro ⟨⟨β,s⟩,w⟩ ⟨hβ,hw⟩
+  exact source_live_of_bridge hR.le hKconn (hball (Metric.ball_subset_closedBall hw)) hp hKH hβ
+
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+abbrev SourceDiskParameter := (C(upperHalfPlane,ℂ) × ℂ) × (ℂ × ℝ)
+
+lemma source_physical_disk_kernel_open :
+  IsOpen {p : SourceDiskParameter × C(ℝ≥0,ℂ) |
+    ∃ t : ℝ≥0, p.1.1.2 ∈ survivingDomain p.2 t ∧
+      ∃ h : p.2 t ∈ upperHalfPlane,
+        p.1.2.2 < ‖p.1.1.1 ⟨p.2 t,h⟩ - p.1.2.1‖} := by
+  let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+  have ht (t : ℝ≥0) :
+      IsOpen {p : SourceDiskParameter × C(ℝ≥0,ℂ) |
+        p.1.1.2 ∈ survivingDomain p.2 t ∧
+          ∃ h : p.2 t ∈ upperHalfPlane,
+            p.1.2.2 < ‖p.1.1.1 ⟨p.2 t,h⟩ - p.1.2.1‖} := by
+    let S : Set (SourceDiskParameter × C(ℝ≥0,ℂ)) := {p | p.2 t ∈ upperHalfPlane}
+    have hS : IsOpen S := isOpen_upperHalfPlane.preimage
+      ((continuous_eval_const t).comp continuous_snd)
+    let Z : S → upperHalfPlane := fun p => ⟨p.1.2 t,p.2⟩
+    have hZ : Continuous Z := Continuous.subtype_mk
+      ((continuous_eval_const t).comp (continuous_snd.comp continuous_subtype_val)) _
+    have hF : Continuous (fun p : S => p.1.1.1.1 (Z p)) :=
+      continuous_eval.comp
+        ((continuous_fst.fst.fst.comp continuous_subtype_val).prodMk hZ)
+    have hLive : IsOpen {p : S | p.1.1.1.2 ∈ survivingDomain p.1.2 t} :=
+      source_survival_open.preimage
+        (((continuous_snd.comp continuous_subtype_val).prodMk continuous_const).prodMk
+          (continuous_fst.fst.snd.comp continuous_subtype_val))
+    have hExit : IsOpen {p : S | p.1.1.2.2 < ‖p.1.1.1.1 (Z p) - p.1.1.2.1‖} :=
+      isOpen_lt (continuous_fst.snd.snd.comp continuous_subtype_val)
+        ((hF.sub (continuous_fst.snd.fst.comp continuous_subtype_val)).norm)
+    have hO := hS.isOpenMap_subtype_val _ (hLive.inter hExit)
+    convert hO using 1
+    ext p
+    constructor
+    · rintro ⟨hl,h,he⟩
+      exact ⟨⟨p,h⟩,⟨hl,he⟩,rfl⟩
+    · rintro ⟨q,⟨hl,he⟩,rfl⟩
+      exact ⟨hl,q.2,he⟩
+  simpa only [ofPred_exists] using isOpen_iUnion ht
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma source_high_vertical_survives {γ : ℝ≥0 → ℂ} {t : ℝ≥0} {R : ℝ}
+    (hR : 0 ≤ R) (hb : ∀ s ∈ Icc 0 t, ‖γ s‖ ≤ R) {y : ℝ} (hy : R < y) :
+    (⟨0,y⟩ : ℂ) ∈ survivingDomain γ t := by
+  let A : Set ℂ := {z | R < z.im}
+  have hA : A ⊆ upperHalfPlane \ γ '' Icc 0 t := by
+    intro z hz
+    refine ⟨lt_of_le_of_lt hR hz, ?_⟩
+    rintro ⟨s,hs,rfl⟩
+    have hh := (le_abs_self (γ s).im).trans ((Complex.abs_im_le_norm _).trans (hb s hs))
+    exact (not_lt_of_ge hh) hz
+  refine ⟨hA hy, ?_⟩
+  intro hbd
+  obtain ⟨C,hC⟩ := (hbd.subset ((convex_halfSpace_im_gt R).isPreconnected.subset_connectedComponentIn hy hA)).exists_norm_le
+  let v : ℝ := max C R + 1
+  have hv : R < v := by dsimp [v]; linarith [le_max_right C R]
+  have hvC : C < v := by dsimp [v]; linarith [le_max_left C R]
+  have hh := hC (⟨0,v⟩ : ℂ) hv
+  have hh' := Complex.abs_im_le_norm (⟨0,v⟩ : ℂ)
+  change |v| ≤ _ at hh'
+  linarith [le_abs_self v]
+
+lemma IsCapacityTwoWitness.source_later_image_unbounded
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) :
+    ¬ Bornology.IsBounded ((fun z => G T z - (U T : ℂ)) '' survivingDomain γ (T+t)) := by
+  intro hbd
+  obtain ⟨C,hC⟩ := hbd.exists_norm_le
+  obtain ⟨R,hR⟩ := (isCompact_Icc.image h.1).isBounded.exists_norm_le
+  have hb : ∀ s ∈ Icc 0 (T+t), ‖γ s‖ ≤ max R 0 := by
+    intro s hs
+    exact (hR _ ⟨s,hs,rfl⟩).trans (le_max_left _ _)
+  have hlim := h.2.2.2.2.2.2.2.2.2.2.2.1 T
+  have hnorm := hlim.norm
+  have he : ∀ᶠ z in cocompact ℂ, z ∈ survivingDomain γ T →
+      ‖z * (G T z - z)‖ < ‖(2 : ℂ)*(T : ℂ)‖ + 1 := by
+    apply eventually_inf_principal.mp
+    exact (tendsto_order.mp hnorm).2 _ (lt_add_one _)
+  obtain ⟨K,hK,hKe⟩ := mem_cocompact.mp he
+  obtain ⟨D,hD⟩ := hK.isBounded.exists_norm_le
+  let A : ℝ := ‖(2 : ℂ)*(T : ℂ)‖ + 1
+  let y : ℝ := max (max (max R 0) D) (max A (C + |U T|)) + 2
+  have hyR : max R 0 < y := by dsimp [y]; linarith [le_max_left (max R 0) D,le_max_left (max (max R 0) D) (max A (C+|U T|))]
+  have hyD : D < y := by dsimp [y]; linarith [le_max_right (max R 0) D,le_max_left (max (max R 0) D) (max A (C+|U T|))]
+  have hyA : A < y := by dsimp [y]; linarith [le_max_left A (C+|U T|),le_max_right (max (max R 0) D) (max A (C+|U T|))]
+  have hyC : C + |U T| + 1 < y := by dsimp [y]; linarith [le_max_right A (C+|U T|),le_max_right (max (max R 0) D) (max A (C+|U T|))]
+  have hy0 : 0 < y := lt_of_le_of_lt (le_max_right R 0) hyR
+  let z : ℂ := ⟨0,y⟩
+  have hnz : ‖z‖ = y := by simp [z,Complex.norm_def,Real.sqrt_mul_self hy0.le]
+  have hz : z ∈ survivingDomain γ (T+t) := source_high_vertical_survives (le_max_right R 0) hb hyR
+  have hzt : z ∈ survivingDomain γ T := survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le)) hz
+  have hzK : z ∈ Kᶜ := by intro h'; have := hD z h'; rw [hnz] at this; linarith
+  have hsmall := hKe hzK hzt
+  have hG : ‖G T z‖ ≤ C + |U T| := by
+    calc
+      ‖G T z‖ = ‖(G T z-(U T : ℂ))+(U T : ℂ)‖ := by rw [sub_add_cancel]
+      _ ≤ ‖G T z-(U T : ℂ)‖+‖(U T : ℂ)‖ := norm_add_le _ _
+      _ ≤ C + |U T| := by gcongr; exact hC _ ⟨z,hz,rfl⟩; simp
+  have htri := norm_sub_norm_le z (G T z)
+  rw [hnz,norm_sub_rev] at htri
+  rw [norm_mul,hnz] at hsmall
+  change y*‖G T z-z‖ < A at hsmall
+  have hpos : 0 < A := by dsimp [A]; positivity
+  nlinarith
+
+lemma IsCapacityTwoWitness.source_surviving_preconnected
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) :
+    IsPreconnected (survivingDomain γ t) := by
+  have he : F t '' upperHalfPlane = survivingDomain γ t := by
+    apply Subset.antisymm
+    · rintro _ ⟨w,hw,rfl⟩
+      exact h.2.2.2.2.2.2.1 t hw
+    · intro z hz
+      exact ⟨G t z,h.2.2.2.2.2.1 t hz,h.2.2.2.2.2.2.2.1 t z hz⟩
+  rw [← he]
+  exact (convex_halfSpace_im_gt 0).isPreconnected.image _ (h.2.2.2.2.1 t).continuousOn
+
+lemma source_stopped_live_transfer
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) (β : ℝ≥0 → ℂ)
+    (hid : ∀ s ∈ Icc 0 t, β s ∈ upperHalfPlane →
+      γ (T+s) = F T (β s + (U T : ℂ)))
+    (z : ℂ) (hz : z ∈ survivingDomain γ (T+t)) :
+    G T z - (U T : ℂ) ∈ survivingDomain β t := by
+  let f : ℂ → ℂ := fun z => G T z - (U T : ℂ)
+  let D := survivingDomain γ (T+t)
+  have hsub : D ⊆ survivingDomain γ T :=
+    survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le))
+  have hmap : f '' D ⊆ upperHalfPlane \ β '' Icc 0 t := by
+    rintro w ⟨v,hv,rfl⟩
+    have hvT := hsub hv
+    have him : f v ∈ upperHalfPlane := by
+      simpa only [f,upperHalfPlane,mem_ofPred_eq,sub_im,ofReal_im,sub_zero]
+        using h.2.2.2.2.2.1 T hvT
+    refine ⟨him,?_⟩
+    rintro ⟨s,hs,he⟩
+    have hβ : β s ∈ upperHalfPlane := he ▸ him
+    have hv' : γ (T+s) = v := by
+      rw [hid s hs hβ,he]
+      simp only [f,sub_add_cancel,h.2.2.2.2.2.2.2.1 T v hvT]
+    exact hv.1.2 ⟨T+s,⟨bot_le,by simpa only [add_comm] using add_le_add_left hs.2 T⟩,hv'⟩
+  have hconn : IsPreconnected (f '' D) :=
+    (h.source_surviving_preconnected (T+t)).image _
+      (((h.2.2.2.1 T).continuousOn.mono hsub).sub continuousOn_const)
+  have hmem : f z ∈ f '' D := ⟨z,hz,rfl⟩
+  refine ⟨hmap hmem,?_⟩
+  intro hb
+  exact h.source_later_image_unbounded T t
+    (hb.subset (hconn.subset_connectedComponentIn hmem hmap))
+
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory Complex TopologicalSpace
+open scoped Topology NNReal
+namespace SLEExactGauge
+noncomputable section
+local instance : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+def SourcePathInteriorDense (β : C(ℝ≥0,ℂ)) : Prop :=
+  ∀ s t : ℝ≥0, s < t → ∃ u ∈ Ioo s t, 0 < (β u).im
+
+lemma source_path_interior_dense_measurable :
+    MeasurableSet {β : C(ℝ≥0,ℂ) | SourcePathInteriorDense β} := by
+  have he : {β : C(ℝ≥0,ℂ) | SourcePathInteriorDense β} =
+      ⋂ n : ℕ, ⋂ m : ℕ, {β | denseSeq ℝ≥0 n < denseSeq ℝ≥0 m →
+        ∃ u ∈ Ioo (denseSeq ℝ≥0 n) (denseSeq ℝ≥0 m), 0 < (β u).im} := by
+    ext β
+    simp only [mem_iInter,mem_ofPred_eq]
+    constructor
+    · exact fun h _ _ => h _ _
+    · intro h s t hst
+      obtain ⟨a,ha⟩ := (denseRange_denseSeq ℝ≥0).exists_mem_open isOpen_Ioo (nonempty_Ioo.mpr hst)
+      obtain ⟨b,hb⟩ := (denseRange_denseSeq ℝ≥0).exists_mem_open isOpen_Ioo (nonempty_Ioo.mpr ha.2)
+      obtain ⟨u,hu,hui⟩ := h a b hb.1
+      exact ⟨u,⟨ha.1.trans hu.1,hu.2.trans hb.2⟩,hui⟩
+  rw [he]
+  apply MeasurableSet.iInter
+  intro n
+  apply MeasurableSet.iInter
+  intro m
+  by_cases hnm : denseSeq ℝ≥0 n < denseSeq ℝ≥0 m
+  · simp only [hnm,true_implies,ofPred_exists]
+    apply IsOpen.measurableSet
+    apply isOpen_iUnion
+    intro u
+    by_cases hu : u ∈ Ioo (denseSeq ℝ≥0 n) (denseSeq ℝ≥0 m)
+    · simpa only [hu,true_and,Function.comp_apply] using isOpen_lt continuous_const (Complex.continuous_im.comp (continuous_eval_const u))
+    · simp [hu]
+  · simp [hnm]
+end
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma IsCapacityTwoWitness.radius_le_of_domain_subset
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (s t : ℝ≥0) (hsub : survivingDomain γ s ⊆ survivingDomain γ t)
+    (z : ℂ) (hz : z ∈ survivingDomain γ s) :
+    fullRadius G s z ≤ fullRadius G t z := by
+  have hγ := h.1
+  have ht : z ∈ survivingDomain γ t := hsub hz
+  have hgs := h.derivative_norm_positive hU s z hz
+  have hgt := h.derivative_norm_positive hU t z ht
+  have hys := h.imag_positive s z hz
+  rcases h with ⟨_,h0,hupper,hG,hF,hGm,hFm,hinv,hinv',hinit,hode,hnorm,htip⟩
+  have hfs := ((hF s).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hGm s hz))).hasDerivAt
+  have hgsd := ((hG s).differentiableAt ((isOpen_survivingDomain hγ s).mem_nhds hz)).hasDerivAt
+  have hgtd := ((hG t).differentiableAt ((isOpen_survivingDomain hγ t).mem_nhds ht)).hasDerivAt
+  have hinvder : deriv (F s) (G s z)*deriv (G s) z = 1 := by
+    have he : (fun w => F s (G s w)) =ᶠ[𝓝 z] (fun w => w) := by
+      filter_upwards [(isOpen_survivingDomain hγ s).mem_nhds hz] with w hw
+      exact hinv s w hw
+    exact ((hfs.comp z hgsd).congr_of_eventuallyEq he.symm).unique (hasDerivAt_id z)
+  have hnp : ‖deriv (F s) (G s z)‖*‖deriv (G s) z‖ = 1 := by
+    simpa only [norm_mul,norm_one] using congrArg norm hinvder
+  let f : ℂ → ℂ := fun w => G t (F s (diskToHalfPlane (G s z) w))
+  have hfd : DifferentiableOn ℂ f (Metric.ball 0 1) :=
+    (hG t).comp ((hF s).comp (diskToHalfPlane_differentiable _) (diskToHalfPlane_maps hys))
+      (fun w hw => hsub (hFm s (diskToHalfPlane_maps hys hw)))
+  have hfm : MapsTo f (Metric.ball 0 1) upperHalfPlane :=
+    fun w hw => hGm t (hsub (hFm s (diskToHalfPlane_maps hys hw)))
+  have hf0 : f 0 = G t z := by simp only [f,diskToHalfPlane_zero,hinv s z hz]
+  have hder : deriv f 0 = deriv (G t) z*deriv (F s) (G s z)*(G s z-star (G s z)) := by
+    have hfs' : HasDerivAt (F s) (deriv (F s) (G s z)) (diskToHalfPlane (G s z) 0) := by
+      simpa only [diskToHalfPlane_zero] using hfs
+    have hgt' : HasDerivAt (G t) (deriv (G t) z) (F s (diskToHalfPlane (G s z) 0)) := by
+      simpa only [diskToHalfPlane_zero,hinv s z hz] using hgtd
+    have hh := hgt'.comp 0 (hfs'.comp 0 (diskToHalfPlane_deriv_zero (G s z)))
+    simpa only [f,Function.comp_def,mul_assoc] using hh.deriv
+  have hn : ‖deriv f 0‖ = ‖deriv (G t) z‖*fullRadius G s z := by
+    rw [hder,norm_mul,norm_mul,norm_sub_conj_of_im_pos hys,fullRadius]
+    have hi : ‖deriv (F s) (G s z)‖ = 1/‖deriv (G s) z‖ := (eq_div_iff hgs.ne').mpr hnp
+    rw [hi]
+    ring
+  have hpos : 0 < ‖deriv f 0‖ := by
+    rw [hn,fullRadius]
+    positivity
+  have hbound := radius_of_halfplane_map (by norm_num : (0:ℝ) < 1) hfd hfm
+    (norm_pos_iff.mp hpos)
+  rw [hf0,hn] at hbound
+  have hy : 0 < (G t z).im := hGm t ht
+  have hh := (le_div_iff₀ (show 0 < ‖deriv (G t) z‖*fullRadius G s z by rwa [← hn])).mp hbound
+  change _ ≤ 2*(G t z).im/‖deriv (G t) z‖
+  exact (le_div_iff₀ hgt).mpr (by nlinarith [hh])
+
+end SLEExactGauge
+namespace SLEExactGauge
+ 
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/ClippedCoordinates.lean b/lean/OAI/Probability/SLE/ClippedCoordinates.lean
index 5c0c570423f4e5791df8dbfa0a06ee371a210e02..d8a925b64238994290ad82802e6ff16382958165 100644
--- a/lean/OAI/Probability/SLE/ClippedCoordinates.lean
+++ b/lean/OAI/Probability/SLE/ClippedCoordinates.lean
@@ -1,905 +1,905 @@
-import Mathlib
-import OAI.Probability.SLE.OptionalStopping
-
-namespace OAI
-
-/-! Clipped Coordinates for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-lemma clippedCenter_norm_lower (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
-    Real.exp lo ≤ ‖clippedCenter lo hi hle u q‖ := by
-  calc
-    _ ≤ clippedHeight lo hi hle q := (clippedHeight_bounds lo hi hle q).1
-    _ = (clippedCenter lo hi hle u q).im := (clippedCenter_im lo hi hle u q).symm
-    _ ≤ ‖clippedCenter lo hi hle u q‖ := Complex.im_le_norm _
-
-lemma clippedCenter_inv_norm_bound (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
-    ‖(clippedCenter lo hi hle u q)⁻¹‖ ≤ (Real.exp lo)⁻¹ := by
-  rw [norm_inv]
-  exact inv_anti₀ (Real.exp_pos lo) (clippedCenter_norm_lower lo hi hle u q)
-
-lemma clippedCenter_dist (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q r : ℂ × ℝ) :
-    dist (clippedCenter lo hi hle u q) (clippedCenter lo hi hle u r) ≤
-      (1 + Real.exp hi) * dist q r := by
-  have he := (exp_clamp_lipschitz lo hi hle).dist_le_mul q.1.im r.1.im
-  have hY : |clippedHeight lo hi hle q - clippedHeight lo hi hle r| ≤
-      Real.exp hi * ‖q.1 - r.1‖ := by
-    change |Real.exp (projIcc lo hi hle q.1.im) - Real.exp (projIcc lo hi hle r.1.im)| ≤ _
-    change |Real.exp (projIcc lo hi hle q.1.im) - Real.exp (projIcc lo hi hle r.1.im)| ≤
-      Real.exp hi * |q.1.im - r.1.im| at he
-    exact he.trans
-      (mul_le_mul_of_nonneg_left (show |q.1.im - r.1.im| ≤ ‖q.1 - r.1‖ from
-        Complex.abs_im_le_norm (q.1 - r.1)) (Real.exp_pos hi).le)
-  have hX : |q.1.re - r.1.re| ≤ ‖q.1 - r.1‖ := Complex.abs_re_le_norm (q.1 - r.1)
-  have hsub : clippedCenter lo hi hle u q - clippedCenter lo hi hle u r =
-      ((q.1.re - r.1.re : ℝ) : ℂ) +
-        ((clippedHeight lo hi hle q - clippedHeight lo hi hle r : ℝ) : ℂ) * Complex.I := by
-    simp only [clippedCenter, Complex.ofReal_sub]
-    ring
-  rw [dist_eq_norm, hsub]
-  calc
-    _ ≤ |q.1.re - r.1.re| + |clippedHeight lo hi hle q - clippedHeight lo hi hle r| := by
-      simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, mul_one] using
-        norm_add_le ((q.1.re - r.1.re : ℝ) : ℂ)
-          (((clippedHeight lo hi hle q - clippedHeight lo hi hle r : ℝ) : ℂ) * Complex.I)
-    _ ≤ (1 + Real.exp hi) * ‖q.1 - r.1‖ := by linarith
-    _ ≤ (1 + Real.exp hi) * dist q r := by
-      apply mul_le_mul_of_nonneg_left _ (by positivity)
-      rw [dist_eq_norm]
-      exact norm_fst_le (q - r)
-
-lemma clippedCenter_inv_dist (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q r : ℂ × ℝ) :
-    dist (clippedCenter lo hi hle u q)⁻¹ (clippedCenter lo hi hle u r)⁻¹ ≤
-      ((Real.exp lo)⁻¹ ^ 2 * (1 + Real.exp hi)) * dist q r := by
-  let b := (Real.exp lo)⁻¹
-  have hb : 0 ≤ b := by dsimp [b]; positivity
-  rw [dist_eq_norm, inv_sub_inv' (clippedCenter_ne_zero lo hi hle u q)
-    (clippedCenter_ne_zero lo hi hle u r), norm_mul, norm_mul]
-  calc
-    _ ≤ b * ((1 + Real.exp hi) * dist q r) * b := by
-      gcongr
-      · exact clippedCenter_inv_norm_bound lo hi hle u q
-      · simpa only [← dist_eq_norm, dist_comm] using clippedCenter_dist lo hi hle u q r
-      · exact clippedCenter_inv_norm_bound lo hi hle u r
-    _ = _ := by dsimp [b]; ring
-
-noncomputable def logLoewnerVector (v : ℂ) : ℂ × ℝ :=
-  (((2 * v.re : ℝ) : ℂ) + ((-2 * Complex.normSq v : ℝ) : ℂ) * Complex.I,
-    (-2 * v ^ 2).re)
-
-lemma bounded_normSq_dist {v w : ℂ} {b : ℝ} (hv : ‖v‖ ≤ b) (hw : ‖w‖ ≤ b) :
-    |Complex.normSq v - Complex.normSq w| ≤ 2 * b * dist v w := by
-  rw [Complex.normSq_eq_norm_sq, Complex.normSq_eq_norm_sq,
-    show ‖v‖ ^ 2 - ‖w‖ ^ 2 = (‖v‖ - ‖w‖) * (‖v‖ + ‖w‖) by ring,
-    abs_mul, abs_of_nonneg (add_nonneg (norm_nonneg v) (norm_nonneg w))]
-  calc
-    _ ≤ ‖v - w‖ * (‖v‖ + ‖w‖) :=
-      mul_le_mul_of_nonneg_right (abs_norm_sub_norm_le v w)
-        (add_nonneg (norm_nonneg v) (norm_nonneg w))
-    _ ≤ ‖v - w‖ * (b + b) := by gcongr
-    _ = _ := by rw [dist_eq_norm]; ring
-
-lemma bounded_sq_dist {v w : ℂ} {b : ℝ} (hv : ‖v‖ ≤ b) (hw : ‖w‖ ≤ b) :
-    ‖v ^ 2 - w ^ 2‖ ≤ 2 * b * dist v w := by
-  rw [show v ^ 2 - w ^ 2 = (v - w) * (v + w) by ring, norm_mul]
-  calc
-    _ ≤ ‖v - w‖ * (‖v‖ + ‖w‖) := mul_le_mul_of_nonneg_left (norm_add_le v w) (norm_nonneg _)
-    _ ≤ ‖v - w‖ * (b + b) := by gcongr
-    _ = _ := by rw [dist_eq_norm]; ring
-
-lemma logLoewnerVector_dist {v w : ℂ} {b : ℝ} (hv : ‖v‖ ≤ b) (hw : ‖w‖ ≤ b) :
-    dist (logLoewnerVector v) (logLoewnerVector w) ≤ (2 + 4 * b) * dist v w := by
-  rw [dist_eq_norm]
-  apply norm_prod_le_iff.mpr
-  constructor
-  · have hr : ((logLoewnerVector v - logLoewnerVector w).1).re = 2 * (v.re - w.re) := by
-      simp [logLoewnerVector]
-      ring
-    have hi : ((logLoewnerVector v - logLoewnerVector w).1).im =
-        -2 * (Complex.normSq v - Complex.normSq w) := by
-      simp [logLoewnerVector]
-      ring
-    have hRe : |v.re - w.re| ≤ dist v w := by
-      simpa only [dist_eq_norm, Complex.sub_re] using Complex.abs_re_le_norm (v - w)
-    calc
-      _ ≤ |((logLoewnerVector v - logLoewnerVector w).1).re| +
-          |((logLoewnerVector v - logLoewnerVector w).1).im| := Complex.norm_le_abs_re_add_abs_im _
-      _ = 2 * |v.re - w.re| + 2 * |Complex.normSq v - Complex.normSq w| := by
-        rw [hr, hi, abs_mul, abs_mul]
-        norm_num
-      _ ≤ 2 * dist v w + 2 * (2 * b * dist v w) := by
-        gcongr
-        exact bounded_normSq_dist hv hw
-      _ = _ := by ring
-  · have he : (logLoewnerVector v - logLoewnerVector w).2 = (-2 * (v ^ 2 - w ^ 2)).re := by
-      simp only [logLoewnerVector, Prod.snd_sub, mul_sub, Complex.sub_re]
-    rw [he, Real.norm_eq_abs]
-    calc
-      _ ≤ ‖-2 * (v ^ 2 - w ^ 2)‖ := Complex.abs_re_le_norm _
-      _ = 2 * ‖v ^ 2 - w ^ 2‖ := by rw [norm_mul]; norm_num
-      _ ≤ 2 * (2 * b * dist v w) := by gcongr; exact bounded_sq_dist hv hw
-      _ ≤ (2 + 4 * b) * dist v w := by nlinarith [show 0 ≤ dist v w from dist_nonneg]
-
-lemma logLoewnerVector_norm {v : ℂ} {b : ℝ} (hb : 0 ≤ b) (hv : ‖v‖ ≤ b) :
-    ‖logLoewnerVector v‖ ≤ (2 + 4 * b) * b := by
-  have hh := logLoewnerVector_dist hv (show ‖(0 : ℂ)‖ ≤ b by simpa using hb)
-  have h0 : logLoewnerVector 0 = 0 := by simp [logLoewnerVector]
-  rw [h0, dist_zero_right, dist_zero_right] at hh
-  exact hh.trans (mul_le_mul_of_nonneg_left hv (by positivity))
-
-lemma clippedLogLoewnerField_bounds (lo hi : ℝ) (hle : lo ≤ hi) :
-    ∃ (L K : ℝ≥0),
-      (∀ u q, ‖clippedLogLoewnerField lo hi hle u q‖ ≤ L) ∧
-      (∀ u, LipschitzWith K (clippedLogLoewnerField lo hi hle u)) := by
-  let b : ℝ := (Real.exp lo)⁻¹
-  have hb : 0 ≤ b := by dsimp [b]; positivity
-  let L : ℝ≥0 := ⟨(2 + 4 * b) * b, by positivity⟩
-  let K : ℝ≥0 := ⟨(2 + 4 * b) * (b ^ 2 * (1 + Real.exp hi)), by positivity⟩
-  refine ⟨L, K, ?_, ?_⟩
-  · intro u q
-    exact logLoewnerVector_norm hb (clippedCenter_inv_norm_bound lo hi hle u q)
-  · intro u
-    apply LipschitzWith.of_dist_le_mul
-    intro q r
-    change dist (logLoewnerVector (clippedCenter lo hi hle u q)⁻¹)
-      (logLoewnerVector (clippedCenter lo hi hle u r)⁻¹) ≤ _
-    calc
-      _ ≤ (2 + 4 * b) * dist (clippedCenter lo hi hle u q)⁻¹ (clippedCenter lo hi hle u r)⁻¹ :=
-        logLoewnerVector_dist (clippedCenter_inv_norm_bound lo hi hle u q)
-          (clippedCenter_inv_norm_bound lo hi hle u r)
-      _ ≤ (2 + 4 * b) * ((b ^ 2 * (1 + Real.exp hi)) * dist q r) :=
-        mul_le_mul_of_nonneg_left (clippedCenter_inv_dist lo hi hle u q r) (by positivity)
-      _ = _ := by
-        change (2 + 4 * b) * (b ^ 2 * (1 + Real.exp hi) * dist q r) =
-          ((2 + 4 * b) * (b ^ 2 * (1 + Real.exp hi))) * dist q r
-        ring
-
-lemma continuous_clippedLogLoewnerField (lo hi : ℝ) (hle : lo ≤ hi) :
-    Continuous (fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) := by
-  have hZ : Continuous (fun p : ℝ × (ℂ × ℝ) => clippedCenter lo hi hle p.1 p.2) := by
-    dsimp only [clippedCenter, clippedHeight]
-    fun_prop
-  have hv : Continuous (fun p : ℝ × (ℂ × ℝ) => (clippedCenter lo hi hle p.1 p.2)⁻¹) :=
-    hZ.inv₀ (fun p => clippedCenter_ne_zero lo hi hle p.1 p.2)
-  have hF : Continuous logLoewnerVector := by fun_prop [logLoewnerVector]
-  exact hF.comp hv
-
- 
-
-theorem clipped_log_loewner_solution
-    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ → Ω → ℝ)
-    (hU : Measurable (fun q : ℝ × Ω => U q.1 q.2))
-    (hc : ∀ ω, Continuous (fun t : ℝ => U t ω))
-    (lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) (T : ℝ≥0) :
-    ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ × ℝ),
-      Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧
-      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
-      (∀ ω, Q 0 ω = q₀) ∧
-      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
-        HasDerivWithinAt (fun s => Q s ω)
-          (clippedLogLoewnerField lo hi hle (U t ω) (Q t ω))
-          (Icc 0 (T : ℝ)) t) := by
-  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) (q : ℂ × ℝ) := clippedLogLoewnerField lo hi hle (U t ω) q
-  have hF := continuous_clippedLogLoewnerField lo hi hle
-  have hm : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => V q.1.1 q.1.2 q.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U q.1.1 q.1.2, q.2)))
-    apply hF.measurable.comp
-    exact (hU.comp measurable_fst).prodMk measurable_snd
-  have hcont (ω : Ω) (q : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω q) := by
-    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun t : ℝ => (U t ω, q)))
-    exact hF.comp ((hc ω).prodMk continuous_const)
-  obtain ⟨Q, hQ⟩ := exists_measurable_bounded_ode V q₀ T L K hm hcont
-    (fun ω t => hlip (U t ω)) (fun ω t q => hbound (U t ω) q)
-  exact ⟨L, Q, hQ⟩
-
-theorem exists_measurable_bounded_ode_picard
-    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
-    (V : ℝ → Ω → E → E) (x₀ : E) (T L K : ℝ≥0)
-    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2))
-    (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x))
-    (hl : ∀ ω t, LipschitzWith K (V t ω))
-    (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) :
-    ∃ X : ℝ → Ω → E,
-      Measurable (fun q : ℝ × Ω => X q.1 q.2) ∧
-      (∀ ω, LipschitzWith L (fun t => X t ω)) ∧
-      (∀ ω, X 0 ω = x₀) ∧
-      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
-        HasDerivWithinAt (fun u => X u ω) (V t ω (X t ω)) (Icc 0 (T : ℝ)) t) ∧
-      (∀ t ω, Tendsto (fun n => picardApprox V x₀ n (projIcc 0 (T : ℝ) T.coe_nonneg t) ω) atTop (𝓝 (X t ω))) := by
-  let t₀ : Icc (0 : ℝ) T := ⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩
-  have hPL (ω : Ω) : IsPicardLindelof (fun t x => V t ω x) t₀ x₀ (L * T) 0 L K := by
-    refine ⟨fun t ht => (hl ω t).lipschitzOnWith, fun x hx => (hc ω x).continuousOn,
-      fun t ht x hx => hb ω t x, ?_⟩
-    simp only [t₀, sub_zero, sub_self, max_eq_left T.coe_nonneg, NNReal.coe_mul,
-      NNReal.coe_zero, le_refl]
-  have hx : x₀ ∈ Metric.closedBall x₀ (0 : ℝ) := Metric.mem_closedBall_self le_rfl
-  choose α hα using fun ω => ODE.FunSpace.exists_isFixedPt_next (hPL ω) hx
-  let X (t : ℝ) (ω : Ω) := (α ω).compProj t
-  let α₀ : ODE.FunSpace t₀ x₀ 0 L :=
-    ⟨fun _ => x₀, (LipschitzWith.const x₀).weaken (show (0 : ℝ≥0) ≤ L from bot_le), hx⟩
-  have hα₀ (t : Icc (0 : ℝ) T) : α₀ t = x₀ := rfl
-  have hconv (t : Icc (0 : ℝ) T) (ω : Ω) :
-      Tendsto (fun n => picardApprox V x₀ n t ω) atTop (𝓝 ((α ω) t)) := by
-    apply tendsto_iff_dist_tendsto_zero.mpr
-    apply squeeze_zero (fun _ => dist_nonneg)
-      (fun n => ?_) (by simpa only [zero_mul] using (FloorSemiring.tendsto_pow_div_factorial_atTop
-        ((K : ℝ) * |(t : ℝ)|)).mul_const (dist α₀ (α ω)))
-    have hh := ODE.FunSpace.dist_iterate_next_apply_le (hPL ω) hx α₀ (α ω) n t
-    rw [(hα ω).iterate n] at hh
-    simpa only [t₀, sub_zero, picardApprox_eq_iterate ω (hPL ω) α₀ hα₀ n t] using hh
-  refine ⟨X, ?_, ?_, ?_, ?_, ?_⟩
-  · apply measurable_of_tendsto_metrizable
-      (f := fun n q => picardApprox V x₀ n (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2)
-    · intro n
-      exact (picardApprox_measurable V x₀ hV n).comp
-        (((continuous_subtype_val.comp (continuous_projIcc (h := T.coe_nonneg))).measurable.comp measurable_fst).prodMk measurable_snd)
-    · apply tendsto_pi_nhds.mpr
-      intro q
-      exact hconv (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2
-  · intro ω
-    simpa only [mul_one, X, ODE.FunSpace.compProj, Function.comp_def] using (α ω).lipschitzWith.comp (LipschitzWith.projIcc T.coe_nonneg)
-  · intro ω
-    change (α ω).compProj 0 = x₀
-    rw [ODE.FunSpace.compProj_of_mem t₀.property]
-    exact ODE.FunSpace.apply_of_zero (α ω)
-  · intro ω t ht
-    apply ODE.hasDerivWithinAt_picard_Icc t₀.property (hPL ω).continuousOn_uncurry
-      (α ω).continuous_compProj.continuousOn
-      (fun s hs => (α ω).compProj_mem_closedBall (t := s) (hPL ω).mul_max_le) x₀ ht
-      |>.congr_of_mem _ ht
-    intro s hs
-    change (α ω).compProj s = ODE.picard (fun t x => V t ω x) t₀ x₀ (α ω).compProj s
-    rw [ODE.FunSpace.compProj_of_mem hs]
-    nth_rw 1 [← hα ω]
-    exact ODE.FunSpace.next_apply (hPL ω) hx (α ω)
-
-  · intro t ω
-    exact hconv (projIcc 0 (T : ℝ) T.coe_nonneg t) ω
-
-lemma picardApprox_congr_on
-    {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    {V W : ℝ → Ω → E → E} (x₀ : E) (t : ℝ)
-    (hVW : ∀ s ∈ Icc 0 t, ∀ ω x, V s ω x = W s ω x) (n : ℕ) :
-    ∀ s ∈ Icc 0 t, ∀ ω, picardApprox V x₀ n s ω = picardApprox W x₀ n s ω := by
-  induction n with
-  | zero => intro s hs ω; rfl
-  | succ n ih =>
-    intro s hs ω
-    dsimp only [picardApprox]
-    congr 1
-    apply setIntegral_congr_fun measurableSet_Ioc
-    intro u hu
-    have hut : u ∈ Icc 0 t := ⟨hu.1.le, hu.2.trans hs.2⟩
-    dsimp only
-    rw [ih u hut ω, hVW u hut]
-
-lemma measurable_driver_frozen
-    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ)
-    (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2))
-    (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω)) (t : ℝ≥0) :
-    @Measurable (ℝ × Ω) ℝ
-      ((borel ℝ).prod (Filtration.natural U
-        (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable) t))
-      (borel ℝ) (fun q => U (min q.1.toNNReal t) q.2) := by
-  let hm := fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable
-  have hp := (Filtration.stronglyAdapted_natural hm).isStronglyProgressive_of_continuous hc
-  have hpt := (hp t).measurable
-  let : MeasurableSpace Ω := Filtration.natural U hm t
-  have htime : Measurable (fun s : ℝ => (⟨min s.toNNReal t, by change min s.toNNReal t ≤ t; exact min_le_right s.toNNReal t⟩ : Iic t)) :=
-    ((continuous_real_toNNReal.min continuous_const).measurable).subtype_mk
-  exact hpt.comp ((htime.comp measurable_fst).prodMk measurable_snd)
-
- 
-
-theorem clipped_log_loewner_adapted_solution
-    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ)
-    (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2))
-    (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω))
-    (lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) (T : ℝ≥0) :
-    ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ × ℝ),
-      Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧
-      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
-      (∀ ω, Q 0 ω = q₀) ∧
-      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
-        HasDerivWithinAt (fun s => Q s ω)
-          (clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (Q t ω))
-          (Icc 0 (T : ℝ)) t) ∧
-      Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable))
-        (fun t : ℝ≥0 => Q t) := by
-  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) (q : ℂ × ℝ) := clippedLogLoewnerField lo hi hle (U t.toNNReal ω) q
-  have hF := continuous_clippedLogLoewnerField lo hi hle
-  have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) :=
-    hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hm : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => V q.1.1 q.1.2 q.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U q.1.1.toNNReal q.1.2, q.2)))
-    exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd)
-  have hcont (ω : Ω) (q : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω q) := by
-    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun t : ℝ => (U t.toNNReal ω, q)))
-    exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const)
-  obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ :=
-    exists_measurable_bounded_ode_picard V q₀ T L K hm hcont
-      (fun ω t => hlip (U t.toNNReal ω)) (fun ω t q => hbound (U t.toNNReal ω) q)
-  refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩
-  intro t
-  have hUfrozen := measurable_driver_frozen U hU hc t
-  let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t
-  let : MeasurableSpace Ω := m
-  let W (s : ℝ) (ω : Ω) (q : ℂ × ℝ) :=
-    clippedLogLoewnerField lo hi hle (U (min s.toNNReal t) ω) q
-  have hW : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => W q.1.1 q.1.2 q.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U (min q.1.1.toNNReal t) q.1.2, q.2)))
-    exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd)
-  let c : ℝ := projIcc 0 (T : ℝ) T.coe_nonneg (t : ℝ)
-  have hc0 : 0 ≤ c := (projIcc 0 (T : ℝ) T.coe_nonneg (t : ℝ)).property.1
-  have hct : c ≤ (t : ℝ) := by
-    dsimp [c]
-    simp only [projIcc]
-    exact max_le t.coe_nonneg (min_le_right _ _)
-  have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n c ω = picardApprox V q₀ n c ω := by
-    apply picardApprox_congr_on q₀ c _ n c ⟨hc0, le_rfl⟩ ω
-    intro s hs ω q
-    have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr (hs.2.trans hct)
-    simp only [W, V, min_eq_left hh]
-  apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n c ω)
-  · intro n
-    exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left
-  · apply tendsto_pi_nhds.mpr
-    intro ω
-    simpa only [heq] using hQconv t ω
-
- 
-
-theorem exists_measurable_global_bounded_ode_picard
-    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
-    (V : ℝ → Ω → E → E) (x₀ : E) (L K : ℝ≥0)
-    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2))
-    (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x))
-    (hl : ∀ ω t, LipschitzWith K (V t ω))
-    (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) :
-    ∃ X : ℝ → Ω → E,
-      Measurable (fun q : ℝ × Ω => X q.1 q.2) ∧
-      (∀ ω, LipschitzWith L (fun t => X t ω)) ∧
-      (∀ ω, X 0 ω = x₀) ∧
-      (∀ ω t, 0 ≤ t →
-        HasDerivWithinAt (fun u => X u ω) (V t ω (X t ω)) (Ici 0) t) ∧
-      (∀ t ω, Tendsto (fun n => picardApprox V x₀ n (max 0 t) ω) atTop (𝓝 (X t ω))) := by
-  classical
-  have hex (t : ℝ) (ω : Ω) : ∃ v : E,
-      Tendsto (fun n => picardApprox V x₀ n (max 0 t) ω) atTop (𝓝 v) := by
-    let T : ℝ≥0 := ⟨max 0 t, le_max_left _ _⟩
-    obtain ⟨X, _, _, _, _, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ T L K hV hc hl hb
-    refine ⟨X t ω, ?_⟩
-    have hc0 := hconv t ω
-    change Tendsto (fun n => picardApprox V x₀ n (max 0 (min (max 0 t) t)) ω) atTop (𝓝 (X t ω)) at hc0
-    simpa only [min_eq_right (le_max_right (0 : ℝ) t)] using hc0
-  let Q (t : ℝ) (ω : Ω) := limUnder atTop (fun n => picardApprox V x₀ n (max 0 t) ω)
-  have hlim (t : ℝ) (ω : Ω) :
-      Tendsto (fun n => picardApprox V x₀ n (max 0 t) ω) atTop (𝓝 (Q t ω)) :=
-    tendsto_nhds_limUnder (hex t ω)
-  have hmatch (T : ℝ≥0) (X : ℝ → Ω → E)
-      (hconv : ∀ t ω, Tendsto (fun n => picardApprox V x₀ n
-        (projIcc 0 (T : ℝ) T.coe_nonneg t) ω) atTop (𝓝 (X t ω)))
-      (t : ℝ) (ht : t ≤ T) (ω : Ω) : Q t ω = X t ω := by
-    apply tendsto_nhds_unique (hlim t ω)
-    simpa only [projIcc, min_eq_right ht] using hconv t ω
-  refine ⟨Q, ?_, ?_, ?_, ?_, hlim⟩
-  · apply measurable_of_tendsto_metrizable (f := fun n (q : ℝ × Ω) => picardApprox V x₀ n (max 0 q.1) q.2)
-    · intro n
-      exact (picardApprox_measurable V x₀ hV n).comp
-        ((measurable_const.max measurable_fst).prodMk measurable_snd)
-    · exact tendsto_pi_nhds.mpr fun q => hlim q.1 q.2
-  · intro ω
-    apply LipschitzWith.of_dist_le_mul
-    intro s t
-    let T : ℝ≥0 := ⟨max (max 0 s) t, (le_max_left 0 s).trans (le_max_left _ _)⟩
-    obtain ⟨X, _, hXl, _, _, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ T L K hV hc hl hb
-    rw [hmatch T X hconv s ((le_max_right 0 s).trans (le_max_left _ _)) ω,
-      hmatch T X hconv t (le_max_right _ _) ω]
-    exact (hXl ω).dist_le_mul s t
-  · intro ω
-    obtain ⟨X, _, _, hX0, _, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ 0 L K hV hc hl hb
-    rw [hmatch 0 X hconv 0 le_rfl ω, hX0 ω]
-  · intro ω t ht
-    let T : ℝ≥0 := ⟨t + 1, by linarith⟩
-    have hlt : t < (T : ℝ) := by change t < t + 1; linarith
-    obtain ⟨X, _, _, _, hXd, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ T L K hV hc hl hb
-    have heq : (fun s => Q s ω) =ᶠ[𝓝 t] (fun s => X s ω) := by
-      filter_upwards [Iic_mem_nhds hlt] with s hs
-      exact hmatch T X hconv s hs ω
-    have hd := hXd ω t ⟨ht, hlt.le⟩
-    have hsets : Icc 0 (T : ℝ) =ᶠ[𝓝 t] Ici 0 := by
-      filter_upwards [Iic_mem_nhds hlt] with s hs
-      change (0 ≤ s ∧ s ≤ (T : ℝ)) = (0 ≤ s)
-      exact propext ⟨And.left, fun h => ⟨h, hs⟩⟩
-    have hd' := hd.congr_set hsets
-    rw [← hmatch T X hconv t hlt.le ω] at hd'
-    exact hd'.congr_of_eventuallyEq (heq.filter_mono nhdsWithin_le_nhds) (heq.eq_of_nhds)
-
-theorem clipped_log_loewner_global_adapted_solution
-    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ)
-    (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2))
-    (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω))
-    (lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) :
-    ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ × ℝ),
-      Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧
-      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
-      (∀ ω, Q 0 ω = q₀) ∧
-      (∀ ω t, 0 ≤ t →
-        HasDerivWithinAt (fun s => Q s ω)
-          (clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (Q t ω))
-          (Ici 0) t) ∧
-      Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable))
-        (fun t : ℝ≥0 => Q t) := by
-  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) (q : ℂ × ℝ) := clippedLogLoewnerField lo hi hle (U t.toNNReal ω) q
-  have hF := continuous_clippedLogLoewnerField lo hi hle
-  have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) :=
-    hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hm : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => V q.1.1 q.1.2 q.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U q.1.1.toNNReal q.1.2, q.2)))
-    exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd)
-  have hcont (ω : Ω) (q : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω q) := by
-    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun t : ℝ => (U t.toNNReal ω, q)))
-    exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const)
-  obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ :=
-    exists_measurable_global_bounded_ode_picard V q₀ L K hm hcont
-      (fun ω t => hlip (U t.toNNReal ω)) (fun ω t q => hbound (U t.toNNReal ω) q)
-  refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩
-  intro t
-  have hUfrozen := measurable_driver_frozen U hU hc t
-  let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t
-  let : MeasurableSpace Ω := m
-  let W (s : ℝ) (ω : Ω) (q : ℂ × ℝ) :=
-    clippedLogLoewnerField lo hi hle (U (min s.toNNReal t) ω) q
-  have hW : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => W q.1.1 q.1.2 q.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U (min q.1.1.toNNReal t) q.1.2, q.2)))
-    exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd)
-  have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n (t : ℝ) ω = picardApprox V q₀ n (t : ℝ) ω := by
-    apply picardApprox_congr_on q₀ (t : ℝ) _ n (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩ ω
-    intro s hs ω q
-    have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hs.2
-    simp only [W, V, min_eq_left hh]
-  apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n (t : ℝ) ω)
-  · intro n
-    exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left
-  · apply tendsto_pi_nhds.mpr
-    intro ω
-    simpa only [heq, max_eq_right t.coe_nonneg] using hQconv t ω
-
-noncomputable def spatialPartial {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (φ : E × ℝ → ℝ) : ℕ → E × ℝ → ℝ
-  | 0 => φ
-  | n + 1 => fun p => fderiv ℝ (spatialPartial φ n) p (0, 1)
-
-lemma spatialPartial_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (n : ℕ) :
-    ContDiff ℝ (⊤ : ℕ∞) (spatialPartial φ n) := by
-  induction n with
-  | zero => exact hφ
-  | succ n ih =>
-    exact (ih.fderiv_right (by simp)).clm_apply contDiff_const
-
-lemma spatialPartial_compact {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (φ : E × ℝ → ℝ) (hφ : HasCompactSupport φ) (n : ℕ) :
-    HasCompactSupport (spatialPartial φ n) := by
-  induction n with
-  | zero => exact hφ
-  | succ n ih =>
-    exact (ih.fderiv ℝ).comp_left (g := fun D : (E × ℝ) →L[ℝ] ℝ => D (0, 1)) (by simp)
-
-lemma spatialPartial_eq_iteratedDeriv {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (n : ℕ) (q : E) (x : ℝ) :
-    spatialPartial φ n (q, x) = iteratedDeriv n (fun y => φ (q, y)) x := by
-  induction n generalizing x with
-  | zero => rfl
-  | succ n ih =>
-    have hd : HasDerivAt (fun y => spatialPartial φ n (q, y))
-        (fderiv ℝ (spatialPartial φ n) (q, x) (0, 1)) x := by
-      exact (((spatialPartial_smooth φ hφ n).differentiable (by simp)) (q, x)).hasFDerivAt.comp_hasDerivAt x
-        ((hasDerivAt_const x q).prodMk (hasDerivAt_id x))
-    simpa only [spatialPartial, iteratedDeriv_succ, ← funext ih] using hd.deriv.symm
-
-lemma compact_smooth_bounded_lipschitz
-    {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [NormedAddCommGroup F] [NormedSpace ℝ F]
-    (φ : E → F) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hφc : HasCompactSupport φ) :
-    ∃ C : ℝ≥0, (∀ p, ‖φ p‖ ≤ C) ∧ LipschitzWith C φ := by
-  obtain ⟨A, hA⟩ := hφ.continuous.bounded_above_of_compact_support hφc
-  obtain ⟨D, hD⟩ := (hφ.continuous_fderiv (by simp)).bounded_above_of_compact_support (hφc.fderiv ℝ)
-  let C := Real.toNNReal (max A D)
-  have hAC : A ≤ (C : ℝ) := (le_max_left _ _).trans (Real.le_coe_toNNReal _)
-  have hDC : D ≤ (C : ℝ) := (le_max_right _ _).trans (Real.le_coe_toNNReal _)
-  refine ⟨C, fun p => (hA p).trans hAC, ?_⟩
-  apply lipschitzWith_of_nnnorm_fderiv_le (hφ.differentiable (by simp))
-  intro p
-  exact_mod_cast (hD p).trans hDC
-
- 
-
-theorem compact_state_smooth_brownian_field
-    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (Q V : ℝ → Ω → E) (L : ℝ≥0)
-    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => Q t))
-    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
-    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
-    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
-    (hQd : ∀ t ω, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) (V t ω) (Ici 0) t)
-    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hφc : HasCompactSupport φ) :
-    ∃ K : ℝ≥0, SmoothTimeBrownianField B hm K
-      (fun t ω x => φ (Q t ω, x))
-      (fun t ω x => fderiv ℝ φ (Q t ω, x) (V t ω, 0)) := by
-
-  classical
-  have hr (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial φ j)
-    (spatialPartial_smooth φ hφ j) (spatialPartial_compact φ hφc j)
-  choose C hCb hCl using hr
-  let C₀ : ℝ≥0 := ∑ j ∈ Finset.range 4, C j
-  have hC (j : ℕ) (hj : j ≤ 3) : C j ≤ C₀ :=
-    Finset.single_le_sum (fun i _ => (zero_le : (0 : ℝ≥0) ≤ C i)) (Finset.mem_range.mpr (by omega))
-  have hCb' (j : ℕ) (hj : j ≤ 3) (p : E × ℝ) : |spatialPartial φ j p| ≤ C₀ := by
-    have hh := (hCb j p).trans (show (C j : ℝ) ≤ C₀ by exact_mod_cast hC j hj)
-    simpa only [Real.norm_eq_abs] using hh
-  have hC2 : LipschitzWith C₀ (spatialPartial φ 2) := (hCl 2).weaken (hC 2 (by decide))
-  obtain ⟨D, hDb, hDl⟩ := compact_smooth_bounded_lipschitz (fderiv ℝ φ)
-    (hφ.fderiv_right (by simp)) (hφc.fderiv ℝ)
-  let K : ℝ≥0 := (C₀ + D) * (L + 1)
-  have hKC : (C₀ : ℝ) ≤ (K : ℝ) := by
-    change (C₀ : ℝ) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
-    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
-  have hKDL : (D : ℝ) * L ≤ (K : ℝ) := by
-    change (D : ℝ) * L ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
-    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
-  have hKCL : (C₀ : ℝ) * ((L : ℝ) + 1) ≤ (K : ℝ) := by
-    change (C₀ : ℝ) * ((L : ℝ) + 1) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
-    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
-  have hQn : Measurable (fun p : ℝ≥0 × Ω => Q p.1 p.2) :=
-    hQm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hVn : Measurable (fun p : ℝ≥0 × Ω => V p.1 p.2) :=
-    hVm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
-  have happ : Continuous (fun p : (E × ℝ) × (E × ℝ) => fderiv ℝ φ p.1 p.2) :=
-    ((hφ.continuous_fderiv (by simp)).comp continuous_fst).clm_apply continuous_snd
-  have hameas : Measurable (fun p : (ℝ≥0 × Ω) × ℝ => fderiv ℝ φ (Q p.1.1 p.1.2, p.2) (V p.1.1 p.1.2, 0)) :=
-    happ.measurable.comp (((hQn.comp measurable_fst).prodMk measurable_snd).prodMk
-      ((hVn.comp measurable_fst).prodMk measurable_const))
-  have habound (t : ℝ≥0) (ω : Ω) (x : ℝ) :
-      |fderiv ℝ φ (Q t ω, x) (V t ω, 0)| ≤ K := by
-    calc
-      _ = ‖fderiv ℝ φ (Q t ω, x) (V t ω, 0)‖ := (Real.norm_eq_abs _).symm
-      _ ≤ ‖fderiv ℝ φ (Q t ω, x)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
-      _ = ‖fderiv ℝ φ (Q t ω, x)‖ * ‖V t ω‖ := by simp
-      _ ≤ (D : ℝ) * L := mul_le_mul (hDb _) (hVb t ω) (norm_nonneg _) D.coe_nonneg
-      _ ≤ K := hKDL
-  have haLip (t : ℝ≥0) (ω : Ω) (x y : ℝ) :
-      |fderiv ℝ φ (Q t ω, x) (V t ω, 0) - fderiv ℝ φ (Q t ω, y) (V t ω, 0)| ≤
-        (K : ℝ) * |x - y| := by
-    have hd : ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ ≤ (D : ℝ) * |x - y| := by
-      simpa [dist_eq_norm, Prod.norm_def, Real.norm_eq_abs] using hDl.dist_le_mul (Q t ω, x) (Q t ω, y)
-    calc
-      _ = ‖(fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)) (V t ω, 0)‖ := by
-        rw [sub_apply, Real.norm_eq_abs]
-      _ ≤ ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
-      _ = ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖V t ω‖ := by simp
-      _ ≤ ((D : ℝ) * |x - y|) * L := mul_le_mul hd (hVb t ω) (norm_nonneg _) (by positivity)
-      _ = ((D : ℝ) * L) * |x - y| := by ring
-      _ ≤ (K : ℝ) * |x - y| := mul_le_mul_of_nonneg_right hKDL (abs_nonneg _)
-  refine ⟨K, ⟨?_, ?_, ?_, hameas, ?_, habound, haLip, ?_, ?_⟩⟩
-  · intro t ω
-    have hc : ContDiff ℝ (⊤ : ℕ∞) (fun x : ℝ => φ (Q t ω, x)) :=
-      hφ.comp (contDiff_const.prodMk contDiff_id)
-    exact hc.of_le (WithTop.coe_le_coe.mpr (show (3 : ℕ∞) ≤ ⊤ from le_top))
-  · intro t j hj
-    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
-    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
-      (((hQa t).comp measurable_fst).prodMk measurable_snd)
-  · intro j hj
-    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
-    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
-      ((hQn.comp measurable_fst).prodMk measurable_snd)
-  · intro t ω x j hj
-    rw [← spatialPartial_eq_iteratedDeriv φ hφ]
-    exact (hCb' j hj _).trans hKC
-  · intro s t ω x y
-    rw [← spatialPartial_eq_iteratedDeriv φ hφ, ← spatialPartial_eq_iteratedDeriv φ hφ]
-    have hQdist := (hQl ω).dist_le_mul (t : ℝ) (s : ℝ)
-    rw [Real.dist_eq] at hQdist
-    calc
-      _ = dist (spatialPartial φ 2 (Q t ω, x)) (spatialPartial φ 2 (Q s ω, y)) := (Real.dist_eq _ _).symm
-      _ ≤ (C₀ : ℝ) * dist (Q t ω, x) (Q s ω, y) := hC2.dist_le_mul _ _
-      _ ≤ (C₀ : ℝ) * (dist (Q t ω) (Q s ω) + |x - y|) := by
-        apply mul_le_mul_of_nonneg_left _ C₀.coe_nonneg
-        rw [Prod.dist_eq, Real.dist_eq]
-        exact max_le (le_add_of_nonneg_right (abs_nonneg _)) (le_add_of_nonneg_left dist_nonneg)
-      _ ≤ (C₀ : ℝ) * ((L : ℝ) * |(t : ℝ) - s| + |x - y|) :=
-        mul_le_mul_of_nonneg_left (show dist (Q (t : ℝ) ω) (Q (s : ℝ) ω) + |x - y| ≤
-          (L : ℝ) * |(t : ℝ) - s| + |x - y| by linarith) C₀.coe_nonneg
-      _ ≤ ((C₀ : ℝ) * ((L : ℝ) + 1)) * (|(t : ℝ) - s| + |x - y|) := by
-        nlinarith [mul_nonneg C₀.coe_nonneg (abs_nonneg ((t : ℝ) - s)),
-          mul_nonneg (mul_nonneg C₀.coe_nonneg L.coe_nonneg) (abs_nonneg (x - y))]
-      _ ≤ (K : ℝ) * (|(t : ℝ) - s| + |x - y|) :=
-        mul_le_mul_of_nonneg_right hKCL (by positivity)
-  · intro s t hst ω x
-    have hst' : (s : ℝ) ≤ (t : ℝ) := by exact_mod_cast hst
-    symm
-    apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le (f := fun u : ℝ => φ (Q u ω, x))
-      (f' := fun u : ℝ => fderiv ℝ φ (Q u.toNNReal ω, x) (V u.toNNReal ω, 0)) hst'
-    · exact (hφ.continuous.comp ((hQl ω).continuous.prodMk continuous_const)).continuousOn
-    · intro u hu
-      have hu0 : 0 < u := lt_of_le_of_lt s.coe_nonneg hu.1
-      have hd := (hQd u ω hu0.le).hasDerivAt (Ici_mem_nhds hu0)
-      have hpair := hd.prodMk (hasDerivAt_const u x)
-      have hc := ((hφ.differentiable (by simp)) (Q u ω, x)).hasFDerivAt.comp_hasDerivAt u hpair
-      simpa only [Function.comp_def, Real.coe_toNNReal _ hu0.le] using hc
-    · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hst']
-      have hmA : Measurable (fun u : ℝ => fderiv ℝ φ (Q u.toNNReal ω, x) (V u.toNNReal ω, 0)) :=
-        hameas.comp ((continuous_real_toNNReal.measurable.prodMk measurable_const).prodMk measurable_const)
-      apply Integrable.of_bound hmA.aestronglyMeasurable (K : ℝ)
-      exact Eventually.of_forall fun u => by simpa only [Real.norm_eq_abs] using habound u.toNNReal ω x
-
-noncomputable def dimension (κ : ℝ) : ℝ := 1 + κ / 8
-noncomputable def codimension (κ : ℝ) : ℝ := ((2 : ℕ) : ℝ) - dimension κ
-noncomputable def greenExponent (κ : ℝ) : ℝ := 8 / κ - 1
-
- 
-
-noncomputable def greenLogNormSq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  (q.1.re - Real.sqrt κ * x) ^ 2 + Real.exp (2 * q.1.im)
-noncomputable def greenLogDensity (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  Real.exp (-codimension κ * Real.log 2 + codimension κ * q.2 +
-    (greenExponent κ - codimension κ) * q.1.im -
-      (greenExponent κ / 2) * Real.log (greenLogNormSq κ q x))
-noncomputable def physicalLogLoewnerField (u : ℝ) (q : ℂ × ℝ) : ℂ × ℝ :=
-  let Z := ((q.1.re - u : ℝ) : ℂ) + (Real.exp q.1.im : ℂ) * Complex.I
-  let v := Z⁻¹
-  (((2 * v.re : ℝ) : ℂ) + ((-2 * Complex.normSq v : ℝ) : ℂ) * Complex.I,
-    (-2 * v ^ 2).re)
-
-lemma greenLogNormSq_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    0 < greenLogNormSq κ q x :=
-  add_pos_of_nonneg_of_pos (sq_nonneg _) (Real.exp_pos _)
-
-lemma greenLogDensity_contDiff (κ : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) := by
-  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
-    Complex.reCLM.contDiff.comp contDiff_fst.fst
-  have him : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
-    Complex.imCLM.contDiff.comp contDiff_fst.fst
-  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) := by
-    exact ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add
-      ((contDiff_const.mul him).exp)
-  have hlog := hN.log (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
-  unfold greenLogDensity
-  apply ContDiff.exp
-  exact (((contDiff_const.add (contDiff_const.mul contDiff_fst.snd)).add
-    (contDiff_const.mul him)).sub (contDiff_const.mul hlog))
-
-lemma greenLogDensity_spatial_derivative (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    HasDerivAt (greenLogDensity κ q)
-      (greenLogDensity κ q x * (greenExponent κ * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
-        greenLogNormSq κ q x)) x := by
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
-    simp
-  have hlog := hN.log (greenLogNormSq_pos κ q x).ne'
-  have hd := ((hasDerivAt_const x (-codimension κ * Real.log 2 + codimension κ * q.2 +
-    (greenExponent κ - codimension κ) * q.1.im)).sub
-      (hlog.const_mul (greenExponent κ / 2))).exp
-  convert hd using 1 <;> try rfl
-  dsimp only [greenLogDensity, Pi.sub_apply, Pi.mul_apply, Function.comp_def]
-  ring
-
-lemma greenLogDensity_spatial_second (κ : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
-    iteratedDeriv 2 (greenLogDensity κ q) x =
-      greenLogDensity κ q x * κ * greenExponent κ *
-        ((greenExponent κ + 1) * (q.1.re - Real.sqrt κ * x) ^ 2 - Real.exp (2 * q.1.im)) /
-        (greenLogNormSq κ q x) ^ 2 := by
-  have hd : deriv (greenLogDensity κ q) = fun x =>
-      greenLogDensity κ q x * (greenExponent κ * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
-        greenLogNormSq κ q x) := funext fun x => (greenLogDensity_spatial_derivative κ q x).deriv
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
-    simp
-  have hcoef := (hX.const_mul (greenExponent κ * Real.sqrt κ)).div hN
-    (greenLogNormSq_pos κ q x).ne'
-  have hsecond := (greenLogDensity_spatial_derivative κ q x).mul hcoef
-  rw [show (2 : ℕ) = 1 + 1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
-  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, id_eq] at hsecond
-  change HasDerivAt (fun y => greenLogDensity κ q y *
-    (greenExponent κ * Real.sqrt κ * (q.1.re - Real.sqrt κ * y) / greenLogNormSq κ q y)) _ x at hsecond
-  rw [hsecond.deriv]
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  dsimp only [greenLogNormSq]
-  generalize greenLogDensity κ q x = f
-  generalize greenExponent κ = a
-  generalize q.1.re - Real.sqrt κ * x = X
-  generalize Real.exp (2 * q.1.im) = Y
-  have hs := Real.sq_sqrt hκ
-  generalize Real.sqrt κ = r at *
-  rw [← hs]
-  field_simp
-  ring
-
-lemma greenLogDensity_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q) v =
-      greenLogDensity κ q x * (codimension κ * v.2 +
-        (greenExponent κ - codimension κ) * v.1.im - (greenExponent κ / 2) *
-          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
-            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x)) := by
-  have hzre : HasDerivAt (fun t : ℝ => (q + t • v).1.re) v.1.re 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
-  have hzim : HasDerivAt (fun t : ℝ => (q + t • v).1.im) v.1.im 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
-  have hj : HasDerivAt (fun t : ℝ => (q + t • v).2) v.2 0 := by
-    simpa only [Prod.snd_add, Prod.smul_snd, smul_eq_mul, id_eq, one_mul]
-      using ((hasDerivAt_id (0 : ℝ)).mul_const v.2).const_add q.2
-  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q + t • v) x)
-      (2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
-        2 * Real.exp (2 * q.1.im) * v.1.im) 0 := by
-    convert ((hzre.sub_const (Real.sqrt κ * x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
-    simp only [zero_smul, add_zero, Nat.cast_ofNat]
-    ring
-  have hlog := hN.log (by simpa only [zero_smul, add_zero] using (greenLogNormSq_pos κ q x).ne')
-  have hcalc := (((hj.const_mul (codimension κ)).const_add (-codimension κ * Real.log 2)).add
-    (hzim.const_mul (greenExponent κ - codimension κ))).sub
-      (hlog.const_mul (greenExponent κ / 2))
-  have hline : HasDerivAt (fun t : ℝ => greenLogDensity κ (q + t • v) x)
-      (greenLogDensity κ q x * (codimension κ * v.2 +
-        (greenExponent κ - codimension κ) * v.1.im - (greenExponent κ / 2) *
-          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
-            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x))) 0 := by
-    convert hcalc.exp using 1 <;> try rfl
-    simp only [Pi.sub_apply, Pi.add_apply, zero_smul, add_zero]
-    simp only [greenLogDensity]
-  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => greenLogDensity κ r x) :=
-    (greenLogDensity_contDiff κ).comp (contDiff_id.prodMk contDiff_const)
-  have haff : HasDerivAt (fun t : ℝ => q + t • v) v 0 := by
-    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
-  have hfd : HasFDerivAt (fun r : ℂ × ℝ => greenLogDensity κ r x)
-      (fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q) (q + (0 : ℝ) • v) := by
-    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
-  have hchain := hfd.comp_hasDerivAt 0 haff
-  simpa only [Function.comp_def] using hchain.unique hline
-
-lemma physicalLogLoewnerField_components (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    (physicalLogLoewnerField (Real.sqrt κ * x) q).1.re =
-      2 * (q.1.re - Real.sqrt κ * x) / greenLogNormSq κ q x ∧
-    (physicalLogLoewnerField (Real.sqrt κ * x) q).1.im =
-      -2 / greenLogNormSq κ q x ∧
-    (physicalLogLoewnerField (Real.sqrt κ * x) q).2 =
-      -2 * ((q.1.re - Real.sqrt κ * x) ^ 2 - Real.exp (2 * q.1.im)) /
-        (greenLogNormSq κ q x) ^ 2 := by
-  let Z : ℂ := ((q.1.re - Real.sqrt κ * x : ℝ) : ℂ) + (Real.exp q.1.im : ℂ) * Complex.I
-  have hre : Z.re = q.1.re - Real.sqrt κ * x := by simp [Z]
-  have him : Z.im = Real.exp q.1.im := by
-    simp only [Z, Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im,
-      Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add, add_zero]
-  have he : Real.exp q.1.im * Real.exp q.1.im = Real.exp (2 * q.1.im) := by
-    rw [← Real.exp_add]; congr 1; ring
-  have hn : Complex.normSq Z = greenLogNormSq κ q x := by
-    rw [Complex.normSq_apply, hre, him, he]
-    simp only [greenLogNormSq, pow_two]
-  change (((2 * Z⁻¹.re : ℝ) : ℂ) + ((-2 * Complex.normSq Z⁻¹ : ℝ) : ℂ) * Complex.I).re = _ ∧
-    (((2 * Z⁻¹.re : ℝ) : ℂ) + ((-2 * Complex.normSq Z⁻¹ : ℝ) : ℂ) * Complex.I).im = _ ∧
-    (-2 * Z⁻¹ ^ 2).re = _
-  constructor
-  · simp [Complex.inv_re, hre, hn, mul_div_assoc]
-  constructor
-  · simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re,
-      Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add, add_zero,
-      Complex.normSq_inv, hn, div_eq_mul_inv]
-  · simp only [pow_two, Complex.mul_re, Complex.mul_im, Complex.neg_re, Complex.neg_im,
-      Complex.re_ofNat, Complex.im_ofNat, neg_zero, zero_mul, sub_zero,
-      Complex.inv_re, Complex.inv_im, hre, him, hn]
-    have he2 : Real.exp q.1.im ^ 2 = Real.exp (2 * q.1.im) := by simpa only [pow_two] using he
-    rw [← he2]
-    field_simp
-
- 
-theorem green_physical_generator_zero (κ : ℝ) (hκ : 0 < κ) (_hκ8 : κ < 8)
-    (q : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q)
-      (physicalLogLoewnerField (Real.sqrt κ * x) q) +
-      (1 / 2 : ℝ) * iteratedDeriv 2 (greenLogDensity κ q) x = 0 := by
-  rw [greenLogDensity_directional, greenLogDensity_spatial_second κ hκ.le]
-  obtain ⟨hr, hi, hj⟩ := physicalLogLoewnerField_components κ q x
-  rw [hr, hi, hj]
-  have ha : κ * (greenExponent κ + 1) = 8 := by
-    simp only [greenExponent, sub_add_cancel]
-    field_simp
-  have hp : κ * greenExponent κ = 8 * codimension κ := by
-    dsimp only [greenExponent, codimension, dimension]
-    field_simp
-    ring
-  have hc1 : -4 * greenExponent κ + κ * greenExponent κ * (greenExponent κ + 1) / 2 = 0 := by
-    linear_combination (greenExponent κ / 2) * ha
-  have hc2 : 4 * codimension κ - κ * greenExponent κ / 2 = 0 := by
-    linarith
-  calc
-    _ = greenLogDensity κ q x *
-      ((-4 * greenExponent κ + κ * greenExponent κ * (greenExponent κ + 1) / 2) *
-        (q.1.re - Real.sqrt κ * x) ^ 2 +
-        (4 * codimension κ - κ * greenExponent κ / 2) * Real.exp (2 * q.1.im)) /
-      (greenLogNormSq κ q x) ^ 2 := by
-        have hn := (greenLogNormSq_pos κ q x).ne'
-        field_simp
-        dsimp only [greenLogNormSq]
-        ring
-    _ = 0 := by rw [hc1, hc2]; ring
-
-lemma greenLogDensity_pair_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (q, x) (v, 0) =
-      fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q v := by
-  have hd := ((greenLogDensity_contDiff κ).differentiable (by simp)) (q, x)
-  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
-  have hc := (hd.hasFDerivAt.comp q hp).fderiv
-  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) hc
-  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
-    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
-
- 
-
-lemma exists_compact_green_field (κ R : ℝ) (hR : 0 ≤ R) :
-    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
-      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = greenLogDensity κ q x) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R →
-        fderiv ℝ φ (q, x) = fderiv ℝ (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (q, x)) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
-        iteratedDeriv 2 (greenLogDensity κ q) x) := by
-  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
-  let φ (p : (ℂ × ℝ) × ℝ) : ℝ := b p * greenLogDensity κ p.1 p.2
-  have hnear (p : (ℂ × ℝ) × ℝ) (hp : ‖p‖ ≤ R) :
-      φ =ᶠ[𝓝 p] (fun r : (ℂ × ℝ) × ℝ => greenLogDensity κ r.1 r.2) := by
-    have hball : p ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
-      simp only [Metric.mem_ball, dist_zero_right]
-      change ‖p‖ < R + 1
-      linarith
-    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with r hr
-    simp only [φ, hr, Pi.one_apply, one_mul]
-  refine ⟨φ, b.contDiff.mul (greenLogDensity_contDiff κ), b.hasCompactSupport.mul_right,
-    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
-  · intro q x h
-    exact (hnear (q, x) h).fderiv_eq
-  · intro q x h
-    have he := (hnear (q, x) h).comp_tendsto
-      ((continuous_const.prodMk continuous_id).tendsto x)
-    exact he.iteratedDeriv_eq 2
-
-lemma clippedLogLoewnerField_eq_physical (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ)
-    (q : ℂ × ℝ) (hq : q.1.im ∈ Icc lo hi) :
-    clippedLogLoewnerField lo hi hle u q = physicalLogLoewnerField u q := by
-  simp only [clippedLogLoewnerField, clippedCenter, clippedHeight, projIcc_of_mem hle hq,
-    physicalLogLoewnerField]
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.OptionalStopping
+
+namespace OAI
+
+/-! Clipped Coordinates for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+lemma clippedCenter_norm_lower (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
+    Real.exp lo ≤ ‖clippedCenter lo hi hle u q‖ := by
+  calc
+    _ ≤ clippedHeight lo hi hle q := (clippedHeight_bounds lo hi hle q).1
+    _ = (clippedCenter lo hi hle u q).im := (clippedCenter_im lo hi hle u q).symm
+    _ ≤ ‖clippedCenter lo hi hle u q‖ := Complex.im_le_norm _
+
+lemma clippedCenter_inv_norm_bound (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
+    ‖(clippedCenter lo hi hle u q)⁻¹‖ ≤ (Real.exp lo)⁻¹ := by
+  rw [norm_inv]
+  exact inv_anti₀ (Real.exp_pos lo) (clippedCenter_norm_lower lo hi hle u q)
+
+lemma clippedCenter_dist (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q r : ℂ × ℝ) :
+    dist (clippedCenter lo hi hle u q) (clippedCenter lo hi hle u r) ≤
+      (1 + Real.exp hi) * dist q r := by
+  have he := (exp_clamp_lipschitz lo hi hle).dist_le_mul q.1.im r.1.im
+  have hY : |clippedHeight lo hi hle q - clippedHeight lo hi hle r| ≤
+      Real.exp hi * ‖q.1 - r.1‖ := by
+    change |Real.exp (projIcc lo hi hle q.1.im) - Real.exp (projIcc lo hi hle r.1.im)| ≤ _
+    change |Real.exp (projIcc lo hi hle q.1.im) - Real.exp (projIcc lo hi hle r.1.im)| ≤
+      Real.exp hi * |q.1.im - r.1.im| at he
+    exact he.trans
+      (mul_le_mul_of_nonneg_left (show |q.1.im - r.1.im| ≤ ‖q.1 - r.1‖ from
+        Complex.abs_im_le_norm (q.1 - r.1)) (Real.exp_pos hi).le)
+  have hX : |q.1.re - r.1.re| ≤ ‖q.1 - r.1‖ := Complex.abs_re_le_norm (q.1 - r.1)
+  have hsub : clippedCenter lo hi hle u q - clippedCenter lo hi hle u r =
+      ((q.1.re - r.1.re : ℝ) : ℂ) +
+        ((clippedHeight lo hi hle q - clippedHeight lo hi hle r : ℝ) : ℂ) * Complex.I := by
+    simp only [clippedCenter, Complex.ofReal_sub]
+    ring
+  rw [dist_eq_norm, hsub]
+  calc
+    _ ≤ |q.1.re - r.1.re| + |clippedHeight lo hi hle q - clippedHeight lo hi hle r| := by
+      simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, mul_one] using
+        norm_add_le ((q.1.re - r.1.re : ℝ) : ℂ)
+          (((clippedHeight lo hi hle q - clippedHeight lo hi hle r : ℝ) : ℂ) * Complex.I)
+    _ ≤ (1 + Real.exp hi) * ‖q.1 - r.1‖ := by linarith
+    _ ≤ (1 + Real.exp hi) * dist q r := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      rw [dist_eq_norm]
+      exact norm_fst_le (q - r)
+
+lemma clippedCenter_inv_dist (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q r : ℂ × ℝ) :
+    dist (clippedCenter lo hi hle u q)⁻¹ (clippedCenter lo hi hle u r)⁻¹ ≤
+      ((Real.exp lo)⁻¹ ^ 2 * (1 + Real.exp hi)) * dist q r := by
+  let b := (Real.exp lo)⁻¹
+  have hb : 0 ≤ b := by dsimp [b]; positivity
+  rw [dist_eq_norm, inv_sub_inv' (clippedCenter_ne_zero lo hi hle u q)
+    (clippedCenter_ne_zero lo hi hle u r), norm_mul, norm_mul]
+  calc
+    _ ≤ b * ((1 + Real.exp hi) * dist q r) * b := by
+      gcongr
+      · exact clippedCenter_inv_norm_bound lo hi hle u q
+      · simpa only [← dist_eq_norm, dist_comm] using clippedCenter_dist lo hi hle u q r
+      · exact clippedCenter_inv_norm_bound lo hi hle u r
+    _ = _ := by dsimp [b]; ring
+
+noncomputable def logLoewnerVector (v : ℂ) : ℂ × ℝ :=
+  (((2 * v.re : ℝ) : ℂ) + ((-2 * Complex.normSq v : ℝ) : ℂ) * Complex.I,
+    (-2 * v ^ 2).re)
+
+lemma bounded_normSq_dist {v w : ℂ} {b : ℝ} (hv : ‖v‖ ≤ b) (hw : ‖w‖ ≤ b) :
+    |Complex.normSq v - Complex.normSq w| ≤ 2 * b * dist v w := by
+  rw [Complex.normSq_eq_norm_sq, Complex.normSq_eq_norm_sq,
+    show ‖v‖ ^ 2 - ‖w‖ ^ 2 = (‖v‖ - ‖w‖) * (‖v‖ + ‖w‖) by ring,
+    abs_mul, abs_of_nonneg (add_nonneg (norm_nonneg v) (norm_nonneg w))]
+  calc
+    _ ≤ ‖v - w‖ * (‖v‖ + ‖w‖) :=
+      mul_le_mul_of_nonneg_right (abs_norm_sub_norm_le v w)
+        (add_nonneg (norm_nonneg v) (norm_nonneg w))
+    _ ≤ ‖v - w‖ * (b + b) := by gcongr
+    _ = _ := by rw [dist_eq_norm]; ring
+
+lemma bounded_sq_dist {v w : ℂ} {b : ℝ} (hv : ‖v‖ ≤ b) (hw : ‖w‖ ≤ b) :
+    ‖v ^ 2 - w ^ 2‖ ≤ 2 * b * dist v w := by
+  rw [show v ^ 2 - w ^ 2 = (v - w) * (v + w) by ring, norm_mul]
+  calc
+    _ ≤ ‖v - w‖ * (‖v‖ + ‖w‖) := mul_le_mul_of_nonneg_left (norm_add_le v w) (norm_nonneg _)
+    _ ≤ ‖v - w‖ * (b + b) := by gcongr
+    _ = _ := by rw [dist_eq_norm]; ring
+
+lemma logLoewnerVector_dist {v w : ℂ} {b : ℝ} (hv : ‖v‖ ≤ b) (hw : ‖w‖ ≤ b) :
+    dist (logLoewnerVector v) (logLoewnerVector w) ≤ (2 + 4 * b) * dist v w := by
+  rw [dist_eq_norm]
+  apply norm_prod_le_iff.mpr
+  constructor
+  · have hr : ((logLoewnerVector v - logLoewnerVector w).1).re = 2 * (v.re - w.re) := by
+      simp [logLoewnerVector]
+      ring
+    have hi : ((logLoewnerVector v - logLoewnerVector w).1).im =
+        -2 * (Complex.normSq v - Complex.normSq w) := by
+      simp [logLoewnerVector]
+      ring
+    have hRe : |v.re - w.re| ≤ dist v w := by
+      simpa only [dist_eq_norm, Complex.sub_re] using Complex.abs_re_le_norm (v - w)
+    calc
+      _ ≤ |((logLoewnerVector v - logLoewnerVector w).1).re| +
+          |((logLoewnerVector v - logLoewnerVector w).1).im| := Complex.norm_le_abs_re_add_abs_im _
+      _ = 2 * |v.re - w.re| + 2 * |Complex.normSq v - Complex.normSq w| := by
+        rw [hr, hi, abs_mul, abs_mul]
+        norm_num
+      _ ≤ 2 * dist v w + 2 * (2 * b * dist v w) := by
+        gcongr
+        exact bounded_normSq_dist hv hw
+      _ = _ := by ring
+  · have he : (logLoewnerVector v - logLoewnerVector w).2 = (-2 * (v ^ 2 - w ^ 2)).re := by
+      simp only [logLoewnerVector, Prod.snd_sub, mul_sub, Complex.sub_re]
+    rw [he, Real.norm_eq_abs]
+    calc
+      _ ≤ ‖-2 * (v ^ 2 - w ^ 2)‖ := Complex.abs_re_le_norm _
+      _ = 2 * ‖v ^ 2 - w ^ 2‖ := by rw [norm_mul]; norm_num
+      _ ≤ 2 * (2 * b * dist v w) := by gcongr; exact bounded_sq_dist hv hw
+      _ ≤ (2 + 4 * b) * dist v w := by nlinarith [show 0 ≤ dist v w from dist_nonneg]
+
+lemma logLoewnerVector_norm {v : ℂ} {b : ℝ} (hb : 0 ≤ b) (hv : ‖v‖ ≤ b) :
+    ‖logLoewnerVector v‖ ≤ (2 + 4 * b) * b := by
+  have hh := logLoewnerVector_dist hv (show ‖(0 : ℂ)‖ ≤ b by simpa using hb)
+  have h0 : logLoewnerVector 0 = 0 := by simp [logLoewnerVector]
+  rw [h0, dist_zero_right, dist_zero_right] at hh
+  exact hh.trans (mul_le_mul_of_nonneg_left hv (by positivity))
+
+lemma clippedLogLoewnerField_bounds (lo hi : ℝ) (hle : lo ≤ hi) :
+    ∃ (L K : ℝ≥0),
+      (∀ u q, ‖clippedLogLoewnerField lo hi hle u q‖ ≤ L) ∧
+      (∀ u, LipschitzWith K (clippedLogLoewnerField lo hi hle u)) := by
+  let b : ℝ := (Real.exp lo)⁻¹
+  have hb : 0 ≤ b := by dsimp [b]; positivity
+  let L : ℝ≥0 := ⟨(2 + 4 * b) * b, by positivity⟩
+  let K : ℝ≥0 := ⟨(2 + 4 * b) * (b ^ 2 * (1 + Real.exp hi)), by positivity⟩
+  refine ⟨L, K, ?_, ?_⟩
+  · intro u q
+    exact logLoewnerVector_norm hb (clippedCenter_inv_norm_bound lo hi hle u q)
+  · intro u
+    apply LipschitzWith.of_dist_le_mul
+    intro q r
+    change dist (logLoewnerVector (clippedCenter lo hi hle u q)⁻¹)
+      (logLoewnerVector (clippedCenter lo hi hle u r)⁻¹) ≤ _
+    calc
+      _ ≤ (2 + 4 * b) * dist (clippedCenter lo hi hle u q)⁻¹ (clippedCenter lo hi hle u r)⁻¹ :=
+        logLoewnerVector_dist (clippedCenter_inv_norm_bound lo hi hle u q)
+          (clippedCenter_inv_norm_bound lo hi hle u r)
+      _ ≤ (2 + 4 * b) * ((b ^ 2 * (1 + Real.exp hi)) * dist q r) :=
+        mul_le_mul_of_nonneg_left (clippedCenter_inv_dist lo hi hle u q r) (by positivity)
+      _ = _ := by
+        change (2 + 4 * b) * (b ^ 2 * (1 + Real.exp hi) * dist q r) =
+          ((2 + 4 * b) * (b ^ 2 * (1 + Real.exp hi))) * dist q r
+        ring
+
+lemma continuous_clippedLogLoewnerField (lo hi : ℝ) (hle : lo ≤ hi) :
+    Continuous (fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) := by
+  have hZ : Continuous (fun p : ℝ × (ℂ × ℝ) => clippedCenter lo hi hle p.1 p.2) := by
+    dsimp only [clippedCenter, clippedHeight]
+    fun_prop
+  have hv : Continuous (fun p : ℝ × (ℂ × ℝ) => (clippedCenter lo hi hle p.1 p.2)⁻¹) :=
+    hZ.inv₀ (fun p => clippedCenter_ne_zero lo hi hle p.1 p.2)
+  have hF : Continuous logLoewnerVector := by fun_prop [logLoewnerVector]
+  exact hF.comp hv
+
+ 
+
+theorem clipped_log_loewner_solution
+    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ → Ω → ℝ)
+    (hU : Measurable (fun q : ℝ × Ω => U q.1 q.2))
+    (hc : ∀ ω, Continuous (fun t : ℝ => U t ω))
+    (lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) (T : ℝ≥0) :
+    ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ × ℝ),
+      Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧
+      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
+      (∀ ω, Q 0 ω = q₀) ∧
+      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
+        HasDerivWithinAt (fun s => Q s ω)
+          (clippedLogLoewnerField lo hi hle (U t ω) (Q t ω))
+          (Icc 0 (T : ℝ)) t) := by
+  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) (q : ℂ × ℝ) := clippedLogLoewnerField lo hi hle (U t ω) q
+  have hF := continuous_clippedLogLoewnerField lo hi hle
+  have hm : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => V q.1.1 q.1.2 q.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U q.1.1 q.1.2, q.2)))
+    apply hF.measurable.comp
+    exact (hU.comp measurable_fst).prodMk measurable_snd
+  have hcont (ω : Ω) (q : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω q) := by
+    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun t : ℝ => (U t ω, q)))
+    exact hF.comp ((hc ω).prodMk continuous_const)
+  obtain ⟨Q, hQ⟩ := exists_measurable_bounded_ode V q₀ T L K hm hcont
+    (fun ω t => hlip (U t ω)) (fun ω t q => hbound (U t ω) q)
+  exact ⟨L, Q, hQ⟩
+
+theorem exists_measurable_bounded_ode_picard
+    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
+    (V : ℝ → Ω → E → E) (x₀ : E) (T L K : ℝ≥0)
+    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2))
+    (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x))
+    (hl : ∀ ω t, LipschitzWith K (V t ω))
+    (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) :
+    ∃ X : ℝ → Ω → E,
+      Measurable (fun q : ℝ × Ω => X q.1 q.2) ∧
+      (∀ ω, LipschitzWith L (fun t => X t ω)) ∧
+      (∀ ω, X 0 ω = x₀) ∧
+      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
+        HasDerivWithinAt (fun u => X u ω) (V t ω (X t ω)) (Icc 0 (T : ℝ)) t) ∧
+      (∀ t ω, Tendsto (fun n => picardApprox V x₀ n (projIcc 0 (T : ℝ) T.coe_nonneg t) ω) atTop (𝓝 (X t ω))) := by
+  let t₀ : Icc (0 : ℝ) T := ⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩
+  have hPL (ω : Ω) : IsPicardLindelof (fun t x => V t ω x) t₀ x₀ (L * T) 0 L K := by
+    refine ⟨fun t ht => (hl ω t).lipschitzOnWith, fun x hx => (hc ω x).continuousOn,
+      fun t ht x hx => hb ω t x, ?_⟩
+    simp only [t₀, sub_zero, sub_self, max_eq_left T.coe_nonneg, NNReal.coe_mul,
+      NNReal.coe_zero, le_refl]
+  have hx : x₀ ∈ Metric.closedBall x₀ (0 : ℝ) := Metric.mem_closedBall_self le_rfl
+  choose α hα using fun ω => ODE.FunSpace.exists_isFixedPt_next (hPL ω) hx
+  let X (t : ℝ) (ω : Ω) := (α ω).compProj t
+  let α₀ : ODE.FunSpace t₀ x₀ 0 L :=
+    ⟨fun _ => x₀, (LipschitzWith.const x₀).weaken (show (0 : ℝ≥0) ≤ L from bot_le), hx⟩
+  have hα₀ (t : Icc (0 : ℝ) T) : α₀ t = x₀ := rfl
+  have hconv (t : Icc (0 : ℝ) T) (ω : Ω) :
+      Tendsto (fun n => picardApprox V x₀ n t ω) atTop (𝓝 ((α ω) t)) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun _ => dist_nonneg)
+      (fun n => ?_) (by simpa only [zero_mul] using (FloorSemiring.tendsto_pow_div_factorial_atTop
+        ((K : ℝ) * |(t : ℝ)|)).mul_const (dist α₀ (α ω)))
+    have hh := ODE.FunSpace.dist_iterate_next_apply_le (hPL ω) hx α₀ (α ω) n t
+    rw [(hα ω).iterate n] at hh
+    simpa only [t₀, sub_zero, picardApprox_eq_iterate ω (hPL ω) α₀ hα₀ n t] using hh
+  refine ⟨X, ?_, ?_, ?_, ?_, ?_⟩
+  · apply measurable_of_tendsto_metrizable
+      (f := fun n q => picardApprox V x₀ n (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2)
+    · intro n
+      exact (picardApprox_measurable V x₀ hV n).comp
+        (((continuous_subtype_val.comp (continuous_projIcc (h := T.coe_nonneg))).measurable.comp measurable_fst).prodMk measurable_snd)
+    · apply tendsto_pi_nhds.mpr
+      intro q
+      exact hconv (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2
+  · intro ω
+    simpa only [mul_one, X, ODE.FunSpace.compProj, Function.comp_def] using (α ω).lipschitzWith.comp (LipschitzWith.projIcc T.coe_nonneg)
+  · intro ω
+    change (α ω).compProj 0 = x₀
+    rw [ODE.FunSpace.compProj_of_mem t₀.property]
+    exact ODE.FunSpace.apply_of_zero (α ω)
+  · intro ω t ht
+    apply ODE.hasDerivWithinAt_picard_Icc t₀.property (hPL ω).continuousOn_uncurry
+      (α ω).continuous_compProj.continuousOn
+      (fun s hs => (α ω).compProj_mem_closedBall (t := s) (hPL ω).mul_max_le) x₀ ht
+      |>.congr_of_mem _ ht
+    intro s hs
+    change (α ω).compProj s = ODE.picard (fun t x => V t ω x) t₀ x₀ (α ω).compProj s
+    rw [ODE.FunSpace.compProj_of_mem hs]
+    nth_rw 1 [← hα ω]
+    exact ODE.FunSpace.next_apply (hPL ω) hx (α ω)
+
+  · intro t ω
+    exact hconv (projIcc 0 (T : ℝ) T.coe_nonneg t) ω
+
+lemma picardApprox_congr_on
+    {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {V W : ℝ → Ω → E → E} (x₀ : E) (t : ℝ)
+    (hVW : ∀ s ∈ Icc 0 t, ∀ ω x, V s ω x = W s ω x) (n : ℕ) :
+    ∀ s ∈ Icc 0 t, ∀ ω, picardApprox V x₀ n s ω = picardApprox W x₀ n s ω := by
+  induction n with
+  | zero => intro s hs ω; rfl
+  | succ n ih =>
+    intro s hs ω
+    dsimp only [picardApprox]
+    congr 1
+    apply setIntegral_congr_fun measurableSet_Ioc
+    intro u hu
+    have hut : u ∈ Icc 0 t := ⟨hu.1.le, hu.2.trans hs.2⟩
+    dsimp only
+    rw [ih u hut ω, hVW u hut]
+
+lemma measurable_driver_frozen
+    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ)
+    (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2))
+    (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω)) (t : ℝ≥0) :
+    @Measurable (ℝ × Ω) ℝ
+      ((borel ℝ).prod (Filtration.natural U
+        (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable) t))
+      (borel ℝ) (fun q => U (min q.1.toNNReal t) q.2) := by
+  let hm := fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable
+  have hp := (Filtration.stronglyAdapted_natural hm).isStronglyProgressive_of_continuous hc
+  have hpt := (hp t).measurable
+  let : MeasurableSpace Ω := Filtration.natural U hm t
+  have htime : Measurable (fun s : ℝ => (⟨min s.toNNReal t, by change min s.toNNReal t ≤ t; exact min_le_right s.toNNReal t⟩ : Iic t)) :=
+    ((continuous_real_toNNReal.min continuous_const).measurable).subtype_mk
+  exact hpt.comp ((htime.comp measurable_fst).prodMk measurable_snd)
+
+ 
+
+theorem clipped_log_loewner_adapted_solution
+    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ)
+    (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2))
+    (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω))
+    (lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) (T : ℝ≥0) :
+    ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ × ℝ),
+      Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧
+      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
+      (∀ ω, Q 0 ω = q₀) ∧
+      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
+        HasDerivWithinAt (fun s => Q s ω)
+          (clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (Q t ω))
+          (Icc 0 (T : ℝ)) t) ∧
+      Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable))
+        (fun t : ℝ≥0 => Q t) := by
+  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) (q : ℂ × ℝ) := clippedLogLoewnerField lo hi hle (U t.toNNReal ω) q
+  have hF := continuous_clippedLogLoewnerField lo hi hle
+  have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) :=
+    hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hm : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => V q.1.1 q.1.2 q.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U q.1.1.toNNReal q.1.2, q.2)))
+    exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd)
+  have hcont (ω : Ω) (q : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω q) := by
+    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun t : ℝ => (U t.toNNReal ω, q)))
+    exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const)
+  obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ :=
+    exists_measurable_bounded_ode_picard V q₀ T L K hm hcont
+      (fun ω t => hlip (U t.toNNReal ω)) (fun ω t q => hbound (U t.toNNReal ω) q)
+  refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩
+  intro t
+  have hUfrozen := measurable_driver_frozen U hU hc t
+  let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t
+  let : MeasurableSpace Ω := m
+  let W (s : ℝ) (ω : Ω) (q : ℂ × ℝ) :=
+    clippedLogLoewnerField lo hi hle (U (min s.toNNReal t) ω) q
+  have hW : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => W q.1.1 q.1.2 q.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U (min q.1.1.toNNReal t) q.1.2, q.2)))
+    exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd)
+  let c : ℝ := projIcc 0 (T : ℝ) T.coe_nonneg (t : ℝ)
+  have hc0 : 0 ≤ c := (projIcc 0 (T : ℝ) T.coe_nonneg (t : ℝ)).property.1
+  have hct : c ≤ (t : ℝ) := by
+    dsimp [c]
+    simp only [projIcc]
+    exact max_le t.coe_nonneg (min_le_right _ _)
+  have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n c ω = picardApprox V q₀ n c ω := by
+    apply picardApprox_congr_on q₀ c _ n c ⟨hc0, le_rfl⟩ ω
+    intro s hs ω q
+    have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr (hs.2.trans hct)
+    simp only [W, V, min_eq_left hh]
+  apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n c ω)
+  · intro n
+    exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left
+  · apply tendsto_pi_nhds.mpr
+    intro ω
+    simpa only [heq] using hQconv t ω
+
+ 
+
+theorem exists_measurable_global_bounded_ode_picard
+    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
+    (V : ℝ → Ω → E → E) (x₀ : E) (L K : ℝ≥0)
+    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2))
+    (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x))
+    (hl : ∀ ω t, LipschitzWith K (V t ω))
+    (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) :
+    ∃ X : ℝ → Ω → E,
+      Measurable (fun q : ℝ × Ω => X q.1 q.2) ∧
+      (∀ ω, LipschitzWith L (fun t => X t ω)) ∧
+      (∀ ω, X 0 ω = x₀) ∧
+      (∀ ω t, 0 ≤ t →
+        HasDerivWithinAt (fun u => X u ω) (V t ω (X t ω)) (Ici 0) t) ∧
+      (∀ t ω, Tendsto (fun n => picardApprox V x₀ n (max 0 t) ω) atTop (𝓝 (X t ω))) := by
+  classical
+  have hex (t : ℝ) (ω : Ω) : ∃ v : E,
+      Tendsto (fun n => picardApprox V x₀ n (max 0 t) ω) atTop (𝓝 v) := by
+    let T : ℝ≥0 := ⟨max 0 t, le_max_left _ _⟩
+    obtain ⟨X, _, _, _, _, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ T L K hV hc hl hb
+    refine ⟨X t ω, ?_⟩
+    have hc0 := hconv t ω
+    change Tendsto (fun n => picardApprox V x₀ n (max 0 (min (max 0 t) t)) ω) atTop (𝓝 (X t ω)) at hc0
+    simpa only [min_eq_right (le_max_right (0 : ℝ) t)] using hc0
+  let Q (t : ℝ) (ω : Ω) := limUnder atTop (fun n => picardApprox V x₀ n (max 0 t) ω)
+  have hlim (t : ℝ) (ω : Ω) :
+      Tendsto (fun n => picardApprox V x₀ n (max 0 t) ω) atTop (𝓝 (Q t ω)) :=
+    tendsto_nhds_limUnder (hex t ω)
+  have hmatch (T : ℝ≥0) (X : ℝ → Ω → E)
+      (hconv : ∀ t ω, Tendsto (fun n => picardApprox V x₀ n
+        (projIcc 0 (T : ℝ) T.coe_nonneg t) ω) atTop (𝓝 (X t ω)))
+      (t : ℝ) (ht : t ≤ T) (ω : Ω) : Q t ω = X t ω := by
+    apply tendsto_nhds_unique (hlim t ω)
+    simpa only [projIcc, min_eq_right ht] using hconv t ω
+  refine ⟨Q, ?_, ?_, ?_, ?_, hlim⟩
+  · apply measurable_of_tendsto_metrizable (f := fun n (q : ℝ × Ω) => picardApprox V x₀ n (max 0 q.1) q.2)
+    · intro n
+      exact (picardApprox_measurable V x₀ hV n).comp
+        ((measurable_const.max measurable_fst).prodMk measurable_snd)
+    · exact tendsto_pi_nhds.mpr fun q => hlim q.1 q.2
+  · intro ω
+    apply LipschitzWith.of_dist_le_mul
+    intro s t
+    let T : ℝ≥0 := ⟨max (max 0 s) t, (le_max_left 0 s).trans (le_max_left _ _)⟩
+    obtain ⟨X, _, hXl, _, _, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ T L K hV hc hl hb
+    rw [hmatch T X hconv s ((le_max_right 0 s).trans (le_max_left _ _)) ω,
+      hmatch T X hconv t (le_max_right _ _) ω]
+    exact (hXl ω).dist_le_mul s t
+  · intro ω
+    obtain ⟨X, _, _, hX0, _, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ 0 L K hV hc hl hb
+    rw [hmatch 0 X hconv 0 le_rfl ω, hX0 ω]
+  · intro ω t ht
+    let T : ℝ≥0 := ⟨t + 1, by linarith⟩
+    have hlt : t < (T : ℝ) := by change t < t + 1; linarith
+    obtain ⟨X, _, _, _, hXd, hconv⟩ := exists_measurable_bounded_ode_picard V x₀ T L K hV hc hl hb
+    have heq : (fun s => Q s ω) =ᶠ[𝓝 t] (fun s => X s ω) := by
+      filter_upwards [Iic_mem_nhds hlt] with s hs
+      exact hmatch T X hconv s hs ω
+    have hd := hXd ω t ⟨ht, hlt.le⟩
+    have hsets : Icc 0 (T : ℝ) =ᶠ[𝓝 t] Ici 0 := by
+      filter_upwards [Iic_mem_nhds hlt] with s hs
+      change (0 ≤ s ∧ s ≤ (T : ℝ)) = (0 ≤ s)
+      exact propext ⟨And.left, fun h => ⟨h, hs⟩⟩
+    have hd' := hd.congr_set hsets
+    rw [← hmatch T X hconv t hlt.le ω] at hd'
+    exact hd'.congr_of_eventuallyEq (heq.filter_mono nhdsWithin_le_nhds) (heq.eq_of_nhds)
+
+theorem clipped_log_loewner_global_adapted_solution
+    {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ)
+    (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2))
+    (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω))
+    (lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ℂ × ℝ) :
+    ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ × ℝ),
+      Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧
+      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
+      (∀ ω, Q 0 ω = q₀) ∧
+      (∀ ω t, 0 ≤ t →
+        HasDerivWithinAt (fun s => Q s ω)
+          (clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (Q t ω))
+          (Ici 0) t) ∧
+      Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable))
+        (fun t : ℝ≥0 => Q t) := by
+  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) (q : ℂ × ℝ) := clippedLogLoewnerField lo hi hle (U t.toNNReal ω) q
+  have hF := continuous_clippedLogLoewnerField lo hi hle
+  have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) :=
+    hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hm : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => V q.1.1 q.1.2 q.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U q.1.1.toNNReal q.1.2, q.2)))
+    exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd)
+  have hcont (ω : Ω) (q : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω q) := by
+    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun t : ℝ => (U t.toNNReal ω, q)))
+    exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const)
+  obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ :=
+    exists_measurable_global_bounded_ode_picard V q₀ L K hm hcont
+      (fun ω t => hlip (U t.toNNReal ω)) (fun ω t q => hbound (U t.toNNReal ω) q)
+  refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩
+  intro t
+  have hUfrozen := measurable_driver_frozen U hU hc t
+  let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t
+  let : MeasurableSpace Ω := m
+  let W (s : ℝ) (ω : Ω) (q : ℂ × ℝ) :=
+    clippedLogLoewnerField lo hi hle (U (min s.toNNReal t) ω) q
+  have hW : Measurable (fun q : (ℝ × Ω) × (ℂ × ℝ) => W q.1.1 q.1.2 q.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun q : (ℝ × Ω) × (ℂ × ℝ) => (U (min q.1.1.toNNReal t) q.1.2, q.2)))
+    exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd)
+  have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n (t : ℝ) ω = picardApprox V q₀ n (t : ℝ) ω := by
+    apply picardApprox_congr_on q₀ (t : ℝ) _ n (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩ ω
+    intro s hs ω q
+    have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hs.2
+    simp only [W, V, min_eq_left hh]
+  apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n (t : ℝ) ω)
+  · intro n
+    exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left
+  · apply tendsto_pi_nhds.mpr
+    intro ω
+    simpa only [heq, max_eq_right t.coe_nonneg] using hQconv t ω
+
+noncomputable def spatialPartial {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (φ : E × ℝ → ℝ) : ℕ → E × ℝ → ℝ
+  | 0 => φ
+  | n + 1 => fun p => fderiv ℝ (spatialPartial φ n) p (0, 1)
+
+lemma spatialPartial_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (n : ℕ) :
+    ContDiff ℝ (⊤ : ℕ∞) (spatialPartial φ n) := by
+  induction n with
+  | zero => exact hφ
+  | succ n ih =>
+    exact (ih.fderiv_right (by simp)).clm_apply contDiff_const
+
+lemma spatialPartial_compact {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (φ : E × ℝ → ℝ) (hφ : HasCompactSupport φ) (n : ℕ) :
+    HasCompactSupport (spatialPartial φ n) := by
+  induction n with
+  | zero => exact hφ
+  | succ n ih =>
+    exact (ih.fderiv ℝ).comp_left (g := fun D : (E × ℝ) →L[ℝ] ℝ => D (0, 1)) (by simp)
+
+lemma spatialPartial_eq_iteratedDeriv {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (n : ℕ) (q : E) (x : ℝ) :
+    spatialPartial φ n (q, x) = iteratedDeriv n (fun y => φ (q, y)) x := by
+  induction n generalizing x with
+  | zero => rfl
+  | succ n ih =>
+    have hd : HasDerivAt (fun y => spatialPartial φ n (q, y))
+        (fderiv ℝ (spatialPartial φ n) (q, x) (0, 1)) x := by
+      exact (((spatialPartial_smooth φ hφ n).differentiable (by simp)) (q, x)).hasFDerivAt.comp_hasDerivAt x
+        ((hasDerivAt_const x q).prodMk (hasDerivAt_id x))
+    simpa only [spatialPartial, iteratedDeriv_succ, ← funext ih] using hd.deriv.symm
+
+lemma compact_smooth_bounded_lipschitz
+    {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [NormedAddCommGroup F] [NormedSpace ℝ F]
+    (φ : E → F) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hφc : HasCompactSupport φ) :
+    ∃ C : ℝ≥0, (∀ p, ‖φ p‖ ≤ C) ∧ LipschitzWith C φ := by
+  obtain ⟨A, hA⟩ := hφ.continuous.bounded_above_of_compact_support hφc
+  obtain ⟨D, hD⟩ := (hφ.continuous_fderiv (by simp)).bounded_above_of_compact_support (hφc.fderiv ℝ)
+  let C := Real.toNNReal (max A D)
+  have hAC : A ≤ (C : ℝ) := (le_max_left _ _).trans (Real.le_coe_toNNReal _)
+  have hDC : D ≤ (C : ℝ) := (le_max_right _ _).trans (Real.le_coe_toNNReal _)
+  refine ⟨C, fun p => (hA p).trans hAC, ?_⟩
+  apply lipschitzWith_of_nnnorm_fderiv_le (hφ.differentiable (by simp))
+  intro p
+  exact_mod_cast (hD p).trans hDC
+
+ 
+
+theorem compact_state_smooth_brownian_field
+    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (Q V : ℝ → Ω → E) (L : ℝ≥0)
+    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => Q t))
+    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
+    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
+    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
+    (hQd : ∀ t ω, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) (V t ω) (Ici 0) t)
+    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hφc : HasCompactSupport φ) :
+    ∃ K : ℝ≥0, SmoothTimeBrownianField B hm K
+      (fun t ω x => φ (Q t ω, x))
+      (fun t ω x => fderiv ℝ φ (Q t ω, x) (V t ω, 0)) := by
+
+  classical
+  have hr (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial φ j)
+    (spatialPartial_smooth φ hφ j) (spatialPartial_compact φ hφc j)
+  choose C hCb hCl using hr
+  let C₀ : ℝ≥0 := ∑ j ∈ Finset.range 4, C j
+  have hC (j : ℕ) (hj : j ≤ 3) : C j ≤ C₀ :=
+    Finset.single_le_sum (fun i _ => (zero_le : (0 : ℝ≥0) ≤ C i)) (Finset.mem_range.mpr (by omega))
+  have hCb' (j : ℕ) (hj : j ≤ 3) (p : E × ℝ) : |spatialPartial φ j p| ≤ C₀ := by
+    have hh := (hCb j p).trans (show (C j : ℝ) ≤ C₀ by exact_mod_cast hC j hj)
+    simpa only [Real.norm_eq_abs] using hh
+  have hC2 : LipschitzWith C₀ (spatialPartial φ 2) := (hCl 2).weaken (hC 2 (by decide))
+  obtain ⟨D, hDb, hDl⟩ := compact_smooth_bounded_lipschitz (fderiv ℝ φ)
+    (hφ.fderiv_right (by simp)) (hφc.fderiv ℝ)
+  let K : ℝ≥0 := (C₀ + D) * (L + 1)
+  have hKC : (C₀ : ℝ) ≤ (K : ℝ) := by
+    change (C₀ : ℝ) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
+    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
+  have hKDL : (D : ℝ) * L ≤ (K : ℝ) := by
+    change (D : ℝ) * L ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
+    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
+  have hKCL : (C₀ : ℝ) * ((L : ℝ) + 1) ≤ (K : ℝ) := by
+    change (C₀ : ℝ) * ((L : ℝ) + 1) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
+    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
+  have hQn : Measurable (fun p : ℝ≥0 × Ω => Q p.1 p.2) :=
+    hQm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hVn : Measurable (fun p : ℝ≥0 × Ω => V p.1 p.2) :=
+    hVm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
+  have happ : Continuous (fun p : (E × ℝ) × (E × ℝ) => fderiv ℝ φ p.1 p.2) :=
+    ((hφ.continuous_fderiv (by simp)).comp continuous_fst).clm_apply continuous_snd
+  have hameas : Measurable (fun p : (ℝ≥0 × Ω) × ℝ => fderiv ℝ φ (Q p.1.1 p.1.2, p.2) (V p.1.1 p.1.2, 0)) :=
+    happ.measurable.comp (((hQn.comp measurable_fst).prodMk measurable_snd).prodMk
+      ((hVn.comp measurable_fst).prodMk measurable_const))
+  have habound (t : ℝ≥0) (ω : Ω) (x : ℝ) :
+      |fderiv ℝ φ (Q t ω, x) (V t ω, 0)| ≤ K := by
+    calc
+      _ = ‖fderiv ℝ φ (Q t ω, x) (V t ω, 0)‖ := (Real.norm_eq_abs _).symm
+      _ ≤ ‖fderiv ℝ φ (Q t ω, x)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
+      _ = ‖fderiv ℝ φ (Q t ω, x)‖ * ‖V t ω‖ := by simp
+      _ ≤ (D : ℝ) * L := mul_le_mul (hDb _) (hVb t ω) (norm_nonneg _) D.coe_nonneg
+      _ ≤ K := hKDL
+  have haLip (t : ℝ≥0) (ω : Ω) (x y : ℝ) :
+      |fderiv ℝ φ (Q t ω, x) (V t ω, 0) - fderiv ℝ φ (Q t ω, y) (V t ω, 0)| ≤
+        (K : ℝ) * |x - y| := by
+    have hd : ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ ≤ (D : ℝ) * |x - y| := by
+      simpa [dist_eq_norm, Prod.norm_def, Real.norm_eq_abs] using hDl.dist_le_mul (Q t ω, x) (Q t ω, y)
+    calc
+      _ = ‖(fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)) (V t ω, 0)‖ := by
+        rw [sub_apply, Real.norm_eq_abs]
+      _ ≤ ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
+      _ = ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖V t ω‖ := by simp
+      _ ≤ ((D : ℝ) * |x - y|) * L := mul_le_mul hd (hVb t ω) (norm_nonneg _) (by positivity)
+      _ = ((D : ℝ) * L) * |x - y| := by ring
+      _ ≤ (K : ℝ) * |x - y| := mul_le_mul_of_nonneg_right hKDL (abs_nonneg _)
+  refine ⟨K, ⟨?_, ?_, ?_, hameas, ?_, habound, haLip, ?_, ?_⟩⟩
+  · intro t ω
+    have hc : ContDiff ℝ (⊤ : ℕ∞) (fun x : ℝ => φ (Q t ω, x)) :=
+      hφ.comp (contDiff_const.prodMk contDiff_id)
+    exact hc.of_le (WithTop.coe_le_coe.mpr (show (3 : ℕ∞) ≤ ⊤ from le_top))
+  · intro t j hj
+    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
+    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
+      (((hQa t).comp measurable_fst).prodMk measurable_snd)
+  · intro j hj
+    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
+    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
+      ((hQn.comp measurable_fst).prodMk measurable_snd)
+  · intro t ω x j hj
+    rw [← spatialPartial_eq_iteratedDeriv φ hφ]
+    exact (hCb' j hj _).trans hKC
+  · intro s t ω x y
+    rw [← spatialPartial_eq_iteratedDeriv φ hφ, ← spatialPartial_eq_iteratedDeriv φ hφ]
+    have hQdist := (hQl ω).dist_le_mul (t : ℝ) (s : ℝ)
+    rw [Real.dist_eq] at hQdist
+    calc
+      _ = dist (spatialPartial φ 2 (Q t ω, x)) (spatialPartial φ 2 (Q s ω, y)) := (Real.dist_eq _ _).symm
+      _ ≤ (C₀ : ℝ) * dist (Q t ω, x) (Q s ω, y) := hC2.dist_le_mul _ _
+      _ ≤ (C₀ : ℝ) * (dist (Q t ω) (Q s ω) + |x - y|) := by
+        apply mul_le_mul_of_nonneg_left _ C₀.coe_nonneg
+        rw [Prod.dist_eq, Real.dist_eq]
+        exact max_le (le_add_of_nonneg_right (abs_nonneg _)) (le_add_of_nonneg_left dist_nonneg)
+      _ ≤ (C₀ : ℝ) * ((L : ℝ) * |(t : ℝ) - s| + |x - y|) :=
+        mul_le_mul_of_nonneg_left (show dist (Q (t : ℝ) ω) (Q (s : ℝ) ω) + |x - y| ≤
+          (L : ℝ) * |(t : ℝ) - s| + |x - y| by linarith) C₀.coe_nonneg
+      _ ≤ ((C₀ : ℝ) * ((L : ℝ) + 1)) * (|(t : ℝ) - s| + |x - y|) := by
+        nlinarith [mul_nonneg C₀.coe_nonneg (abs_nonneg ((t : ℝ) - s)),
+          mul_nonneg (mul_nonneg C₀.coe_nonneg L.coe_nonneg) (abs_nonneg (x - y))]
+      _ ≤ (K : ℝ) * (|(t : ℝ) - s| + |x - y|) :=
+        mul_le_mul_of_nonneg_right hKCL (by positivity)
+  · intro s t hst ω x
+    have hst' : (s : ℝ) ≤ (t : ℝ) := by exact_mod_cast hst
+    symm
+    apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le (f := fun u : ℝ => φ (Q u ω, x))
+      (f' := fun u : ℝ => fderiv ℝ φ (Q u.toNNReal ω, x) (V u.toNNReal ω, 0)) hst'
+    · exact (hφ.continuous.comp ((hQl ω).continuous.prodMk continuous_const)).continuousOn
+    · intro u hu
+      have hu0 : 0 < u := lt_of_le_of_lt s.coe_nonneg hu.1
+      have hd := (hQd u ω hu0.le).hasDerivAt (Ici_mem_nhds hu0)
+      have hpair := hd.prodMk (hasDerivAt_const u x)
+      have hc := ((hφ.differentiable (by simp)) (Q u ω, x)).hasFDerivAt.comp_hasDerivAt u hpair
+      simpa only [Function.comp_def, Real.coe_toNNReal _ hu0.le] using hc
+    · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hst']
+      have hmA : Measurable (fun u : ℝ => fderiv ℝ φ (Q u.toNNReal ω, x) (V u.toNNReal ω, 0)) :=
+        hameas.comp ((continuous_real_toNNReal.measurable.prodMk measurable_const).prodMk measurable_const)
+      apply Integrable.of_bound hmA.aestronglyMeasurable (K : ℝ)
+      exact Eventually.of_forall fun u => by simpa only [Real.norm_eq_abs] using habound u.toNNReal ω x
+
+noncomputable def dimension (κ : ℝ) : ℝ := 1 + κ / 8
+noncomputable def codimension (κ : ℝ) : ℝ := ((2 : ℕ) : ℝ) - dimension κ
+noncomputable def greenExponent (κ : ℝ) : ℝ := 8 / κ - 1
+
+ 
+
+noncomputable def greenLogNormSq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  (q.1.re - Real.sqrt κ * x) ^ 2 + Real.exp (2 * q.1.im)
+noncomputable def greenLogDensity (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  Real.exp (-codimension κ * Real.log 2 + codimension κ * q.2 +
+    (greenExponent κ - codimension κ) * q.1.im -
+      (greenExponent κ / 2) * Real.log (greenLogNormSq κ q x))
+noncomputable def physicalLogLoewnerField (u : ℝ) (q : ℂ × ℝ) : ℂ × ℝ :=
+  let Z := ((q.1.re - u : ℝ) : ℂ) + (Real.exp q.1.im : ℂ) * Complex.I
+  let v := Z⁻¹
+  (((2 * v.re : ℝ) : ℂ) + ((-2 * Complex.normSq v : ℝ) : ℂ) * Complex.I,
+    (-2 * v ^ 2).re)
+
+lemma greenLogNormSq_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    0 < greenLogNormSq κ q x :=
+  add_pos_of_nonneg_of_pos (sq_nonneg _) (Real.exp_pos _)
+
+lemma greenLogDensity_contDiff (κ : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) := by
+  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
+    Complex.reCLM.contDiff.comp contDiff_fst.fst
+  have him : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
+    Complex.imCLM.contDiff.comp contDiff_fst.fst
+  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) := by
+    exact ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add
+      ((contDiff_const.mul him).exp)
+  have hlog := hN.log (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
+  unfold greenLogDensity
+  apply ContDiff.exp
+  exact (((contDiff_const.add (contDiff_const.mul contDiff_fst.snd)).add
+    (contDiff_const.mul him)).sub (contDiff_const.mul hlog))
+
+lemma greenLogDensity_spatial_derivative (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    HasDerivAt (greenLogDensity κ q)
+      (greenLogDensity κ q x * (greenExponent κ * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
+        greenLogNormSq κ q x)) x := by
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
+    simp
+  have hlog := hN.log (greenLogNormSq_pos κ q x).ne'
+  have hd := ((hasDerivAt_const x (-codimension κ * Real.log 2 + codimension κ * q.2 +
+    (greenExponent κ - codimension κ) * q.1.im)).sub
+      (hlog.const_mul (greenExponent κ / 2))).exp
+  convert hd using 1 <;> try rfl
+  dsimp only [greenLogDensity, Pi.sub_apply, Pi.mul_apply, Function.comp_def]
+  ring
+
+lemma greenLogDensity_spatial_second (κ : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
+    iteratedDeriv 2 (greenLogDensity κ q) x =
+      greenLogDensity κ q x * κ * greenExponent κ *
+        ((greenExponent κ + 1) * (q.1.re - Real.sqrt κ * x) ^ 2 - Real.exp (2 * q.1.im)) /
+        (greenLogNormSq κ q x) ^ 2 := by
+  have hd : deriv (greenLogDensity κ q) = fun x =>
+      greenLogDensity κ q x * (greenExponent κ * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
+        greenLogNormSq κ q x) := funext fun x => (greenLogDensity_spatial_derivative κ q x).deriv
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
+    simp
+  have hcoef := (hX.const_mul (greenExponent κ * Real.sqrt κ)).div hN
+    (greenLogNormSq_pos κ q x).ne'
+  have hsecond := (greenLogDensity_spatial_derivative κ q x).mul hcoef
+  rw [show (2 : ℕ) = 1 + 1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
+  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, id_eq] at hsecond
+  change HasDerivAt (fun y => greenLogDensity κ q y *
+    (greenExponent κ * Real.sqrt κ * (q.1.re - Real.sqrt κ * y) / greenLogNormSq κ q y)) _ x at hsecond
+  rw [hsecond.deriv]
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  dsimp only [greenLogNormSq]
+  generalize greenLogDensity κ q x = f
+  generalize greenExponent κ = a
+  generalize q.1.re - Real.sqrt κ * x = X
+  generalize Real.exp (2 * q.1.im) = Y
+  have hs := Real.sq_sqrt hκ
+  generalize Real.sqrt κ = r at *
+  rw [← hs]
+  field_simp
+  ring
+
+lemma greenLogDensity_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q) v =
+      greenLogDensity κ q x * (codimension κ * v.2 +
+        (greenExponent κ - codimension κ) * v.1.im - (greenExponent κ / 2) *
+          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
+            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x)) := by
+  have hzre : HasDerivAt (fun t : ℝ => (q + t • v).1.re) v.1.re 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
+  have hzim : HasDerivAt (fun t : ℝ => (q + t • v).1.im) v.1.im 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
+  have hj : HasDerivAt (fun t : ℝ => (q + t • v).2) v.2 0 := by
+    simpa only [Prod.snd_add, Prod.smul_snd, smul_eq_mul, id_eq, one_mul]
+      using ((hasDerivAt_id (0 : ℝ)).mul_const v.2).const_add q.2
+  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q + t • v) x)
+      (2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
+        2 * Real.exp (2 * q.1.im) * v.1.im) 0 := by
+    convert ((hzre.sub_const (Real.sqrt κ * x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
+    simp only [zero_smul, add_zero, Nat.cast_ofNat]
+    ring
+  have hlog := hN.log (by simpa only [zero_smul, add_zero] using (greenLogNormSq_pos κ q x).ne')
+  have hcalc := (((hj.const_mul (codimension κ)).const_add (-codimension κ * Real.log 2)).add
+    (hzim.const_mul (greenExponent κ - codimension κ))).sub
+      (hlog.const_mul (greenExponent κ / 2))
+  have hline : HasDerivAt (fun t : ℝ => greenLogDensity κ (q + t • v) x)
+      (greenLogDensity κ q x * (codimension κ * v.2 +
+        (greenExponent κ - codimension κ) * v.1.im - (greenExponent κ / 2) *
+          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
+            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x))) 0 := by
+    convert hcalc.exp using 1 <;> try rfl
+    simp only [Pi.sub_apply, Pi.add_apply, zero_smul, add_zero]
+    simp only [greenLogDensity]
+  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => greenLogDensity κ r x) :=
+    (greenLogDensity_contDiff κ).comp (contDiff_id.prodMk contDiff_const)
+  have haff : HasDerivAt (fun t : ℝ => q + t • v) v 0 := by
+    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
+  have hfd : HasFDerivAt (fun r : ℂ × ℝ => greenLogDensity κ r x)
+      (fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q) (q + (0 : ℝ) • v) := by
+    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
+  have hchain := hfd.comp_hasDerivAt 0 haff
+  simpa only [Function.comp_def] using hchain.unique hline
+
+lemma physicalLogLoewnerField_components (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    (physicalLogLoewnerField (Real.sqrt κ * x) q).1.re =
+      2 * (q.1.re - Real.sqrt κ * x) / greenLogNormSq κ q x ∧
+    (physicalLogLoewnerField (Real.sqrt κ * x) q).1.im =
+      -2 / greenLogNormSq κ q x ∧
+    (physicalLogLoewnerField (Real.sqrt κ * x) q).2 =
+      -2 * ((q.1.re - Real.sqrt κ * x) ^ 2 - Real.exp (2 * q.1.im)) /
+        (greenLogNormSq κ q x) ^ 2 := by
+  let Z : ℂ := ((q.1.re - Real.sqrt κ * x : ℝ) : ℂ) + (Real.exp q.1.im : ℂ) * Complex.I
+  have hre : Z.re = q.1.re - Real.sqrt κ * x := by simp [Z]
+  have him : Z.im = Real.exp q.1.im := by
+    simp only [Z, Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im,
+      Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add, add_zero]
+  have he : Real.exp q.1.im * Real.exp q.1.im = Real.exp (2 * q.1.im) := by
+    rw [← Real.exp_add]; congr 1; ring
+  have hn : Complex.normSq Z = greenLogNormSq κ q x := by
+    rw [Complex.normSq_apply, hre, him, he]
+    simp only [greenLogNormSq, pow_two]
+  change (((2 * Z⁻¹.re : ℝ) : ℂ) + ((-2 * Complex.normSq Z⁻¹ : ℝ) : ℂ) * Complex.I).re = _ ∧
+    (((2 * Z⁻¹.re : ℝ) : ℂ) + ((-2 * Complex.normSq Z⁻¹ : ℝ) : ℂ) * Complex.I).im = _ ∧
+    (-2 * Z⁻¹ ^ 2).re = _
+  constructor
+  · simp [Complex.inv_re, hre, hn, mul_div_assoc]
+  constructor
+  · simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re,
+      Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add, add_zero,
+      Complex.normSq_inv, hn, div_eq_mul_inv]
+  · simp only [pow_two, Complex.mul_re, Complex.mul_im, Complex.neg_re, Complex.neg_im,
+      Complex.re_ofNat, Complex.im_ofNat, neg_zero, zero_mul, sub_zero,
+      Complex.inv_re, Complex.inv_im, hre, him, hn]
+    have he2 : Real.exp q.1.im ^ 2 = Real.exp (2 * q.1.im) := by simpa only [pow_two] using he
+    rw [← he2]
+    field_simp
+
+ 
+theorem green_physical_generator_zero (κ : ℝ) (hκ : 0 < κ) (_hκ8 : κ < 8)
+    (q : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q)
+      (physicalLogLoewnerField (Real.sqrt κ * x) q) +
+      (1 / 2 : ℝ) * iteratedDeriv 2 (greenLogDensity κ q) x = 0 := by
+  rw [greenLogDensity_directional, greenLogDensity_spatial_second κ hκ.le]
+  obtain ⟨hr, hi, hj⟩ := physicalLogLoewnerField_components κ q x
+  rw [hr, hi, hj]
+  have ha : κ * (greenExponent κ + 1) = 8 := by
+    simp only [greenExponent, sub_add_cancel]
+    field_simp
+  have hp : κ * greenExponent κ = 8 * codimension κ := by
+    dsimp only [greenExponent, codimension, dimension]
+    field_simp
+    ring
+  have hc1 : -4 * greenExponent κ + κ * greenExponent κ * (greenExponent κ + 1) / 2 = 0 := by
+    linear_combination (greenExponent κ / 2) * ha
+  have hc2 : 4 * codimension κ - κ * greenExponent κ / 2 = 0 := by
+    linarith
+  calc
+    _ = greenLogDensity κ q x *
+      ((-4 * greenExponent κ + κ * greenExponent κ * (greenExponent κ + 1) / 2) *
+        (q.1.re - Real.sqrt κ * x) ^ 2 +
+        (4 * codimension κ - κ * greenExponent κ / 2) * Real.exp (2 * q.1.im)) /
+      (greenLogNormSq κ q x) ^ 2 := by
+        have hn := (greenLogNormSq_pos κ q x).ne'
+        field_simp
+        dsimp only [greenLogNormSq]
+        ring
+    _ = 0 := by rw [hc1, hc2]; ring
+
+lemma greenLogDensity_pair_directional (κ : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (q, x) (v, 0) =
+      fderiv ℝ (fun r : ℂ × ℝ => greenLogDensity κ r x) q v := by
+  have hd := ((greenLogDensity_contDiff κ).differentiable (by simp)) (q, x)
+  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
+  have hc := (hd.hasFDerivAt.comp q hp).fderiv
+  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) hc
+  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
+    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
+
+ 
+
+lemma exists_compact_green_field (κ R : ℝ) (hR : 0 ≤ R) :
+    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
+      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = greenLogDensity κ q x) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R →
+        fderiv ℝ φ (q, x) = fderiv ℝ (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (q, x)) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
+        iteratedDeriv 2 (greenLogDensity κ q) x) := by
+  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let φ (p : (ℂ × ℝ) × ℝ) : ℝ := b p * greenLogDensity κ p.1 p.2
+  have hnear (p : (ℂ × ℝ) × ℝ) (hp : ‖p‖ ≤ R) :
+      φ =ᶠ[𝓝 p] (fun r : (ℂ × ℝ) × ℝ => greenLogDensity κ r.1 r.2) := by
+    have hball : p ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
+      simp only [Metric.mem_ball, dist_zero_right]
+      change ‖p‖ < R + 1
+      linarith
+    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with r hr
+    simp only [φ, hr, Pi.one_apply, one_mul]
+  refine ⟨φ, b.contDiff.mul (greenLogDensity_contDiff κ), b.hasCompactSupport.mul_right,
+    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
+  · intro q x h
+    exact (hnear (q, x) h).fderiv_eq
+  · intro q x h
+    have he := (hnear (q, x) h).comp_tendsto
+      ((continuous_const.prodMk continuous_id).tendsto x)
+    exact he.iteratedDeriv_eq 2
+
+lemma clippedLogLoewnerField_eq_physical (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ)
+    (q : ℂ × ℝ) (hq : q.1.im ∈ Icc lo hi) :
+    clippedLogLoewnerField lo hi hle u q = physicalLogLoewnerField u q := by
+  simp only [clippedLogLoewnerField, clippedCenter, clippedHeight, projIcc_of_mem hle hq,
+    physicalLogLoewnerField]
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/DiskHalfPlane.lean b/lean/OAI/Probability/SLE/DiskHalfPlane.lean
index 16d0c64ecb6ebd6c49876c6b50119de6596fb487..78a74d0778b2ed2938c2254a3781acc799b40259 100644
--- a/lean/OAI/Probability/SLE/DiskHalfPlane.lean
+++ b/lean/OAI/Probability/SLE/DiskHalfPlane.lean
@@ -1,813 +1,813 @@
-import Mathlib
-import OAI.Probability.SLE.BrownianStrongMarkov
-
-namespace OAI
-
-/-! Disk Half Plane for chordal SLE. -/
-
-section
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-noncomputable def diskToHalfPlane (w ζ : ℂ) : ℂ := (w - star w * ζ) / (1 - ζ)
-
-lemma disk_denominator_ne {ζ : ℂ} (hζ : ζ ∈ ball (0 : ℂ) 1) : 1 - ζ ≠ 0 := by
-  intro he
-  have hh := mem_ball_zero_iff.mp hζ
-  rw [← sub_eq_zero.mp he] at hh
-  simp at hh
-
-lemma diskToHalfPlane_im (w ζ : ℂ) :
-    (diskToHalfPlane w ζ).im = w.im * (1 - Complex.normSq ζ) / Complex.normSq (1 - ζ) := by
-  simp only [diskToHalfPlane, Complex.div_im, Complex.sub_im, Complex.sub_re, Complex.mul_im,
-    Complex.mul_re, Complex.star_def, Complex.conj_re, Complex.conj_im, Complex.one_im,
-    Complex.one_re, Complex.normSq_apply]
-  ring
-
-lemma diskToHalfPlane_maps {w : ℂ} (hw : 0 < w.im) :
-    MapsTo (diskToHalfPlane w) (ball 0 1) upperHalfPlane := by
-  intro ζ hζ
-  have hn : ‖ζ‖ < 1 := mem_ball_zero_iff.mp hζ
-  have hn2 : Complex.normSq ζ < 1 := by rw [Complex.normSq_eq_norm_sq]; nlinarith [norm_nonneg ζ]
-  have hp : 0 < Complex.normSq (1 - ζ) := Complex.normSq_pos.mpr (disk_denominator_ne hζ)
-  change 0 < (diskToHalfPlane w ζ).im
-  rw [diskToHalfPlane_im]
-  exact div_pos (mul_pos hw (sub_pos.mpr hn2)) hp
-
-lemma diskToHalfPlane_differentiable (w : ℂ) :
-    DifferentiableOn ℂ (diskToHalfPlane w) (ball 0 1) :=
-  ((differentiableOn_id.const_mul (star w)).const_sub w).div
-    (differentiableOn_id.const_sub 1) (fun _ hζ => disk_denominator_ne hζ)
-
-@[simp] lemma diskToHalfPlane_zero (w : ℂ) : diskToHalfPlane w 0 = w := by simp [diskToHalfPlane]
-
-lemma diskToHalfPlane_deriv_zero (w : ℂ) :
-    HasDerivAt (diskToHalfPlane w) (w - star w) 0 := by
-  have hh := ((hasDerivAt_const (0 : ℂ) w).sub ((hasDerivAt_id 0).const_mul (star w))).div
-    ((hasDerivAt_const (0 : ℂ) 1).sub (hasDerivAt_id 0)) (by norm_num : (1 : ℂ) - 0 ≠ 0)
-  convert hh using 1 <;> first | rfl | (dsimp; ring)
-
-lemma diskToHalfPlane_injOn {w : ℂ} (hw : 0 < w.im) : InjOn (diskToHalfPlane w) (ball 0 1) := by
-  intro ζ hζ ξ hξ he
-  have hc := (div_eq_div_iff (disk_denominator_ne hζ) (disk_denominator_ne hξ)).mp he
-  have hp : (w - star w) * (ζ - ξ) = 0 := by linear_combination hc
-  have hn : w - star w ≠ 0 := by
-    intro he
-    have him := congrArg Complex.im he
-    simp at him
-    linarith
-  exact sub_eq_zero.mp ((mul_eq_zero.mp hp).resolve_left hn)
-
-lemma norm_sub_conj_of_im_pos {w : ℂ} (hw : 0 < w.im) : ‖w - star w‖ = 2 * w.im := by
-  rw [Complex.star_def, Complex.sub_conj, norm_mul, Complex.norm_real, Real.norm_eq_abs,
-    abs_of_pos (mul_pos (by norm_num) hw), Complex.norm_I, mul_one]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma IsCapacityTwoWitness.radius_le_omitted_distance
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (_hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t)
-    (a : ℂ) (ha : a ∉ survivingDomain γ t) :
-    fullRadius G t z ≤ (128 * Real.pi) * dist z a := by
-  have hγ := h.1
-  rcases h with ⟨_, hγ0, hγim, hG, hF, hGmap, hFmap, hinv, hinv', h0, hode, hnorm, htip⟩
-  have hy : 0 < (G t z).im := hGmap t hz
-  have hg := ((hG t).differentiableAt ((isOpen_survivingDomain hγ t).mem_nhds hz)).hasDerivAt
-  have hf := ((hF t).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hGmap t hz))).hasDerivAt
-  have hident : (fun v => F t (G t v)) =ᶠ[𝓝 z] (fun v => v) := by
-    filter_upwards [(isOpen_survivingDomain hγ t).mem_nhds hz] with v hv
-    exact hinv t v hv
-  have hprod : deriv (F t) (G t z) * deriv (G t) z = 1 :=
-    ((hf.comp z hg).congr_of_eventuallyEq hident.symm).unique (hasDerivAt_id z)
-  have hnormprod : ‖deriv (F t) (G t z)‖ * ‖deriv (G t) z‖ = 1 := by
-    simpa only [norm_mul, norm_one] using congrArg norm hprod
-  have hgpos : 0 < ‖deriv (G t) z‖ := by
-    by_contra hn
-    have he : ‖deriv (G t) z‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
-    rw [he, mul_zero] at hnormprod
-    norm_num at hnormprod
-  let f : ℂ → ℂ := F t ∘ diskToHalfPlane (G t z)
-  have hfd : DifferentiableOn ℂ f (ball 0 1) :=
-    (hF t).comp (diskToHalfPlane_differentiable _) (diskToHalfPlane_maps hy)
-  have hfi : InjOn f (ball 0 1) := by
-    intro v hv w hw he
-    apply diskToHalfPlane_injOn hy hv hw
-    have he' := congrArg (G t) he
-    change G t (F t (diskToHalfPlane (G t z) v)) = G t (F t (diskToHalfPlane (G t z) w)) at he'
-    rwa [hinv' t _ (diskToHalfPlane_maps hy hv), hinv' t _ (diskToHalfPlane_maps hy hw)] at he'
-  have hfimage : f '' ball 0 1 ⊆ survivingDomain γ t := by
-    rintro _ ⟨v, hv, rfl⟩
-    exact hFmap t (diskToHalfPlane_maps hy hv)
-  have hfv0 : f 0 = z := by simp only [f, Function.comp_apply, diskToHalfPlane_zero, hinv t z hz]
-  have hfq : HasDerivAt (F t) (deriv (F t) (G t z)) (diskToHalfPlane (G t z) 0) := by simpa using hf
-  have hfder : deriv f 0 = deriv (F t) (G t z) * (G t z - star (G t z)) :=
-    (hfq.comp 0 (diskToHalfPlane_deriv_zero _)).deriv
-  have hfrad : ‖deriv f 0‖ = fullRadius G t z := by
-    rw [hfder, norm_mul, norm_sub_conj_of_im_pos hy, fullRadius]
-    apply (eq_div_iff hgpos.ne').mpr
-    nlinarith [hnormprod]
-  have hh := univalent_omitted_value_bound hfd hfi a (fun ha' => ha (hfimage ha'))
-  rwa [hfrad, hfv0, ← dist_eq_norm] at hh
-
-lemma IsCapacityTwoWitness.radius_le_past_distance
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t)
-    (s : ℝ≥0) (hs : s ≤ t) :
-    fullRadius G t z ≤ (128 * Real.pi) * dist z (γ s) :=
-  h.radius_le_omitted_distance hU t z hz (γ s)
-    (fun hw => hw.1.2 ⟨s, ⟨zero_le, hs⟩, rfl⟩)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma IsCapacityTwoWitness.radius_approach_of_trace_approach
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
-    (T : ℝ≥0) (hT : dist (γ T) z < r) :
-    ∃ s : ℝ≥0, s ≤ T ∧ z ∈ survivingDomain γ s ∧ fullRadius G s z < (128 * Real.pi) * r := by
-  let ρ : ℝ := (r + dist (γ T) z) / 2
-  have hρ : 0 < ρ := by dsimp [ρ]; linarith [(dist_nonneg : 0 ≤ dist (γ T) z)]
-  have hρr : ρ < r := by dsimp [ρ]; linarith
-  have hTρ : dist (γ T) z < ρ := by dsimp [ρ]; linarith
-  have hγ : Continuous γ := h.1
-  let S : Set ℝ≥0 := Icc 0 T ∩ {s | dist (γ s) z ≤ ρ}
-  have hSc : IsCompact S := isCompact_Icc.inter_right (isClosed_le (hγ.dist continuous_const) continuous_const)
-  have hSn : S.Nonempty := ⟨T, ⟨⟨zero_le, le_rfl⟩, hTρ.le⟩⟩
-  obtain ⟨τ, hτ, hmin⟩ := hSc.exists_isMinOn hSn continuous_id.continuousOn
-  have hτ0 : 0 < τ := by
-    by_contra hn
-    have he : τ = 0 := le_antisymm (le_of_not_gt hn) zero_le
-    have hh := hτ.2
-    change dist (γ τ) z ≤ ρ at hh
-    rw [he, h.2.1, dist_zero_left] at hh
-    have hz := Complex.im_le_norm z
-    linarith
-  have hprior (s : ℝ≥0) (hs : s < τ) : ρ < dist (γ s) z := by
-    by_contra hn
-    have hmem : s ∈ S := ⟨⟨zero_le, hs.le.trans hτ.1.2⟩, le_of_not_gt hn⟩
-    exact not_le_of_gt hs (hmin hmem)
-  have hτcl : τ ∈ closure (Ioo (0 : ℝ≥0) τ) := by
-    rw [closure_Ioo (ne_of_lt hτ0)]
-    exact ⟨hτ0.le, le_rfl⟩
-  have hO : IsOpen {s : ℝ≥0 | dist (γ s) z < r} := isOpen_lt (hγ.dist continuous_const) continuous_const
-  have hτO : τ ∈ {s : ℝ≥0 | dist (γ s) z < r} := lt_of_le_of_lt hτ.2 hρr
-  obtain ⟨s, hsnear, hs⟩ := mem_closure_iff.mp hτcl _ hO hτO
-  have hsT : s ≤ T := hs.2.le.trans hτ.1.2
-  have hfree : ball z ρ ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by
-    intro w hw
-    refine ⟨ball_subset_upperHalfPlane (hρr.le.trans hy) hw, ?_⟩
-    rintro ⟨u, hu, rfl⟩
-    exact not_lt_of_ge (hprior u (hu.2.trans_lt hs.2)).le (mem_ball.mp hw)
-  have hsz := h.surviving_of_free_ball_future_tip s T hsT z ρ hρ hfree (mem_ball.mpr hTρ)
-  refine ⟨s, hsT, hsz, ?_⟩
-  have hh := h.radius_le_past_distance hU s z hsz s le_rfl
-  rw [dist_comm z (γ s)] at hh
-  exact hh.trans_lt (mul_lt_mul_of_pos_left hsnear (mul_pos (by norm_num) Real.pi_pos))
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma logRadiusClock_logCoordinates
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t) :
-    logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) =
-      Real.log ((2 * z.im) / fullRadius G t z) := by
-  have hy := h.imag_positive t z hz
-  have hj := h.derivative_norm_positive hU t z hz
-  have hR := h.fullRadius_positive hU t z hz
-  rw [logRadiusClock, logCoordinates_im, logCoordinates_snd,
-    Real.log_div (mul_ne_zero (by norm_num) hz.1.1.ne') hR.ne',
-    Real.log_mul (by norm_num) hz.1.1.ne', fullRadius,
-    Real.log_div (mul_ne_zero (by norm_num) hy.ne') hj.ne',
-    Real.log_mul (by norm_num) hy.ne']
-  ring
-
-lemma fullRadius_eq_logClock
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t) :
-    fullRadius G t z = (2 * z.im) *
-      Real.exp (-logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t)) := by
-  rw [logRadiusClock_logCoordinates hU h z t hz, Real.exp_neg,
-    Real.exp_log (div_pos (mul_pos (by norm_num) hz.1.1) (h.fullRadius_positive hU t z hz))]
-  field_simp [(show 0 < z.im from hz.1.1).ne']
-
-lemma fullRadius_level_iff_logClock
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t) (u : ℝ) :
-    fullRadius G t z = (2 * z.im) * Real.exp (-u) ↔
-      logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u := by
-  rw [fullRadius_eq_logClock hU h z t hz,
-    mul_right_inj' (mul_ne_zero (by norm_num) hz.1.1.ne'), Real.exp_eq_exp, neg_inj]
-
-lemma IsCapacityTwoWitness.radius_clock_strictMonoOn
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (T : ℝ≥0) (hz : z ∈ survivingDomain γ T) :
-    StrictMonoOn (fun t : ℝ => Real.log ((2 * z.im) / fullRadius G t.toNNReal z))
-      (Icc 0 (T : ℝ)) := by
-  have hd := loewner_radius_clock_derivative hU h T z hz
-  apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc _ _)
-    (fun t ht => (hd t ht).continuousWithinAt)
-    (fun t ht => (hd t (interior_subset ht)).mono interior_subset)
-  intro t ht
-  have hy := h.imag_positive t.toNNReal z (h.live_at_real_time T z hz t (interior_subset ht).2)
-  have hn := Complex.normSq_pos.mpr (h.centered_ne_zero t.toNNReal z
-    (h.live_at_real_time T z hz t (interior_subset ht).2))
-  positivity
-
-lemma live_radius_unique
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (s t : ℝ≥0) (hs : z ∈ survivingDomain γ s)
-    (ht : z ∈ survivingDomain γ t) (he : fullRadius G s z = fullRadius G t z) : s = t := by
-  have hmax : z ∈ survivingDomain γ (max s t) := by
-    rcases le_total s t with hh | hh
-    · simpa only [max_eq_right hh] using ht
-    · simpa only [max_eq_left hh] using hs
-  have hi := h.radius_clock_strictMonoOn hU z (max s t) hmax
-  apply NNReal.coe_injective
-  apply hi.injOn
-  · exact ⟨s.coe_nonneg, by exact_mod_cast le_max_left s t⟩
-  · exact ⟨t.coe_nonneg, by exact_mod_cast le_max_right s t⟩
-  · simp only [Real.toNNReal_coe, he]
-
-lemma live_clock_le_time
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (s t : ℝ≥0) (hs : z ∈ survivingDomain γ s)
-    (_ht : z ∈ survivingDomain γ t)
-    (he : Real.log ((2 * z.im) / fullRadius G s z) ≤
-      Real.log ((2 * z.im) / fullRadius G t z)) : s ≤ t := by
-  by_contra hn
-  have hts := not_le.mp hn
-  have hi := h.radius_clock_strictMonoOn hU z s hs
-  have hh := hi (show (t : ℝ) ∈ Icc 0 (s : ℝ) from ⟨t.coe_nonneg, by exact_mod_cast hts.le⟩)
-    (show (s : ℝ) ∈ Icc 0 (s : ℝ) from ⟨s.coe_nonneg, le_rfl⟩) (by exact_mod_cast hts)
-  simp only [Real.toNNReal_coe] at hh
-  exact (not_lt_of_ge he) hh
-
-lemma live_angleSq_lower_bound
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (T : ℝ≥0) (hz : z ∈ survivingDomain γ T) :
-    ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ≥0, t ≤ T →
-      δ ≤ (Real.sin (Complex.arg (G t z - (U t : ℂ)))) ^ 2 := by
-  let A (t : ℝ) := (Real.sin (Complex.arg (G t.toNNReal z - (U t.toNNReal : ℂ)))) ^ 2
-  have hn (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
-      ‖G t.toNNReal z - (U t.toNNReal : ℂ)‖ ≠ 0 :=
-    norm_ne_zero_iff.mpr (h.centered_ne_zero _ z (h.live_at_real_time T z hz t ht.2))
-  have hc : ContinuousOn A (Icc 0 (T : ℝ)) := by
-    have hGc := h.time_continuousOn T z hz
-    have hUc := Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal)
-    have hZc := hGc.sub hUc.continuousOn
-    simp only [A, Complex.sin_arg]
-    exact ((Complex.continuous_im.comp_continuousOn hZc).div hZc.norm hn).pow 2
-  obtain ⟨t, ht, hmin⟩ := (isCompact_Icc : IsCompact (Icc 0 (T : ℝ))).exists_isMinOn
-    ⟨0, le_rfl, T.coe_nonneg⟩ hc
-  have hp : 0 < A t := by
-    have hy := h.imag_positive t.toNNReal z (h.live_at_real_time T z hz t ht.2)
-    dsimp only [A]
-    rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero]
-    exact sq_pos_of_pos (div_pos hy (lt_of_le_of_ne (norm_nonneg _) (hn t ht).symm))
-  refine ⟨A t, hp, fun s hs => ?_⟩
-  have hh := hmin (show (s : ℝ) ∈ Icc 0 (T : ℝ) from ⟨s.coe_nonneg, by exact_mod_cast hs⟩)
-  change A t ≤ A (s : ℝ) at hh
-  simpa only [A, Real.toNNReal_coe] using hh
-
-lemma radiusLevelTime_eq_of_live_level
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (u : ℝ) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t)
-    (hR : fullRadius G t z = (2 * z.im) * Real.exp (-u)) :
-    radiusLevelTime γ G z u = (t : ℝ≥0∞) := by
-  have he : {s : ℝ≥0∞ | s < ∞ ∧ z ∈ survivingDomain γ s.toNNReal ∧
-      fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u)} = {(t : ℝ≥0∞)} := by
-    ext s
-    constructor
-    · rintro ⟨hs, hsl, hsR⟩
-      have hh : s.toNNReal = t := live_radius_unique hU h z s.toNNReal t hsl ht (hsR.trans hR.symm)
-      have hse : (s.toNNReal : ℝ≥0∞) = s := ENNReal.coe_toNNReal hs.ne
-      rw [Set.mem_singleton_iff, ← hse, hh]
-    · intro hs
-      rw [Set.mem_singleton_iff] at hs
-      subst s
-      simpa only [Set.mem_ofPred_eq, ENNReal.coe_lt_top, ENNReal.toNNReal_coe, true_and] using And.intro ht hR
-  rw [radiusLevelTime, he, sInf_singleton]
-
-lemma stoppedGreen_eq_of_live_level
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (κ : ℝ) (z : ℂ) (u : ℝ) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t)
-    (hR : fullRadius G t z = (2 * z.im) * Real.exp (-u)) :
-    stoppedGreen κ U γ G z u = greenDensity κ U γ G t z := by
-  rw [stoppedGreen, radiusLevelTime_eq_of_live_level hU h z u t ht hR]
-  simp only [ENNReal.coe_lt_top, ↓reduceIte, ENNReal.toNNReal_coe]
-
-lemma stoppedGreen_eq_zero_of_no_level
-    (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ)
-    (z : ℂ) (u : ℝ)
-    (hn : ¬ ∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧
-      fullRadius G t z = (2 * z.im) * Real.exp (-u)) :
-    stoppedGreen κ U γ G z u = 0 := by
-  have he : {s : ℝ≥0∞ | s < ∞ ∧ z ∈ survivingDomain γ s.toNNReal ∧
-      fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u)} = ∅ := by
-    apply Set.eq_empty_iff_forall_notMem.mpr
-    rintro s ⟨_, hsl, hsR⟩
-    exact hn ⟨s.toNNReal, hsl, hsR⟩
-  simp only [stoppedGreen, radiusLevelTime, he, sInf_empty, lt_self_iff_false, ↓reduceIte]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma IsCapacityTwoWitness.fullRadius_initial
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (hz : z ∈ survivingDomain γ 0) :
-    fullRadius G 0 z = 2 * z.im := by
-  rw [fullRadius, h.initial_value z hz.1.1,
-    loewner_spatial_derivative hU h 0 z hz]
-  simp
-
- 
-
-lemma radiusLevelTime_finite_of_live_radius_lt
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0)
-    (ht : z ∈ survivingDomain γ t)
-    (hR : fullRadius G t z < (2 * z.im) * Real.exp (-u)) :
-    radiusLevelTime γ G z u < ∞ := by
-  let L : ℝ → ℝ := fun s => Real.log ((2 * z.im) / fullRadius G s.toNNReal z)
-  have hy : 0 < 2 * z.im := mul_pos (by norm_num) ht.1.1
-  have hz0 : z ∈ survivingDomain γ 0 := survivingDomain_antitone γ zero_le ht
-  have hL0 : L 0 = 0 := by
-    simp only [L, Real.toNNReal_zero, h.fullRadius_initial hU z hz0,
-      div_self hy.ne', Real.log_one]
-  have hLt : u < L t := by
-    have he := fullRadius_eq_logClock hU h z t ht
-    rw [he] at hR
-    have hc := (mul_lt_mul_iff_right₀ hy).mp hR
-    rw [Real.exp_lt_exp] at hc
-    dsimp only [L]
-    rw [Real.toNNReal_coe, ← logRadiusClock_logCoordinates hU h z t ht]
-    linarith
-  have hLc : ContinuousOn L (Icc 0 (t : ℝ)) :=
-    fun s hs => (loewner_radius_clock_derivative hU h t z ht s hs).continuousWithinAt
-  obtain ⟨s, hs, hsu⟩ := intermediate_value_Icc t.coe_nonneg hLc
-    (show u ∈ Icc (L 0) (L t) from ⟨by rwa [hL0], hLt.le⟩)
-  have hsl : z ∈ survivingDomain γ s.toNNReal := h.live_at_real_time t z ht s hs.2
-  have hsr : fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u) := by
-    apply (fullRadius_level_iff_logClock hU h z s.toNNReal hsl u).mpr
-    rw [logRadiusClock_logCoordinates hU h z s.toNNReal hsl]
-    exact hsu
-  rw [radiusLevelTime_eq_of_live_level hU h z u s.toNNReal hsl hsr]
-  exact ENNReal.coe_lt_top
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-structure FiniteIncreasingClock (Ω : Type*) where
-  S : ℝ → Ω → ℝ
-  T : Ω → ℝ≥0
-  zero : ∀ ω, S 0 ω = 0
-  continuous : ∀ ω, Continuous (fun t => S t ω)
-  strict : ∀ ω, StrictMonoOn (fun t => S t ω) (Icc 0 (T ω : ℝ))
-
-namespace FiniteIncreasingClock
-variable {Ω : Type*} (c : FiniteIncreasingClock Ω)
-
-lemma terminal_nonneg (ω : Ω) : 0 ≤ c.S (c.T ω) ω := by
-  rw [← c.zero ω]
-  exact (c.strict ω).monotoneOn ⟨le_rfl, (c.T ω).coe_nonneg⟩
-    ⟨(c.T ω).coe_nonneg, le_rfl⟩ (c.T ω).coe_nonneg
-
-noncomputable def iso (ω : Ω) : Icc (0 : ℝ) (c.T ω : ℝ) ≃o Icc (0 : ℝ) (c.S (c.T ω) ω) := by
-  let f : Icc (0 : ℝ) (c.T ω : ℝ) → Icc (0 : ℝ) (c.S (c.T ω) ω) := fun t =>
-    ⟨c.S t ω, by
-      constructor
-      · simpa only [c.zero ω] using
-          ((c.strict ω).monotoneOn ⟨le_rfl, (c.T ω).coe_nonneg⟩ t.property t.property.1)
-      · exact (c.strict ω).monotoneOn t.property ⟨(c.T ω).coe_nonneg, le_rfl⟩ t.property.2⟩
-  apply StrictMono.orderIsoOfSurjective f
-  · intro s t hst
-    exact c.strict ω s.property t.property hst
-  · intro s
-    obtain ⟨t, ht, he⟩ := intermediate_value_Icc (c.T ω).coe_nonneg
-      (c.continuous ω).continuousOn (show (s : ℝ) ∈ Icc (c.S 0 ω) (c.S (c.T ω) ω) by
-        simpa only [c.zero ω] using s.property)
-    exact ⟨⟨t, ht⟩, Subtype.ext he⟩
-
-lemma iso_apply (ω : Ω) (t : Icc (0 : ℝ) (c.T ω : ℝ)) :
-    (c.iso ω t : ℝ) = c.S t ω := rfl
-
-noncomputable def inverse (s : ℝ) (ω : Ω) : ℝ :=
-  ((c.iso ω).symm ⟨min (max s 0) (c.S (c.T ω) ω),
-    ⟨le_min (le_max_right _ _) (c.terminal_nonneg ω), min_le_right _ _⟩⟩ : ℝ)
-
-lemma inverse_mem (s : ℝ) (ω : Ω) : c.inverse s ω ∈ Icc (0 : ℝ) (c.T ω : ℝ) :=
-  ((c.iso ω).symm _).property
-
-lemma clock_inverse (s : ℝ) (ω : Ω) :
-    c.S (c.inverse s ω) ω = min (max s 0) (c.S (c.T ω) ω) := by
-  exact congrArg Subtype.val ((c.iso ω).apply_symm_apply _)
-
-lemma inverse_clock (ω : Ω) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) (c.T ω : ℝ)) :
-    c.inverse (c.S t ω) ω = t := by
-  apply (c.strict ω).injOn (c.inverse_mem _ ω) ht
-  dsimp only
-  rw [c.clock_inverse]
-  have h0 : 0 ≤ c.S t ω := by
-    rw [← c.zero ω]
-    exact (c.strict ω).monotoneOn ⟨le_rfl, (c.T ω).coe_nonneg⟩ ht ht.1
-  have hT := (c.strict ω).monotoneOn ht ⟨(c.T ω).coe_nonneg, le_rfl⟩ ht.2
-  rw [max_eq_left h0, min_eq_left hT]
-
-lemma continuous_inverse (ω : Ω) : Continuous (fun s => c.inverse s ω) := by
-  apply continuous_subtype_val.comp
-  apply (c.iso ω).symm.continuous.comp
-  exact Continuous.subtype_mk ((continuous_id.max continuous_const).min continuous_const) _
-
-lemma inverse_le_iff (s : ℝ) (ω : Ω) (t : ℝ) (ht : 0 ≤ t) :
-    c.inverse s ω ≤ t ↔ (c.T ω : ℝ) ≤ t ∨
-      min (max s 0) (c.S (c.T ω) ω) ≤ c.S t ω := by
-  by_cases hT : (c.T ω : ℝ) ≤ t
-  · exact iff_of_true ((c.inverse_mem s ω).2.trans hT) (Or.inl hT)
-  · have htm : t ∈ Icc (0 : ℝ) (c.T ω : ℝ) := ⟨ht, (lt_of_not_ge hT).le⟩
-    rw [or_iff_right hT, ← c.clock_inverse]
-    exact ((c.strict ω).le_iff_le (c.inverse_mem s ω) htm).symm
-
-lemma measurable_inverse [MeasurableSpace Ω]
-    (hS : Measurable (Function.uncurry c.S)) (hT : Measurable c.T) :
-    Measurable (Function.uncurry c.inverse) := by
-  apply measurable_of_Iic
-  intro t
-  by_cases ht : 0 ≤ t
-  · have hterminal : Measurable (fun ω => c.S (c.T ω) ω) :=
-      hS.comp ((measurable_coe_nnreal_real.comp hT).prodMk measurable_id)
-    have he : (Function.uncurry c.inverse) ⁻¹' Iic t =
-        {p : ℝ × Ω | (c.T p.2 : ℝ) ≤ t} ∪
-        {p : ℝ × Ω | min (max p.1 0) (c.S (c.T p.2) p.2) ≤ c.S t p.2} := by
-      ext p
-      exact c.inverse_le_iff p.1 p.2 t ht
-    rw [he]
-    exact (measurableSet_le ((measurable_coe_nnreal_real.comp hT).comp measurable_snd) measurable_const).union
-      (measurableSet_le ((measurable_fst.max measurable_const).min (hterminal.comp measurable_snd))
-        (hS.comp (measurable_const.prodMk measurable_snd)))
-  · have he : (Function.uncurry c.inverse) ⁻¹' Iic t = ∅ := by
-      apply Set.eq_empty_iff_forall_notMem.mpr
-      intro p hp
-      exact ht ((c.inverse_mem p.1 p.2).1.trans hp)
-    rw [he]
-    exact MeasurableSet.empty
-
-end FiniteIncreasingClock
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma powerLogDensity_pos (κ p α : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    0 < powerLogDensity κ p α q x := Real.exp_pos _
-
-lemma powerLogDensity_le_radius
-    (κ p α η₀ u : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ α) (q : ℂ × ℝ) (x : ℝ)
-    (hs : logRadiusClock η₀ q ≤ u) :
-    powerLogDensity κ p α q x ≤ Real.exp (p * (u - Real.log 2 - η₀)) := by
-  rw [powerLogDensity_radius_angle κ p α η₀ q x]
-  have ha' : (logAngleSq κ q x) ^ (α / 2) ≤ 1 := by
-    simpa only [Real.one_rpow] using Real.rpow_le_rpow (logAngleSq_pos κ q x).le
-      (logAngleSq_le_one κ q x) (div_nonneg ha (by norm_num))
-  calc
-    _ ≤ Real.exp (p * (logRadiusClock η₀ q - Real.log 2 - η₀)) * 1 :=
-      mul_le_mul_of_nonneg_left ha' (Real.exp_pos _).le
-    _ ≤ _ := by rw [mul_one]; apply Real.exp_le_exp.mpr; gcongr
-
-lemma powerLogGenerator_abs_bound
-    (κ p α lo : ℝ) (q : ℂ × ℝ) (x : ℝ) (hlo : lo ≤ q.1.im) :
-    |powerLogGenerator κ p α q x| ≤
-      powerLogDensity κ p α q x *
-        (|(-4 * α + κ * α * (α + 1) / 2)| + |(4 * p - κ * α / 2)|) /
-          Real.exp (2 * lo) := by
-  let A := -4 * α + κ * α * (α + 1) / 2
-  let D := 4 * p - κ * α / 2
-  let X := q.1.re - Real.sqrt κ * x
-  let Y := Real.exp (2 * q.1.im)
-  let N := greenLogNormSq κ q x
-  let φ := powerLogDensity κ p α q x
-  have hN : 0 < N := greenLogNormSq_pos κ q x
-  have hY : 0 < Y := Real.exp_pos _
-  have hφ : 0 ≤ φ := (powerLogDensity_pos κ p α q x).le
-  have hX : X ^ 2 ≤ N := le_add_of_nonneg_right hY.le
-  have hYN : Y ≤ N := le_add_of_nonneg_left (sq_nonneg _)
-  have hlow : Real.exp (2 * lo) ≤ N :=
-    (Real.exp_le_exp.mpr (by linarith)).trans hYN
-  have hnum : |A * X ^ 2 + D * Y| ≤ (|A| + |D|) * N := by
-    calc
-      _ ≤ |A * X ^ 2| + |D * Y| := abs_add_le _ _
-      _ = |A| * X ^ 2 + |D| * Y := by
-        rw [abs_mul, abs_mul, abs_of_nonneg (sq_nonneg X), abs_of_pos hY]
-      _ ≤ |A| * N + |D| * N := add_le_add
-        (mul_le_mul_of_nonneg_left hX (abs_nonneg A))
-        (mul_le_mul_of_nonneg_left hYN (abs_nonneg D))
-      _ = _ := by ring
-  change |φ * (A * X ^ 2 + D * Y) / N ^ 2| ≤ φ * (|A| + |D|) / _
-  rw [abs_div, abs_mul, abs_of_nonneg hφ, abs_of_nonneg (sq_nonneg N)]
-  calc
-    _ ≤ φ * ((|A| + |D|) * N) / N ^ 2 := by gcongr
-    _ = φ * (|A| + |D|) / N := by field_simp
-    _ ≤ _ := div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) hlow
-
-lemma powerLogGenerator_le_radius_height
-    (κ p α η₀ u lo : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ α) (q : ℂ × ℝ) (x : ℝ)
-    (hs : logRadiusClock η₀ q ≤ u) (hlo : lo ≤ q.1.im) :
-    |powerLogGenerator κ p α q x| ≤
-      Real.exp (p * (u - Real.log 2 - η₀)) *
-        (|(-4 * α + κ * α * (α + 1) / 2)| + |(4 * p - κ * α / 2)|) /
-          Real.exp (2 * lo) := by
-  apply (powerLogGenerator_abs_bound κ p α lo q x hlo).trans
-  apply div_le_div_of_nonneg_right _ (Real.exp_pos _).le
-  exact mul_le_mul_of_nonneg_right (powerLogDensity_le_radius κ p α η₀ u hp ha q x hs)
-    (add_nonneg (abs_nonneg _) (abs_nonneg _))
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem exists_finite_angular_kernel
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
-    (ε : ℝ) (hε : 0 < ε) (hε0 : ε ≤ logAngleSq κ (initialLogCoordinates z) 0) :
-    ∃ f : Ω → ℝ,
-      Integrable f P ∧ (∫ ω, f ω ∂P) = initialGreen κ z ∧
-      (∀ ω, 0 ≤ f ω ∧ f ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))) ∧
-      ∀ (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
-        IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F →
-        ∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧
-          Real.log ((2 * z.im) / fullRadius G t z) ≤ u ∧
-          (Real.log ((2 * z.im) / fullRadius G t z) = u ∨
-            (Real.sin (Complex.arg (G t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 = ε) ∧
-          f ω = greenDensity κ (fun t => Real.sqrt κ * B t ω) γ G t z ∧
-          ((Real.sin (Complex.arg (G t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 = ε →
-            f ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) *
-              ε ^ (greenExponent κ / 2)) := by
-  let η₀ := Real.log z.im
-  let lo := η₀ - u / (2 * ε) - 1
-  let hi := η₀ + 1
-  have hule : 0 ≤ u / (2 * ε) := by positivity
-  have hle : lo ≤ hi := by dsimp [lo, hi]; linarith
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  obtain ⟨c⟩ := exists_clippedLogSolution B hm hBm hBc κ lo hi hle (initialLogCoordinates z)
-  have hη : (initialLogCoordinates z).1.im = η₀ := by
-    simp only [initialLogCoordinates, Complex.add_im, Complex.ofReal_im, Complex.mul_im,
-      Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, zero_add, add_zero]
-    rfl
-  let a := 4 * ε ^ 2 / Real.exp (2 * η₀)
-  have ha : 0 < a := by dsimp [a]; positivity
-  obtain ⟨N, hN⟩ := exists_nat_gt (u / a)
-  have hT : u < (4 * ε ^ 2 / Real.exp (2 * (initialLogCoordinates z).1.im)) * ((N : ℝ≥0) : ℝ) := by
-    rw [hη]
-    have hh := (div_lt_iff₀ ha).mp hN
-    simpa only [NNReal.coe_natCast, mul_comm (N : ℝ) a] using hh
-  obtain ⟨τ, hτ, ht, hin, hpre, hbound, hfi, hfm⟩ := c.finite_angular_law
-    hB hBm hBc hB0 hκ hκ8 rfl u ε hu hε hε0
-    (by rw [hη]; dsimp [lo]; linarith) (by rw [hη]; dsimp [hi]; linarith) (N : ℝ≥0) hT
-  let f (ω : Ω) := greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)
-  refine ⟨f, hfi, hfm.trans (greenLogDensity_initial κ z hz), ?_, ?_⟩
-  · intro ω
-    refine ⟨greenLogDensity_nonneg _ _ _, ?_⟩
-    have hh := greenLogDensity_le_radius κ (initialLogCoordinates z).1.im u hκ hκ8 (c.Q (τ ω) ω) (B (τ ω) ω)
-      (hin ω (τ ω) le_rfl).1
-    simpa only [hη] using hh
-  · intro ω γ G F hw
-    have hU : Continuous (fun t => Real.sqrt κ * B t ω) := (hBc ω).const_mul _
-    have hQd (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) := (c.derivative ω s hs.1).mono
-      (show Icc (0 : ℝ) (τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-    have hQb (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) : (c.Q s ω).1.im ∈ Icc lo hi := by
-      have hh := (hin ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.2
-      simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
-    obtain ⟨hlive, heq⟩ := clipped_log_identification hU hw z hz (τ ω) lo hi hle
-      (fun s => c.Q s ω) (c.initial ω) hQd hQb
-    have he : c.Q (τ ω) ω = logLoewnerCoordinates G z (τ ω) :=
-      heq ⟨(τ ω).coe_nonneg, le_rfl⟩
-    have hr : logRadiusClock (initialLogCoordinates z).1.im (c.Q (τ ω) ω) =
-        Real.log ((2 * z.im) / fullRadius G (τ ω) z) := by
-      rw [he, hη]
-      exact logRadiusClock_logCoordinates hU hw z (τ ω) hlive
-    have ha' : logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) =
-        (Real.sin (Complex.arg (G (τ ω) z - ((Real.sqrt κ * B (τ ω) ω : ℝ) : ℂ)))) ^ 2 := by
-      rw [he]
-      exact logAngleSq_logCoordinates κ _ G z _ (hw.imag_positive _ z hlive)
-    refine ⟨τ ω, hlive, ?_, ?_, ?_, ?_⟩
-    · simpa only [hr] using (hin ω (τ ω) le_rfl).1
-    · simpa only [hr, ha'] using hbound ω
-    · dsimp only [f]
-      rw [he]
-      exact greenLogDensity_logCoordinates hU hw κ _ z _ hlive rfl
-    · intro hA
-      have hh := greenLogDensity_le_at_angular_exit κ (initialLogCoordinates z).1.im u ε hκ hκ8
-        (c.Q (τ ω) ω) (B (τ ω) ω) (hin ω (τ ω) le_rfl).1 (by rw [ha', hA])
-      simpa only [hη] using hh
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-structure AngularLocalization {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (κ : ℝ)
-    (z : ℂ) (u ε : ℝ) where
-  lo : ℝ
-  hi : ℝ
-  le : lo ≤ hi
-  solution : ClippedLogSolution B hm κ lo hi le (initialLogCoordinates z)
-  T : ℝ≥0
-  τ : Ω → ℝ≥0
-  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-    (fun ω => (τ ω : WithTop ℝ≥0))
-  bounded : ∀ ω, τ ω ≤ T
-  inside : ∀ ω t, t ≤ τ ω →
-    logRadiusClock (Real.log z.im) (solution.Q t ω) ≤ u ∧
-    ε ≤ logAngleSq κ (solution.Q t ω) (B t ω) ∧
-    lo ≤ (solution.Q t ω).1.im ∧ (solution.Q t ω).1.im ≤ hi
-  before : ∀ ω t, t < τ ω →
-    logRadiusClock (Real.log z.im) (solution.Q t ω) < u ∧
-    ε < logAngleSq κ (solution.Q t ω) (B t ω)
-  boundary : ∀ ω,
-    logRadiusClock (Real.log z.im) (solution.Q (τ ω) ω) = u ∨
-    logAngleSq κ (solution.Q (τ ω) ω) (B (τ ω) ω) = ε
-
-lemma initialLogCoordinates_im (z : ℂ) :
-    (initialLogCoordinates z).1.im = Real.log z.im := by
-  simp [initialLogCoordinates]
-
-lemma exists_angularLocalization
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ) (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
-    (hε0 : ε ≤ logAngleSq κ (initialLogCoordinates z) 0) :
-    Nonempty (AngularLocalization B hm κ z u ε) := by
-  let η₀ := Real.log z.im
-  let lo := η₀ - u / (2 * ε) - 1
-  let hi := η₀ + 1
-  have hule : 0 ≤ u / (2 * ε) := by positivity
-  have hle : lo ≤ hi := by dsimp [lo, hi]; linarith
-  obtain ⟨c⟩ := exists_clippedLogSolution B hm hBm hBc κ lo hi hle (initialLogCoordinates z)
-  let a := 4 * ε ^ 2 / Real.exp (2 * η₀)
-  have ha : 0 < a := by dsimp [a]; positivity
-  obtain ⟨N, hN⟩ := exists_nat_gt (u / a)
-  have hT : u < (4 * ε ^ 2 / Real.exp (2 * (initialLogCoordinates z).1.im)) * ((N : ℝ≥0) : ℝ) := by
-    rw [initialLogCoordinates_im]
-    have hh := (div_lt_iff₀ ha).mp hN
-    simpa only [NNReal.coe_natCast, mul_comm (N : ℝ) a] using hh
-  obtain ⟨τ, hτ, ht, hin, hpre, hbound, _, _⟩ := c.finite_angular_law
-    hB hBm hBc hB0 hκ hκ8 rfl u ε hu hε hε0
-    (by rw [initialLogCoordinates_im]; dsimp [lo, η₀]; linarith)
-    (by rw [initialLogCoordinates_im]; dsimp [hi, η₀]; linarith) (N : ℝ≥0) hT
-  exact ⟨⟨lo, hi, hle, c, N, τ, hτ, fun ω => (ht ω).le,
-    by simpa only [initialLogCoordinates_im] using hin,
-    by simpa only [initialLogCoordinates_im] using hpre,
-    by simpa only [initialLogCoordinates_im] using hbound⟩⟩
-
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ}
-  (a : AngularLocalization B hm κ z u ε)
-
-lemma chart_identification
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F) :
-    z ∈ survivingDomain γ (a.τ ω) ∧
-      ∀ t : ℝ≥0, t ≤ a.τ ω → a.solution.Q t ω = logLoewnerCoordinates G z t := by
-  have hQd (s : ℝ) (hs : s ∈ Icc 0 (a.τ ω : ℝ)) := (a.solution.derivative ω s hs.1).mono
-    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-  have hQb (s : ℝ) (hs : s ∈ Icc 0 (a.τ ω : ℝ)) : (a.solution.Q s ω).1.im ∈ Icc a.lo a.hi := by
-    have hh := (a.inside ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.2
-    simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
-  obtain ⟨hlive, heq⟩ := clipped_log_identification ((hBc ω).const_mul _) hw z hz (a.τ ω)
-    a.lo a.hi a.le (fun s => a.solution.Q s ω) (a.solution.initial ω) hQd hQb
-  exact ⟨hlive, fun t ht => by simpa only [Real.toNNReal_coe] using heq ⟨t.coe_nonneg, ht⟩⟩
-
-lemma clock_eq_chart
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F)
-    (t : ℝ≥0) (ht : t ≤ a.τ ω) :
-    logRadiusClock (Real.log z.im) (a.solution.Q t ω) =
-      Real.log ((2 * z.im) / fullRadius G t z) := by
-  obtain ⟨hl, he⟩ := a.chart_identification hBc hz ω γ G F hw
-  rw [he t ht]
-  exact logRadiusClock_logCoordinates ((hBc ω).const_mul _) hw z t (survivingDomain_antitone γ ht hl)
-
-lemma clock_stopped_measurable :
-    Measurable[a.stopping.measurableSpace]
-      (fun ω => logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω)) := by
-  have hQa : StronglyAdapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => a.solution.Q t) := fun t => (a.solution.adapted t).stronglyMeasurable
-  have hQc (ω : Ω) : Continuous (fun t : ℝ≥0 => a.solution.Q t ω) :=
-    (a.solution.lipschitz ω).continuous.comp NNReal.continuous_coe
-  have hQ := measurable_stoppedValue (hQa.isStronglyProgressive_of_continuous hQc) a.stopping
-  have hmQ : Measurable[a.stopping.measurableSpace] (fun ω => a.solution.Q (a.τ ω) ω) := by
-    exact hQ
-  exact (continuous_logRadiusClock _).measurable.comp hmQ
-
-lemma mean_green
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane) :
-    Integrable (fun ω => greenLogDensity κ (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
-      (∫ ω, greenLogDensity κ (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) = initialGreen κ z := by
-  obtain ⟨hi, he⟩ := a.solution.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
-    (Real.log z.im) u a.τ a.stopping a.T a.bounded
-    (fun ω t ht => ⟨(a.inside ω t ht).1, (a.inside ω t ht).2.2⟩)
-  exact ⟨hi, he.trans (greenLogDensity_initial κ z hz)⟩
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.BrownianStrongMarkov
+
+namespace OAI
+
+/-! Disk Half Plane for chordal SLE. -/
+
+section
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+noncomputable def diskToHalfPlane (w ζ : ℂ) : ℂ := (w - star w * ζ) / (1 - ζ)
+
+lemma disk_denominator_ne {ζ : ℂ} (hζ : ζ ∈ ball (0 : ℂ) 1) : 1 - ζ ≠ 0 := by
+  intro he
+  have hh := mem_ball_zero_iff.mp hζ
+  rw [← sub_eq_zero.mp he] at hh
+  simp at hh
+
+lemma diskToHalfPlane_im (w ζ : ℂ) :
+    (diskToHalfPlane w ζ).im = w.im * (1 - Complex.normSq ζ) / Complex.normSq (1 - ζ) := by
+  simp only [diskToHalfPlane, Complex.div_im, Complex.sub_im, Complex.sub_re, Complex.mul_im,
+    Complex.mul_re, Complex.star_def, Complex.conj_re, Complex.conj_im, Complex.one_im,
+    Complex.one_re, Complex.normSq_apply]
+  ring
+
+lemma diskToHalfPlane_maps {w : ℂ} (hw : 0 < w.im) :
+    MapsTo (diskToHalfPlane w) (ball 0 1) upperHalfPlane := by
+  intro ζ hζ
+  have hn : ‖ζ‖ < 1 := mem_ball_zero_iff.mp hζ
+  have hn2 : Complex.normSq ζ < 1 := by rw [Complex.normSq_eq_norm_sq]; nlinarith [norm_nonneg ζ]
+  have hp : 0 < Complex.normSq (1 - ζ) := Complex.normSq_pos.mpr (disk_denominator_ne hζ)
+  change 0 < (diskToHalfPlane w ζ).im
+  rw [diskToHalfPlane_im]
+  exact div_pos (mul_pos hw (sub_pos.mpr hn2)) hp
+
+lemma diskToHalfPlane_differentiable (w : ℂ) :
+    DifferentiableOn ℂ (diskToHalfPlane w) (ball 0 1) :=
+  ((differentiableOn_id.const_mul (star w)).const_sub w).div
+    (differentiableOn_id.const_sub 1) (fun _ hζ => disk_denominator_ne hζ)
+
+@[simp] lemma diskToHalfPlane_zero (w : ℂ) : diskToHalfPlane w 0 = w := by simp [diskToHalfPlane]
+
+lemma diskToHalfPlane_deriv_zero (w : ℂ) :
+    HasDerivAt (diskToHalfPlane w) (w - star w) 0 := by
+  have hh := ((hasDerivAt_const (0 : ℂ) w).sub ((hasDerivAt_id 0).const_mul (star w))).div
+    ((hasDerivAt_const (0 : ℂ) 1).sub (hasDerivAt_id 0)) (by norm_num : (1 : ℂ) - 0 ≠ 0)
+  convert hh using 1 <;> first | rfl | (dsimp; ring)
+
+lemma diskToHalfPlane_injOn {w : ℂ} (hw : 0 < w.im) : InjOn (diskToHalfPlane w) (ball 0 1) := by
+  intro ζ hζ ξ hξ he
+  have hc := (div_eq_div_iff (disk_denominator_ne hζ) (disk_denominator_ne hξ)).mp he
+  have hp : (w - star w) * (ζ - ξ) = 0 := by linear_combination hc
+  have hn : w - star w ≠ 0 := by
+    intro he
+    have him := congrArg Complex.im he
+    simp at him
+    linarith
+  exact sub_eq_zero.mp ((mul_eq_zero.mp hp).resolve_left hn)
+
+lemma norm_sub_conj_of_im_pos {w : ℂ} (hw : 0 < w.im) : ‖w - star w‖ = 2 * w.im := by
+  rw [Complex.star_def, Complex.sub_conj, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_pos (mul_pos (by norm_num) hw), Complex.norm_I, mul_one]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma IsCapacityTwoWitness.radius_le_omitted_distance
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (_hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t)
+    (a : ℂ) (ha : a ∉ survivingDomain γ t) :
+    fullRadius G t z ≤ (128 * Real.pi) * dist z a := by
+  have hγ := h.1
+  rcases h with ⟨_, hγ0, hγim, hG, hF, hGmap, hFmap, hinv, hinv', h0, hode, hnorm, htip⟩
+  have hy : 0 < (G t z).im := hGmap t hz
+  have hg := ((hG t).differentiableAt ((isOpen_survivingDomain hγ t).mem_nhds hz)).hasDerivAt
+  have hf := ((hF t).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hGmap t hz))).hasDerivAt
+  have hident : (fun v => F t (G t v)) =ᶠ[𝓝 z] (fun v => v) := by
+    filter_upwards [(isOpen_survivingDomain hγ t).mem_nhds hz] with v hv
+    exact hinv t v hv
+  have hprod : deriv (F t) (G t z) * deriv (G t) z = 1 :=
+    ((hf.comp z hg).congr_of_eventuallyEq hident.symm).unique (hasDerivAt_id z)
+  have hnormprod : ‖deriv (F t) (G t z)‖ * ‖deriv (G t) z‖ = 1 := by
+    simpa only [norm_mul, norm_one] using congrArg norm hprod
+  have hgpos : 0 < ‖deriv (G t) z‖ := by
+    by_contra hn
+    have he : ‖deriv (G t) z‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
+    rw [he, mul_zero] at hnormprod
+    norm_num at hnormprod
+  let f : ℂ → ℂ := F t ∘ diskToHalfPlane (G t z)
+  have hfd : DifferentiableOn ℂ f (ball 0 1) :=
+    (hF t).comp (diskToHalfPlane_differentiable _) (diskToHalfPlane_maps hy)
+  have hfi : InjOn f (ball 0 1) := by
+    intro v hv w hw he
+    apply diskToHalfPlane_injOn hy hv hw
+    have he' := congrArg (G t) he
+    change G t (F t (diskToHalfPlane (G t z) v)) = G t (F t (diskToHalfPlane (G t z) w)) at he'
+    rwa [hinv' t _ (diskToHalfPlane_maps hy hv), hinv' t _ (diskToHalfPlane_maps hy hw)] at he'
+  have hfimage : f '' ball 0 1 ⊆ survivingDomain γ t := by
+    rintro _ ⟨v, hv, rfl⟩
+    exact hFmap t (diskToHalfPlane_maps hy hv)
+  have hfv0 : f 0 = z := by simp only [f, Function.comp_apply, diskToHalfPlane_zero, hinv t z hz]
+  have hfq : HasDerivAt (F t) (deriv (F t) (G t z)) (diskToHalfPlane (G t z) 0) := by simpa using hf
+  have hfder : deriv f 0 = deriv (F t) (G t z) * (G t z - star (G t z)) :=
+    (hfq.comp 0 (diskToHalfPlane_deriv_zero _)).deriv
+  have hfrad : ‖deriv f 0‖ = fullRadius G t z := by
+    rw [hfder, norm_mul, norm_sub_conj_of_im_pos hy, fullRadius]
+    apply (eq_div_iff hgpos.ne').mpr
+    nlinarith [hnormprod]
+  have hh := univalent_omitted_value_bound hfd hfi a (fun ha' => ha (hfimage ha'))
+  rwa [hfrad, hfv0, ← dist_eq_norm] at hh
+
+lemma IsCapacityTwoWitness.radius_le_past_distance
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t)
+    (s : ℝ≥0) (hs : s ≤ t) :
+    fullRadius G t z ≤ (128 * Real.pi) * dist z (γ s) :=
+  h.radius_le_omitted_distance hU t z hz (γ s)
+    (fun hw => hw.1.2 ⟨s, ⟨zero_le, hs⟩, rfl⟩)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma IsCapacityTwoWitness.radius_approach_of_trace_approach
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
+    (T : ℝ≥0) (hT : dist (γ T) z < r) :
+    ∃ s : ℝ≥0, s ≤ T ∧ z ∈ survivingDomain γ s ∧ fullRadius G s z < (128 * Real.pi) * r := by
+  let ρ : ℝ := (r + dist (γ T) z) / 2
+  have hρ : 0 < ρ := by dsimp [ρ]; linarith [(dist_nonneg : 0 ≤ dist (γ T) z)]
+  have hρr : ρ < r := by dsimp [ρ]; linarith
+  have hTρ : dist (γ T) z < ρ := by dsimp [ρ]; linarith
+  have hγ : Continuous γ := h.1
+  let S : Set ℝ≥0 := Icc 0 T ∩ {s | dist (γ s) z ≤ ρ}
+  have hSc : IsCompact S := isCompact_Icc.inter_right (isClosed_le (hγ.dist continuous_const) continuous_const)
+  have hSn : S.Nonempty := ⟨T, ⟨⟨zero_le, le_rfl⟩, hTρ.le⟩⟩
+  obtain ⟨τ, hτ, hmin⟩ := hSc.exists_isMinOn hSn continuous_id.continuousOn
+  have hτ0 : 0 < τ := by
+    by_contra hn
+    have he : τ = 0 := le_antisymm (le_of_not_gt hn) zero_le
+    have hh := hτ.2
+    change dist (γ τ) z ≤ ρ at hh
+    rw [he, h.2.1, dist_zero_left] at hh
+    have hz := Complex.im_le_norm z
+    linarith
+  have hprior (s : ℝ≥0) (hs : s < τ) : ρ < dist (γ s) z := by
+    by_contra hn
+    have hmem : s ∈ S := ⟨⟨zero_le, hs.le.trans hτ.1.2⟩, le_of_not_gt hn⟩
+    exact not_le_of_gt hs (hmin hmem)
+  have hτcl : τ ∈ closure (Ioo (0 : ℝ≥0) τ) := by
+    rw [closure_Ioo (ne_of_lt hτ0)]
+    exact ⟨hτ0.le, le_rfl⟩
+  have hO : IsOpen {s : ℝ≥0 | dist (γ s) z < r} := isOpen_lt (hγ.dist continuous_const) continuous_const
+  have hτO : τ ∈ {s : ℝ≥0 | dist (γ s) z < r} := lt_of_le_of_lt hτ.2 hρr
+  obtain ⟨s, hsnear, hs⟩ := mem_closure_iff.mp hτcl _ hO hτO
+  have hsT : s ≤ T := hs.2.le.trans hτ.1.2
+  have hfree : ball z ρ ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by
+    intro w hw
+    refine ⟨ball_subset_upperHalfPlane (hρr.le.trans hy) hw, ?_⟩
+    rintro ⟨u, hu, rfl⟩
+    exact not_lt_of_ge (hprior u (hu.2.trans_lt hs.2)).le (mem_ball.mp hw)
+  have hsz := h.surviving_of_free_ball_future_tip s T hsT z ρ hρ hfree (mem_ball.mpr hTρ)
+  refine ⟨s, hsT, hsz, ?_⟩
+  have hh := h.radius_le_past_distance hU s z hsz s le_rfl
+  rw [dist_comm z (γ s)] at hh
+  exact hh.trans_lt (mul_lt_mul_of_pos_left hsnear (mul_pos (by norm_num) Real.pi_pos))
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma logRadiusClock_logCoordinates
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t) :
+    logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) =
+      Real.log ((2 * z.im) / fullRadius G t z) := by
+  have hy := h.imag_positive t z hz
+  have hj := h.derivative_norm_positive hU t z hz
+  have hR := h.fullRadius_positive hU t z hz
+  rw [logRadiusClock, logCoordinates_im, logCoordinates_snd,
+    Real.log_div (mul_ne_zero (by norm_num) hz.1.1.ne') hR.ne',
+    Real.log_mul (by norm_num) hz.1.1.ne', fullRadius,
+    Real.log_div (mul_ne_zero (by norm_num) hy.ne') hj.ne',
+    Real.log_mul (by norm_num) hy.ne']
+  ring
+
+lemma fullRadius_eq_logClock
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t) :
+    fullRadius G t z = (2 * z.im) *
+      Real.exp (-logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t)) := by
+  rw [logRadiusClock_logCoordinates hU h z t hz, Real.exp_neg,
+    Real.exp_log (div_pos (mul_pos (by norm_num) hz.1.1) (h.fullRadius_positive hU t z hz))]
+  field_simp [(show 0 < z.im from hz.1.1).ne']
+
+lemma fullRadius_level_iff_logClock
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t) (u : ℝ) :
+    fullRadius G t z = (2 * z.im) * Real.exp (-u) ↔
+      logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u := by
+  rw [fullRadius_eq_logClock hU h z t hz,
+    mul_right_inj' (mul_ne_zero (by norm_num) hz.1.1.ne'), Real.exp_eq_exp, neg_inj]
+
+lemma IsCapacityTwoWitness.radius_clock_strictMonoOn
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (T : ℝ≥0) (hz : z ∈ survivingDomain γ T) :
+    StrictMonoOn (fun t : ℝ => Real.log ((2 * z.im) / fullRadius G t.toNNReal z))
+      (Icc 0 (T : ℝ)) := by
+  have hd := loewner_radius_clock_derivative hU h T z hz
+  apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc _ _)
+    (fun t ht => (hd t ht).continuousWithinAt)
+    (fun t ht => (hd t (interior_subset ht)).mono interior_subset)
+  intro t ht
+  have hy := h.imag_positive t.toNNReal z (h.live_at_real_time T z hz t (interior_subset ht).2)
+  have hn := Complex.normSq_pos.mpr (h.centered_ne_zero t.toNNReal z
+    (h.live_at_real_time T z hz t (interior_subset ht).2))
+  positivity
+
+lemma live_radius_unique
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (s t : ℝ≥0) (hs : z ∈ survivingDomain γ s)
+    (ht : z ∈ survivingDomain γ t) (he : fullRadius G s z = fullRadius G t z) : s = t := by
+  have hmax : z ∈ survivingDomain γ (max s t) := by
+    rcases le_total s t with hh | hh
+    · simpa only [max_eq_right hh] using ht
+    · simpa only [max_eq_left hh] using hs
+  have hi := h.radius_clock_strictMonoOn hU z (max s t) hmax
+  apply NNReal.coe_injective
+  apply hi.injOn
+  · exact ⟨s.coe_nonneg, by exact_mod_cast le_max_left s t⟩
+  · exact ⟨t.coe_nonneg, by exact_mod_cast le_max_right s t⟩
+  · simp only [Real.toNNReal_coe, he]
+
+lemma live_clock_le_time
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (s t : ℝ≥0) (hs : z ∈ survivingDomain γ s)
+    (_ht : z ∈ survivingDomain γ t)
+    (he : Real.log ((2 * z.im) / fullRadius G s z) ≤
+      Real.log ((2 * z.im) / fullRadius G t z)) : s ≤ t := by
+  by_contra hn
+  have hts := not_le.mp hn
+  have hi := h.radius_clock_strictMonoOn hU z s hs
+  have hh := hi (show (t : ℝ) ∈ Icc 0 (s : ℝ) from ⟨t.coe_nonneg, by exact_mod_cast hts.le⟩)
+    (show (s : ℝ) ∈ Icc 0 (s : ℝ) from ⟨s.coe_nonneg, le_rfl⟩) (by exact_mod_cast hts)
+  simp only [Real.toNNReal_coe] at hh
+  exact (not_lt_of_ge he) hh
+
+lemma live_angleSq_lower_bound
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (T : ℝ≥0) (hz : z ∈ survivingDomain γ T) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ≥0, t ≤ T →
+      δ ≤ (Real.sin (Complex.arg (G t z - (U t : ℂ)))) ^ 2 := by
+  let A (t : ℝ) := (Real.sin (Complex.arg (G t.toNNReal z - (U t.toNNReal : ℂ)))) ^ 2
+  have hn (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
+      ‖G t.toNNReal z - (U t.toNNReal : ℂ)‖ ≠ 0 :=
+    norm_ne_zero_iff.mpr (h.centered_ne_zero _ z (h.live_at_real_time T z hz t ht.2))
+  have hc : ContinuousOn A (Icc 0 (T : ℝ)) := by
+    have hGc := h.time_continuousOn T z hz
+    have hUc := Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal)
+    have hZc := hGc.sub hUc.continuousOn
+    simp only [A, Complex.sin_arg]
+    exact ((Complex.continuous_im.comp_continuousOn hZc).div hZc.norm hn).pow 2
+  obtain ⟨t, ht, hmin⟩ := (isCompact_Icc : IsCompact (Icc 0 (T : ℝ))).exists_isMinOn
+    ⟨0, le_rfl, T.coe_nonneg⟩ hc
+  have hp : 0 < A t := by
+    have hy := h.imag_positive t.toNNReal z (h.live_at_real_time T z hz t ht.2)
+    dsimp only [A]
+    rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero]
+    exact sq_pos_of_pos (div_pos hy (lt_of_le_of_ne (norm_nonneg _) (hn t ht).symm))
+  refine ⟨A t, hp, fun s hs => ?_⟩
+  have hh := hmin (show (s : ℝ) ∈ Icc 0 (T : ℝ) from ⟨s.coe_nonneg, by exact_mod_cast hs⟩)
+  change A t ≤ A (s : ℝ) at hh
+  simpa only [A, Real.toNNReal_coe] using hh
+
+lemma radiusLevelTime_eq_of_live_level
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (u : ℝ) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t)
+    (hR : fullRadius G t z = (2 * z.im) * Real.exp (-u)) :
+    radiusLevelTime γ G z u = (t : ℝ≥0∞) := by
+  have he : {s : ℝ≥0∞ | s < ∞ ∧ z ∈ survivingDomain γ s.toNNReal ∧
+      fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u)} = {(t : ℝ≥0∞)} := by
+    ext s
+    constructor
+    · rintro ⟨hs, hsl, hsR⟩
+      have hh : s.toNNReal = t := live_radius_unique hU h z s.toNNReal t hsl ht (hsR.trans hR.symm)
+      have hse : (s.toNNReal : ℝ≥0∞) = s := ENNReal.coe_toNNReal hs.ne
+      rw [Set.mem_singleton_iff, ← hse, hh]
+    · intro hs
+      rw [Set.mem_singleton_iff] at hs
+      subst s
+      simpa only [Set.mem_ofPred_eq, ENNReal.coe_lt_top, ENNReal.toNNReal_coe, true_and] using And.intro ht hR
+  rw [radiusLevelTime, he, sInf_singleton]
+
+lemma stoppedGreen_eq_of_live_level
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (κ : ℝ) (z : ℂ) (u : ℝ) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t)
+    (hR : fullRadius G t z = (2 * z.im) * Real.exp (-u)) :
+    stoppedGreen κ U γ G z u = greenDensity κ U γ G t z := by
+  rw [stoppedGreen, radiusLevelTime_eq_of_live_level hU h z u t ht hR]
+  simp only [ENNReal.coe_lt_top, ↓reduceIte, ENNReal.toNNReal_coe]
+
+lemma stoppedGreen_eq_zero_of_no_level
+    (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ)
+    (z : ℂ) (u : ℝ)
+    (hn : ¬ ∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧
+      fullRadius G t z = (2 * z.im) * Real.exp (-u)) :
+    stoppedGreen κ U γ G z u = 0 := by
+  have he : {s : ℝ≥0∞ | s < ∞ ∧ z ∈ survivingDomain γ s.toNNReal ∧
+      fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u)} = ∅ := by
+    apply Set.eq_empty_iff_forall_notMem.mpr
+    rintro s ⟨_, hsl, hsR⟩
+    exact hn ⟨s.toNNReal, hsl, hsR⟩
+  simp only [stoppedGreen, radiusLevelTime, he, sInf_empty, lt_self_iff_false, ↓reduceIte]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma IsCapacityTwoWitness.fullRadius_initial
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (hz : z ∈ survivingDomain γ 0) :
+    fullRadius G 0 z = 2 * z.im := by
+  rw [fullRadius, h.initial_value z hz.1.1,
+    loewner_spatial_derivative hU h 0 z hz]
+  simp
+
+ 
+
+lemma radiusLevelTime_finite_of_live_radius_lt
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0)
+    (ht : z ∈ survivingDomain γ t)
+    (hR : fullRadius G t z < (2 * z.im) * Real.exp (-u)) :
+    radiusLevelTime γ G z u < ∞ := by
+  let L : ℝ → ℝ := fun s => Real.log ((2 * z.im) / fullRadius G s.toNNReal z)
+  have hy : 0 < 2 * z.im := mul_pos (by norm_num) ht.1.1
+  have hz0 : z ∈ survivingDomain γ 0 := survivingDomain_antitone γ zero_le ht
+  have hL0 : L 0 = 0 := by
+    simp only [L, Real.toNNReal_zero, h.fullRadius_initial hU z hz0,
+      div_self hy.ne', Real.log_one]
+  have hLt : u < L t := by
+    have he := fullRadius_eq_logClock hU h z t ht
+    rw [he] at hR
+    have hc := (mul_lt_mul_iff_right₀ hy).mp hR
+    rw [Real.exp_lt_exp] at hc
+    dsimp only [L]
+    rw [Real.toNNReal_coe, ← logRadiusClock_logCoordinates hU h z t ht]
+    linarith
+  have hLc : ContinuousOn L (Icc 0 (t : ℝ)) :=
+    fun s hs => (loewner_radius_clock_derivative hU h t z ht s hs).continuousWithinAt
+  obtain ⟨s, hs, hsu⟩ := intermediate_value_Icc t.coe_nonneg hLc
+    (show u ∈ Icc (L 0) (L t) from ⟨by rwa [hL0], hLt.le⟩)
+  have hsl : z ∈ survivingDomain γ s.toNNReal := h.live_at_real_time t z ht s hs.2
+  have hsr : fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u) := by
+    apply (fullRadius_level_iff_logClock hU h z s.toNNReal hsl u).mpr
+    rw [logRadiusClock_logCoordinates hU h z s.toNNReal hsl]
+    exact hsu
+  rw [radiusLevelTime_eq_of_live_level hU h z u s.toNNReal hsl hsr]
+  exact ENNReal.coe_lt_top
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+structure FiniteIncreasingClock (Ω : Type*) where
+  S : ℝ → Ω → ℝ
+  T : Ω → ℝ≥0
+  zero : ∀ ω, S 0 ω = 0
+  continuous : ∀ ω, Continuous (fun t => S t ω)
+  strict : ∀ ω, StrictMonoOn (fun t => S t ω) (Icc 0 (T ω : ℝ))
+
+namespace FiniteIncreasingClock
+variable {Ω : Type*} (c : FiniteIncreasingClock Ω)
+
+lemma terminal_nonneg (ω : Ω) : 0 ≤ c.S (c.T ω) ω := by
+  rw [← c.zero ω]
+  exact (c.strict ω).monotoneOn ⟨le_rfl, (c.T ω).coe_nonneg⟩
+    ⟨(c.T ω).coe_nonneg, le_rfl⟩ (c.T ω).coe_nonneg
+
+noncomputable def iso (ω : Ω) : Icc (0 : ℝ) (c.T ω : ℝ) ≃o Icc (0 : ℝ) (c.S (c.T ω) ω) := by
+  let f : Icc (0 : ℝ) (c.T ω : ℝ) → Icc (0 : ℝ) (c.S (c.T ω) ω) := fun t =>
+    ⟨c.S t ω, by
+      constructor
+      · simpa only [c.zero ω] using
+          ((c.strict ω).monotoneOn ⟨le_rfl, (c.T ω).coe_nonneg⟩ t.property t.property.1)
+      · exact (c.strict ω).monotoneOn t.property ⟨(c.T ω).coe_nonneg, le_rfl⟩ t.property.2⟩
+  apply StrictMono.orderIsoOfSurjective f
+  · intro s t hst
+    exact c.strict ω s.property t.property hst
+  · intro s
+    obtain ⟨t, ht, he⟩ := intermediate_value_Icc (c.T ω).coe_nonneg
+      (c.continuous ω).continuousOn (show (s : ℝ) ∈ Icc (c.S 0 ω) (c.S (c.T ω) ω) by
+        simpa only [c.zero ω] using s.property)
+    exact ⟨⟨t, ht⟩, Subtype.ext he⟩
+
+lemma iso_apply (ω : Ω) (t : Icc (0 : ℝ) (c.T ω : ℝ)) :
+    (c.iso ω t : ℝ) = c.S t ω := rfl
+
+noncomputable def inverse (s : ℝ) (ω : Ω) : ℝ :=
+  ((c.iso ω).symm ⟨min (max s 0) (c.S (c.T ω) ω),
+    ⟨le_min (le_max_right _ _) (c.terminal_nonneg ω), min_le_right _ _⟩⟩ : ℝ)
+
+lemma inverse_mem (s : ℝ) (ω : Ω) : c.inverse s ω ∈ Icc (0 : ℝ) (c.T ω : ℝ) :=
+  ((c.iso ω).symm _).property
+
+lemma clock_inverse (s : ℝ) (ω : Ω) :
+    c.S (c.inverse s ω) ω = min (max s 0) (c.S (c.T ω) ω) := by
+  exact congrArg Subtype.val ((c.iso ω).apply_symm_apply _)
+
+lemma inverse_clock (ω : Ω) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) (c.T ω : ℝ)) :
+    c.inverse (c.S t ω) ω = t := by
+  apply (c.strict ω).injOn (c.inverse_mem _ ω) ht
+  dsimp only
+  rw [c.clock_inverse]
+  have h0 : 0 ≤ c.S t ω := by
+    rw [← c.zero ω]
+    exact (c.strict ω).monotoneOn ⟨le_rfl, (c.T ω).coe_nonneg⟩ ht ht.1
+  have hT := (c.strict ω).monotoneOn ht ⟨(c.T ω).coe_nonneg, le_rfl⟩ ht.2
+  rw [max_eq_left h0, min_eq_left hT]
+
+lemma continuous_inverse (ω : Ω) : Continuous (fun s => c.inverse s ω) := by
+  apply continuous_subtype_val.comp
+  apply (c.iso ω).symm.continuous.comp
+  exact Continuous.subtype_mk ((continuous_id.max continuous_const).min continuous_const) _
+
+lemma inverse_le_iff (s : ℝ) (ω : Ω) (t : ℝ) (ht : 0 ≤ t) :
+    c.inverse s ω ≤ t ↔ (c.T ω : ℝ) ≤ t ∨
+      min (max s 0) (c.S (c.T ω) ω) ≤ c.S t ω := by
+  by_cases hT : (c.T ω : ℝ) ≤ t
+  · exact iff_of_true ((c.inverse_mem s ω).2.trans hT) (Or.inl hT)
+  · have htm : t ∈ Icc (0 : ℝ) (c.T ω : ℝ) := ⟨ht, (lt_of_not_ge hT).le⟩
+    rw [or_iff_right hT, ← c.clock_inverse]
+    exact ((c.strict ω).le_iff_le (c.inverse_mem s ω) htm).symm
+
+lemma measurable_inverse [MeasurableSpace Ω]
+    (hS : Measurable (Function.uncurry c.S)) (hT : Measurable c.T) :
+    Measurable (Function.uncurry c.inverse) := by
+  apply measurable_of_Iic
+  intro t
+  by_cases ht : 0 ≤ t
+  · have hterminal : Measurable (fun ω => c.S (c.T ω) ω) :=
+      hS.comp ((measurable_coe_nnreal_real.comp hT).prodMk measurable_id)
+    have he : (Function.uncurry c.inverse) ⁻¹' Iic t =
+        {p : ℝ × Ω | (c.T p.2 : ℝ) ≤ t} ∪
+        {p : ℝ × Ω | min (max p.1 0) (c.S (c.T p.2) p.2) ≤ c.S t p.2} := by
+      ext p
+      exact c.inverse_le_iff p.1 p.2 t ht
+    rw [he]
+    exact (measurableSet_le ((measurable_coe_nnreal_real.comp hT).comp measurable_snd) measurable_const).union
+      (measurableSet_le ((measurable_fst.max measurable_const).min (hterminal.comp measurable_snd))
+        (hS.comp (measurable_const.prodMk measurable_snd)))
+  · have he : (Function.uncurry c.inverse) ⁻¹' Iic t = ∅ := by
+      apply Set.eq_empty_iff_forall_notMem.mpr
+      intro p hp
+      exact ht ((c.inverse_mem p.1 p.2).1.trans hp)
+    rw [he]
+    exact MeasurableSet.empty
+
+end FiniteIncreasingClock
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma powerLogDensity_pos (κ p α : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    0 < powerLogDensity κ p α q x := Real.exp_pos _
+
+lemma powerLogDensity_le_radius
+    (κ p α η₀ u : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ α) (q : ℂ × ℝ) (x : ℝ)
+    (hs : logRadiusClock η₀ q ≤ u) :
+    powerLogDensity κ p α q x ≤ Real.exp (p * (u - Real.log 2 - η₀)) := by
+  rw [powerLogDensity_radius_angle κ p α η₀ q x]
+  have ha' : (logAngleSq κ q x) ^ (α / 2) ≤ 1 := by
+    simpa only [Real.one_rpow] using Real.rpow_le_rpow (logAngleSq_pos κ q x).le
+      (logAngleSq_le_one κ q x) (div_nonneg ha (by norm_num))
+  calc
+    _ ≤ Real.exp (p * (logRadiusClock η₀ q - Real.log 2 - η₀)) * 1 :=
+      mul_le_mul_of_nonneg_left ha' (Real.exp_pos _).le
+    _ ≤ _ := by rw [mul_one]; apply Real.exp_le_exp.mpr; gcongr
+
+lemma powerLogGenerator_abs_bound
+    (κ p α lo : ℝ) (q : ℂ × ℝ) (x : ℝ) (hlo : lo ≤ q.1.im) :
+    |powerLogGenerator κ p α q x| ≤
+      powerLogDensity κ p α q x *
+        (|(-4 * α + κ * α * (α + 1) / 2)| + |(4 * p - κ * α / 2)|) /
+          Real.exp (2 * lo) := by
+  let A := -4 * α + κ * α * (α + 1) / 2
+  let D := 4 * p - κ * α / 2
+  let X := q.1.re - Real.sqrt κ * x
+  let Y := Real.exp (2 * q.1.im)
+  let N := greenLogNormSq κ q x
+  let φ := powerLogDensity κ p α q x
+  have hN : 0 < N := greenLogNormSq_pos κ q x
+  have hY : 0 < Y := Real.exp_pos _
+  have hφ : 0 ≤ φ := (powerLogDensity_pos κ p α q x).le
+  have hX : X ^ 2 ≤ N := le_add_of_nonneg_right hY.le
+  have hYN : Y ≤ N := le_add_of_nonneg_left (sq_nonneg _)
+  have hlow : Real.exp (2 * lo) ≤ N :=
+    (Real.exp_le_exp.mpr (by linarith)).trans hYN
+  have hnum : |A * X ^ 2 + D * Y| ≤ (|A| + |D|) * N := by
+    calc
+      _ ≤ |A * X ^ 2| + |D * Y| := abs_add_le _ _
+      _ = |A| * X ^ 2 + |D| * Y := by
+        rw [abs_mul, abs_mul, abs_of_nonneg (sq_nonneg X), abs_of_pos hY]
+      _ ≤ |A| * N + |D| * N := add_le_add
+        (mul_le_mul_of_nonneg_left hX (abs_nonneg A))
+        (mul_le_mul_of_nonneg_left hYN (abs_nonneg D))
+      _ = _ := by ring
+  change |φ * (A * X ^ 2 + D * Y) / N ^ 2| ≤ φ * (|A| + |D|) / _
+  rw [abs_div, abs_mul, abs_of_nonneg hφ, abs_of_nonneg (sq_nonneg N)]
+  calc
+    _ ≤ φ * ((|A| + |D|) * N) / N ^ 2 := by gcongr
+    _ = φ * (|A| + |D|) / N := by field_simp
+    _ ≤ _ := div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) hlow
+
+lemma powerLogGenerator_le_radius_height
+    (κ p α η₀ u lo : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ α) (q : ℂ × ℝ) (x : ℝ)
+    (hs : logRadiusClock η₀ q ≤ u) (hlo : lo ≤ q.1.im) :
+    |powerLogGenerator κ p α q x| ≤
+      Real.exp (p * (u - Real.log 2 - η₀)) *
+        (|(-4 * α + κ * α * (α + 1) / 2)| + |(4 * p - κ * α / 2)|) /
+          Real.exp (2 * lo) := by
+  apply (powerLogGenerator_abs_bound κ p α lo q x hlo).trans
+  apply div_le_div_of_nonneg_right _ (Real.exp_pos _).le
+  exact mul_le_mul_of_nonneg_right (powerLogDensity_le_radius κ p α η₀ u hp ha q x hs)
+    (add_nonneg (abs_nonneg _) (abs_nonneg _))
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem exists_finite_angular_kernel
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
+    (ε : ℝ) (hε : 0 < ε) (hε0 : ε ≤ logAngleSq κ (initialLogCoordinates z) 0) :
+    ∃ f : Ω → ℝ,
+      Integrable f P ∧ (∫ ω, f ω ∂P) = initialGreen κ z ∧
+      (∀ ω, 0 ≤ f ω ∧ f ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))) ∧
+      ∀ (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
+        IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F →
+        ∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧
+          Real.log ((2 * z.im) / fullRadius G t z) ≤ u ∧
+          (Real.log ((2 * z.im) / fullRadius G t z) = u ∨
+            (Real.sin (Complex.arg (G t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 = ε) ∧
+          f ω = greenDensity κ (fun t => Real.sqrt κ * B t ω) γ G t z ∧
+          ((Real.sin (Complex.arg (G t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 = ε →
+            f ω ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) *
+              ε ^ (greenExponent κ / 2)) := by
+  let η₀ := Real.log z.im
+  let lo := η₀ - u / (2 * ε) - 1
+  let hi := η₀ + 1
+  have hule : 0 ≤ u / (2 * ε) := by positivity
+  have hle : lo ≤ hi := by dsimp [lo, hi]; linarith
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  obtain ⟨c⟩ := exists_clippedLogSolution B hm hBm hBc κ lo hi hle (initialLogCoordinates z)
+  have hη : (initialLogCoordinates z).1.im = η₀ := by
+    simp only [initialLogCoordinates, Complex.add_im, Complex.ofReal_im, Complex.mul_im,
+      Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, zero_add, add_zero]
+    rfl
+  let a := 4 * ε ^ 2 / Real.exp (2 * η₀)
+  have ha : 0 < a := by dsimp [a]; positivity
+  obtain ⟨N, hN⟩ := exists_nat_gt (u / a)
+  have hT : u < (4 * ε ^ 2 / Real.exp (2 * (initialLogCoordinates z).1.im)) * ((N : ℝ≥0) : ℝ) := by
+    rw [hη]
+    have hh := (div_lt_iff₀ ha).mp hN
+    simpa only [NNReal.coe_natCast, mul_comm (N : ℝ) a] using hh
+  obtain ⟨τ, hτ, ht, hin, hpre, hbound, hfi, hfm⟩ := c.finite_angular_law
+    hB hBm hBc hB0 hκ hκ8 rfl u ε hu hε hε0
+    (by rw [hη]; dsimp [lo]; linarith) (by rw [hη]; dsimp [hi]; linarith) (N : ℝ≥0) hT
+  let f (ω : Ω) := greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)
+  refine ⟨f, hfi, hfm.trans (greenLogDensity_initial κ z hz), ?_, ?_⟩
+  · intro ω
+    refine ⟨greenLogDensity_nonneg _ _ _, ?_⟩
+    have hh := greenLogDensity_le_radius κ (initialLogCoordinates z).1.im u hκ hκ8 (c.Q (τ ω) ω) (B (τ ω) ω)
+      (hin ω (τ ω) le_rfl).1
+    simpa only [hη] using hh
+  · intro ω γ G F hw
+    have hU : Continuous (fun t => Real.sqrt κ * B t ω) := (hBc ω).const_mul _
+    have hQd (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) := (c.derivative ω s hs.1).mono
+      (show Icc (0 : ℝ) (τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+    have hQb (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) : (c.Q s ω).1.im ∈ Icc lo hi := by
+      have hh := (hin ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.2
+      simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
+    obtain ⟨hlive, heq⟩ := clipped_log_identification hU hw z hz (τ ω) lo hi hle
+      (fun s => c.Q s ω) (c.initial ω) hQd hQb
+    have he : c.Q (τ ω) ω = logLoewnerCoordinates G z (τ ω) :=
+      heq ⟨(τ ω).coe_nonneg, le_rfl⟩
+    have hr : logRadiusClock (initialLogCoordinates z).1.im (c.Q (τ ω) ω) =
+        Real.log ((2 * z.im) / fullRadius G (τ ω) z) := by
+      rw [he, hη]
+      exact logRadiusClock_logCoordinates hU hw z (τ ω) hlive
+    have ha' : logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) =
+        (Real.sin (Complex.arg (G (τ ω) z - ((Real.sqrt κ * B (τ ω) ω : ℝ) : ℂ)))) ^ 2 := by
+      rw [he]
+      exact logAngleSq_logCoordinates κ _ G z _ (hw.imag_positive _ z hlive)
+    refine ⟨τ ω, hlive, ?_, ?_, ?_, ?_⟩
+    · simpa only [hr] using (hin ω (τ ω) le_rfl).1
+    · simpa only [hr, ha'] using hbound ω
+    · dsimp only [f]
+      rw [he]
+      exact greenLogDensity_logCoordinates hU hw κ _ z _ hlive rfl
+    · intro hA
+      have hh := greenLogDensity_le_at_angular_exit κ (initialLogCoordinates z).1.im u ε hκ hκ8
+        (c.Q (τ ω) ω) (B (τ ω) ω) (hin ω (τ ω) le_rfl).1 (by rw [ha', hA])
+      simpa only [hη] using hh
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+structure AngularLocalization {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (κ : ℝ)
+    (z : ℂ) (u ε : ℝ) where
+  lo : ℝ
+  hi : ℝ
+  le : lo ≤ hi
+  solution : ClippedLogSolution B hm κ lo hi le (initialLogCoordinates z)
+  T : ℝ≥0
+  τ : Ω → ℝ≥0
+  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+    (fun ω => (τ ω : WithTop ℝ≥0))
+  bounded : ∀ ω, τ ω ≤ T
+  inside : ∀ ω t, t ≤ τ ω →
+    logRadiusClock (Real.log z.im) (solution.Q t ω) ≤ u ∧
+    ε ≤ logAngleSq κ (solution.Q t ω) (B t ω) ∧
+    lo ≤ (solution.Q t ω).1.im ∧ (solution.Q t ω).1.im ≤ hi
+  before : ∀ ω t, t < τ ω →
+    logRadiusClock (Real.log z.im) (solution.Q t ω) < u ∧
+    ε < logAngleSq κ (solution.Q t ω) (B t ω)
+  boundary : ∀ ω,
+    logRadiusClock (Real.log z.im) (solution.Q (τ ω) ω) = u ∨
+    logAngleSq κ (solution.Q (τ ω) ω) (B (τ ω) ω) = ε
+
+lemma initialLogCoordinates_im (z : ℂ) :
+    (initialLogCoordinates z).1.im = Real.log z.im := by
+  simp [initialLogCoordinates]
+
+lemma exists_angularLocalization
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ) (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
+    (hε0 : ε ≤ logAngleSq κ (initialLogCoordinates z) 0) :
+    Nonempty (AngularLocalization B hm κ z u ε) := by
+  let η₀ := Real.log z.im
+  let lo := η₀ - u / (2 * ε) - 1
+  let hi := η₀ + 1
+  have hule : 0 ≤ u / (2 * ε) := by positivity
+  have hle : lo ≤ hi := by dsimp [lo, hi]; linarith
+  obtain ⟨c⟩ := exists_clippedLogSolution B hm hBm hBc κ lo hi hle (initialLogCoordinates z)
+  let a := 4 * ε ^ 2 / Real.exp (2 * η₀)
+  have ha : 0 < a := by dsimp [a]; positivity
+  obtain ⟨N, hN⟩ := exists_nat_gt (u / a)
+  have hT : u < (4 * ε ^ 2 / Real.exp (2 * (initialLogCoordinates z).1.im)) * ((N : ℝ≥0) : ℝ) := by
+    rw [initialLogCoordinates_im]
+    have hh := (div_lt_iff₀ ha).mp hN
+    simpa only [NNReal.coe_natCast, mul_comm (N : ℝ) a] using hh
+  obtain ⟨τ, hτ, ht, hin, hpre, hbound, _, _⟩ := c.finite_angular_law
+    hB hBm hBc hB0 hκ hκ8 rfl u ε hu hε hε0
+    (by rw [initialLogCoordinates_im]; dsimp [lo, η₀]; linarith)
+    (by rw [initialLogCoordinates_im]; dsimp [hi, η₀]; linarith) (N : ℝ≥0) hT
+  exact ⟨⟨lo, hi, hle, c, N, τ, hτ, fun ω => (ht ω).le,
+    by simpa only [initialLogCoordinates_im] using hin,
+    by simpa only [initialLogCoordinates_im] using hpre,
+    by simpa only [initialLogCoordinates_im] using hbound⟩⟩
+
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ}
+  (a : AngularLocalization B hm κ z u ε)
+
+lemma chart_identification
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F) :
+    z ∈ survivingDomain γ (a.τ ω) ∧
+      ∀ t : ℝ≥0, t ≤ a.τ ω → a.solution.Q t ω = logLoewnerCoordinates G z t := by
+  have hQd (s : ℝ) (hs : s ∈ Icc 0 (a.τ ω : ℝ)) := (a.solution.derivative ω s hs.1).mono
+    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+  have hQb (s : ℝ) (hs : s ∈ Icc 0 (a.τ ω : ℝ)) : (a.solution.Q s ω).1.im ∈ Icc a.lo a.hi := by
+    have hh := (a.inside ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.2
+    simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
+  obtain ⟨hlive, heq⟩ := clipped_log_identification ((hBc ω).const_mul _) hw z hz (a.τ ω)
+    a.lo a.hi a.le (fun s => a.solution.Q s ω) (a.solution.initial ω) hQd hQb
+  exact ⟨hlive, fun t ht => by simpa only [Real.toNNReal_coe] using heq ⟨t.coe_nonneg, ht⟩⟩
+
+lemma clock_eq_chart
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F)
+    (t : ℝ≥0) (ht : t ≤ a.τ ω) :
+    logRadiusClock (Real.log z.im) (a.solution.Q t ω) =
+      Real.log ((2 * z.im) / fullRadius G t z) := by
+  obtain ⟨hl, he⟩ := a.chart_identification hBc hz ω γ G F hw
+  rw [he t ht]
+  exact logRadiusClock_logCoordinates ((hBc ω).const_mul _) hw z t (survivingDomain_antitone γ ht hl)
+
+lemma clock_stopped_measurable :
+    Measurable[a.stopping.measurableSpace]
+      (fun ω => logRadiusClock (Real.log z.im) (a.solution.Q (a.τ ω) ω)) := by
+  have hQa : StronglyAdapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => a.solution.Q t) := fun t => (a.solution.adapted t).stronglyMeasurable
+  have hQc (ω : Ω) : Continuous (fun t : ℝ≥0 => a.solution.Q t ω) :=
+    (a.solution.lipschitz ω).continuous.comp NNReal.continuous_coe
+  have hQ := measurable_stoppedValue (hQa.isStronglyProgressive_of_continuous hQc) a.stopping
+  have hmQ : Measurable[a.stopping.measurableSpace] (fun ω => a.solution.Q (a.τ ω) ω) := by
+    exact hQ
+  exact (continuous_logRadiusClock _).measurable.comp hmQ
+
+lemma mean_green
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane) :
+    Integrable (fun ω => greenLogDensity κ (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
+      (∫ ω, greenLogDensity κ (a.solution.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) = initialGreen κ z := by
+  obtain ⟨hi, he⟩ := a.solution.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
+    (Real.log z.im) u a.τ a.stopping a.T a.bounded
+    (fun ω t ht => ⟨(a.inside ω t ht).1, (a.inside ω t ht).2.2⟩)
+  exact ⟨hi, he.trans (greenLogDensity_initial κ z hz)⟩
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/DyadicStrips.lean b/lean/OAI/Probability/SLE/DyadicStrips.lean
index 5d2717ac3ea9cbb58c8625c5ed749cf3457845f8..9af5dbe5033ffbc1c6137ba9671408850bb9f406 100644
--- a/lean/OAI/Probability/SLE/DyadicStrips.lean
+++ b/lean/OAI/Probability/SLE/DyadicStrips.lean
@@ -1,697 +1,697 @@
-import Mathlib
-import OAI.Probability.SLE.RectangleMeans
-import OAI.Probability.SLE.SausagePointBound
-
-namespace OAI
-
-/-! Dyadic Strips for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma inv_pow_rpow_two (k : ℕ) (α : ℝ) :
-    (1/(2:ℝ)^k)^α = (2:ℝ)^(-α*(k:ℝ)) := by
-  rw [one_div,Real.inv_rpow (by positivity)]
-  rw [← Real.rpow_natCast (2:ℝ) k,← Real.rpow_mul (by norm_num : 0 ≤ (2:ℝ)),
-    ← Real.rpow_neg (by norm_num : 0 ≤ (2:ℝ))]
-  congr 1
-  ring
-
- 
-
-theorem sle_dyadic_strip_rectangle_hit
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (j : ℤ) (k : ℕ) (i m : Fin (2^k)) :
-    P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' dyadicStripRectangle j k i m} ≤
-      ENNReal.ofReal (stripHittingConstant κ *
-        (2:ℝ)^(-annularApproachPower κ*(k:ℝ))) := by
-  have hk : 0 < (2:ℝ)^k := by positivity
-  have hm : (m:ℝ) ≤ (2:ℝ)^k := by
-    exact_mod_cast m.isLt.le
-  have hz : 0 ≤ Real.pi*((m:ℝ)/(2:ℝ)^k) := by positivity
-  have hp : Real.pi*((m:ℝ)/(2:ℝ)^k) ≤ Real.pi := by
-    exact mul_le_of_le_one_right Real.pi_pos.le ((div_le_one hk).mpr hm)
-  have hh := sle_exp_strip_rectangle_hit κ hκ hκ8 γ hSLE
-    ((j:ℝ)+(i:ℝ)/(2:ℝ)^k) (Real.pi*((m:ℝ)/(2:ℝ)^k))
-    (1/(2:ℝ)^k) hz hp (by positivity)
-  simpa only [dyadicStripRectangle,inv_pow_rpow_two] using hh
-
- 
-
-theorem ordinary_sle_strip_hitting_constants
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∃ C α : ℝ, 0 < C ∧ 0 < α ∧
-      ∀ {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-        (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-        ∀ (j : ℤ) (k : ℕ) (i m : Fin (2^k)),
-        P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' dyadicStripRectangle j k i m} ≤
-          ENNReal.ofReal (C*(2:ℝ)^(-α*(k:ℝ))) := by
-  refine ⟨stripHittingConstant κ,annularApproachPower κ,
-    stripHittingConstant_pos κ hκ hκ8,annularApproachPower_pos κ hκ hκ8,?_⟩
-  intro Ω _ P γ hγ j k i m
-  exact sle_dyadic_strip_rectangle_hit κ hκ hκ8 γ hγ j k i m
-
-end SLEExactGauge
-end
-
-section
-
-open Set Filter MeasureTheory
-open scoped Topology ContDiff ENNReal BigOperators NNReal
-namespace SLEExactGauge
-
- 
-def IsStripWeakGradient (u : ℂ → ℝ) (D : ℂ → ℂ) : Prop :=
-  MemLp u 2 (volume.restrict openUnitStrip) ∧
-  MemLp D 2 (volume.restrict openUnitStrip) ∧
-  ∀ φ : ℂ → ℝ, ContDiff ℝ ⊤ φ → HasCompactSupport φ → tsupport φ ⊆ openUnitStrip →
-    (∫ z in openUnitStrip, u z * (fderiv ℝ φ z) (1:ℂ)) =
-      -(∫ z in openUnitStrip, (D z).re * φ z) ∧
-    (∫ z in openUnitStrip, u z * (fderiv ℝ φ z) Complex.I) =
-      -(∫ z in openUnitStrip, (D z).im * φ z)
-
-noncomputable def annulusKernel (q : ℂ) (r R : ℝ) (w : ℂ) : ℝ≥0∞ :=
-  if r < ‖w-q‖ ∧ ‖w-q‖ < R then ENNReal.ofReal (‖w-q‖^2)⁻¹ else 0
-
-lemma measurable_annulusKernel (q : ℂ) (r R : ℝ) : Measurable (annulusKernel q r R) := by
-  unfold annulusKernel
-  exact Measurable.ite ((isOpen_lt continuous_const (continuous_id.sub continuous_const).norm).measurableSet.inter
-    (isOpen_lt (continuous_id.sub continuous_const).norm continuous_const).measurableSet)
-    (ENNReal.measurable_ofReal.comp ((((measurable_id.sub measurable_const).norm).pow_const 2).inv))
-    measurable_const
-
-lemma lintegral_inv_Ioo (r R : ℝ) (hr : 0 < r) (hrR : r < R) :
-    (∫⁻ x in Ioo r R, ENNReal.ofReal x⁻¹) = ENNReal.ofReal (Real.log (R/r)) := by
-  have hc : ContinuousOn (fun x : ℝ => x⁻¹) (Icc r R) :=
-    continuousOn_id.inv₀ (fun x hx => ne_of_gt (hr.trans_le hx.1))
-  have hi : IntegrableOn (fun x : ℝ => x⁻¹) (Ioo r R) :=
-    hc.integrableOn_Icc.mono_set Ioo_subset_Icc_self
-  rw [← ofReal_integral_eq_lintegral_ofReal hi]
-  · congr 1
-    change (∫ x, x⁻¹ ∂volume.restrict (Ioo r R)) = _
-    rw [restrict_Ioo_eq_restrict_Ioc,← intervalIntegral.integral_of_le hrR.le]
-    apply integral_inv
-    rw [uIcc_of_le hrR.le]
-    intro h
-    exact not_le_of_gt hr h.1
-  · filter_upwards [ae_restrict_mem measurableSet_Ioo] with x hx
-    exact inv_nonneg.mpr (hr.trans hx.1).le
-
-lemma annulusKernel_polar (r R : ℝ) (p : ℝ × ℝ) (hp : 0 < p.1) :
-    ENNReal.ofReal p.1 * annulusKernel 0 r R (Complex.polarCoord.symm p) =
-      (Ioo r R).indicator (fun x : ℝ => ENNReal.ofReal x⁻¹) p.1 := by
-  simp only [annulusKernel,sub_zero,Complex.norm_polarCoord_symm,abs_of_pos hp]
-  by_cases h : r < p.1 ∧ p.1 < R
-  · simp only [Set.indicator,Set.mem_Ioo,ite_eq_left h]
-    rw [← ENNReal.ofReal_mul hp.le]
-    congr 1
-    field_simp
-  · simp only [Set.indicator,Set.mem_Ioo,ite_eq_right h,mul_zero]
-
- 
-
-theorem annulusKernel_integral (q : ℂ) (r R : ℝ) (hr : 0 < r) (hrR : r < R) :
-    (∫⁻ w, annulusKernel q r R w) = ENNReal.ofReal (2*Real.pi*Real.log (R/r)) := by
-  have hcenter : (∫⁻ w, annulusKernel q r R w) = ∫⁻ w, annulusKernel 0 r R w := by
-    simpa only [annulusKernel,sub_zero] using
-      (lintegral_sub_right_eq_self (μ := (volume : Measure ℂ)) (annulusKernel 0 r R) q)
-  rw [hcenter,← Complex.lintegral_comp_polarCoord_symm]
-  let f : ℝ → ℝ≥0∞ := (Ioo r R).indicator (fun x : ℝ => ENNReal.ofReal x⁻¹)
-  have hf : Measurable f := (ENNReal.measurable_ofReal.comp measurable_inv).indicator measurableSet_Ioo
-  have he : (∫⁻ p in polarCoord.target, ENNReal.ofReal p.1 •
-      annulusKernel 0 r R (Complex.polarCoord.symm p)) =
-      ∫⁻ p in Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi, f p.1 := by
-    apply setLIntegral_congr_fun (measurableSet_Ioi.prod measurableSet_Ioo)
-    intro p hp
-    exact annulusKernel_polar r R p hp.1
-  rw [he]
-  change (∫⁻ p in Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi, f p.1 ∂volume.prod volume) = _
-  rw [setLIntegral_prod (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ))
-    (fun p : ℝ × ℝ => f p.1) ((hf.comp measurable_fst).aemeasurable)]
-  simp only [setLIntegral_const,Real.volume_Ioo,sub_neg_eq_add]
-  rw [lintegral_mul_const' _ _ ENNReal.ofReal_ne_top]
-  have hsub : Ioo r R ⊆ Ioi (0:ℝ) := fun x hx => hr.trans hx.1
-  have hfint : (∫⁻ x in Ioi (0:ℝ), f x) = ENNReal.ofReal (Real.log (R/r)) := by
-    dsimp [f]
-    rw [setLIntegral_indicator measurableSet_Ioo,inter_eq_left.mpr hsub,
-      lintegral_inv_Ioo r R hr hrR]
-  rw [hfint,← ENNReal.ofReal_mul]
-  · congr 1
-    ring
-  · exact Real.log_nonneg ((one_le_div hr).mpr hrR.le)
-
-lemma complex_deriv_real_det (d : ℂ) :
-    ((ContinuousLinearMap.toSpanSingleton ℂ d).restrictScalars ℝ).det = ‖d‖^2 := by
-  simp [ContinuousLinearMap.det,LinearMap.det_restrictScalars,
-    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
-
- 
-
-theorem univalent_annulus_dirichlet_bound {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F {z | 0 < z.im})
-    (hi : InjOn F {z | 0 < z.im}) (q : ℂ) (r R : ℝ) (hr : 0 < r) (hrR : r < R) :
-    (∫⁻ z in {z : ℂ | 0 < z.im}, ENNReal.ofReal (‖deriv F z‖^2) *
-      annulusKernel q r R (F z)) ≤ ENNReal.ofReal (2*Real.pi*Real.log (R/r)) := by
-  have ho : IsOpen {z : ℂ | 0 < z.im} := isOpen_lt continuous_const Complex.continuous_im
-  let D : ℂ → ℂ →L[ℝ] ℂ := fun z => (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
-  have hD : ∀ z ∈ {z : ℂ | 0 < z.im}, HasFDerivWithinAt F (D z) {z : ℂ | 0 < z.im} z := by
-    intro z hz
-    exact ((hF.differentiableAt (ho.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
-  have he := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume ho.measurableSet hD hi (annulusKernel q r R)
-  have hdet (z : ℂ) : |(D z).det| = ‖deriv F z‖^2 := by
-    rw [show D z = (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ from rfl,
-      complex_deriv_real_det,abs_of_nonneg (sq_nonneg _)]
-  simp only [hdet] at he
-  rw [← he]
-  calc
-    _ ≤ ∫⁻ w, annulusKernel q r R w := setLIntegral_le_lintegral _ _
-    _ = _ := annulusKernel_integral q r R hr hrR
-
-noncomputable def truncatedConformalLog (F : ℂ → ℂ) (q : ℂ) (r H : ℝ) (z : ℂ) : ℝ :=
-  min H (max 0 (Real.log (‖F z-q‖/r)))
-
-lemma lipschitz_clip (H : ℝ) : LipschitzWith 1 (fun x : ℝ => min H (max 0 x)) := by
-  have hmax : LipschitzWith 1 (fun x : ℝ => max 0 x) := by
-    simpa [Function.comp_def] using lipschitzWith_max.comp ((LipschitzWith.const (0:ℝ)).prodMk LipschitzWith.id)
-  simpa [Function.comp_def] using lipschitzWith_min.comp ((LipschitzWith.const H).prodMk hmax)
-
-lemma norm_fderiv_clip_le {v : ℂ → ℝ} {z : ℂ} (hv : ContDiffAt ℝ 1 v z) (H : ℝ) :
-    ‖fderiv ℝ (fun w => min H (max 0 (v w))) z‖ ≤ ‖fderiv ℝ v z‖ := by
-  apply le_of_forall_gt_imp_ge_of_dense
-  intro K hK
-  have hK0 : 0 ≤ K := (norm_nonneg _).trans hK.le
-  obtain ⟨s,hs,hvs⟩ := hv.exists_lipschitzOnWith_of_nnnorm_lt ⟨K,hK0⟩ hK
-  have hc := (lipschitz_clip H).comp_lipschitzOnWith hvs
-  have he : (1 : ℝ≥0)*⟨K,hK0⟩ = ⟨K,hK0⟩ := one_mul _
-  rw [he] at hc
-  exact norm_fderiv_le_of_lipschitzOn ℝ hs hc
-
-lemma norm_fderiv_log_norm_div_le {F : ℂ → ℂ} {z : ℂ}
-    (hF : DifferentiableAt ℂ F z) (q : ℂ) {r : ℝ} (hr : 0 < r)
-    (h0 : F z-q ≠ 0) :
-    ‖fderiv ℝ (fun w => Real.log (‖F w-q‖/r)) z‖ ≤ ‖deriv F z‖ / ‖F z-q‖ := by
-  let A : ℂ →L[ℝ] ℂ := (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
-  have hA : HasFDerivAt (fun w => F w-q) A z :=
-    ((hF.hasDerivAt.hasFDerivAt.restrictScalars ℝ).sub_const q)
-  have hn := (differentiableAt_id.norm ℝ h0).hasFDerivAt
-  have hnorm : ‖fderiv ℝ (fun w : ℂ => ‖w‖) (F z-q)‖ = 1 :=
-    norm_fderiv_norm (differentiableAt_id.norm ℝ h0)
-  have hn0 : 0 < ‖F z-q‖ := norm_pos_iff.mpr h0
-  have hl := ((hn.comp z hA).log (norm_ne_zero_iff.mpr h0)).sub_const (Real.log r)
-  have he : (fun w => Real.log (‖F w-q‖/r)) =ᶠ[𝓝 z]
-      (fun w => Real.log ‖F w-q‖ - Real.log r) := by
-    filter_upwards [(hA.continuousAt.norm).eventually (Ioi_mem_nhds hn0)] with w hw
-    exact Real.log_div (ne_of_gt hw) hr.ne'
-  have hd := hl.congr_of_eventuallyEq he
-  rw [hd.fderiv,norm_smul,Real.norm_eq_abs]
-  simp only [Function.comp_apply,id_eq] at *
-  rw [abs_of_pos (inv_pos.mpr hn0)]
-  have hAnorm : ‖A‖ = ‖deriv F z‖ := by
-    simp [A]
-  have hb := (fderiv ℝ (fun w : ℂ => ‖w‖) (F z-q)).opNorm_comp_le A
-  rw [hnorm,one_mul,hAnorm] at hb
-  simpa [div_eq_mul_inv,mul_comm] using mul_le_mul_of_nonneg_left hb (inv_nonneg.mpr hn0.le)
-
-lemma truncatedConformalLog_eq_zero {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
-    (hr : 0 < r) (hH : 0 ≤ H) (hl : ‖F z-q‖ ≤ r) :
-    truncatedConformalLog F q r H z = 0 := by
-  have hb : Real.log (‖F z-q‖/r) ≤ 0 := Real.log_nonpos
-    (div_nonneg (norm_nonneg _) hr.le) ((div_le_one hr).mpr hl)
-  simp [truncatedConformalLog,max_eq_left hb,min_eq_right hH]
-
-lemma truncatedConformalLog_eq_height {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
-    (hr : 0 < r) (hu : Real.exp H*r ≤ ‖F z-q‖) :
-    truncatedConformalLog F q r H z = H := by
-  have hn : 0 < ‖F z-q‖ := (mul_pos (Real.exp_pos _) hr).trans_le hu
-  have hb : H ≤ Real.log (‖F z-q‖/r) :=
-    (Real.le_log_iff_exp_le (div_pos hn hr)).mpr ((le_div_iff₀ hr).mpr hu)
-  exact min_eq_left (hb.trans (le_max_right _ _))
-
-lemma fderiv_truncated_eq_zero_of_low {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
-    (hF : ContinuousAt F z) (hr : 0 < r) (hH : 0 ≤ H) (hl : ‖F z-q‖ < r) :
-    fderiv ℝ (truncatedConformalLog F q r H) z = 0 := by
-  have he : truncatedConformalLog F q r H =ᶠ[𝓝 z] (fun _ => (0:ℝ)) := by
-    filter_upwards [((hF.sub_const q).norm).eventually (Iio_mem_nhds hl)] with w hw
-    exact truncatedConformalLog_eq_zero hr hH hw.le
-  exact ((hasFDerivAt_const (0:ℝ) z).congr_of_eventuallyEq he).fderiv
-
-lemma fderiv_truncated_eq_zero_of_high {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
-    (hF : ContinuousAt F z) (hr : 0 < r) (hu : Real.exp H*r < ‖F z-q‖) :
-    fderiv ℝ (truncatedConformalLog F q r H) z = 0 := by
-  have he : truncatedConformalLog F q r H =ᶠ[𝓝 z] (fun _ => H) := by
-    filter_upwards [((hF.sub_const q).norm).eventually (Ioi_mem_nhds hu)] with w hw
-    exact truncatedConformalLog_eq_height hr hw.le
-  exact ((hasFDerivAt_const H z).congr_of_eventuallyEq he).fderiv
-
-lemma norm_fderiv_truncated_le {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) {z : ℂ} (hz : z ∈ D) (q : ℂ) {r H : ℝ}
-    (hr : 0 < r) (h0 : F z-q ≠ 0) :
-    ‖fderiv ℝ (truncatedConformalLog F q r H) z‖ ≤ ‖deriv F z‖ / ‖F z-q‖ := by
-  have hcF : ContDiffAt ℝ 1 F z := ((hF.analyticAt (hD.mem_nhds hz)).restrictScalars (𝕜 := ℝ)).contDiffAt
-  have hv : ContDiffAt ℝ 1 (fun w => Real.log (‖F w-q‖/r)) z :=
-    (((hcF.sub contDiffAt_const).norm ℝ h0).div_const r).log
-      (div_ne_zero (norm_ne_zero_iff.mpr h0) hr.ne')
-  exact (norm_fderiv_clip_le hv H).trans
-    (norm_fderiv_log_norm_div_le (hF.differentiableAt (hD.mem_nhds hz)) q hr h0)
-
- 
-
-lemma truncated_log_energy_pointwise {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) {z : ℂ} (hz : z ∈ D) (q : ℂ)
-    {r H a b : ℝ} (hr : 0 < r) (hH : 0 < H) (ha : 0 < a) (har : a < r)
-    (hRb : Real.exp H*r < b) :
-    ENNReal.ofReal (‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2) ≤
-      ENNReal.ofReal (‖deriv F z‖^2) * annulusKernel q a b (F z) := by
-  have hc := (hF.differentiableAt (hD.mem_nhds hz)).continuousAt
-  by_cases hband : a < ‖F z-q‖ ∧ ‖F z-q‖ < b
-  · have hn : 0 < ‖F z-q‖ := ha.trans hband.1
-    have hb := norm_fderiv_truncated_le hD hF hz q hr (norm_ne_zero_iff.mp hn.ne') (H := H)
-    have hs : ‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2 ≤
-        ‖deriv F z‖^2 * (‖F z-q‖^2)⁻¹ := by
-      calc
-        _ ≤ (‖deriv F z‖/‖F z-q‖)^2 := pow_le_pow_left₀ (norm_nonneg _) hb 2
-        _ = _ := by rw [div_pow,div_eq_mul_inv]
-    simpa only [annulusKernel,ite_eq_left hband,← ENNReal.ofReal_mul (sq_nonneg _)] using
-      ENNReal.ofReal_le_ofReal hs
-  · have he : fderiv ℝ (truncatedConformalLog F q r H) z = 0 := by
-      rcases not_and_or.mp hband with hl | hu
-      · exact fderiv_truncated_eq_zero_of_low hc hr hH.le ((le_of_not_gt hl).trans_lt har)
-      · exact fderiv_truncated_eq_zero_of_high hc hr (hRb.trans_le (le_of_not_gt hu))
-    simp [he]
-
-lemma truncated_log_lipschitz_near {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) {z : ℂ} (hz : z ∈ D) (q : ℂ) {r H : ℝ}
-    (hr : 0 < r) (hH : 0 < H) :
-    ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ z ∈ U ∧
-      LipschitzOnWith K (truncatedConformalLog F q r H) U := by
-  by_cases h0 : F z-q = 0
-  · have hn : ‖F z-q‖ < r := by simpa [h0] using hr
-    have he := (((hF.differentiableAt (hD.mem_nhds hz)).continuousAt.sub_const q).norm).eventually
-      (Iio_mem_nhds hn)
-    obtain ⟨U,hU,hUo,hzU⟩ := mem_nhds_iff.mp he
-    refine ⟨0,U,hUo,hzU,?_⟩
-    intro x hx y hy
-    rw [truncatedConformalLog_eq_zero hr hH.le (hU hx).le,
-      truncatedConformalLog_eq_zero hr hH.le (hU hy).le]
-    simp
-  · have hcF : ContDiffAt ℝ 1 F z :=
-      ((hF.analyticAt (hD.mem_nhds hz)).restrictScalars (𝕜 := ℝ)).contDiffAt
-    have hv : ContDiffAt ℝ 1 (fun w => Real.log (‖F w-q‖/r)) z :=
-      (((hcF.sub contDiffAt_const).norm ℝ h0).div_const r).log
-        (div_ne_zero (norm_ne_zero_iff.mpr h0) hr.ne')
-    let K : ℝ≥0 := ‖fderiv ℝ (fun w => Real.log (‖F w-q‖/r)) z‖₊ + 1
-    have hK : ‖fderiv ℝ (fun w => Real.log (‖F w-q‖/r)) z‖₊ < K := by
-      dsimp [K]; simp
-    obtain ⟨U,hU,hLip⟩ := hv.exists_lipschitzOnWith_of_nnnorm_lt K hK
-    refine ⟨1*K,interior U,isOpen_interior,mem_interior_iff_mem_nhds.mpr hU,?_⟩
-    exact ((lipschitz_clip H).comp_lipschitzOnWith hLip).mono interior_subset
-
- 
-theorem truncated_log_ae_differentiable {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    ∀ᵐ z ∂volume, z ∈ D → DifferentiableAt ℝ (truncatedConformalLog F q r H) z := by
-  classical
-  have hloc (x : D) := truncated_log_lipschitz_near hD hF x.2 q hr hH
-  choose K U hUo hxU hULip using hloc
-  have hcov : D ⊆ ⋃ x : D, U x := fun z hz => mem_iUnion.mpr ⟨⟨z,hz⟩,hxU ⟨z,hz⟩⟩
-  obtain ⟨t,ht,htcov⟩ := (HereditarilyLindelofSpace.isLindelof D).elim_countable_subcover U hUo hcov
-  have ha (x : D) : ∀ᵐ z ∂volume, z ∈ U x →
-      DifferentiableAt ℝ (truncatedConformalLog F q r H) z := by
-    filter_upwards [(hULip x).ae_differentiableWithinAt_of_mem (μ := volume)] with z hz hzU
-    exact (hz hzU).differentiableAt ((hUo x).mem_nhds hzU)
-  filter_upwards [(ae_ball_iff ht).mpr (fun x _ => ha x)] with z hz hzD
-  obtain ⟨x,htx,hzx⟩ := mem_iUnion₂.mp (htcov hzD)
-  exact hz x htx hzx
-
-lemma local_lipschitz_weak_derivative {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
-    (hlocal : ∀ x ∈ D, ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ x ∈ U ∧ LipschitzOnWith K u U)
-    (φ : ℂ → ℝ) (hφ : ContDiff ℝ ⊤ φ) (hcφ : HasCompactSupport φ)
-    (hsφ : tsupport φ ⊆ D) (v : ℂ) :
-    (∫ z in D, u z * (fderiv ℝ φ z) v) =
-      -(∫ z in D, (fderiv ℝ u z) v * φ z) := by
-  obtain ⟨S,hSc,hφS,hSD⟩ := exists_compact_between hcφ hD hsφ
-  have huS : LocallyLipschitzOn S u := by
-    intro x hx
-    obtain ⟨K,U,hUo,hxU,hU⟩ := hlocal x (hSD hx)
-    exact ⟨K,U,mem_nhdsWithin_of_mem_nhds (hUo.mem_nhds hxU),hU⟩
-  obtain ⟨K,hK⟩ := huS.exists_lipschitzOnWith_of_compact hSc
-  obtain ⟨g,hglip,hug⟩ := hK.extend_real
-  obtain ⟨C,hφlip⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hcφ hφ (by simp)
-  have hφdiff : ∀ z, DifferentiableAt ℝ φ z := fun z => (hφ.differentiable (by simp)).differentiableAt
-  have ht (z : ℂ) (hz : z ∈ tsupport φ) : u =ᶠ[𝓝 z] g := by
-    filter_upwards [mem_interior_iff_mem_nhds.mp (hφS hz)] with w hw
-    exact hug hw
-  have hid := hglip.integral_lineDeriv_mul_eq (μ := volume) hφlip hcφ v
-  have hfull : (∫ z, (fderiv ℝ g z) v * φ z) = -(∫ z, g z * (fderiv ℝ φ z) v) := by
-    calc
-      _ = ∫ z, lineDeriv ℝ g z v * φ z := by
-        apply integral_congr_ae
-        filter_upwards [hglip.ae_differentiableAt (μ := volume)] with z hz
-        rw [hz.lineDeriv_eq_fderiv]
-      _ = ∫ z, lineDeriv ℝ φ z (-v) * g z := hid
-      _ = _ := by
-        simp_rw [(hφdiff _).lineDeriv_eq_fderiv,map_neg,neg_mul,mul_comm _ (g _)]
-        rw [integral_neg]
-  have he1 : (fun z => u z * (fderiv ℝ φ z) v) = (fun z => g z * (fderiv ℝ φ z) v) := by
-    funext z
-    by_cases hz : z ∈ tsupport φ
-    · rw [hug (interior_subset (hφS hz))]
-    · simp [fderiv_of_notMem_tsupport ℝ hz]
-  have he2 : (fun z => (fderiv ℝ u z) v * φ z) = (fun z => (fderiv ℝ g z) v * φ z) := by
-    funext z
-    by_cases hz : z ∈ tsupport φ
-    · rw [(ht z hz).fderiv_eq]
-    · simp [image_eq_zero_of_notMem_tsupport hz]
-  have hs1 : (∫ z in D, u z * (fderiv ℝ φ z) v) = ∫ z, u z * (fderiv ℝ φ z) v := by
-    apply setIntegral_eq_integral_of_forall_compl_eq_zero
-    intro z hz
-    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
-    simp [fderiv_of_notMem_tsupport ℝ hn]
-  have hs2 : (∫ z in D, (fderiv ℝ u z) v * φ z) = ∫ z, (fderiv ℝ u z) v * φ z := by
-    apply setIntegral_eq_integral_of_forall_compl_eq_zero
-    intro z hz
-    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
-    simp [image_eq_zero_of_notMem_tsupport hn]
-  rw [hs1,hs2,he1,he2,hfull]
-  simp
-
-theorem truncated_log_weak_derivative {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H)
-    (φ : ℂ → ℝ) (hφ : ContDiff ℝ ⊤ φ) (hcφ : HasCompactSupport φ)
-    (hsφ : tsupport φ ⊆ D) (v : ℂ) :
-    (∫ z in D, truncatedConformalLog F q r H z * (fderiv ℝ φ z) v) =
-      -(∫ z in D, (fderiv ℝ (truncatedConformalLog F q r H) z) v * φ z) :=
-  local_lipschitz_weak_derivative hD (fun _ hx => truncated_log_lipschitz_near hD hF hx q hr hH)
-    φ hφ hcφ hsφ v
-
-lemma open_univalent_weighted_area {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) (hi : InjOn F D) (q : ℂ) (a b : ℝ)
-    (ha : 0 < a) (hab : a < b) :
-    (∫⁻ z in D, ENNReal.ofReal (‖deriv F z‖^2)*annulusKernel q a b (F z)) ≤
-      ENNReal.ofReal (2*Real.pi*Real.log (b/a)) := by
-  let B : ℂ → ℂ →L[ℝ] ℂ := fun z =>
-    (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
-  have hB : ∀ z ∈ D, HasFDerivWithinAt F (B z) D z := by
-    intro z hz
-    exact ((hF.differentiableAt (hD.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
-  have he := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hD.measurableSet hB hi (annulusKernel q a b)
-  have hdet (z : ℂ) : |(B z).det| = ‖deriv F z‖^2 := by
-    rw [show B z = (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ from rfl,
-      complex_deriv_real_det,abs_of_nonneg (sq_nonneg _)]
-  simp only [hdet] at he
-  rw [← he]
-  exact (setLIntegral_le_lintegral _ _).trans_eq (annulusKernel_integral q a b ha hab)
-
- 
-
-theorem truncated_log_dirichlet_bound {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) (hi : InjOn F D)
-    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    (∫⁻ z in D, ENNReal.ofReal (‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2)) ≤
-      ENNReal.ofReal (2*Real.pi*H) := by
-  have hbound (ε : ℝ) (hε : 0 < ε) :
-      (∫⁻ z in D, ENNReal.ofReal (‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2)) ≤
-        ENNReal.ofReal (2*Real.pi*(H+2*ε)) := by
-    let a := r * Real.exp (-ε)
-    let b := Real.exp (H+ε)*r
-    have ha : 0 < a := mul_pos hr (Real.exp_pos _)
-    have har : a < r := by
-      dsimp [a]
-      exact (mul_lt_mul_of_pos_left (Real.exp_lt_one_iff.mpr (neg_lt_zero.mpr hε)) hr).trans_eq (mul_one r)
-    have hrR : r < Real.exp H*r := by
-      simpa using mul_lt_mul_of_pos_right (Real.one_lt_exp_iff.mpr hH) hr
-    have hRb : Real.exp H*r < b := by
-      exact mul_lt_mul_of_pos_right (Real.exp_lt_exp.mpr (lt_add_of_pos_right _ hε)) hr
-    have hab : a < b := har.trans (hrR.trans hRb)
-    have hlog : Real.log (b/a) = H+2*ε := by
-      have hid : b/a = Real.exp (H+2*ε) := by
-        dsimp [a,b]
-        rw [show H+2*ε = (H+ε)+ε by ring,Real.exp_add (H+ε) ε,Real.exp_neg]
-        field_simp
-      rw [hid,Real.log_exp]
-    calc
-      _ ≤ ∫⁻ z in D, ENNReal.ofReal (‖deriv F z‖^2)*annulusKernel q a b (F z) := by
-        apply setLIntegral_mono' hD.measurableSet
-        intro z hz
-        exact truncated_log_energy_pointwise hD hF hz q hr hH ha har hRb
-      _ ≤ ENNReal.ofReal (2*Real.pi*Real.log (b/a)) := open_univalent_weighted_area hD hF hi q a b ha hab
-      _ = _ := by rw [hlog]
-  have ht : Tendsto (fun ε : ℝ => ENNReal.ofReal (2*Real.pi*(H+2*ε))) (𝓝[>] 0)
-      (𝓝 (ENNReal.ofReal (2*Real.pi*H))) := by
-    have hc : Continuous (fun ε : ℝ => ENNReal.ofReal (2*Real.pi*(H+2*ε))) := ENNReal.continuous_ofReal.comp (by fun_prop)
-    simpa using (hc.tendsto 0).mono_left (nhdsWithin_le_nhds : 𝓝[>] (0:ℝ) ≤ 𝓝 0)
-  exact ge_of_tendsto ht (by filter_upwards [self_mem_nhdsWithin] with ε hε using hbound ε hε)
-
-lemma measurable_real_gradient (u : ℂ → ℝ) : Measurable (gradient u) :=
-  (InnerProductSpace.toDual ℝ ℂ).symm.continuous.measurable.comp (measurable_fderiv ℝ u)
-
-lemma norm_real_gradient (u : ℂ → ℝ) (z : ℂ) : ‖gradient u z‖ = ‖fderiv ℝ u z‖ := by
-  simp [gradient]
-
-theorem truncated_log_sobolev_energy {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) (hi : InjOn F D)
-    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    MemLp (gradient (truncatedConformalLog F q r H)) 2 (volume.restrict D) ∧
-    (∫ z in D, ‖gradient (truncatedConformalLog F q r H) z‖^2) ≤ 2*Real.pi*H ∧
-    ∀ (φ : ℂ → ℝ), ContDiff ℝ ⊤ φ → HasCompactSupport φ → tsupport φ ⊆ D →
-      ((∫ z in D, truncatedConformalLog F q r H z * (fderiv ℝ φ z) (1:ℂ)) =
-        -(∫ z in D, (gradient (truncatedConformalLog F q r H) z).re * φ z)) ∧
-      ((∫ z in D, truncatedConformalLog F q r H z * (fderiv ℝ φ z) Complex.I) =
-        -(∫ z in D, (gradient (truncatedConformalLog F q r H) z).im * φ z)) := by
-  let u := truncatedConformalLog F q r H
-  have hgm : AEStronglyMeasurable (gradient u) (volume.restrict D) :=
-    (measurable_real_gradient u).aestronglyMeasurable
-  have hsq : AEStronglyMeasurable (fun z => ‖gradient u z‖^2) (volume.restrict D) := hgm.norm.pow 2
-  have hb : (∫⁻ z in D, ENNReal.ofReal (‖gradient u z‖^2)) ≤ ENNReal.ofReal (2*Real.pi*H) := by
-    simpa only [norm_real_gradient] using truncated_log_dirichlet_bound hD hF hi q r H hr hH
-  have hint : Integrable (fun z => ‖gradient u z‖^2) (volume.restrict D) :=
-    (lintegral_ofReal_ne_top_iff_integrable hsq (ae_of_all _ fun z => sq_nonneg _)).mp
-      (ne_of_lt (hb.trans_lt ENNReal.ofReal_lt_top))
-  have henergy : (∫ z in D, ‖gradient u z‖^2) ≤ 2*Real.pi*H := by
-    have he := ofReal_integral_eq_lintegral_ofReal hint (ae_of_all _ fun z => sq_nonneg _)
-    rw [← he] at hb
-    exact (ENNReal.ofReal_le_ofReal_iff (by positivity : 0 ≤ 2*Real.pi*H)).mp hb
-  refine ⟨(memLp_two_iff_integrable_sq_norm hgm).mpr hint,henergy,?_⟩
-  intro φ hφ hcφ hsφ
-  constructor
-  · simpa only [real_gradient_re] using truncated_log_weak_derivative hD hF q r H hr hH φ hφ hcφ hsφ (1:ℂ)
-  · simpa only [real_gradient_im] using truncated_log_weak_derivative hD hF q r H hr hH φ hφ hcφ hsφ Complex.I
-
-noncomputable def conformalStripLog (F : ℂ → ℂ) (q : ℂ) (r H : ℝ) (z : ℂ) : ℝ :=
-  min H (max 0 (Real.log (‖F (Complex.exp z)-q‖/r)))
-
-lemma exp_strip_halfplane : MapsTo Complex.exp {z : ℂ | 0 < z.im ∧ z.im < Real.pi}
-    {z : ℂ | 0 < z.im} := by
-  intro z hz
-  change 0 < (Complex.exp z).im
-  rw [Complex.exp_im]
-  exact mul_pos (Real.exp_pos _) (Real.sin_pos_of_pos_of_lt_pi hz.1 hz.2)
-
-lemma exp_injective_strip : InjOn Complex.exp {z : ℂ | 0 < z.im ∧ z.im < Real.pi} := by
-  intro x hx y hy hxy
-  exact Complex.exp_inj_of_neg_pi_lt_of_le_pi
-    ((neg_neg_of_pos Real.pi_pos).trans hx.1) hx.2.le
-    ((neg_neg_of_pos Real.pi_pos).trans hy.1) hy.2.le hxy
-
- 
-theorem conformal_strip_total_energy {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F {z | 0 < z.im}) (hi : InjOn F {z | 0 < z.im})
-    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    MemLp (gradient (conformalStripLog F q r H)) 2
-      (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) ∧
-    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
-      ‖gradient (conformalStripLog F q r H) z‖^2) ≤ 2*Real.pi*H ∧
-    ∀ (φ : ℂ → ℝ), ContDiff ℝ ⊤ φ → HasCompactSupport φ →
-      tsupport φ ⊆ {z : ℂ | 0 < z.im ∧ z.im < Real.pi} →
-      ((∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
-          conformalStripLog F q r H z * (fderiv ℝ φ z) (1:ℂ)) =
-        -(∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
-          (gradient (conformalStripLog F q r H) z).re * φ z)) ∧
-      ((∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
-          conformalStripLog F q r H z * (fderiv ℝ φ z) Complex.I) =
-        -(∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
-          (gradient (conformalStripLog F q r H) z).im * φ z)) := by
-  have hS : IsOpen {z : ℂ | 0 < z.im ∧ z.im < Real.pi} :=
-    (isOpen_lt continuous_const Complex.continuous_im).inter
-      (isOpen_lt Complex.continuous_im continuous_const)
-  have hcomp : DifferentiableOn ℂ (F ∘ Complex.exp) {z : ℂ | 0 < z.im ∧ z.im < Real.pi} :=
-    hF.comp Complex.differentiable_exp.differentiableOn exp_strip_halfplane
-  have hicomp : InjOn (F ∘ Complex.exp) {z : ℂ | 0 < z.im ∧ z.im < Real.pi} :=
-    hi.comp exp_injective_strip exp_strip_halfplane
-  exact truncated_log_sobolev_energy hS hcomp hicomp q r H hr hH
-
-lemma truncated_log_continuousOn {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F D) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    ContinuousOn (truncatedConformalLog F q r H) D := by
-  apply LocallyLipschitzOn.continuousOn
-  intro z hz
-  obtain ⟨K,U,hU,hzU,hlip⟩ := truncated_log_lipschitz_near hD hF hz q hr hH
-  exact ⟨K,U,mem_nhdsWithin_of_mem_nhds (hU.mem_nhds hzU),hlip⟩
-
-lemma truncated_log_bounds (F : ℂ → ℂ) (q : ℂ) (r H : ℝ) (hH : 0 ≤ H) (z : ℂ) :
-    0 ≤ truncatedConformalLog F q r H z ∧ truncatedConformalLog F q r H z ≤ H := by
-  exact ⟨le_min hH (le_max_left _ _),min_le_left _ _⟩
-
-theorem conformal_unit_strip_weak_gradient {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F {z | 0 < z.im}) (hi : InjOn F {z | 0 < z.im})
-    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    IsStripWeakGradient (conformalStripLog F q r H)
-      (gradient (conformalStripLog F q r H)) ∧
-    (∫ z in openUnitStrip, ‖gradient (conformalStripLog F q r H) z‖^2) ≤
-      2*Real.pi*H := by
-  have hsub : openUnitStrip ⊆ {z : ℂ | 0 < z.im ∧ z.im < Real.pi} := fun z hz => hz.2.2
-  have hcm : DifferentiableOn ℂ (F ∘ Complex.exp) openUnitStrip :=
-    (hF.comp Complex.differentiable_exp.differentiableOn exp_strip_halfplane).mono hsub
-  have hicm : InjOn (F ∘ Complex.exp) openUnitStrip :=
-    (hi.comp exp_injective_strip exp_strip_halfplane).mono hsub
-  obtain ⟨hgrad,henergy,hweak⟩ :=
-    truncated_log_sobolev_energy openUnitStrip_isOpen hcm hicm q r H hr hH
-  have : IsFiniteMeasure (volume.restrict openUnitStrip) :=
-    isFiniteMeasure_restrict.mpr openUnitStrip_measure_lt_top.ne
-  have hu : MemLp (truncatedConformalLog (F ∘ Complex.exp) q r H) 2
-      (volume.restrict openUnitStrip) :=
-    memLp_of_bounded (ae_of_all _ fun z => truncated_log_bounds _ _ _ _ hH.le z)
-      ((truncated_log_continuousOn openUnitStrip_isOpen hcm q r H hr hH).aestronglyMeasurable
-        openUnitStrip_isOpen.measurableSet) 2
-  exact ⟨⟨hu,hgrad,hweak⟩,henergy⟩
-
- 
-noncomputable def sphericalTruncatedLog (q : ℂ) (r H : ℝ) : OnePoint ℂ → ℝ :=
-  OnePoint.rec H (fun z => min H (max 0 (Real.log (‖z-q‖/r))))
-
-@[simp] lemma sphericalTruncatedLog_coe (q z : ℂ) (r H : ℝ) :
-    sphericalTruncatedLog q r H (z : OnePoint ℂ) =
-      truncatedConformalLog id q r H z := rfl
-
-@[simp] lemma sphericalTruncatedLog_infty (q : ℂ) (r H : ℝ) :
-    sphericalTruncatedLog q r H OnePoint.infty = H := rfl
-
-lemma sphericalTruncatedLog_continuous (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    Continuous (sphericalTruncatedLog q r H) := by
-  apply continuous_iff_continuousAt.mpr
-  intro x
-  induction x using OnePoint.rec with
-  | infty =>
-    apply OnePoint.continuousAt_infty.mpr
-    intro S hS
-    refine ⟨Metric.closedBall q (Real.exp H*r),Metric.isClosed_closedBall,isCompact_closedBall _ _,?_⟩
-    intro z hz
-    have hb : Real.exp H*r ≤ ‖z-q‖ := by
-      rw [mem_compl_iff,Metric.mem_closedBall,dist_eq_norm] at hz
-      exact (lt_of_not_ge hz).le
-    have he : sphericalTruncatedLog q r H (z : OnePoint ℂ) = H := by
-      rw [sphericalTruncatedLog_coe]
-      exact truncatedConformalLog_eq_height (F := id) hr hb
-    change sphericalTruncatedLog q r H (z : OnePoint ℂ) ∈ S
-    rw [he]
-    exact mem_of_mem_nhds hS
-  | coe z =>
-    rw [OnePoint.continuousAt_coe]
-    change ContinuousAt (truncatedConformalLog id q r H) z
-    have hc := truncated_log_continuousOn isOpen_univ
-      (differentiable_id.differentiableOn : DifferentiableOn ℂ id Set.univ) q r H hr hH
-    exact (continuousOn_univ.mp hc).continuousAt
-
- 
-theorem spherical_strip_log_continuous {Fbar : ℂ → OnePoint ℂ}
-    (hFbar : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    ContinuousOn (fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z)))
-      {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} := by
-  have hmap : MapsTo Complex.exp {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi}
-      {z : ℂ | 0 ≤ z.im} := by
-    intro z hz
-    change 0 ≤ (Complex.exp z).im
-    rw [Complex.exp_im]
-    exact mul_nonneg (Real.exp_pos _).le (Real.sin_nonneg_of_nonneg_of_le_pi hz.1 hz.2)
-  exact (sphericalTruncatedLog_continuous q r H hr hH).continuousOn.comp
-    (hFbar.comp Complex.continuous_exp.continuousOn hmap) (mapsTo_univ _ _)
-
- 
-
-theorem source_strip_log_regularity {F : ℂ → ℂ} {Fbar : ℂ → OnePoint ℂ}
-    (hF : DifferentiableOn ℂ F {z | 0 < z.im}) (hi : InjOn F {z | 0 < z.im})
-    (hbar : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    (hext : ∀ z : ℂ, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    let U := fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))
-    ContinuousOn U {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} ∧
-    (∀ z, 0 ≤ U z ∧ U z ≤ H) ∧
-    IsStripWeakGradient U (gradient U) ∧
-    MemLp (gradient U) 2 (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) ∧
-    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi}, ‖gradient U z‖^2) ≤
-      2*Real.pi*H := by
-  let U := fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))
-  let V := conformalStripLog F q r H
-  let S := {z : ℂ | 0 < z.im ∧ z.im < Real.pi}
-  have hS : IsOpen S := (isOpen_lt continuous_const Complex.continuous_im).inter
-    (isOpen_lt Complex.continuous_im continuous_const)
-  have heq : EqOn U V S := by
-    intro z hz
-    dsimp [U]
-    rw [hext (Complex.exp z) (exp_strip_halfplane hz),sphericalTruncatedLog_coe]
-    rfl
-  have hgeq : EqOn (gradient U) (gradient V) S := by
-    intro z hz
-    have he : U =ᶠ[𝓝 z] V := by
-      filter_upwards [hS.mem_nhds hz] with w hw
-      exact heq hw
-    simp only [gradient,he.fderiv_eq]
-  have heae : U =ᵐ[volume.restrict S] V := by
-    filter_upwards [ae_restrict_mem hS.measurableSet] with z hz
-    exact heq hz
-  have hgae : gradient U =ᵐ[volume.restrict S] gradient V := by
-    filter_upwards [ae_restrict_mem hS.measurableSet] with z hz
-    exact hgeq hz
-  have hsub : openUnitStrip ⊆ S := fun z hz => hz.2.2
-  have heu : U =ᵐ[volume.restrict openUnitStrip] V :=
-    heae.filter_mono (ae_mono (Measure.restrict_mono hsub le_rfl))
-  have hgu : gradient U =ᵐ[volume.restrict openUnitStrip] gradient V :=
-    hgae.filter_mono (ae_mono (Measure.restrict_mono hsub le_rfl))
-  obtain ⟨hV,-⟩ := conformal_unit_strip_weak_gradient hF hi q r H hr hH
-  obtain ⟨hgrad,henergy,-⟩ := conformal_strip_total_energy hF hi q r H hr hH
-  refine ⟨spherical_strip_log_continuous hbar q r H hr hH,?_,?_,(memLp_congr_ae hgae).mpr hgrad,?_⟩
-  · intro z
-    change 0 ≤ sphericalTruncatedLog q r H (Fbar (Complex.exp z)) ∧
-      sphericalTruncatedLog q r H (Fbar (Complex.exp z)) ≤ H
-    generalize Fbar (Complex.exp z) = x
-    induction x using OnePoint.rec with
-    | infty => simpa using And.intro hH.le (le_refl H)
-    | coe w => exact truncated_log_bounds id q r H hH.le w
-  · refine ⟨(memLp_congr_ae heu).mpr hV.1,(memLp_congr_ae hgu).mpr hV.2.1,?_⟩
-    intro φ hφ hc hs
-    obtain ⟨hrx,hix⟩ := hV.2.2 φ hφ hc hs
-    have h1 : (∫ z in openUnitStrip, U z * (fderiv ℝ φ z) (1:ℂ)) =
-        (∫ z in openUnitStrip, V z * (fderiv ℝ φ z) (1:ℂ)) :=
-      integral_congr_ae (heu.mono fun z hz => by dsimp only; rw [hz])
-    have h2 : (∫ z in openUnitStrip, U z * (fderiv ℝ φ z) Complex.I) =
-        (∫ z in openUnitStrip, V z * (fderiv ℝ φ z) Complex.I) :=
-      integral_congr_ae (heu.mono fun z hz => by dsimp only; rw [hz])
-    have h3 : (∫ z in openUnitStrip, (gradient U z).re * φ z) =
-        (∫ z in openUnitStrip, (gradient V z).re * φ z) :=
-      integral_congr_ae (hgu.mono fun z hz => by dsimp only; rw [hz])
-    have h4 : (∫ z in openUnitStrip, (gradient U z).im * φ z) =
-        (∫ z in openUnitStrip, (gradient V z).im * φ z) :=
-      integral_congr_ae (hgu.mono fun z hz => by dsimp only; rw [hz])
-    exact ⟨h1.trans (hrx.trans (congrArg Neg.neg h3.symm)),
-      h2.trans (hix.trans (congrArg Neg.neg h4.symm))⟩
-  · calc
-      _ = ∫ z in S, ‖gradient V z‖^2 :=
-        integral_congr_ae (hgae.mono fun z hz => by dsimp only; rw [hz])
-      _ ≤ _ := henergy
-
- 
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.RectangleMeans
+import OAI.Probability.SLE.SausagePointBound
+
+namespace OAI
+
+/-! Dyadic Strips for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma inv_pow_rpow_two (k : ℕ) (α : ℝ) :
+    (1/(2:ℝ)^k)^α = (2:ℝ)^(-α*(k:ℝ)) := by
+  rw [one_div,Real.inv_rpow (by positivity)]
+  rw [← Real.rpow_natCast (2:ℝ) k,← Real.rpow_mul (by norm_num : 0 ≤ (2:ℝ)),
+    ← Real.rpow_neg (by norm_num : 0 ≤ (2:ℝ))]
+  congr 1
+  ring
+
+ 
+
+theorem sle_dyadic_strip_rectangle_hit
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (j : ℤ) (k : ℕ) (i m : Fin (2^k)) :
+    P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' dyadicStripRectangle j k i m} ≤
+      ENNReal.ofReal (stripHittingConstant κ *
+        (2:ℝ)^(-annularApproachPower κ*(k:ℝ))) := by
+  have hk : 0 < (2:ℝ)^k := by positivity
+  have hm : (m:ℝ) ≤ (2:ℝ)^k := by
+    exact_mod_cast m.isLt.le
+  have hz : 0 ≤ Real.pi*((m:ℝ)/(2:ℝ)^k) := by positivity
+  have hp : Real.pi*((m:ℝ)/(2:ℝ)^k) ≤ Real.pi := by
+    exact mul_le_of_le_one_right Real.pi_pos.le ((div_le_one hk).mpr hm)
+  have hh := sle_exp_strip_rectangle_hit κ hκ hκ8 γ hSLE
+    ((j:ℝ)+(i:ℝ)/(2:ℝ)^k) (Real.pi*((m:ℝ)/(2:ℝ)^k))
+    (1/(2:ℝ)^k) hz hp (by positivity)
+  simpa only [dyadicStripRectangle,inv_pow_rpow_two] using hh
+
+ 
+
+theorem ordinary_sle_strip_hitting_constants
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∃ C α : ℝ, 0 < C ∧ 0 < α ∧
+      ∀ {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+        (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+        ∀ (j : ℤ) (k : ℕ) (i m : Fin (2^k)),
+        P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' dyadicStripRectangle j k i m} ≤
+          ENNReal.ofReal (C*(2:ℝ)^(-α*(k:ℝ))) := by
+  refine ⟨stripHittingConstant κ,annularApproachPower κ,
+    stripHittingConstant_pos κ hκ hκ8,annularApproachPower_pos κ hκ hκ8,?_⟩
+  intro Ω _ P γ hγ j k i m
+  exact sle_dyadic_strip_rectangle_hit κ hκ hκ8 γ hγ j k i m
+
+end SLEExactGauge
+end
+
+section
+
+open Set Filter MeasureTheory
+open scoped Topology ContDiff ENNReal BigOperators NNReal
+namespace SLEExactGauge
+
+ 
+def IsStripWeakGradient (u : ℂ → ℝ) (D : ℂ → ℂ) : Prop :=
+  MemLp u 2 (volume.restrict openUnitStrip) ∧
+  MemLp D 2 (volume.restrict openUnitStrip) ∧
+  ∀ φ : ℂ → ℝ, ContDiff ℝ ⊤ φ → HasCompactSupport φ → tsupport φ ⊆ openUnitStrip →
+    (∫ z in openUnitStrip, u z * (fderiv ℝ φ z) (1:ℂ)) =
+      -(∫ z in openUnitStrip, (D z).re * φ z) ∧
+    (∫ z in openUnitStrip, u z * (fderiv ℝ φ z) Complex.I) =
+      -(∫ z in openUnitStrip, (D z).im * φ z)
+
+noncomputable def annulusKernel (q : ℂ) (r R : ℝ) (w : ℂ) : ℝ≥0∞ :=
+  if r < ‖w-q‖ ∧ ‖w-q‖ < R then ENNReal.ofReal (‖w-q‖^2)⁻¹ else 0
+
+lemma measurable_annulusKernel (q : ℂ) (r R : ℝ) : Measurable (annulusKernel q r R) := by
+  unfold annulusKernel
+  exact Measurable.ite ((isOpen_lt continuous_const (continuous_id.sub continuous_const).norm).measurableSet.inter
+    (isOpen_lt (continuous_id.sub continuous_const).norm continuous_const).measurableSet)
+    (ENNReal.measurable_ofReal.comp ((((measurable_id.sub measurable_const).norm).pow_const 2).inv))
+    measurable_const
+
+lemma lintegral_inv_Ioo (r R : ℝ) (hr : 0 < r) (hrR : r < R) :
+    (∫⁻ x in Ioo r R, ENNReal.ofReal x⁻¹) = ENNReal.ofReal (Real.log (R/r)) := by
+  have hc : ContinuousOn (fun x : ℝ => x⁻¹) (Icc r R) :=
+    continuousOn_id.inv₀ (fun x hx => ne_of_gt (hr.trans_le hx.1))
+  have hi : IntegrableOn (fun x : ℝ => x⁻¹) (Ioo r R) :=
+    hc.integrableOn_Icc.mono_set Ioo_subset_Icc_self
+  rw [← ofReal_integral_eq_lintegral_ofReal hi]
+  · congr 1
+    change (∫ x, x⁻¹ ∂volume.restrict (Ioo r R)) = _
+    rw [restrict_Ioo_eq_restrict_Ioc,← intervalIntegral.integral_of_le hrR.le]
+    apply integral_inv
+    rw [uIcc_of_le hrR.le]
+    intro h
+    exact not_le_of_gt hr h.1
+  · filter_upwards [ae_restrict_mem measurableSet_Ioo] with x hx
+    exact inv_nonneg.mpr (hr.trans hx.1).le
+
+lemma annulusKernel_polar (r R : ℝ) (p : ℝ × ℝ) (hp : 0 < p.1) :
+    ENNReal.ofReal p.1 * annulusKernel 0 r R (Complex.polarCoord.symm p) =
+      (Ioo r R).indicator (fun x : ℝ => ENNReal.ofReal x⁻¹) p.1 := by
+  simp only [annulusKernel,sub_zero,Complex.norm_polarCoord_symm,abs_of_pos hp]
+  by_cases h : r < p.1 ∧ p.1 < R
+  · simp only [Set.indicator,Set.mem_Ioo,ite_eq_left h]
+    rw [← ENNReal.ofReal_mul hp.le]
+    congr 1
+    field_simp
+  · simp only [Set.indicator,Set.mem_Ioo,ite_eq_right h,mul_zero]
+
+ 
+
+theorem annulusKernel_integral (q : ℂ) (r R : ℝ) (hr : 0 < r) (hrR : r < R) :
+    (∫⁻ w, annulusKernel q r R w) = ENNReal.ofReal (2*Real.pi*Real.log (R/r)) := by
+  have hcenter : (∫⁻ w, annulusKernel q r R w) = ∫⁻ w, annulusKernel 0 r R w := by
+    simpa only [annulusKernel,sub_zero] using
+      (lintegral_sub_right_eq_self (μ := (volume : Measure ℂ)) (annulusKernel 0 r R) q)
+  rw [hcenter,← Complex.lintegral_comp_polarCoord_symm]
+  let f : ℝ → ℝ≥0∞ := (Ioo r R).indicator (fun x : ℝ => ENNReal.ofReal x⁻¹)
+  have hf : Measurable f := (ENNReal.measurable_ofReal.comp measurable_inv).indicator measurableSet_Ioo
+  have he : (∫⁻ p in polarCoord.target, ENNReal.ofReal p.1 •
+      annulusKernel 0 r R (Complex.polarCoord.symm p)) =
+      ∫⁻ p in Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi, f p.1 := by
+    apply setLIntegral_congr_fun (measurableSet_Ioi.prod measurableSet_Ioo)
+    intro p hp
+    exact annulusKernel_polar r R p hp.1
+  rw [he]
+  change (∫⁻ p in Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi, f p.1 ∂volume.prod volume) = _
+  rw [setLIntegral_prod (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ))
+    (fun p : ℝ × ℝ => f p.1) ((hf.comp measurable_fst).aemeasurable)]
+  simp only [setLIntegral_const,Real.volume_Ioo,sub_neg_eq_add]
+  rw [lintegral_mul_const' _ _ ENNReal.ofReal_ne_top]
+  have hsub : Ioo r R ⊆ Ioi (0:ℝ) := fun x hx => hr.trans hx.1
+  have hfint : (∫⁻ x in Ioi (0:ℝ), f x) = ENNReal.ofReal (Real.log (R/r)) := by
+    dsimp [f]
+    rw [setLIntegral_indicator measurableSet_Ioo,inter_eq_left.mpr hsub,
+      lintegral_inv_Ioo r R hr hrR]
+  rw [hfint,← ENNReal.ofReal_mul]
+  · congr 1
+    ring
+  · exact Real.log_nonneg ((one_le_div hr).mpr hrR.le)
+
+lemma complex_deriv_real_det (d : ℂ) :
+    ((ContinuousLinearMap.toSpanSingleton ℂ d).restrictScalars ℝ).det = ‖d‖^2 := by
+  simp [ContinuousLinearMap.det,LinearMap.det_restrictScalars,
+    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
+
+ 
+
+theorem univalent_annulus_dirichlet_bound {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F {z | 0 < z.im})
+    (hi : InjOn F {z | 0 < z.im}) (q : ℂ) (r R : ℝ) (hr : 0 < r) (hrR : r < R) :
+    (∫⁻ z in {z : ℂ | 0 < z.im}, ENNReal.ofReal (‖deriv F z‖^2) *
+      annulusKernel q r R (F z)) ≤ ENNReal.ofReal (2*Real.pi*Real.log (R/r)) := by
+  have ho : IsOpen {z : ℂ | 0 < z.im} := isOpen_lt continuous_const Complex.continuous_im
+  let D : ℂ → ℂ →L[ℝ] ℂ := fun z => (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
+  have hD : ∀ z ∈ {z : ℂ | 0 < z.im}, HasFDerivWithinAt F (D z) {z : ℂ | 0 < z.im} z := by
+    intro z hz
+    exact ((hF.differentiableAt (ho.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
+  have he := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume ho.measurableSet hD hi (annulusKernel q r R)
+  have hdet (z : ℂ) : |(D z).det| = ‖deriv F z‖^2 := by
+    rw [show D z = (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ from rfl,
+      complex_deriv_real_det,abs_of_nonneg (sq_nonneg _)]
+  simp only [hdet] at he
+  rw [← he]
+  calc
+    _ ≤ ∫⁻ w, annulusKernel q r R w := setLIntegral_le_lintegral _ _
+    _ = _ := annulusKernel_integral q r R hr hrR
+
+noncomputable def truncatedConformalLog (F : ℂ → ℂ) (q : ℂ) (r H : ℝ) (z : ℂ) : ℝ :=
+  min H (max 0 (Real.log (‖F z-q‖/r)))
+
+lemma lipschitz_clip (H : ℝ) : LipschitzWith 1 (fun x : ℝ => min H (max 0 x)) := by
+  have hmax : LipschitzWith 1 (fun x : ℝ => max 0 x) := by
+    simpa [Function.comp_def] using lipschitzWith_max.comp ((LipschitzWith.const (0:ℝ)).prodMk LipschitzWith.id)
+  simpa [Function.comp_def] using lipschitzWith_min.comp ((LipschitzWith.const H).prodMk hmax)
+
+lemma norm_fderiv_clip_le {v : ℂ → ℝ} {z : ℂ} (hv : ContDiffAt ℝ 1 v z) (H : ℝ) :
+    ‖fderiv ℝ (fun w => min H (max 0 (v w))) z‖ ≤ ‖fderiv ℝ v z‖ := by
+  apply le_of_forall_gt_imp_ge_of_dense
+  intro K hK
+  have hK0 : 0 ≤ K := (norm_nonneg _).trans hK.le
+  obtain ⟨s,hs,hvs⟩ := hv.exists_lipschitzOnWith_of_nnnorm_lt ⟨K,hK0⟩ hK
+  have hc := (lipschitz_clip H).comp_lipschitzOnWith hvs
+  have he : (1 : ℝ≥0)*⟨K,hK0⟩ = ⟨K,hK0⟩ := one_mul _
+  rw [he] at hc
+  exact norm_fderiv_le_of_lipschitzOn ℝ hs hc
+
+lemma norm_fderiv_log_norm_div_le {F : ℂ → ℂ} {z : ℂ}
+    (hF : DifferentiableAt ℂ F z) (q : ℂ) {r : ℝ} (hr : 0 < r)
+    (h0 : F z-q ≠ 0) :
+    ‖fderiv ℝ (fun w => Real.log (‖F w-q‖/r)) z‖ ≤ ‖deriv F z‖ / ‖F z-q‖ := by
+  let A : ℂ →L[ℝ] ℂ := (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
+  have hA : HasFDerivAt (fun w => F w-q) A z :=
+    ((hF.hasDerivAt.hasFDerivAt.restrictScalars ℝ).sub_const q)
+  have hn := (differentiableAt_id.norm ℝ h0).hasFDerivAt
+  have hnorm : ‖fderiv ℝ (fun w : ℂ => ‖w‖) (F z-q)‖ = 1 :=
+    norm_fderiv_norm (differentiableAt_id.norm ℝ h0)
+  have hn0 : 0 < ‖F z-q‖ := norm_pos_iff.mpr h0
+  have hl := ((hn.comp z hA).log (norm_ne_zero_iff.mpr h0)).sub_const (Real.log r)
+  have he : (fun w => Real.log (‖F w-q‖/r)) =ᶠ[𝓝 z]
+      (fun w => Real.log ‖F w-q‖ - Real.log r) := by
+    filter_upwards [(hA.continuousAt.norm).eventually (Ioi_mem_nhds hn0)] with w hw
+    exact Real.log_div (ne_of_gt hw) hr.ne'
+  have hd := hl.congr_of_eventuallyEq he
+  rw [hd.fderiv,norm_smul,Real.norm_eq_abs]
+  simp only [Function.comp_apply,id_eq] at *
+  rw [abs_of_pos (inv_pos.mpr hn0)]
+  have hAnorm : ‖A‖ = ‖deriv F z‖ := by
+    simp [A]
+  have hb := (fderiv ℝ (fun w : ℂ => ‖w‖) (F z-q)).opNorm_comp_le A
+  rw [hnorm,one_mul,hAnorm] at hb
+  simpa [div_eq_mul_inv,mul_comm] using mul_le_mul_of_nonneg_left hb (inv_nonneg.mpr hn0.le)
+
+lemma truncatedConformalLog_eq_zero {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
+    (hr : 0 < r) (hH : 0 ≤ H) (hl : ‖F z-q‖ ≤ r) :
+    truncatedConformalLog F q r H z = 0 := by
+  have hb : Real.log (‖F z-q‖/r) ≤ 0 := Real.log_nonpos
+    (div_nonneg (norm_nonneg _) hr.le) ((div_le_one hr).mpr hl)
+  simp [truncatedConformalLog,max_eq_left hb,min_eq_right hH]
+
+lemma truncatedConformalLog_eq_height {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
+    (hr : 0 < r) (hu : Real.exp H*r ≤ ‖F z-q‖) :
+    truncatedConformalLog F q r H z = H := by
+  have hn : 0 < ‖F z-q‖ := (mul_pos (Real.exp_pos _) hr).trans_le hu
+  have hb : H ≤ Real.log (‖F z-q‖/r) :=
+    (Real.le_log_iff_exp_le (div_pos hn hr)).mpr ((le_div_iff₀ hr).mpr hu)
+  exact min_eq_left (hb.trans (le_max_right _ _))
+
+lemma fderiv_truncated_eq_zero_of_low {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
+    (hF : ContinuousAt F z) (hr : 0 < r) (hH : 0 ≤ H) (hl : ‖F z-q‖ < r) :
+    fderiv ℝ (truncatedConformalLog F q r H) z = 0 := by
+  have he : truncatedConformalLog F q r H =ᶠ[𝓝 z] (fun _ => (0:ℝ)) := by
+    filter_upwards [((hF.sub_const q).norm).eventually (Iio_mem_nhds hl)] with w hw
+    exact truncatedConformalLog_eq_zero hr hH hw.le
+  exact ((hasFDerivAt_const (0:ℝ) z).congr_of_eventuallyEq he).fderiv
+
+lemma fderiv_truncated_eq_zero_of_high {F : ℂ → ℂ} {q z : ℂ} {r H : ℝ}
+    (hF : ContinuousAt F z) (hr : 0 < r) (hu : Real.exp H*r < ‖F z-q‖) :
+    fderiv ℝ (truncatedConformalLog F q r H) z = 0 := by
+  have he : truncatedConformalLog F q r H =ᶠ[𝓝 z] (fun _ => H) := by
+    filter_upwards [((hF.sub_const q).norm).eventually (Ioi_mem_nhds hu)] with w hw
+    exact truncatedConformalLog_eq_height hr hw.le
+  exact ((hasFDerivAt_const H z).congr_of_eventuallyEq he).fderiv
+
+lemma norm_fderiv_truncated_le {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) {z : ℂ} (hz : z ∈ D) (q : ℂ) {r H : ℝ}
+    (hr : 0 < r) (h0 : F z-q ≠ 0) :
+    ‖fderiv ℝ (truncatedConformalLog F q r H) z‖ ≤ ‖deriv F z‖ / ‖F z-q‖ := by
+  have hcF : ContDiffAt ℝ 1 F z := ((hF.analyticAt (hD.mem_nhds hz)).restrictScalars (𝕜 := ℝ)).contDiffAt
+  have hv : ContDiffAt ℝ 1 (fun w => Real.log (‖F w-q‖/r)) z :=
+    (((hcF.sub contDiffAt_const).norm ℝ h0).div_const r).log
+      (div_ne_zero (norm_ne_zero_iff.mpr h0) hr.ne')
+  exact (norm_fderiv_clip_le hv H).trans
+    (norm_fderiv_log_norm_div_le (hF.differentiableAt (hD.mem_nhds hz)) q hr h0)
+
+ 
+
+lemma truncated_log_energy_pointwise {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) {z : ℂ} (hz : z ∈ D) (q : ℂ)
+    {r H a b : ℝ} (hr : 0 < r) (hH : 0 < H) (ha : 0 < a) (har : a < r)
+    (hRb : Real.exp H*r < b) :
+    ENNReal.ofReal (‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2) ≤
+      ENNReal.ofReal (‖deriv F z‖^2) * annulusKernel q a b (F z) := by
+  have hc := (hF.differentiableAt (hD.mem_nhds hz)).continuousAt
+  by_cases hband : a < ‖F z-q‖ ∧ ‖F z-q‖ < b
+  · have hn : 0 < ‖F z-q‖ := ha.trans hband.1
+    have hb := norm_fderiv_truncated_le hD hF hz q hr (norm_ne_zero_iff.mp hn.ne') (H := H)
+    have hs : ‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2 ≤
+        ‖deriv F z‖^2 * (‖F z-q‖^2)⁻¹ := by
+      calc
+        _ ≤ (‖deriv F z‖/‖F z-q‖)^2 := pow_le_pow_left₀ (norm_nonneg _) hb 2
+        _ = _ := by rw [div_pow,div_eq_mul_inv]
+    simpa only [annulusKernel,ite_eq_left hband,← ENNReal.ofReal_mul (sq_nonneg _)] using
+      ENNReal.ofReal_le_ofReal hs
+  · have he : fderiv ℝ (truncatedConformalLog F q r H) z = 0 := by
+      rcases not_and_or.mp hband with hl | hu
+      · exact fderiv_truncated_eq_zero_of_low hc hr hH.le ((le_of_not_gt hl).trans_lt har)
+      · exact fderiv_truncated_eq_zero_of_high hc hr (hRb.trans_le (le_of_not_gt hu))
+    simp [he]
+
+lemma truncated_log_lipschitz_near {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) {z : ℂ} (hz : z ∈ D) (q : ℂ) {r H : ℝ}
+    (hr : 0 < r) (hH : 0 < H) :
+    ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ z ∈ U ∧
+      LipschitzOnWith K (truncatedConformalLog F q r H) U := by
+  by_cases h0 : F z-q = 0
+  · have hn : ‖F z-q‖ < r := by simpa [h0] using hr
+    have he := (((hF.differentiableAt (hD.mem_nhds hz)).continuousAt.sub_const q).norm).eventually
+      (Iio_mem_nhds hn)
+    obtain ⟨U,hU,hUo,hzU⟩ := mem_nhds_iff.mp he
+    refine ⟨0,U,hUo,hzU,?_⟩
+    intro x hx y hy
+    rw [truncatedConformalLog_eq_zero hr hH.le (hU hx).le,
+      truncatedConformalLog_eq_zero hr hH.le (hU hy).le]
+    simp
+  · have hcF : ContDiffAt ℝ 1 F z :=
+      ((hF.analyticAt (hD.mem_nhds hz)).restrictScalars (𝕜 := ℝ)).contDiffAt
+    have hv : ContDiffAt ℝ 1 (fun w => Real.log (‖F w-q‖/r)) z :=
+      (((hcF.sub contDiffAt_const).norm ℝ h0).div_const r).log
+        (div_ne_zero (norm_ne_zero_iff.mpr h0) hr.ne')
+    let K : ℝ≥0 := ‖fderiv ℝ (fun w => Real.log (‖F w-q‖/r)) z‖₊ + 1
+    have hK : ‖fderiv ℝ (fun w => Real.log (‖F w-q‖/r)) z‖₊ < K := by
+      dsimp [K]; simp
+    obtain ⟨U,hU,hLip⟩ := hv.exists_lipschitzOnWith_of_nnnorm_lt K hK
+    refine ⟨1*K,interior U,isOpen_interior,mem_interior_iff_mem_nhds.mpr hU,?_⟩
+    exact ((lipschitz_clip H).comp_lipschitzOnWith hLip).mono interior_subset
+
+ 
+theorem truncated_log_ae_differentiable {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    ∀ᵐ z ∂volume, z ∈ D → DifferentiableAt ℝ (truncatedConformalLog F q r H) z := by
+  classical
+  have hloc (x : D) := truncated_log_lipschitz_near hD hF x.2 q hr hH
+  choose K U hUo hxU hULip using hloc
+  have hcov : D ⊆ ⋃ x : D, U x := fun z hz => mem_iUnion.mpr ⟨⟨z,hz⟩,hxU ⟨z,hz⟩⟩
+  obtain ⟨t,ht,htcov⟩ := (HereditarilyLindelofSpace.isLindelof D).elim_countable_subcover U hUo hcov
+  have ha (x : D) : ∀ᵐ z ∂volume, z ∈ U x →
+      DifferentiableAt ℝ (truncatedConformalLog F q r H) z := by
+    filter_upwards [(hULip x).ae_differentiableWithinAt_of_mem (μ := volume)] with z hz hzU
+    exact (hz hzU).differentiableAt ((hUo x).mem_nhds hzU)
+  filter_upwards [(ae_ball_iff ht).mpr (fun x _ => ha x)] with z hz hzD
+  obtain ⟨x,htx,hzx⟩ := mem_iUnion₂.mp (htcov hzD)
+  exact hz x htx hzx
+
+lemma local_lipschitz_weak_derivative {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
+    (hlocal : ∀ x ∈ D, ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ x ∈ U ∧ LipschitzOnWith K u U)
+    (φ : ℂ → ℝ) (hφ : ContDiff ℝ ⊤ φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ D) (v : ℂ) :
+    (∫ z in D, u z * (fderiv ℝ φ z) v) =
+      -(∫ z in D, (fderiv ℝ u z) v * φ z) := by
+  obtain ⟨S,hSc,hφS,hSD⟩ := exists_compact_between hcφ hD hsφ
+  have huS : LocallyLipschitzOn S u := by
+    intro x hx
+    obtain ⟨K,U,hUo,hxU,hU⟩ := hlocal x (hSD hx)
+    exact ⟨K,U,mem_nhdsWithin_of_mem_nhds (hUo.mem_nhds hxU),hU⟩
+  obtain ⟨K,hK⟩ := huS.exists_lipschitzOnWith_of_compact hSc
+  obtain ⟨g,hglip,hug⟩ := hK.extend_real
+  obtain ⟨C,hφlip⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hcφ hφ (by simp)
+  have hφdiff : ∀ z, DifferentiableAt ℝ φ z := fun z => (hφ.differentiable (by simp)).differentiableAt
+  have ht (z : ℂ) (hz : z ∈ tsupport φ) : u =ᶠ[𝓝 z] g := by
+    filter_upwards [mem_interior_iff_mem_nhds.mp (hφS hz)] with w hw
+    exact hug hw
+  have hid := hglip.integral_lineDeriv_mul_eq (μ := volume) hφlip hcφ v
+  have hfull : (∫ z, (fderiv ℝ g z) v * φ z) = -(∫ z, g z * (fderiv ℝ φ z) v) := by
+    calc
+      _ = ∫ z, lineDeriv ℝ g z v * φ z := by
+        apply integral_congr_ae
+        filter_upwards [hglip.ae_differentiableAt (μ := volume)] with z hz
+        rw [hz.lineDeriv_eq_fderiv]
+      _ = ∫ z, lineDeriv ℝ φ z (-v) * g z := hid
+      _ = _ := by
+        simp_rw [(hφdiff _).lineDeriv_eq_fderiv,map_neg,neg_mul,mul_comm _ (g _)]
+        rw [integral_neg]
+  have he1 : (fun z => u z * (fderiv ℝ φ z) v) = (fun z => g z * (fderiv ℝ φ z) v) := by
+    funext z
+    by_cases hz : z ∈ tsupport φ
+    · rw [hug (interior_subset (hφS hz))]
+    · simp [fderiv_of_notMem_tsupport ℝ hz]
+  have he2 : (fun z => (fderiv ℝ u z) v * φ z) = (fun z => (fderiv ℝ g z) v * φ z) := by
+    funext z
+    by_cases hz : z ∈ tsupport φ
+    · rw [(ht z hz).fderiv_eq]
+    · simp [image_eq_zero_of_notMem_tsupport hz]
+  have hs1 : (∫ z in D, u z * (fderiv ℝ φ z) v) = ∫ z, u z * (fderiv ℝ φ z) v := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro z hz
+    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
+    simp [fderiv_of_notMem_tsupport ℝ hn]
+  have hs2 : (∫ z in D, (fderiv ℝ u z) v * φ z) = ∫ z, (fderiv ℝ u z) v * φ z := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro z hz
+    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
+    simp [image_eq_zero_of_notMem_tsupport hn]
+  rw [hs1,hs2,he1,he2,hfull]
+  simp
+
+theorem truncated_log_weak_derivative {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H)
+    (φ : ℂ → ℝ) (hφ : ContDiff ℝ ⊤ φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ D) (v : ℂ) :
+    (∫ z in D, truncatedConformalLog F q r H z * (fderiv ℝ φ z) v) =
+      -(∫ z in D, (fderiv ℝ (truncatedConformalLog F q r H) z) v * φ z) :=
+  local_lipschitz_weak_derivative hD (fun _ hx => truncated_log_lipschitz_near hD hF hx q hr hH)
+    φ hφ hcφ hsφ v
+
+lemma open_univalent_weighted_area {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) (hi : InjOn F D) (q : ℂ) (a b : ℝ)
+    (ha : 0 < a) (hab : a < b) :
+    (∫⁻ z in D, ENNReal.ofReal (‖deriv F z‖^2)*annulusKernel q a b (F z)) ≤
+      ENNReal.ofReal (2*Real.pi*Real.log (b/a)) := by
+  let B : ℂ → ℂ →L[ℝ] ℂ := fun z =>
+    (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
+  have hB : ∀ z ∈ D, HasFDerivWithinAt F (B z) D z := by
+    intro z hz
+    exact ((hF.differentiableAt (hD.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
+  have he := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hD.measurableSet hB hi (annulusKernel q a b)
+  have hdet (z : ℂ) : |(B z).det| = ‖deriv F z‖^2 := by
+    rw [show B z = (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ from rfl,
+      complex_deriv_real_det,abs_of_nonneg (sq_nonneg _)]
+  simp only [hdet] at he
+  rw [← he]
+  exact (setLIntegral_le_lintegral _ _).trans_eq (annulusKernel_integral q a b ha hab)
+
+ 
+
+theorem truncated_log_dirichlet_bound {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) (hi : InjOn F D)
+    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    (∫⁻ z in D, ENNReal.ofReal (‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2)) ≤
+      ENNReal.ofReal (2*Real.pi*H) := by
+  have hbound (ε : ℝ) (hε : 0 < ε) :
+      (∫⁻ z in D, ENNReal.ofReal (‖fderiv ℝ (truncatedConformalLog F q r H) z‖^2)) ≤
+        ENNReal.ofReal (2*Real.pi*(H+2*ε)) := by
+    let a := r * Real.exp (-ε)
+    let b := Real.exp (H+ε)*r
+    have ha : 0 < a := mul_pos hr (Real.exp_pos _)
+    have har : a < r := by
+      dsimp [a]
+      exact (mul_lt_mul_of_pos_left (Real.exp_lt_one_iff.mpr (neg_lt_zero.mpr hε)) hr).trans_eq (mul_one r)
+    have hrR : r < Real.exp H*r := by
+      simpa using mul_lt_mul_of_pos_right (Real.one_lt_exp_iff.mpr hH) hr
+    have hRb : Real.exp H*r < b := by
+      exact mul_lt_mul_of_pos_right (Real.exp_lt_exp.mpr (lt_add_of_pos_right _ hε)) hr
+    have hab : a < b := har.trans (hrR.trans hRb)
+    have hlog : Real.log (b/a) = H+2*ε := by
+      have hid : b/a = Real.exp (H+2*ε) := by
+        dsimp [a,b]
+        rw [show H+2*ε = (H+ε)+ε by ring,Real.exp_add (H+ε) ε,Real.exp_neg]
+        field_simp
+      rw [hid,Real.log_exp]
+    calc
+      _ ≤ ∫⁻ z in D, ENNReal.ofReal (‖deriv F z‖^2)*annulusKernel q a b (F z) := by
+        apply setLIntegral_mono' hD.measurableSet
+        intro z hz
+        exact truncated_log_energy_pointwise hD hF hz q hr hH ha har hRb
+      _ ≤ ENNReal.ofReal (2*Real.pi*Real.log (b/a)) := open_univalent_weighted_area hD hF hi q a b ha hab
+      _ = _ := by rw [hlog]
+  have ht : Tendsto (fun ε : ℝ => ENNReal.ofReal (2*Real.pi*(H+2*ε))) (𝓝[>] 0)
+      (𝓝 (ENNReal.ofReal (2*Real.pi*H))) := by
+    have hc : Continuous (fun ε : ℝ => ENNReal.ofReal (2*Real.pi*(H+2*ε))) := ENNReal.continuous_ofReal.comp (by fun_prop)
+    simpa using (hc.tendsto 0).mono_left (nhdsWithin_le_nhds : 𝓝[>] (0:ℝ) ≤ 𝓝 0)
+  exact ge_of_tendsto ht (by filter_upwards [self_mem_nhdsWithin] with ε hε using hbound ε hε)
+
+lemma measurable_real_gradient (u : ℂ → ℝ) : Measurable (gradient u) :=
+  (InnerProductSpace.toDual ℝ ℂ).symm.continuous.measurable.comp (measurable_fderiv ℝ u)
+
+lemma norm_real_gradient (u : ℂ → ℝ) (z : ℂ) : ‖gradient u z‖ = ‖fderiv ℝ u z‖ := by
+  simp [gradient]
+
+theorem truncated_log_sobolev_energy {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) (hi : InjOn F D)
+    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    MemLp (gradient (truncatedConformalLog F q r H)) 2 (volume.restrict D) ∧
+    (∫ z in D, ‖gradient (truncatedConformalLog F q r H) z‖^2) ≤ 2*Real.pi*H ∧
+    ∀ (φ : ℂ → ℝ), ContDiff ℝ ⊤ φ → HasCompactSupport φ → tsupport φ ⊆ D →
+      ((∫ z in D, truncatedConformalLog F q r H z * (fderiv ℝ φ z) (1:ℂ)) =
+        -(∫ z in D, (gradient (truncatedConformalLog F q r H) z).re * φ z)) ∧
+      ((∫ z in D, truncatedConformalLog F q r H z * (fderiv ℝ φ z) Complex.I) =
+        -(∫ z in D, (gradient (truncatedConformalLog F q r H) z).im * φ z)) := by
+  let u := truncatedConformalLog F q r H
+  have hgm : AEStronglyMeasurable (gradient u) (volume.restrict D) :=
+    (measurable_real_gradient u).aestronglyMeasurable
+  have hsq : AEStronglyMeasurable (fun z => ‖gradient u z‖^2) (volume.restrict D) := hgm.norm.pow 2
+  have hb : (∫⁻ z in D, ENNReal.ofReal (‖gradient u z‖^2)) ≤ ENNReal.ofReal (2*Real.pi*H) := by
+    simpa only [norm_real_gradient] using truncated_log_dirichlet_bound hD hF hi q r H hr hH
+  have hint : Integrable (fun z => ‖gradient u z‖^2) (volume.restrict D) :=
+    (lintegral_ofReal_ne_top_iff_integrable hsq (ae_of_all _ fun z => sq_nonneg _)).mp
+      (ne_of_lt (hb.trans_lt ENNReal.ofReal_lt_top))
+  have henergy : (∫ z in D, ‖gradient u z‖^2) ≤ 2*Real.pi*H := by
+    have he := ofReal_integral_eq_lintegral_ofReal hint (ae_of_all _ fun z => sq_nonneg _)
+    rw [← he] at hb
+    exact (ENNReal.ofReal_le_ofReal_iff (by positivity : 0 ≤ 2*Real.pi*H)).mp hb
+  refine ⟨(memLp_two_iff_integrable_sq_norm hgm).mpr hint,henergy,?_⟩
+  intro φ hφ hcφ hsφ
+  constructor
+  · simpa only [real_gradient_re] using truncated_log_weak_derivative hD hF q r H hr hH φ hφ hcφ hsφ (1:ℂ)
+  · simpa only [real_gradient_im] using truncated_log_weak_derivative hD hF q r H hr hH φ hφ hcφ hsφ Complex.I
+
+noncomputable def conformalStripLog (F : ℂ → ℂ) (q : ℂ) (r H : ℝ) (z : ℂ) : ℝ :=
+  min H (max 0 (Real.log (‖F (Complex.exp z)-q‖/r)))
+
+lemma exp_strip_halfplane : MapsTo Complex.exp {z : ℂ | 0 < z.im ∧ z.im < Real.pi}
+    {z : ℂ | 0 < z.im} := by
+  intro z hz
+  change 0 < (Complex.exp z).im
+  rw [Complex.exp_im]
+  exact mul_pos (Real.exp_pos _) (Real.sin_pos_of_pos_of_lt_pi hz.1 hz.2)
+
+lemma exp_injective_strip : InjOn Complex.exp {z : ℂ | 0 < z.im ∧ z.im < Real.pi} := by
+  intro x hx y hy hxy
+  exact Complex.exp_inj_of_neg_pi_lt_of_le_pi
+    ((neg_neg_of_pos Real.pi_pos).trans hx.1) hx.2.le
+    ((neg_neg_of_pos Real.pi_pos).trans hy.1) hy.2.le hxy
+
+ 
+theorem conformal_strip_total_energy {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F {z | 0 < z.im}) (hi : InjOn F {z | 0 < z.im})
+    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    MemLp (gradient (conformalStripLog F q r H)) 2
+      (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) ∧
+    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
+      ‖gradient (conformalStripLog F q r H) z‖^2) ≤ 2*Real.pi*H ∧
+    ∀ (φ : ℂ → ℝ), ContDiff ℝ ⊤ φ → HasCompactSupport φ →
+      tsupport φ ⊆ {z : ℂ | 0 < z.im ∧ z.im < Real.pi} →
+      ((∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
+          conformalStripLog F q r H z * (fderiv ℝ φ z) (1:ℂ)) =
+        -(∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
+          (gradient (conformalStripLog F q r H) z).re * φ z)) ∧
+      ((∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
+          conformalStripLog F q r H z * (fderiv ℝ φ z) Complex.I) =
+        -(∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi},
+          (gradient (conformalStripLog F q r H) z).im * φ z)) := by
+  have hS : IsOpen {z : ℂ | 0 < z.im ∧ z.im < Real.pi} :=
+    (isOpen_lt continuous_const Complex.continuous_im).inter
+      (isOpen_lt Complex.continuous_im continuous_const)
+  have hcomp : DifferentiableOn ℂ (F ∘ Complex.exp) {z : ℂ | 0 < z.im ∧ z.im < Real.pi} :=
+    hF.comp Complex.differentiable_exp.differentiableOn exp_strip_halfplane
+  have hicomp : InjOn (F ∘ Complex.exp) {z : ℂ | 0 < z.im ∧ z.im < Real.pi} :=
+    hi.comp exp_injective_strip exp_strip_halfplane
+  exact truncated_log_sobolev_energy hS hcomp hicomp q r H hr hH
+
+lemma truncated_log_continuousOn {D : Set ℂ} (hD : IsOpen D) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F D) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    ContinuousOn (truncatedConformalLog F q r H) D := by
+  apply LocallyLipschitzOn.continuousOn
+  intro z hz
+  obtain ⟨K,U,hU,hzU,hlip⟩ := truncated_log_lipschitz_near hD hF hz q hr hH
+  exact ⟨K,U,mem_nhdsWithin_of_mem_nhds (hU.mem_nhds hzU),hlip⟩
+
+lemma truncated_log_bounds (F : ℂ → ℂ) (q : ℂ) (r H : ℝ) (hH : 0 ≤ H) (z : ℂ) :
+    0 ≤ truncatedConformalLog F q r H z ∧ truncatedConformalLog F q r H z ≤ H := by
+  exact ⟨le_min hH (le_max_left _ _),min_le_left _ _⟩
+
+theorem conformal_unit_strip_weak_gradient {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F {z | 0 < z.im}) (hi : InjOn F {z | 0 < z.im})
+    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    IsStripWeakGradient (conformalStripLog F q r H)
+      (gradient (conformalStripLog F q r H)) ∧
+    (∫ z in openUnitStrip, ‖gradient (conformalStripLog F q r H) z‖^2) ≤
+      2*Real.pi*H := by
+  have hsub : openUnitStrip ⊆ {z : ℂ | 0 < z.im ∧ z.im < Real.pi} := fun z hz => hz.2.2
+  have hcm : DifferentiableOn ℂ (F ∘ Complex.exp) openUnitStrip :=
+    (hF.comp Complex.differentiable_exp.differentiableOn exp_strip_halfplane).mono hsub
+  have hicm : InjOn (F ∘ Complex.exp) openUnitStrip :=
+    (hi.comp exp_injective_strip exp_strip_halfplane).mono hsub
+  obtain ⟨hgrad,henergy,hweak⟩ :=
+    truncated_log_sobolev_energy openUnitStrip_isOpen hcm hicm q r H hr hH
+  have : IsFiniteMeasure (volume.restrict openUnitStrip) :=
+    isFiniteMeasure_restrict.mpr openUnitStrip_measure_lt_top.ne
+  have hu : MemLp (truncatedConformalLog (F ∘ Complex.exp) q r H) 2
+      (volume.restrict openUnitStrip) :=
+    memLp_of_bounded (ae_of_all _ fun z => truncated_log_bounds _ _ _ _ hH.le z)
+      ((truncated_log_continuousOn openUnitStrip_isOpen hcm q r H hr hH).aestronglyMeasurable
+        openUnitStrip_isOpen.measurableSet) 2
+  exact ⟨⟨hu,hgrad,hweak⟩,henergy⟩
+
+ 
+noncomputable def sphericalTruncatedLog (q : ℂ) (r H : ℝ) : OnePoint ℂ → ℝ :=
+  OnePoint.rec H (fun z => min H (max 0 (Real.log (‖z-q‖/r))))
+
+@[simp] lemma sphericalTruncatedLog_coe (q z : ℂ) (r H : ℝ) :
+    sphericalTruncatedLog q r H (z : OnePoint ℂ) =
+      truncatedConformalLog id q r H z := rfl
+
+@[simp] lemma sphericalTruncatedLog_infty (q : ℂ) (r H : ℝ) :
+    sphericalTruncatedLog q r H OnePoint.infty = H := rfl
+
+lemma sphericalTruncatedLog_continuous (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    Continuous (sphericalTruncatedLog q r H) := by
+  apply continuous_iff_continuousAt.mpr
+  intro x
+  induction x using OnePoint.rec with
+  | infty =>
+    apply OnePoint.continuousAt_infty.mpr
+    intro S hS
+    refine ⟨Metric.closedBall q (Real.exp H*r),Metric.isClosed_closedBall,isCompact_closedBall _ _,?_⟩
+    intro z hz
+    have hb : Real.exp H*r ≤ ‖z-q‖ := by
+      rw [mem_compl_iff,Metric.mem_closedBall,dist_eq_norm] at hz
+      exact (lt_of_not_ge hz).le
+    have he : sphericalTruncatedLog q r H (z : OnePoint ℂ) = H := by
+      rw [sphericalTruncatedLog_coe]
+      exact truncatedConformalLog_eq_height (F := id) hr hb
+    change sphericalTruncatedLog q r H (z : OnePoint ℂ) ∈ S
+    rw [he]
+    exact mem_of_mem_nhds hS
+  | coe z =>
+    rw [OnePoint.continuousAt_coe]
+    change ContinuousAt (truncatedConformalLog id q r H) z
+    have hc := truncated_log_continuousOn isOpen_univ
+      (differentiable_id.differentiableOn : DifferentiableOn ℂ id Set.univ) q r H hr hH
+    exact (continuousOn_univ.mp hc).continuousAt
+
+ 
+theorem spherical_strip_log_continuous {Fbar : ℂ → OnePoint ℂ}
+    (hFbar : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    ContinuousOn (fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z)))
+      {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} := by
+  have hmap : MapsTo Complex.exp {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi}
+      {z : ℂ | 0 ≤ z.im} := by
+    intro z hz
+    change 0 ≤ (Complex.exp z).im
+    rw [Complex.exp_im]
+    exact mul_nonneg (Real.exp_pos _).le (Real.sin_nonneg_of_nonneg_of_le_pi hz.1 hz.2)
+  exact (sphericalTruncatedLog_continuous q r H hr hH).continuousOn.comp
+    (hFbar.comp Complex.continuous_exp.continuousOn hmap) (mapsTo_univ _ _)
+
+ 
+
+theorem source_strip_log_regularity {F : ℂ → ℂ} {Fbar : ℂ → OnePoint ℂ}
+    (hF : DifferentiableOn ℂ F {z | 0 < z.im}) (hi : InjOn F {z | 0 < z.im})
+    (hbar : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    (hext : ∀ z : ℂ, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    let U := fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))
+    ContinuousOn U {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} ∧
+    (∀ z, 0 ≤ U z ∧ U z ≤ H) ∧
+    IsStripWeakGradient U (gradient U) ∧
+    MemLp (gradient U) 2 (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) ∧
+    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi}, ‖gradient U z‖^2) ≤
+      2*Real.pi*H := by
+  let U := fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))
+  let V := conformalStripLog F q r H
+  let S := {z : ℂ | 0 < z.im ∧ z.im < Real.pi}
+  have hS : IsOpen S := (isOpen_lt continuous_const Complex.continuous_im).inter
+    (isOpen_lt Complex.continuous_im continuous_const)
+  have heq : EqOn U V S := by
+    intro z hz
+    dsimp [U]
+    rw [hext (Complex.exp z) (exp_strip_halfplane hz),sphericalTruncatedLog_coe]
+    rfl
+  have hgeq : EqOn (gradient U) (gradient V) S := by
+    intro z hz
+    have he : U =ᶠ[𝓝 z] V := by
+      filter_upwards [hS.mem_nhds hz] with w hw
+      exact heq hw
+    simp only [gradient,he.fderiv_eq]
+  have heae : U =ᵐ[volume.restrict S] V := by
+    filter_upwards [ae_restrict_mem hS.measurableSet] with z hz
+    exact heq hz
+  have hgae : gradient U =ᵐ[volume.restrict S] gradient V := by
+    filter_upwards [ae_restrict_mem hS.measurableSet] with z hz
+    exact hgeq hz
+  have hsub : openUnitStrip ⊆ S := fun z hz => hz.2.2
+  have heu : U =ᵐ[volume.restrict openUnitStrip] V :=
+    heae.filter_mono (ae_mono (Measure.restrict_mono hsub le_rfl))
+  have hgu : gradient U =ᵐ[volume.restrict openUnitStrip] gradient V :=
+    hgae.filter_mono (ae_mono (Measure.restrict_mono hsub le_rfl))
+  obtain ⟨hV,-⟩ := conformal_unit_strip_weak_gradient hF hi q r H hr hH
+  obtain ⟨hgrad,henergy,-⟩ := conformal_strip_total_energy hF hi q r H hr hH
+  refine ⟨spherical_strip_log_continuous hbar q r H hr hH,?_,?_,(memLp_congr_ae hgae).mpr hgrad,?_⟩
+  · intro z
+    change 0 ≤ sphericalTruncatedLog q r H (Fbar (Complex.exp z)) ∧
+      sphericalTruncatedLog q r H (Fbar (Complex.exp z)) ≤ H
+    generalize Fbar (Complex.exp z) = x
+    induction x using OnePoint.rec with
+    | infty => simpa using And.intro hH.le (le_refl H)
+    | coe w => exact truncated_log_bounds id q r H hH.le w
+  · refine ⟨(memLp_congr_ae heu).mpr hV.1,(memLp_congr_ae hgu).mpr hV.2.1,?_⟩
+    intro φ hφ hc hs
+    obtain ⟨hrx,hix⟩ := hV.2.2 φ hφ hc hs
+    have h1 : (∫ z in openUnitStrip, U z * (fderiv ℝ φ z) (1:ℂ)) =
+        (∫ z in openUnitStrip, V z * (fderiv ℝ φ z) (1:ℂ)) :=
+      integral_congr_ae (heu.mono fun z hz => by dsimp only; rw [hz])
+    have h2 : (∫ z in openUnitStrip, U z * (fderiv ℝ φ z) Complex.I) =
+        (∫ z in openUnitStrip, V z * (fderiv ℝ φ z) Complex.I) :=
+      integral_congr_ae (heu.mono fun z hz => by dsimp only; rw [hz])
+    have h3 : (∫ z in openUnitStrip, (gradient U z).re * φ z) =
+        (∫ z in openUnitStrip, (gradient V z).re * φ z) :=
+      integral_congr_ae (hgu.mono fun z hz => by dsimp only; rw [hz])
+    have h4 : (∫ z in openUnitStrip, (gradient U z).im * φ z) =
+        (∫ z in openUnitStrip, (gradient V z).im * φ z) :=
+      integral_congr_ae (hgu.mono fun z hz => by dsimp only; rw [hz])
+    exact ⟨h1.trans (hrx.trans (congrArg Neg.neg h3.symm)),
+      h2.trans (hix.trans (congrArg Neg.neg h4.symm))⟩
+  · calc
+      _ = ∫ z in S, ‖gradient V z‖^2 :=
+        integral_congr_ae (hgae.mono fun z hz => by dsimp only; rw [hz])
+      _ ≤ _ := henergy
+
+ 
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/ExponentialMartingales.lean b/lean/OAI/Probability/SLE/ExponentialMartingales.lean
index 54bb89f2f3204d50eba592b22c88d2da2ad50433..113e87025a9905775ae713bc1644d416bdd7a94c 100644
--- a/lean/OAI/Probability/SLE/ExponentialMartingales.lean
+++ b/lean/OAI/Probability/SLE/ExponentialMartingales.lean
@@ -1,776 +1,776 @@
-import Mathlib
-import OAI.Probability.SLE.SimpleFields
-
-namespace OAI
-
-/-! Exponential Martingales for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-open L2Limit
-lemma finite_uniformIntegrable_one_of_eLpNorm_two_bounded {Ω ι : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {X : ι → Ω → ℝ}
-    (hXm : ∀ i, AEStronglyMeasurable (X i) P) (C : ℝ≥0)
-    (hC : ∀ i, eLpNorm (X i) 2 P ≤ C) : UniformIntegrable X 1 P := by
-  let D : ℝ≥0∞ := C * (P univ) ^ (1/2 : ℝ)
-  have hDt : D ≠ ∞ := ENNReal.mul_ne_top ENNReal.coe_ne_top
-    (ENNReal.rpow_ne_top_of_nonneg (by norm_num) (measure_ne_top P univ))
-  refine ⟨unifIntegrable_iff.mpr ?_, D.toNNReal, ?_⟩
-  · intro bound hbound
-    obtain ⟨small, hsmall, hsmall_bound⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hbound
-    let ε : ℝ := small
-    have hε : 0 < ε := by exact_mod_cast hsmall
-    let d : ℝ≥0 := ⟨ε / ((C : ℝ)+1), div_nonneg hε.le (by positivity)⟩
-    have hd : 0 < (d : ℝ) := div_pos hε (by positivity)
-    refine ⟨ENNReal.ofReal ((d : ℝ)^2), ENNReal.ofReal_pos.mpr (sq_pos_of_pos hd),
-      fun i s hμs => ?_⟩
-    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (μ := P.restrict s)
-      (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i).restrict
-    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
-    rw [Measure.restrict_apply_univ] at hb
-    have hp : (P s) ^ (1/2 : ℝ) ≤ (d : ℝ≥0∞) := by
-      calc
-        _ ≤ (ENNReal.ofReal ((d : ℝ)^2)) ^ (1/2 : ℝ) :=
-          ENNReal.rpow_le_rpow hμs (by norm_num)
-        _ = _ := by
-          rw [ENNReal.ofReal_pow d.coe_nonneg]
-          simp only [ENNReal.ofReal_coe_nnreal]
-          rw [← ENNReal.rpow_natCast, ← ENNReal.rpow_mul]
-          norm_num
-    calc
-      _ ≤ eLpNorm (X i) 2 (P.restrict s) * (P s) ^ (1/2 : ℝ) := hb
-      _ ≤ C * (d : ℝ≥0∞) := mul_le_mul
-        ((eLpNorm_mono_measure _ Measure.restrict_le_self).trans (hC i)) hp (by positivity) (by positivity)
-      _ ≤ ENNReal.ofReal ε := by
-        rw [← ENNReal.coe_mul, ← ENNReal.ofReal_coe_nnreal]
-        simp only [NNReal.coe_mul]
-        apply ENNReal.ofReal_le_ofReal
-        change (C : ℝ) * (ε / ((C : ℝ)+1)) ≤ ε
-        rw [← mul_div_assoc, div_le_iff₀ (by positivity)]
-        nlinarith
-      _ ≤ bound := by
-        simpa only [ε, ENNReal.ofReal_coe_nnreal] using hsmall_bound.le
-  · intro i
-    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i)
-    rw [ENNReal.coe_toNNReal hDt]
-    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
-    exact hb.trans (mul_le_mul (hC i) le_rfl bot_le bot_le)
-
-theorem finite_L1_tendsto_of_square_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ} {Y : Ω → ℝ}
-    (hX : ∀ n, MemLp (X n) 2 P) (A : ℝ)
-    (hA : ∀ n, (∫ ω, X n ω ^ 2 ∂P) ≤ A)
-    (hXY : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) :
-    Integrable Y P ∧ Tendsto (fun n => eLpNorm (X n - Y) 1 P) atTop (𝓝 0) := by
-  let C : ℝ≥0∞ := (ENNReal.ofReal A) ^ (1/2 : ℝ)
-  have hCt : C ≠ ∞ := ENNReal.rpow_ne_top_of_nonneg (by norm_num) ENNReal.ofReal_ne_top
-  have hC (n : ℕ) : eLpNorm (X n) 2 P ≤ C.toNNReal := by
-    rw [ENNReal.coe_toNNReal hCt, eLpNorm_two_eq_integral_sq (hX n)]
-    exact ENNReal.rpow_le_rpow (ENNReal.ofReal_le_ofReal (hA n)) (by norm_num)
-  have hui := finite_uniformIntegrable_one_of_eLpNorm_two_bounded
-    (fun n => (hX n).aestronglyMeasurable) C.toNNReal hC
-  have hY := hui.memLp_of_ae_tendsto hXY
-  exact ⟨memLp_one_iff_integrable.1 hY,
-    tendsto_Lp_finite_of_tendsto_ae (by norm_num : (1 : ℝ≥0∞) ≤ 1)
-      (by norm_num) (fun n => (hX n).aestronglyMeasurable) hY hui.unifIntegrable hXY⟩
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-
-lemma product_memLp_two_of_section_square_bound
-    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν] [SFinite P]
-    {X : Z × Ω → ℝ} (hX : Measurable X)
-    (hX2 : ∀ z, MemLp (fun ω => X (z, ω)) 2 P)
-    (V : ℝ) (hV : ∀ z, (∫ ω, X (z, ω)^2 ∂P) ≤ V) :
-    MemLp X 2 (ν.prod P) ∧
-      (∫ p, X p ^ 2 ∂(ν.prod P)) ≤ V * (ν univ).toReal := by
-  have hm : StronglyMeasurable (fun z => ∫ ω, X (z, ω)^2 ∂P) :=
-    (hX.pow_const 2).stronglyMeasurable.integral_prod_right'
-  have hn (z : Z) : 0 ≤ ∫ ω, X (z, ω)^2 ∂P := integral_nonneg fun ω => sq_nonneg _
-  have hi : Integrable (fun z => ∫ ω, X (z, ω)^2 ∂P) ν := by
-    apply Integrable.of_bound hm.aestronglyMeasurable V
-    exact Eventually.of_forall fun z => by rw [Real.norm_eq_abs, abs_of_nonneg (hn z)]; exact hV z
-  have hisq : Integrable (fun p => X p ^ 2) (ν.prod P) := by
-    apply (integrable_prod_iff (hX.pow_const 2).aestronglyMeasurable).mpr
-    refine ⟨Eventually.of_forall fun z => (hX2 z).integrable_sq, ?_⟩
-    simpa only [Real.norm_eq_abs, abs_sq] using hi
-  refine ⟨(memLp_two_iff_integrable_sq hX.aestronglyMeasurable).mpr hisq, ?_⟩
-  rw [integral_prod _ hisq]
-  calc
-    _ ≤ ∫ _z, V ∂ν := integral_mono hi (integrable_const _) hV
-    _ = _ := by simp [mul_comm, Measure.real]
-
-lemma area_eLpNorm_one_le_product
-    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [SFinite ν] [SFinite P]
-    {X Y : Z × Ω → ℝ} (hX : Integrable X (ν.prod P)) (hY : Integrable Y (ν.prod P)) :
-    eLpNorm (fun ω => (∫ z, X (z, ω) ∂ν) - ∫ z, Y (z, ω) ∂ν) 1 P ≤
-      eLpNorm (X - Y) 1 (ν.prod P) := by
-  rw [eLpNorm_one_eq_lintegral_enorm
-    (f := fun ω => (∫ z, X (z, ω) ∂ν) - ∫ z, Y (z, ω) ∂ν)
-    (hX.integral_prod_right.sub hY.integral_prod_right).aestronglyMeasurable,
-    eLpNorm_one_eq_lintegral_enorm (hX.sub hY).aestronglyMeasurable]
-  calc
-    _ ≤ ∫⁻ ω, ∫⁻ z, ‖X (z, ω) - Y (z, ω)‖ₑ ∂ν ∂P := by
-      apply lintegral_mono_ae
-      filter_upwards [hX.prod_left_ae, hY.prod_left_ae] with ω hx hy
-      rw [← integral_sub hx hy]
-      exact enorm_integral_le_lintegral_enorm _
-    _ = _ := by
-      rw [← lintegral_lintegral_swap (hX.sub hY).aestronglyMeasurable.enorm]
-      exact (lintegral_prod _ (hX.sub hY).aestronglyMeasurable.enorm).symm
-
- 
-
-theorem area_L1_tendsto_of_square_bound
-    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν] [IsProbabilityMeasure P]
-    {X : ℕ → Z × Ω → ℝ} {Y : Z × Ω → ℝ}
-    (hXm : ∀ n, Measurable (X n)) (hYm : Measurable Y)
-    (hX2 : ∀ n z, MemLp (fun ω => X n (z, ω)) 2 P)
-    (V : ℝ) (hV : ∀ n z, (∫ ω, X n (z, ω)^2 ∂P) ≤ V)
-    (hXY : ∀ z, ∀ᵐ ω ∂P, Tendsto (fun n => X n (z, ω)) atTop (𝓝 (Y (z, ω)))) :
-    Integrable Y (ν.prod P) ∧ Tendsto (fun n =>
-      eLpNorm (fun ω => (∫ z, X n (z, ω) ∂ν) - ∫ z, Y (z, ω) ∂ν) 1 P) atTop (𝓝 0) := by
-  have hb (n : ℕ) := product_memLp_two_of_section_square_bound (ν := ν) (P := P) (hXm n) (hX2 n) V (hV n)
-  have hc : ∀ᵐ p ∂(ν.prod P), Tendsto (fun n => X n p) atTop (𝓝 (Y p)) :=
-    (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun hXm hYm)).mpr (Eventually.of_forall hXY)
-  obtain ⟨hi, ht⟩ := finite_L1_tendsto_of_square_bound (fun n => (hb n).1)
-    (V * (ν univ).toReal) (fun n => (hb n).2) hc
-  refine ⟨hi, ?_⟩
-  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht
-    (fun n => bot_le) (fun n => area_eLpNorm_one_le_product ((hb n).1.integrable (by norm_num)) hi)
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-open BrownianPredictable PredictableLimit
-
-lemma area_driver_sum {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {ν : Measure Z} [IsFiniteMeasure ν]
-    (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) (C : ℕ → Z × Ω → ℝ)
-    (hC : ∀ i, Measurable (C i)) (K : ℝ) (hb : ∀ i z ω, |C i (z, ω)| ≤ K)
-    (n : ℕ) (ω : Ω) :
-    (∫ z, driverPredictableSum B t (fun i ω => C i (z, ω)) n ω ∂ν) =
-      driverPredictableSum B t (fun i ω => ∫ z, C i (z, ω) ∂ν) n ω := by
-  have hi (i : ℕ) : Integrable (fun z => C i (z, ω)) ν :=
-    Integrable.of_bound ((hC i).comp (measurable_id.prodMk measurable_const)).aestronglyMeasurable K
-      (Eventually.of_forall fun z => by simpa only [Real.norm_eq_abs] using hb i z ω)
-  simp only [driverPredictableSum]
-  rw [integral_finsetSum _ (fun i _ => (hi i).mul_const _)]
-  exact Finset.sum_congr rfl fun i hi => integral_mul_const _ _
-
- 
-
-theorem brownian_area_sum_L1_limit
-    {Z Ω : Type*} [MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-    (T : ℝ≥0) (C : ℕ → ℕ → Z × Ω → ℝ)
-    (hCm : ∀ n i, Measurable (C n i))
-    (hCa : ∀ n i z, Measurable[Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)] (fun ω => C n i (z, ω)))
-    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ n i z ω, |C n i (z, ω)| ≤ K)
-    (Y : Z × Ω → ℝ) (hYm : Measurable Y)
-    (hlim : ∀ z, ∀ᵐ ω ∂P, Tendsto
-      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z, ω)) (2^n) ω)
-      atTop (𝓝 (Y (z, ω)))) :
-    Integrable Y (ν.prod P) ∧ Tendsto (fun n =>
-      eLpNorm (fun ω => driverPredictableSum B (dyadicTime T n)
-        (fun i ω => ∫ z, C n i (z, ω) ∂ν) (2^n) ω - ∫ z, Y (z, ω) ∂ν) 1 P)
-        atTop (𝓝 0) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let X (n : ℕ) (p : Z × Ω) :=
-    driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (p.1, ω)) (2^n) p.2
-  have hCp (n i : ℕ) (z : Z) : Measurable[pastSigma B (dyadicTime T n i)]
-      (fun ω => C n i (z, ω)) := by
-    exact (hCa n i z).mono
-      (le_of_eq (Filtration.natural_eq_comap B (fun t => (hm t).stronglyMeasurable) _)) le_rfl
-  have hC2 (n i : ℕ) (z : Z) : MemLp (fun ω => C n i (z, ω)) 2 P :=
-    MemLp.of_bound ((hCm n i).comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable K
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hb n i z ω)
-  have hXm (n : ℕ) : Measurable (X n) := by
-    apply Finset.measurable_sum
-    intro i hi
-    exact (hCm n i).mul (((hm _).comp measurable_snd).sub ((hm _).comp measurable_snd))
-  have hX2 (n : ℕ) (z : Z) : MemLp (fun ω => X n (z, ω)) 2 P := by
-    exact dyadicDriverSum_memLp hB T (fun n i ω => C n i (z, ω))
-      (fun n i => hCp n i z) (fun n i => hC2 n i z) n
-  have hV (n : ℕ) (z : Z) : (∫ ω, X n (z, ω)^2 ∂P) ≤ K^2 * (T : ℝ) := by
-    change (∫ ω, (∑ i ∈ Finset.range (2^n), C n i (z, ω) *
-      (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) ^ 2 ∂P) ≤ _
-    rw [brownian_predictableSum_isometry hB _ (dyadicTime_mono T n) _
-      (fun i => hCp n i z) (fun i => hC2 n i z)]
-    calc
-      _ ≤ ∑ i ∈ Finset.range (2^n), K^2 * ((T : ℝ) / (2 : ℝ)^n) := by
-        apply Finset.sum_le_sum
-        intro i hi
-        rw [dyadicTime_step]
-        apply mul_le_mul_of_nonneg_right _ (by positivity)
-        calc
-          _ ≤ ∫ _ω, K^2 ∂P := integral_mono (hC2 n i z).integrable_sq (integrable_const _) fun ω =>
-            by
-              have hh := (sq_le_sq₀ (abs_nonneg (C n i (z, ω))) hK).mpr (hb n i z ω)
-              simpa only [sq_abs] using hh
-          _ = K^2 := by simp
-      _ = K^2 * (T : ℝ) := by
-        simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat]
-        field_simp
-  obtain ⟨hi, ht⟩ := area_L1_tendsto_of_square_bound (ν := ν) hXm hYm hX2 _ hV hlim
-  refine ⟨hi, ?_⟩
-  have he (n : ℕ) (ω : Ω) : (∫ z, X n (z, ω) ∂ν) =
-      driverPredictableSum B (dyadicTime T n) (fun i ω => ∫ z, C n i (z, ω) ∂ν) (2^n) ω :=
-    area_driver_sum B _ _ (hCm n) K (hb n) _ ω
-  simpa only [he] using ht
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-open BrownianPredictable PredictableLimit
-
- 
-
-theorem brownian_area_exponential_limit
-    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-    (T : ℝ≥0) (C : ℕ → ℕ → Z × Ω → ℝ)
-    (hCm : ∀ n i, Measurable (C n i))
-    (hCa : ∀ n i, @Measurable (Z × Ω) ℝ
-      (mZ.prod (Filtration.natural B (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)))
-      (borel ℝ) (C n i))
-    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ n i z ω, |C n i (z, ω)| ≤ K)
-    (Y : Z × Ω → ℝ) (hYm : Measurable Y)
-    (hlim : ∀ z, ∀ᵐ ω ∂P, Tendsto
-      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z, ω)) (2^n) ω)
-      atTop (𝓝 (Y (z, ω))))
-    (Q : Ω → ℝ)
-    (hE : ∀ᵐ ω ∂P, Tendsto (fun n => driverPredictableEnergy (dyadicTime T n)
-      (fun i ω => ∫ z, C n i (z, ω) ∂ν) (2^n) ω) atTop (𝓝 (Q ω)))
-    (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp
-      (θ * (∫ z, Y (z, ω) ∂ν) - θ^2 * Q ω / 2)) ∂P) ≤ 1 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let D (n i : ℕ) (ω : Ω) := ∫ z, C n i (z, ω) ∂ν
-  have hDa (n i : ℕ) : Measurable[Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)] (D n i) := by
-    let : MeasurableSpace Ω := Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)
-    exact (hCa n i).stronglyMeasurable.integral_prod_left'.measurable
-  let X (n : ℕ) := driverPredictableSum B (dyadicTime T n) (D n) (2^n)
-  let E (n : ℕ) := driverPredictableEnergy (dyadicTime T n) (D n) (2^n)
-  have hXm (n : ℕ) : Measurable (X n) :=
-    (driverPredictableSum_measurable_past B hm _ (dyadicTime_mono T n) _ (hDa n) _).mono
-      (Filtration.le _ _) le_rfl
-  have hEm (n : ℕ) : Measurable (E n) :=
-    (driverPredictableEnergy_measurable_past B hm _ (dyadicTime_mono T n) _ (hDa n) _).mono
-      (Filtration.le _ _) le_rfl
-  have hca (n i : ℕ) (z : Z) : Measurable[Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)] (fun ω => C n i (z, ω)) :=
-    (hCa n i).comp (measurable_const.prodMk measurable_id)
-  obtain ⟨hi, ht⟩ := brownian_area_sum_L1_limit (ν := ν) (P := P) hB hm T C hCm hca K hK hb Y hYm hlim
-  have hY : Measurable (fun ω => ∫ z, Y (z, ω) ∂ν) :=
-    hYm.stronglyMeasurable.integral_prod_left'.measurable
-  have hc : TendstoInMeasure P X atTop (fun ω => ∫ z, Y (z, ω) ∂ν) :=
-    tendstoInMeasure_of_tendsto_eLpNorm (by norm_num : (1 : ℝ≥0∞) ≠ 0) ht
-  obtain ⟨ns, hns, hs⟩ := hc.exists_seq_tendsto_ae
-  apply compensated_exponential_limit_energy (fun n => X (ns n)) (fun n => E (ns n))
-    (fun ω => ∫ z, Y (z, ω) ∂ν) Q θ
-    (fun n => (((hXm _).const_mul θ).sub (((hEm _).const_mul (θ^2)).div_const 2)).exp.ennreal_ofReal.aemeasurable)
-    hs
-  · filter_upwards [hE] with ω he
-    exact he.comp hns.tendsto_atTop
-  · intro n
-    exact (driverPredictable_exponential hB hm _ (dyadicTime_mono T (ns n)) _ (hDa _) θ _).le
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField BrownianIto TaylorMeshes
-
-noncomputable def areaPrefixSample {Z : Type*} (f : ℝ × Z → ℝ) (τ : Z → ℝ≥0)
-    (T : ℝ≥0) (n i : ℕ) (z : Z) : ℝ :=
-  if dyadicTime T n i < τ z then f (dyadicTime T n i, z) else 0
-
-lemma areaPrefixSample_measurable {Z : Type*} [MeasurableSpace Z]
-    {f : ℝ × Z → ℝ} (hf : Measurable f) {τ : Z → ℝ≥0} (hτ : Measurable τ)
-    (T : ℝ≥0) (n i : ℕ) : Measurable (areaPrefixSample f τ T n i) :=
-  (hf.comp (measurable_const.prodMk measurable_id)).ite
-    (measurableSet_lt measurable_const hτ) measurable_const
-
-lemma areaPrefixSample_bound {Z : Type*} {f : ℝ × Z → ℝ}
-    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
-    (τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z : Z) :
-    |areaPrefixSample f τ T n i z| ≤ K := by
-  unfold areaPrefixSample
-  split_ifs
-  · exact hb _ _
-  · simpa using hK
-
-lemma prefix_pair_energy_tendsto {Z : Type*} (f : ℝ × Z → ℝ)
-    (hf : ∀ z, Continuous (fun t => f (t,z)))
-    (τ : Z → ℝ≥0) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T) (z w : Z) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (areaPrefixSample f τ T n i z * areaPrefixSample f τ T n i w) *
-        ((T : ℝ) / (2 : ℝ)^n)) atTop
-      (𝓝 (∫ t in (0 : ℝ)..(min (τ z) (τ w) : ℝ≥0), f (t,z) * f (t,w))) := by
-  have hh := stopped_energy_tendsto ((min_le_left (τ z) (τ w)).trans (hτ z))
-    ((hf z).mul (hf w)).continuousOn
-  simp only [Pi.mul_apply] at hh
-  convert hh using 1
-  ext n
-  apply Finset.sum_congr rfl
-  intro i hi
-  rw [dyadicTime_step]
-  simp only [areaPrefixSample, lt_min_iff]
-  split_ifs <;> simp_all
-
-lemma prefix_pair_energy_bound {Z : Type*} {f : ℝ × Z → ℝ}
-    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
-    (τ : Z → ℝ≥0) (T : ℝ≥0) (n : ℕ) (z w : Z) :
-    |∑ i ∈ Finset.range (2^n),
-      (areaPrefixSample f τ T n i z * areaPrefixSample f τ T n i w) *
-        ((T : ℝ) / (2 : ℝ)^n)| ≤ K^2 * (T : ℝ) := by
-  calc
-    _ ≤ ∑ i ∈ Finset.range (2^n),
-        |(areaPrefixSample f τ T n i z * areaPrefixSample f τ T n i w) *
-          ((T : ℝ) / (2 : ℝ)^n)| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ i ∈ Finset.range (2^n), K^2 * ((T : ℝ) / (2 : ℝ)^n) := by
-      apply Finset.sum_le_sum
-      intro i hi
-      rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (T : ℝ) / (2 : ℝ)^n), abs_mul, pow_two]
-      exact mul_le_mul_of_nonneg_right
-        (mul_le_mul (areaPrefixSample_bound hK hb τ T n i z)
-          (areaPrefixSample_bound hK hb τ T n i w) (abs_nonneg _) hK) (by positivity)
-    _ = _ := by
-      simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat]
-      field_simp
-
- 
-
-theorem area_prefix_energy_tendsto {Z : Type*} [MeasurableSpace Z]
-    {ν : Measure Z} [IsFiniteMeasure ν]
-    {f : ℝ × Z → ℝ} (hfm : Measurable f) (hfc : ∀ z, Continuous (fun t => f (t,z)))
-    {τ : Z → ℝ≥0} (hτm : Measurable τ) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
-    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (∫ z, areaPrefixSample f τ T n i z ∂ν)^2 * ((T : ℝ) / (2 : ℝ)^n)) atTop
-      (𝓝 (∫ p : Z × Z, (∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
-        f (t,p.1) * f (t,p.2)) ∂(ν.prod ν))) := by
-  let H (n : ℕ) (p : Z × Z) := ∑ i ∈ Finset.range (2^n),
-    (areaPrefixSample f τ T n i p.1 * areaPrefixSample f τ T n i p.2) *
-      ((T : ℝ) / (2 : ℝ)^n)
-  have hC (n i : ℕ) := areaPrefixSample_measurable hfm hτm T n i
-  have hH (n : ℕ) : Measurable (H n) := by
-    apply Finset.measurable_sum
-    intro i hi
-    exact (((hC n i).comp measurable_fst).mul ((hC n i).comp measurable_snd)).mul_const _
-  have hbH (n : ℕ) (p : Z × Z) : ‖H n p‖ ≤ K^2 * (T : ℝ) := by
-    rw [Real.norm_eq_abs]
-    exact prefix_pair_energy_bound hK hb τ T n p.1 p.2
-  have ht := tendsto_integral_of_dominated_convergence
-    (μ := ν.prod ν) (fun _ => K^2 * (T : ℝ))
-    (fun n => (hH n).aestronglyMeasurable) (integrable_const _)
-    (fun n => Eventually.of_forall (hbH n))
-    (Eventually.of_forall fun p : Z × Z => prefix_pair_energy_tendsto f hfc τ T hτ p.1 p.2)
-  have he (n : ℕ) : (∫ p, H n p ∂(ν.prod ν)) =
-      ∑ i ∈ Finset.range (2^n), (∫ z, areaPrefixSample f τ T n i z ∂ν)^2 *
-        ((T : ℝ) / (2 : ℝ)^n) := by
-    have hi (i : ℕ) : Integrable (areaPrefixSample f τ T n i) ν :=
-      Integrable.of_bound (hC n i).aestronglyMeasurable K
-        (Eventually.of_forall fun z => by rw [Real.norm_eq_abs]; exact areaPrefixSample_bound hK hb τ T n i z)
-    rw [show H n = (fun p : Z × Z => ∑ i ∈ Finset.range (2^n),
-      (areaPrefixSample f τ T n i p.1 * areaPrefixSample f τ T n i p.2) *
-        ((T : ℝ) / (2 : ℝ)^n)) from rfl]
-    rw [integral_finsetSum _ (fun i _ => ((hi i).mul_prod (hi i)).mul_const _)]
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [integral_mul_const, integral_prod_mul, pow_two]
-  simpa only [he] using ht
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma interval_integral_prefix_lt {S T : ℝ≥0} (hST : S ≤ T) (f : ℝ → ℝ) :
-    (∫ t in (0 : ℝ)..(T : ℝ), if t < (S : ℝ) then f t else 0) =
-      ∫ t in (0 : ℝ)..(S : ℝ), f t := by
-  calc
-    _ = ∫ t in (0 : ℝ)..(T : ℝ), indicator {t : ℝ | t ≤ S} f t := by
-      apply intervalIntegral.integral_congr_ae
-      have hn : ∀ᵐ t : ℝ, t ≠ (S : ℝ) := by simp [ae_iff]
-      filter_upwards [hn] with t ht
-      intro _
-      simp only [Set.indicator, Set.mem_ofPred_eq, le_iff_lt_or_eq, ht, or_false]
-    _ = _ := intervalIntegral.integral_indicator ⟨S.coe_nonneg, by exact_mod_cast hST⟩
-
- 
-
-theorem area_prefix_energy_identity {Z : Type*} [MeasurableSpace Z]
-    {ν : Measure Z} [IsFiniteMeasure ν]
-    {f : ℝ × Z → ℝ} (hfm : Measurable f)
-    {τ : Z → ℝ≥0} (hτm : Measurable τ) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
-    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
-    (∫ p : Z × Z, (∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
-        f (t,p.1) * f (t,p.2)) ∂(ν.prod ν)) =
-      ∫ t in (0 : ℝ)..(T : ℝ), (∫ z, if t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
-  let g (p : ℝ × (Z × Z)) := if p.1 < (min (τ p.2.1) (τ p.2.2) : ℝ≥0) then
-    f (p.1,p.2.1) * f (p.1,p.2.2) else 0
-  have htm : Measurable (fun p : ℝ × (Z × Z) => (min (τ p.2.1) (τ p.2.2) : ℝ)) :=
-    NNReal.continuous_coe.measurable.comp (((hτm.comp (measurable_fst.comp measurable_snd))).min
-      (hτm.comp (measurable_snd.comp measurable_snd)))
-  have hgm : Measurable g :=
-    ((hfm.comp (measurable_fst.prodMk (measurable_fst.comp measurable_snd))).mul
-      (hfm.comp (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))).ite
-        (measurableSet_lt measurable_fst htm) measurable_const
-  have hgb (p : ℝ × (Z × Z)) : ‖g p‖ ≤ K^2 := by
-    rw [Real.norm_eq_abs]
-    dsimp only [g]
-    split_ifs
-    · rw [abs_mul, pow_two]
-      exact mul_le_mul (hb _ _) (hb _ _) (abs_nonneg _) hK
-    · simpa using sq_nonneg K
-  have hgi : Integrable g ((volume.restrict (Ioc 0 (T : ℝ))).prod (ν.prod ν)) :=
-    Integrable.of_bound hgm.aestronglyMeasurable (K^2) (Eventually.of_forall hgb)
-  have hp (p : Z × Z) : (∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
-      f (t,p.1) * f (t,p.2)) = ∫ t in Ioc (0 : ℝ) T, g (t,p) := by
-    rw [← intervalIntegral.integral_of_le T.coe_nonneg]
-    exact (interval_integral_prefix_lt ((min_le_left _ _).trans (hτ p.1)) _).symm
-  have hg (t : ℝ) : (∫ p : Z × Z, g (t,p) ∂(ν.prod ν)) =
-      (∫ z, if t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
-    calc
-      _ = ∫ p : Z × Z, (if t < (τ p.1 : ℝ) then f (t,p.1) else 0) *
-          (if t < (τ p.2 : ℝ) then f (t,p.2) else 0) ∂(ν.prod ν) := by
-        apply integral_congr_ae
-        apply Eventually.of_forall
-        intro p
-        simp only [g, NNReal.coe_min, lt_min_iff]
-        split_ifs <;> simp_all
-      _ = _ := by
-        simpa only [pow_two] using (integral_prod_mul (μ := ν) (ν := ν)
-          (fun z => if t < (τ z : ℝ) then f (t,z) else 0)
-          (fun z => if t < (τ z : ℝ) then f (t,z) else 0))
-  simp_rw [hp]
-  rw [← integral_integral_swap hgi]
-  simp only [hg, intervalIntegral.integral_of_le T.coe_nonneg]
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable
-
-noncomputable def areaWindowSample {Z : Type*} (f : ℝ × Z → ℝ)
-    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z : Z) : ℝ :=
-  if σ z ≤ dyadicTime T n i ∧ dyadicTime T n i < τ z then f (dyadicTime T n i,z) else 0
-
-lemma areaWindowSample_measurable {Z : Type*} [MeasurableSpace Z]
-    {f : ℝ × Z → ℝ} (hf : Measurable f) {σ τ : Z → ℝ≥0}
-    (hσ : Measurable σ) (hτ : Measurable τ) (T : ℝ≥0) (n i : ℕ) :
-    Measurable (areaWindowSample f σ τ T n i) :=
-  (hf.comp (measurable_const.prodMk measurable_id)).ite
-    ((measurableSet_le hσ measurable_const).inter (measurableSet_lt measurable_const (NNReal.continuous_coe.measurable.comp hτ))) measurable_const
-
-lemma areaWindowSample_bound {Z : Type*} {f : ℝ × Z → ℝ}
-    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
-    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z : Z) :
-    |areaWindowSample f σ τ T n i z| ≤ K := by
-  unfold areaWindowSample
-  split_ifs
-  · exact hb _ _
-  · simpa using hK
-
-lemma window_pair_eq_prefix_difference {Z : Type*} (f : ℝ × Z → ℝ)
-    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z w : Z) :
-    areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w =
-      (if dyadicTime T n i < min (τ z) (τ w) then
-        f (dyadicTime T n i,z) * f (dyadicTime T n i,w) else 0) -
-      (if dyadicTime T n i < min (min (τ z) (τ w)) (max (σ z) (σ w)) then
-        f (dyadicTime T n i,z) * f (dyadicTime T n i,w) else 0) := by
-  simp only [areaWindowSample, lt_min_iff, lt_max_iff]
-  by_cases hz : σ z ≤ dyadicTime T n i
-  · by_cases hw : σ w ≤ dyadicTime T n i
-    · simp [hz, hw, not_lt.mpr hz, not_lt.mpr hw]
-      split_ifs <;> simp_all
-    · simp only [hw, false_and, ite_false, mul_zero]
-      have ht := lt_of_not_ge hw
-      simp [ht, sub_self]
-  · simp only [hz, false_and, ite_false, zero_mul]
-    have ht := lt_of_not_ge hz
-    simp [ht, sub_self]
-
-lemma window_pair_energy_tendsto {Z : Type*} (f : ℝ × Z → ℝ)
-    (hf : ∀ z, Continuous (fun t => f (t,z)))
-    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T) (z w : Z) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w) *
-        ((T : ℝ) / (2 : ℝ)^n)) atTop
-      (𝓝 ((∫ t in (0 : ℝ)..(min (τ z) (τ w) : ℝ≥0), f (t,z) * f (t,w)) -
-        ∫ t in (0 : ℝ)..(min (min (τ z) (τ w)) (max (σ z) (σ w)) : ℝ≥0), f (t,z) * f (t,w))) := by
-  have hH := stopped_energy_tendsto ((min_le_left (τ z) (τ w)).trans (hτ z))
-    ((hf z).mul (hf w)).continuousOn
-  have hL := stopped_energy_tendsto
-    ((min_le_left (min (τ z) (τ w)) (max (σ z) (σ w))).trans ((min_le_left (τ z) (τ w)).trans (hτ z)))
-    ((hf z).mul (hf w)).continuousOn
-  simp only [Pi.mul_apply] at hH hL
-  convert hH.sub hL using 1
-  ext n
-  rw [← Finset.sum_sub_distrib]
-  apply Finset.sum_congr rfl
-  intro i hi
-  rw [window_pair_eq_prefix_difference, dyadicTime_step, sub_mul]
-
-lemma window_pair_energy_bound {Z : Type*} {f : ℝ × Z → ℝ}
-    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
-    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n : ℕ) (z w : Z) :
-    |∑ i ∈ Finset.range (2^n),
-      (areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w) *
-        ((T : ℝ) / (2 : ℝ)^n)| ≤ K^2 * (T : ℝ) := by
-  calc
-    _ ≤ ∑ i ∈ Finset.range (2^n),
-        |(areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w) *
-          ((T : ℝ) / (2 : ℝ)^n)| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ i ∈ Finset.range (2^n), K^2 * ((T : ℝ) / (2 : ℝ)^n) := by
-      apply Finset.sum_le_sum
-      intro i hi
-      rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (T : ℝ) / (2 : ℝ)^n), abs_mul, pow_two]
-      exact mul_le_mul_of_nonneg_right
-        (mul_le_mul (areaWindowSample_bound hK hb σ τ T n i z)
-          (areaWindowSample_bound hK hb σ τ T n i w) (abs_nonneg _) hK) (by positivity)
-    _ = _ := by
-      simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat]
-      field_simp
-
-lemma area_window_pair_energy_tendsto {Z : Type*} [MeasurableSpace Z]
-    {ν : Measure Z} [IsFiniteMeasure ν]
-    {f : ℝ × Z → ℝ} (hfm : Measurable f) (hfc : ∀ z, Continuous (fun t => f (t,z)))
-    {σ τ : Z → ℝ≥0} (hσm : Measurable σ) (hτm : Measurable τ) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
-    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (∫ z, areaWindowSample f σ τ T n i z ∂ν)^2 * ((T : ℝ) / (2 : ℝ)^n)) atTop
-      (𝓝 (∫ p : Z × Z, ((∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
-        f (t,p.1) * f (t,p.2)) - ∫ t in (0 : ℝ)..
-          (min (min (τ p.1) (τ p.2)) (max (σ p.1) (σ p.2)) : ℝ≥0), f (t,p.1) * f (t,p.2)) ∂(ν.prod ν))) := by
-  let H (n : ℕ) (p : Z × Z) := ∑ i ∈ Finset.range (2^n),
-    (areaWindowSample f σ τ T n i p.1 * areaWindowSample f σ τ T n i p.2) *
-      ((T : ℝ) / (2 : ℝ)^n)
-  have hC (n i : ℕ) := areaWindowSample_measurable hfm hσm hτm T n i
-  have hH (n : ℕ) : Measurable (H n) := by
-    apply Finset.measurable_sum
-    intro i hi
-    exact (((hC n i).comp measurable_fst).mul ((hC n i).comp measurable_snd)).mul_const _
-  have hbH (n : ℕ) (p : Z × Z) : ‖H n p‖ ≤ K^2 * (T : ℝ) := by
-    rw [Real.norm_eq_abs]
-    exact window_pair_energy_bound hK hb σ τ T n p.1 p.2
-  have ht := tendsto_integral_of_dominated_convergence
-    (μ := ν.prod ν) (fun _ => K^2 * (T : ℝ))
-    (fun n => (hH n).aestronglyMeasurable) (integrable_const _)
-    (fun n => Eventually.of_forall (hbH n))
-    (Eventually.of_forall fun p : Z × Z => window_pair_energy_tendsto f hfc σ τ T hτ p.1 p.2)
-  have he (n : ℕ) : (∫ p, H n p ∂(ν.prod ν)) =
-      ∑ i ∈ Finset.range (2^n), (∫ z, areaWindowSample f σ τ T n i z ∂ν)^2 *
-        ((T : ℝ) / (2 : ℝ)^n) := by
-    have hi (i : ℕ) : Integrable (areaWindowSample f σ τ T n i) ν :=
-      Integrable.of_bound (hC n i).aestronglyMeasurable K
-        (Eventually.of_forall fun z => by rw [Real.norm_eq_abs]; exact areaWindowSample_bound hK hb σ τ T n i z)
-    change (∫ p : Z × Z, ∑ i ∈ Finset.range (2^n),
-      (areaWindowSample f σ τ T n i p.1 * areaWindowSample f σ τ T n i p.2) *
-        ((T : ℝ) / (2 : ℝ)^n) ∂(ν.prod ν)) = _
-    rw [integral_finsetSum _ (fun i _ => ((hi i).mul_prod (hi i)).mul_const _)]
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [integral_mul_const, integral_prod_mul, pow_two]
-  simpa only [he] using ht
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma real_window_pair_eq (f : ℝ → ℝ) (a b c d t : ℝ) :
-    (if a ≤ t ∧ t < b ∧ c ≤ t ∧ t < d then f t else 0) =
-      (if t < min b d then f t else 0) -
-      (if t < min (min b d) (max a c) then f t else 0) := by
-  simp only [lt_min_iff, lt_max_iff]
-  by_cases ha : a ≤ t
-  · by_cases hc : c ≤ t
-    · simp [ha, hc, not_lt.mpr ha, not_lt.mpr hc]
-    · simp only [hc, false_and, and_false, ite_false]
-      have ht := lt_of_not_ge hc
-      simp [ht]
-  · simp only [ha, false_and, ite_false]
-    have ht := lt_of_not_ge ha
-    simp [ht]
-
- 
-
-theorem area_window_energy_identity {Z : Type*} [MeasurableSpace Z]
-    {ν : Measure Z} [IsFiniteMeasure ν]
-    {f : ℝ × Z → ℝ} (hfm : Measurable f)
-    {σ τ : Z → ℝ≥0} (hσm : Measurable σ) (hτm : Measurable τ)
-    (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
-    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
-    (∫ p : Z × Z, ((∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
-        f (t,p.1) * f (t,p.2)) - ∫ t in (0 : ℝ)..
-          (min (min (τ p.1) (τ p.2)) (max (σ p.1) (σ p.2)) : ℝ≥0), f (t,p.1) * f (t,p.2)) ∂(ν.prod ν)) =
-      ∫ t in (0 : ℝ)..(T : ℝ),
-        (∫ z, if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
-  let g (p : ℝ × (Z × Z)) := if (σ p.2.1 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2.1 : ℝ) ∧
-      (σ p.2.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2.2 : ℝ) then f (p.1,p.2.1) * f (p.1,p.2.2) else 0
-  have htm (v : Z → ℝ≥0) (hv : Measurable v) : Measurable (fun z => (v z : ℝ)) :=
-    NNReal.continuous_coe.measurable.comp hv
-  have hgm : Measurable g :=
-    ((hfm.comp (measurable_fst.prodMk (measurable_fst.comp measurable_snd))).mul
-      (hfm.comp (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))).ite
-        ((measurableSet_le ((htm σ hσm).comp (measurable_fst.comp measurable_snd)) measurable_fst).inter
-          ((measurableSet_lt measurable_fst ((htm τ hτm).comp (measurable_fst.comp measurable_snd))).inter
-            ((measurableSet_le ((htm σ hσm).comp (measurable_snd.comp measurable_snd)) measurable_fst).inter
-              (measurableSet_lt measurable_fst ((htm τ hτm).comp (measurable_snd.comp measurable_snd)))))) measurable_const
-  have hgb (p : ℝ × (Z × Z)) : ‖g p‖ ≤ K^2 := by
-    rw [Real.norm_eq_abs]
-    dsimp only [g]
-    split_ifs
-    · rw [abs_mul, pow_two]
-      exact mul_le_mul (hb _ _) (hb _ _) (abs_nonneg _) hK
-    · simpa using sq_nonneg K
-  have hgi : Integrable g ((volume.restrict (Ioc 0 (T : ℝ))).prod (ν.prod ν)) :=
-    Integrable.of_bound hgm.aestronglyMeasurable (K^2) (Eventually.of_forall hgb)
-  have hp (p : Z × Z) : ((∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
-      f (t,p.1) * f (t,p.2)) - ∫ t in (0 : ℝ)..
-        (min (min (τ p.1) (τ p.2)) (max (σ p.1) (σ p.2)) : ℝ≥0), f (t,p.1) * f (t,p.2)) =
-        ∫ t in Ioc (0 : ℝ) T, g (t,p) := by
-    let H := min (τ p.1) (τ p.2)
-    let L := min H (max (σ p.1) (σ p.2))
-    have hHT : H ≤ T := (min_le_left _ _).trans (hτ p.1)
-    have hLT : L ≤ T := (min_le_left _ _).trans hHT
-    have hi (S : ℝ≥0) : IntervalIntegrable
-        (fun t : ℝ => if t < (S : ℝ) then f (t,p.1) * f (t,p.2) else 0) volume 0 T := by
-      apply (intervalIntegrable_iff_integrableOn_Icc_of_le T.coe_nonneg).mpr
-      apply Integrable.of_bound (μ := volume.restrict (Icc (0 : ℝ) T)) ?_ (K^2) ?_
-      · exact (((hfm.comp (measurable_id.prodMk measurable_const)).mul
-          (hfm.comp (measurable_id.prodMk measurable_const))).ite
-            (measurableSet_lt measurable_id measurable_const) measurable_const).aestronglyMeasurable
-      · apply Eventually.of_forall
-        intro t
-        rw [Real.norm_eq_abs]
-        split_ifs
-        · rw [abs_mul, pow_two]
-          exact mul_le_mul (hb _ _) (hb _ _) (abs_nonneg _) hK
-        · simpa using sq_nonneg K
-    rw [← interval_integral_prefix_lt hHT, ← interval_integral_prefix_lt hLT,
-      ← intervalIntegral.integral_sub (hi H) (hi L), ← intervalIntegral.integral_of_le T.coe_nonneg]
-    apply intervalIntegral.integral_congr
-    intro t ht
-    dsimp only [g, H, L]
-    simpa only [NNReal.coe_min, NNReal.coe_max] using
-      (real_window_pair_eq (fun t => f (t,p.1) * f (t,p.2)) (σ p.1) (τ p.1) (σ p.2) (τ p.2) t).symm
-  have hg (t : ℝ) : (∫ p : Z × Z, g (t,p) ∂(ν.prod ν)) =
-      (∫ z, if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
-    calc
-      _ = ∫ p : Z × Z,
-          (if (σ p.1 : ℝ) ≤ t ∧ t < (τ p.1 : ℝ) then f (t,p.1) else 0) *
-          (if (σ p.2 : ℝ) ≤ t ∧ t < (τ p.2 : ℝ) then f (t,p.2) else 0) ∂(ν.prod ν) := by
-        apply integral_congr_ae
-        apply Eventually.of_forall
-        intro p
-        dsimp only [g]
-        split_ifs <;> simp_all
-      _ = _ := by
-        simpa only [pow_two] using (integral_prod_mul (μ := ν) (ν := ν)
-          (fun z => if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0)
-          (fun z => if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0))
-  simp_rw [hp]
-  rw [← integral_integral_swap hgi]
-  simp only [hg, intervalIntegral.integral_of_le T.coe_nonneg]
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField
-
- 
-
-end SLEExactGauge
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.SimpleFields
+
+namespace OAI
+
+/-! Exponential Martingales for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+open L2Limit
+lemma finite_uniformIntegrable_one_of_eLpNorm_two_bounded {Ω ι : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {X : ι → Ω → ℝ}
+    (hXm : ∀ i, AEStronglyMeasurable (X i) P) (C : ℝ≥0)
+    (hC : ∀ i, eLpNorm (X i) 2 P ≤ C) : UniformIntegrable X 1 P := by
+  let D : ℝ≥0∞ := C * (P univ) ^ (1/2 : ℝ)
+  have hDt : D ≠ ∞ := ENNReal.mul_ne_top ENNReal.coe_ne_top
+    (ENNReal.rpow_ne_top_of_nonneg (by norm_num) (measure_ne_top P univ))
+  refine ⟨unifIntegrable_iff.mpr ?_, D.toNNReal, ?_⟩
+  · intro bound hbound
+    obtain ⟨small, hsmall, hsmall_bound⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hbound
+    let ε : ℝ := small
+    have hε : 0 < ε := by exact_mod_cast hsmall
+    let d : ℝ≥0 := ⟨ε / ((C : ℝ)+1), div_nonneg hε.le (by positivity)⟩
+    have hd : 0 < (d : ℝ) := div_pos hε (by positivity)
+    refine ⟨ENNReal.ofReal ((d : ℝ)^2), ENNReal.ofReal_pos.mpr (sq_pos_of_pos hd),
+      fun i s hμs => ?_⟩
+    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (μ := P.restrict s)
+      (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i).restrict
+    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
+    rw [Measure.restrict_apply_univ] at hb
+    have hp : (P s) ^ (1/2 : ℝ) ≤ (d : ℝ≥0∞) := by
+      calc
+        _ ≤ (ENNReal.ofReal ((d : ℝ)^2)) ^ (1/2 : ℝ) :=
+          ENNReal.rpow_le_rpow hμs (by norm_num)
+        _ = _ := by
+          rw [ENNReal.ofReal_pow d.coe_nonneg]
+          simp only [ENNReal.ofReal_coe_nnreal]
+          rw [← ENNReal.rpow_natCast, ← ENNReal.rpow_mul]
+          norm_num
+    calc
+      _ ≤ eLpNorm (X i) 2 (P.restrict s) * (P s) ^ (1/2 : ℝ) := hb
+      _ ≤ C * (d : ℝ≥0∞) := mul_le_mul
+        ((eLpNorm_mono_measure _ Measure.restrict_le_self).trans (hC i)) hp (by positivity) (by positivity)
+      _ ≤ ENNReal.ofReal ε := by
+        rw [← ENNReal.coe_mul, ← ENNReal.ofReal_coe_nnreal]
+        simp only [NNReal.coe_mul]
+        apply ENNReal.ofReal_le_ofReal
+        change (C : ℝ) * (ε / ((C : ℝ)+1)) ≤ ε
+        rw [← mul_div_assoc, div_le_iff₀ (by positivity)]
+        nlinarith
+      _ ≤ bound := by
+        simpa only [ε, ENNReal.ofReal_coe_nnreal] using hsmall_bound.le
+  · intro i
+    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i)
+    rw [ENNReal.coe_toNNReal hDt]
+    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
+    exact hb.trans (mul_le_mul (hC i) le_rfl bot_le bot_le)
+
+theorem finite_L1_tendsto_of_square_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ} {Y : Ω → ℝ}
+    (hX : ∀ n, MemLp (X n) 2 P) (A : ℝ)
+    (hA : ∀ n, (∫ ω, X n ω ^ 2 ∂P) ≤ A)
+    (hXY : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) :
+    Integrable Y P ∧ Tendsto (fun n => eLpNorm (X n - Y) 1 P) atTop (𝓝 0) := by
+  let C : ℝ≥0∞ := (ENNReal.ofReal A) ^ (1/2 : ℝ)
+  have hCt : C ≠ ∞ := ENNReal.rpow_ne_top_of_nonneg (by norm_num) ENNReal.ofReal_ne_top
+  have hC (n : ℕ) : eLpNorm (X n) 2 P ≤ C.toNNReal := by
+    rw [ENNReal.coe_toNNReal hCt, eLpNorm_two_eq_integral_sq (hX n)]
+    exact ENNReal.rpow_le_rpow (ENNReal.ofReal_le_ofReal (hA n)) (by norm_num)
+  have hui := finite_uniformIntegrable_one_of_eLpNorm_two_bounded
+    (fun n => (hX n).aestronglyMeasurable) C.toNNReal hC
+  have hY := hui.memLp_of_ae_tendsto hXY
+  exact ⟨memLp_one_iff_integrable.1 hY,
+    tendsto_Lp_finite_of_tendsto_ae (by norm_num : (1 : ℝ≥0∞) ≤ 1)
+      (by norm_num) (fun n => (hX n).aestronglyMeasurable) hY hui.unifIntegrable hXY⟩
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+
+lemma product_memLp_two_of_section_square_bound
+    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν] [SFinite P]
+    {X : Z × Ω → ℝ} (hX : Measurable X)
+    (hX2 : ∀ z, MemLp (fun ω => X (z, ω)) 2 P)
+    (V : ℝ) (hV : ∀ z, (∫ ω, X (z, ω)^2 ∂P) ≤ V) :
+    MemLp X 2 (ν.prod P) ∧
+      (∫ p, X p ^ 2 ∂(ν.prod P)) ≤ V * (ν univ).toReal := by
+  have hm : StronglyMeasurable (fun z => ∫ ω, X (z, ω)^2 ∂P) :=
+    (hX.pow_const 2).stronglyMeasurable.integral_prod_right'
+  have hn (z : Z) : 0 ≤ ∫ ω, X (z, ω)^2 ∂P := integral_nonneg fun ω => sq_nonneg _
+  have hi : Integrable (fun z => ∫ ω, X (z, ω)^2 ∂P) ν := by
+    apply Integrable.of_bound hm.aestronglyMeasurable V
+    exact Eventually.of_forall fun z => by rw [Real.norm_eq_abs, abs_of_nonneg (hn z)]; exact hV z
+  have hisq : Integrable (fun p => X p ^ 2) (ν.prod P) := by
+    apply (integrable_prod_iff (hX.pow_const 2).aestronglyMeasurable).mpr
+    refine ⟨Eventually.of_forall fun z => (hX2 z).integrable_sq, ?_⟩
+    simpa only [Real.norm_eq_abs, abs_sq] using hi
+  refine ⟨(memLp_two_iff_integrable_sq hX.aestronglyMeasurable).mpr hisq, ?_⟩
+  rw [integral_prod _ hisq]
+  calc
+    _ ≤ ∫ _z, V ∂ν := integral_mono hi (integrable_const _) hV
+    _ = _ := by simp [mul_comm, Measure.real]
+
+lemma area_eLpNorm_one_le_product
+    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [SFinite ν] [SFinite P]
+    {X Y : Z × Ω → ℝ} (hX : Integrable X (ν.prod P)) (hY : Integrable Y (ν.prod P)) :
+    eLpNorm (fun ω => (∫ z, X (z, ω) ∂ν) - ∫ z, Y (z, ω) ∂ν) 1 P ≤
+      eLpNorm (X - Y) 1 (ν.prod P) := by
+  rw [eLpNorm_one_eq_lintegral_enorm
+    (f := fun ω => (∫ z, X (z, ω) ∂ν) - ∫ z, Y (z, ω) ∂ν)
+    (hX.integral_prod_right.sub hY.integral_prod_right).aestronglyMeasurable,
+    eLpNorm_one_eq_lintegral_enorm (hX.sub hY).aestronglyMeasurable]
+  calc
+    _ ≤ ∫⁻ ω, ∫⁻ z, ‖X (z, ω) - Y (z, ω)‖ₑ ∂ν ∂P := by
+      apply lintegral_mono_ae
+      filter_upwards [hX.prod_left_ae, hY.prod_left_ae] with ω hx hy
+      rw [← integral_sub hx hy]
+      exact enorm_integral_le_lintegral_enorm _
+    _ = _ := by
+      rw [← lintegral_lintegral_swap (hX.sub hY).aestronglyMeasurable.enorm]
+      exact (lintegral_prod _ (hX.sub hY).aestronglyMeasurable.enorm).symm
+
+ 
+
+theorem area_L1_tendsto_of_square_bound
+    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν] [IsProbabilityMeasure P]
+    {X : ℕ → Z × Ω → ℝ} {Y : Z × Ω → ℝ}
+    (hXm : ∀ n, Measurable (X n)) (hYm : Measurable Y)
+    (hX2 : ∀ n z, MemLp (fun ω => X n (z, ω)) 2 P)
+    (V : ℝ) (hV : ∀ n z, (∫ ω, X n (z, ω)^2 ∂P) ≤ V)
+    (hXY : ∀ z, ∀ᵐ ω ∂P, Tendsto (fun n => X n (z, ω)) atTop (𝓝 (Y (z, ω)))) :
+    Integrable Y (ν.prod P) ∧ Tendsto (fun n =>
+      eLpNorm (fun ω => (∫ z, X n (z, ω) ∂ν) - ∫ z, Y (z, ω) ∂ν) 1 P) atTop (𝓝 0) := by
+  have hb (n : ℕ) := product_memLp_two_of_section_square_bound (ν := ν) (P := P) (hXm n) (hX2 n) V (hV n)
+  have hc : ∀ᵐ p ∂(ν.prod P), Tendsto (fun n => X n p) atTop (𝓝 (Y p)) :=
+    (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun hXm hYm)).mpr (Eventually.of_forall hXY)
+  obtain ⟨hi, ht⟩ := finite_L1_tendsto_of_square_bound (fun n => (hb n).1)
+    (V * (ν univ).toReal) (fun n => (hb n).2) hc
+  refine ⟨hi, ?_⟩
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht
+    (fun n => bot_le) (fun n => area_eLpNorm_one_le_product ((hb n).1.integrable (by norm_num)) hi)
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+open BrownianPredictable PredictableLimit
+
+lemma area_driver_sum {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {ν : Measure Z} [IsFiniteMeasure ν]
+    (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) (C : ℕ → Z × Ω → ℝ)
+    (hC : ∀ i, Measurable (C i)) (K : ℝ) (hb : ∀ i z ω, |C i (z, ω)| ≤ K)
+    (n : ℕ) (ω : Ω) :
+    (∫ z, driverPredictableSum B t (fun i ω => C i (z, ω)) n ω ∂ν) =
+      driverPredictableSum B t (fun i ω => ∫ z, C i (z, ω) ∂ν) n ω := by
+  have hi (i : ℕ) : Integrable (fun z => C i (z, ω)) ν :=
+    Integrable.of_bound ((hC i).comp (measurable_id.prodMk measurable_const)).aestronglyMeasurable K
+      (Eventually.of_forall fun z => by simpa only [Real.norm_eq_abs] using hb i z ω)
+  simp only [driverPredictableSum]
+  rw [integral_finsetSum _ (fun i _ => (hi i).mul_const _)]
+  exact Finset.sum_congr rfl fun i hi => integral_mul_const _ _
+
+ 
+
+theorem brownian_area_sum_L1_limit
+    {Z Ω : Type*} [MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+    (T : ℝ≥0) (C : ℕ → ℕ → Z × Ω → ℝ)
+    (hCm : ∀ n i, Measurable (C n i))
+    (hCa : ∀ n i z, Measurable[Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)] (fun ω => C n i (z, ω)))
+    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ n i z ω, |C n i (z, ω)| ≤ K)
+    (Y : Z × Ω → ℝ) (hYm : Measurable Y)
+    (hlim : ∀ z, ∀ᵐ ω ∂P, Tendsto
+      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z, ω)) (2^n) ω)
+      atTop (𝓝 (Y (z, ω)))) :
+    Integrable Y (ν.prod P) ∧ Tendsto (fun n =>
+      eLpNorm (fun ω => driverPredictableSum B (dyadicTime T n)
+        (fun i ω => ∫ z, C n i (z, ω) ∂ν) (2^n) ω - ∫ z, Y (z, ω) ∂ν) 1 P)
+        atTop (𝓝 0) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let X (n : ℕ) (p : Z × Ω) :=
+    driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (p.1, ω)) (2^n) p.2
+  have hCp (n i : ℕ) (z : Z) : Measurable[pastSigma B (dyadicTime T n i)]
+      (fun ω => C n i (z, ω)) := by
+    exact (hCa n i z).mono
+      (le_of_eq (Filtration.natural_eq_comap B (fun t => (hm t).stronglyMeasurable) _)) le_rfl
+  have hC2 (n i : ℕ) (z : Z) : MemLp (fun ω => C n i (z, ω)) 2 P :=
+    MemLp.of_bound ((hCm n i).comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable K
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hb n i z ω)
+  have hXm (n : ℕ) : Measurable (X n) := by
+    apply Finset.measurable_sum
+    intro i hi
+    exact (hCm n i).mul (((hm _).comp measurable_snd).sub ((hm _).comp measurable_snd))
+  have hX2 (n : ℕ) (z : Z) : MemLp (fun ω => X n (z, ω)) 2 P := by
+    exact dyadicDriverSum_memLp hB T (fun n i ω => C n i (z, ω))
+      (fun n i => hCp n i z) (fun n i => hC2 n i z) n
+  have hV (n : ℕ) (z : Z) : (∫ ω, X n (z, ω)^2 ∂P) ≤ K^2 * (T : ℝ) := by
+    change (∫ ω, (∑ i ∈ Finset.range (2^n), C n i (z, ω) *
+      (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) ^ 2 ∂P) ≤ _
+    rw [brownian_predictableSum_isometry hB _ (dyadicTime_mono T n) _
+      (fun i => hCp n i z) (fun i => hC2 n i z)]
+    calc
+      _ ≤ ∑ i ∈ Finset.range (2^n), K^2 * ((T : ℝ) / (2 : ℝ)^n) := by
+        apply Finset.sum_le_sum
+        intro i hi
+        rw [dyadicTime_step]
+        apply mul_le_mul_of_nonneg_right _ (by positivity)
+        calc
+          _ ≤ ∫ _ω, K^2 ∂P := integral_mono (hC2 n i z).integrable_sq (integrable_const _) fun ω =>
+            by
+              have hh := (sq_le_sq₀ (abs_nonneg (C n i (z, ω))) hK).mpr (hb n i z ω)
+              simpa only [sq_abs] using hh
+          _ = K^2 := by simp
+      _ = K^2 * (T : ℝ) := by
+        simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat]
+        field_simp
+  obtain ⟨hi, ht⟩ := area_L1_tendsto_of_square_bound (ν := ν) hXm hYm hX2 _ hV hlim
+  refine ⟨hi, ?_⟩
+  have he (n : ℕ) (ω : Ω) : (∫ z, X n (z, ω) ∂ν) =
+      driverPredictableSum B (dyadicTime T n) (fun i ω => ∫ z, C n i (z, ω) ∂ν) (2^n) ω :=
+    area_driver_sum B _ _ (hCm n) K (hb n) _ ω
+  simpa only [he] using ht
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+open BrownianPredictable PredictableLimit
+
+ 
+
+theorem brownian_area_exponential_limit
+    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+    (T : ℝ≥0) (C : ℕ → ℕ → Z × Ω → ℝ)
+    (hCm : ∀ n i, Measurable (C n i))
+    (hCa : ∀ n i, @Measurable (Z × Ω) ℝ
+      (mZ.prod (Filtration.natural B (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)))
+      (borel ℝ) (C n i))
+    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ n i z ω, |C n i (z, ω)| ≤ K)
+    (Y : Z × Ω → ℝ) (hYm : Measurable Y)
+    (hlim : ∀ z, ∀ᵐ ω ∂P, Tendsto
+      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z, ω)) (2^n) ω)
+      atTop (𝓝 (Y (z, ω))))
+    (Q : Ω → ℝ)
+    (hE : ∀ᵐ ω ∂P, Tendsto (fun n => driverPredictableEnergy (dyadicTime T n)
+      (fun i ω => ∫ z, C n i (z, ω) ∂ν) (2^n) ω) atTop (𝓝 (Q ω)))
+    (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp
+      (θ * (∫ z, Y (z, ω) ∂ν) - θ^2 * Q ω / 2)) ∂P) ≤ 1 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let D (n i : ℕ) (ω : Ω) := ∫ z, C n i (z, ω) ∂ν
+  have hDa (n i : ℕ) : Measurable[Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)] (D n i) := by
+    let : MeasurableSpace Ω := Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)
+    exact (hCa n i).stronglyMeasurable.integral_prod_left'.measurable
+  let X (n : ℕ) := driverPredictableSum B (dyadicTime T n) (D n) (2^n)
+  let E (n : ℕ) := driverPredictableEnergy (dyadicTime T n) (D n) (2^n)
+  have hXm (n : ℕ) : Measurable (X n) :=
+    (driverPredictableSum_measurable_past B hm _ (dyadicTime_mono T n) _ (hDa n) _).mono
+      (Filtration.le _ _) le_rfl
+  have hEm (n : ℕ) : Measurable (E n) :=
+    (driverPredictableEnergy_measurable_past B hm _ (dyadicTime_mono T n) _ (hDa n) _).mono
+      (Filtration.le _ _) le_rfl
+  have hca (n i : ℕ) (z : Z) : Measurable[Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable) (dyadicTime T n i)] (fun ω => C n i (z, ω)) :=
+    (hCa n i).comp (measurable_const.prodMk measurable_id)
+  obtain ⟨hi, ht⟩ := brownian_area_sum_L1_limit (ν := ν) (P := P) hB hm T C hCm hca K hK hb Y hYm hlim
+  have hY : Measurable (fun ω => ∫ z, Y (z, ω) ∂ν) :=
+    hYm.stronglyMeasurable.integral_prod_left'.measurable
+  have hc : TendstoInMeasure P X atTop (fun ω => ∫ z, Y (z, ω) ∂ν) :=
+    tendstoInMeasure_of_tendsto_eLpNorm (by norm_num : (1 : ℝ≥0∞) ≠ 0) ht
+  obtain ⟨ns, hns, hs⟩ := hc.exists_seq_tendsto_ae
+  apply compensated_exponential_limit_energy (fun n => X (ns n)) (fun n => E (ns n))
+    (fun ω => ∫ z, Y (z, ω) ∂ν) Q θ
+    (fun n => (((hXm _).const_mul θ).sub (((hEm _).const_mul (θ^2)).div_const 2)).exp.ennreal_ofReal.aemeasurable)
+    hs
+  · filter_upwards [hE] with ω he
+    exact he.comp hns.tendsto_atTop
+  · intro n
+    exact (driverPredictable_exponential hB hm _ (dyadicTime_mono T (ns n)) _ (hDa _) θ _).le
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField BrownianIto TaylorMeshes
+
+noncomputable def areaPrefixSample {Z : Type*} (f : ℝ × Z → ℝ) (τ : Z → ℝ≥0)
+    (T : ℝ≥0) (n i : ℕ) (z : Z) : ℝ :=
+  if dyadicTime T n i < τ z then f (dyadicTime T n i, z) else 0
+
+lemma areaPrefixSample_measurable {Z : Type*} [MeasurableSpace Z]
+    {f : ℝ × Z → ℝ} (hf : Measurable f) {τ : Z → ℝ≥0} (hτ : Measurable τ)
+    (T : ℝ≥0) (n i : ℕ) : Measurable (areaPrefixSample f τ T n i) :=
+  (hf.comp (measurable_const.prodMk measurable_id)).ite
+    (measurableSet_lt measurable_const hτ) measurable_const
+
+lemma areaPrefixSample_bound {Z : Type*} {f : ℝ × Z → ℝ}
+    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
+    (τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z : Z) :
+    |areaPrefixSample f τ T n i z| ≤ K := by
+  unfold areaPrefixSample
+  split_ifs
+  · exact hb _ _
+  · simpa using hK
+
+lemma prefix_pair_energy_tendsto {Z : Type*} (f : ℝ × Z → ℝ)
+    (hf : ∀ z, Continuous (fun t => f (t,z)))
+    (τ : Z → ℝ≥0) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T) (z w : Z) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (areaPrefixSample f τ T n i z * areaPrefixSample f τ T n i w) *
+        ((T : ℝ) / (2 : ℝ)^n)) atTop
+      (𝓝 (∫ t in (0 : ℝ)..(min (τ z) (τ w) : ℝ≥0), f (t,z) * f (t,w))) := by
+  have hh := stopped_energy_tendsto ((min_le_left (τ z) (τ w)).trans (hτ z))
+    ((hf z).mul (hf w)).continuousOn
+  simp only [Pi.mul_apply] at hh
+  convert hh using 1
+  ext n
+  apply Finset.sum_congr rfl
+  intro i hi
+  rw [dyadicTime_step]
+  simp only [areaPrefixSample, lt_min_iff]
+  split_ifs <;> simp_all
+
+lemma prefix_pair_energy_bound {Z : Type*} {f : ℝ × Z → ℝ}
+    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
+    (τ : Z → ℝ≥0) (T : ℝ≥0) (n : ℕ) (z w : Z) :
+    |∑ i ∈ Finset.range (2^n),
+      (areaPrefixSample f τ T n i z * areaPrefixSample f τ T n i w) *
+        ((T : ℝ) / (2 : ℝ)^n)| ≤ K^2 * (T : ℝ) := by
+  calc
+    _ ≤ ∑ i ∈ Finset.range (2^n),
+        |(areaPrefixSample f τ T n i z * areaPrefixSample f τ T n i w) *
+          ((T : ℝ) / (2 : ℝ)^n)| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ i ∈ Finset.range (2^n), K^2 * ((T : ℝ) / (2 : ℝ)^n) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (T : ℝ) / (2 : ℝ)^n), abs_mul, pow_two]
+      exact mul_le_mul_of_nonneg_right
+        (mul_le_mul (areaPrefixSample_bound hK hb τ T n i z)
+          (areaPrefixSample_bound hK hb τ T n i w) (abs_nonneg _) hK) (by positivity)
+    _ = _ := by
+      simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat]
+      field_simp
+
+ 
+
+theorem area_prefix_energy_tendsto {Z : Type*} [MeasurableSpace Z]
+    {ν : Measure Z} [IsFiniteMeasure ν]
+    {f : ℝ × Z → ℝ} (hfm : Measurable f) (hfc : ∀ z, Continuous (fun t => f (t,z)))
+    {τ : Z → ℝ≥0} (hτm : Measurable τ) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
+    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (∫ z, areaPrefixSample f τ T n i z ∂ν)^2 * ((T : ℝ) / (2 : ℝ)^n)) atTop
+      (𝓝 (∫ p : Z × Z, (∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
+        f (t,p.1) * f (t,p.2)) ∂(ν.prod ν))) := by
+  let H (n : ℕ) (p : Z × Z) := ∑ i ∈ Finset.range (2^n),
+    (areaPrefixSample f τ T n i p.1 * areaPrefixSample f τ T n i p.2) *
+      ((T : ℝ) / (2 : ℝ)^n)
+  have hC (n i : ℕ) := areaPrefixSample_measurable hfm hτm T n i
+  have hH (n : ℕ) : Measurable (H n) := by
+    apply Finset.measurable_sum
+    intro i hi
+    exact (((hC n i).comp measurable_fst).mul ((hC n i).comp measurable_snd)).mul_const _
+  have hbH (n : ℕ) (p : Z × Z) : ‖H n p‖ ≤ K^2 * (T : ℝ) := by
+    rw [Real.norm_eq_abs]
+    exact prefix_pair_energy_bound hK hb τ T n p.1 p.2
+  have ht := tendsto_integral_of_dominated_convergence
+    (μ := ν.prod ν) (fun _ => K^2 * (T : ℝ))
+    (fun n => (hH n).aestronglyMeasurable) (integrable_const _)
+    (fun n => Eventually.of_forall (hbH n))
+    (Eventually.of_forall fun p : Z × Z => prefix_pair_energy_tendsto f hfc τ T hτ p.1 p.2)
+  have he (n : ℕ) : (∫ p, H n p ∂(ν.prod ν)) =
+      ∑ i ∈ Finset.range (2^n), (∫ z, areaPrefixSample f τ T n i z ∂ν)^2 *
+        ((T : ℝ) / (2 : ℝ)^n) := by
+    have hi (i : ℕ) : Integrable (areaPrefixSample f τ T n i) ν :=
+      Integrable.of_bound (hC n i).aestronglyMeasurable K
+        (Eventually.of_forall fun z => by rw [Real.norm_eq_abs]; exact areaPrefixSample_bound hK hb τ T n i z)
+    rw [show H n = (fun p : Z × Z => ∑ i ∈ Finset.range (2^n),
+      (areaPrefixSample f τ T n i p.1 * areaPrefixSample f τ T n i p.2) *
+        ((T : ℝ) / (2 : ℝ)^n)) from rfl]
+    rw [integral_finsetSum _ (fun i _ => ((hi i).mul_prod (hi i)).mul_const _)]
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [integral_mul_const, integral_prod_mul, pow_two]
+  simpa only [he] using ht
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma interval_integral_prefix_lt {S T : ℝ≥0} (hST : S ≤ T) (f : ℝ → ℝ) :
+    (∫ t in (0 : ℝ)..(T : ℝ), if t < (S : ℝ) then f t else 0) =
+      ∫ t in (0 : ℝ)..(S : ℝ), f t := by
+  calc
+    _ = ∫ t in (0 : ℝ)..(T : ℝ), indicator {t : ℝ | t ≤ S} f t := by
+      apply intervalIntegral.integral_congr_ae
+      have hn : ∀ᵐ t : ℝ, t ≠ (S : ℝ) := by simp [ae_iff]
+      filter_upwards [hn] with t ht
+      intro _
+      simp only [Set.indicator, Set.mem_ofPred_eq, le_iff_lt_or_eq, ht, or_false]
+    _ = _ := intervalIntegral.integral_indicator ⟨S.coe_nonneg, by exact_mod_cast hST⟩
+
+ 
+
+theorem area_prefix_energy_identity {Z : Type*} [MeasurableSpace Z]
+    {ν : Measure Z} [IsFiniteMeasure ν]
+    {f : ℝ × Z → ℝ} (hfm : Measurable f)
+    {τ : Z → ℝ≥0} (hτm : Measurable τ) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
+    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
+    (∫ p : Z × Z, (∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
+        f (t,p.1) * f (t,p.2)) ∂(ν.prod ν)) =
+      ∫ t in (0 : ℝ)..(T : ℝ), (∫ z, if t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
+  let g (p : ℝ × (Z × Z)) := if p.1 < (min (τ p.2.1) (τ p.2.2) : ℝ≥0) then
+    f (p.1,p.2.1) * f (p.1,p.2.2) else 0
+  have htm : Measurable (fun p : ℝ × (Z × Z) => (min (τ p.2.1) (τ p.2.2) : ℝ)) :=
+    NNReal.continuous_coe.measurable.comp (((hτm.comp (measurable_fst.comp measurable_snd))).min
+      (hτm.comp (measurable_snd.comp measurable_snd)))
+  have hgm : Measurable g :=
+    ((hfm.comp (measurable_fst.prodMk (measurable_fst.comp measurable_snd))).mul
+      (hfm.comp (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))).ite
+        (measurableSet_lt measurable_fst htm) measurable_const
+  have hgb (p : ℝ × (Z × Z)) : ‖g p‖ ≤ K^2 := by
+    rw [Real.norm_eq_abs]
+    dsimp only [g]
+    split_ifs
+    · rw [abs_mul, pow_two]
+      exact mul_le_mul (hb _ _) (hb _ _) (abs_nonneg _) hK
+    · simpa using sq_nonneg K
+  have hgi : Integrable g ((volume.restrict (Ioc 0 (T : ℝ))).prod (ν.prod ν)) :=
+    Integrable.of_bound hgm.aestronglyMeasurable (K^2) (Eventually.of_forall hgb)
+  have hp (p : Z × Z) : (∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
+      f (t,p.1) * f (t,p.2)) = ∫ t in Ioc (0 : ℝ) T, g (t,p) := by
+    rw [← intervalIntegral.integral_of_le T.coe_nonneg]
+    exact (interval_integral_prefix_lt ((min_le_left _ _).trans (hτ p.1)) _).symm
+  have hg (t : ℝ) : (∫ p : Z × Z, g (t,p) ∂(ν.prod ν)) =
+      (∫ z, if t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
+    calc
+      _ = ∫ p : Z × Z, (if t < (τ p.1 : ℝ) then f (t,p.1) else 0) *
+          (if t < (τ p.2 : ℝ) then f (t,p.2) else 0) ∂(ν.prod ν) := by
+        apply integral_congr_ae
+        apply Eventually.of_forall
+        intro p
+        simp only [g, NNReal.coe_min, lt_min_iff]
+        split_ifs <;> simp_all
+      _ = _ := by
+        simpa only [pow_two] using (integral_prod_mul (μ := ν) (ν := ν)
+          (fun z => if t < (τ z : ℝ) then f (t,z) else 0)
+          (fun z => if t < (τ z : ℝ) then f (t,z) else 0))
+  simp_rw [hp]
+  rw [← integral_integral_swap hgi]
+  simp only [hg, intervalIntegral.integral_of_le T.coe_nonneg]
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable
+
+noncomputable def areaWindowSample {Z : Type*} (f : ℝ × Z → ℝ)
+    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z : Z) : ℝ :=
+  if σ z ≤ dyadicTime T n i ∧ dyadicTime T n i < τ z then f (dyadicTime T n i,z) else 0
+
+lemma areaWindowSample_measurable {Z : Type*} [MeasurableSpace Z]
+    {f : ℝ × Z → ℝ} (hf : Measurable f) {σ τ : Z → ℝ≥0}
+    (hσ : Measurable σ) (hτ : Measurable τ) (T : ℝ≥0) (n i : ℕ) :
+    Measurable (areaWindowSample f σ τ T n i) :=
+  (hf.comp (measurable_const.prodMk measurable_id)).ite
+    ((measurableSet_le hσ measurable_const).inter (measurableSet_lt measurable_const (NNReal.continuous_coe.measurable.comp hτ))) measurable_const
+
+lemma areaWindowSample_bound {Z : Type*} {f : ℝ × Z → ℝ}
+    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
+    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z : Z) :
+    |areaWindowSample f σ τ T n i z| ≤ K := by
+  unfold areaWindowSample
+  split_ifs
+  · exact hb _ _
+  · simpa using hK
+
+lemma window_pair_eq_prefix_difference {Z : Type*} (f : ℝ × Z → ℝ)
+    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n i : ℕ) (z w : Z) :
+    areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w =
+      (if dyadicTime T n i < min (τ z) (τ w) then
+        f (dyadicTime T n i,z) * f (dyadicTime T n i,w) else 0) -
+      (if dyadicTime T n i < min (min (τ z) (τ w)) (max (σ z) (σ w)) then
+        f (dyadicTime T n i,z) * f (dyadicTime T n i,w) else 0) := by
+  simp only [areaWindowSample, lt_min_iff, lt_max_iff]
+  by_cases hz : σ z ≤ dyadicTime T n i
+  · by_cases hw : σ w ≤ dyadicTime T n i
+    · simp [hz, hw, not_lt.mpr hz, not_lt.mpr hw]
+      split_ifs <;> simp_all
+    · simp only [hw, false_and, ite_false, mul_zero]
+      have ht := lt_of_not_ge hw
+      simp [ht, sub_self]
+  · simp only [hz, false_and, ite_false, zero_mul]
+    have ht := lt_of_not_ge hz
+    simp [ht, sub_self]
+
+lemma window_pair_energy_tendsto {Z : Type*} (f : ℝ × Z → ℝ)
+    (hf : ∀ z, Continuous (fun t => f (t,z)))
+    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T) (z w : Z) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w) *
+        ((T : ℝ) / (2 : ℝ)^n)) atTop
+      (𝓝 ((∫ t in (0 : ℝ)..(min (τ z) (τ w) : ℝ≥0), f (t,z) * f (t,w)) -
+        ∫ t in (0 : ℝ)..(min (min (τ z) (τ w)) (max (σ z) (σ w)) : ℝ≥0), f (t,z) * f (t,w))) := by
+  have hH := stopped_energy_tendsto ((min_le_left (τ z) (τ w)).trans (hτ z))
+    ((hf z).mul (hf w)).continuousOn
+  have hL := stopped_energy_tendsto
+    ((min_le_left (min (τ z) (τ w)) (max (σ z) (σ w))).trans ((min_le_left (τ z) (τ w)).trans (hτ z)))
+    ((hf z).mul (hf w)).continuousOn
+  simp only [Pi.mul_apply] at hH hL
+  convert hH.sub hL using 1
+  ext n
+  rw [← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro i hi
+  rw [window_pair_eq_prefix_difference, dyadicTime_step, sub_mul]
+
+lemma window_pair_energy_bound {Z : Type*} {f : ℝ × Z → ℝ}
+    {K : ℝ} (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K)
+    (σ τ : Z → ℝ≥0) (T : ℝ≥0) (n : ℕ) (z w : Z) :
+    |∑ i ∈ Finset.range (2^n),
+      (areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w) *
+        ((T : ℝ) / (2 : ℝ)^n)| ≤ K^2 * (T : ℝ) := by
+  calc
+    _ ≤ ∑ i ∈ Finset.range (2^n),
+        |(areaWindowSample f σ τ T n i z * areaWindowSample f σ τ T n i w) *
+          ((T : ℝ) / (2 : ℝ)^n)| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ i ∈ Finset.range (2^n), K^2 * ((T : ℝ) / (2 : ℝ)^n) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (T : ℝ) / (2 : ℝ)^n), abs_mul, pow_two]
+      exact mul_le_mul_of_nonneg_right
+        (mul_le_mul (areaWindowSample_bound hK hb σ τ T n i z)
+          (areaWindowSample_bound hK hb σ τ T n i w) (abs_nonneg _) hK) (by positivity)
+    _ = _ := by
+      simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat]
+      field_simp
+
+lemma area_window_pair_energy_tendsto {Z : Type*} [MeasurableSpace Z]
+    {ν : Measure Z} [IsFiniteMeasure ν]
+    {f : ℝ × Z → ℝ} (hfm : Measurable f) (hfc : ∀ z, Continuous (fun t => f (t,z)))
+    {σ τ : Z → ℝ≥0} (hσm : Measurable σ) (hτm : Measurable τ) (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
+    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (∫ z, areaWindowSample f σ τ T n i z ∂ν)^2 * ((T : ℝ) / (2 : ℝ)^n)) atTop
+      (𝓝 (∫ p : Z × Z, ((∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
+        f (t,p.1) * f (t,p.2)) - ∫ t in (0 : ℝ)..
+          (min (min (τ p.1) (τ p.2)) (max (σ p.1) (σ p.2)) : ℝ≥0), f (t,p.1) * f (t,p.2)) ∂(ν.prod ν))) := by
+  let H (n : ℕ) (p : Z × Z) := ∑ i ∈ Finset.range (2^n),
+    (areaWindowSample f σ τ T n i p.1 * areaWindowSample f σ τ T n i p.2) *
+      ((T : ℝ) / (2 : ℝ)^n)
+  have hC (n i : ℕ) := areaWindowSample_measurable hfm hσm hτm T n i
+  have hH (n : ℕ) : Measurable (H n) := by
+    apply Finset.measurable_sum
+    intro i hi
+    exact (((hC n i).comp measurable_fst).mul ((hC n i).comp measurable_snd)).mul_const _
+  have hbH (n : ℕ) (p : Z × Z) : ‖H n p‖ ≤ K^2 * (T : ℝ) := by
+    rw [Real.norm_eq_abs]
+    exact window_pair_energy_bound hK hb σ τ T n p.1 p.2
+  have ht := tendsto_integral_of_dominated_convergence
+    (μ := ν.prod ν) (fun _ => K^2 * (T : ℝ))
+    (fun n => (hH n).aestronglyMeasurable) (integrable_const _)
+    (fun n => Eventually.of_forall (hbH n))
+    (Eventually.of_forall fun p : Z × Z => window_pair_energy_tendsto f hfc σ τ T hτ p.1 p.2)
+  have he (n : ℕ) : (∫ p, H n p ∂(ν.prod ν)) =
+      ∑ i ∈ Finset.range (2^n), (∫ z, areaWindowSample f σ τ T n i z ∂ν)^2 *
+        ((T : ℝ) / (2 : ℝ)^n) := by
+    have hi (i : ℕ) : Integrable (areaWindowSample f σ τ T n i) ν :=
+      Integrable.of_bound (hC n i).aestronglyMeasurable K
+        (Eventually.of_forall fun z => by rw [Real.norm_eq_abs]; exact areaWindowSample_bound hK hb σ τ T n i z)
+    change (∫ p : Z × Z, ∑ i ∈ Finset.range (2^n),
+      (areaWindowSample f σ τ T n i p.1 * areaWindowSample f σ τ T n i p.2) *
+        ((T : ℝ) / (2 : ℝ)^n) ∂(ν.prod ν)) = _
+    rw [integral_finsetSum _ (fun i _ => ((hi i).mul_prod (hi i)).mul_const _)]
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [integral_mul_const, integral_prod_mul, pow_two]
+  simpa only [he] using ht
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma real_window_pair_eq (f : ℝ → ℝ) (a b c d t : ℝ) :
+    (if a ≤ t ∧ t < b ∧ c ≤ t ∧ t < d then f t else 0) =
+      (if t < min b d then f t else 0) -
+      (if t < min (min b d) (max a c) then f t else 0) := by
+  simp only [lt_min_iff, lt_max_iff]
+  by_cases ha : a ≤ t
+  · by_cases hc : c ≤ t
+    · simp [ha, hc, not_lt.mpr ha, not_lt.mpr hc]
+    · simp only [hc, false_and, and_false, ite_false]
+      have ht := lt_of_not_ge hc
+      simp [ht]
+  · simp only [ha, false_and, ite_false]
+    have ht := lt_of_not_ge ha
+    simp [ht]
+
+ 
+
+theorem area_window_energy_identity {Z : Type*} [MeasurableSpace Z]
+    {ν : Measure Z} [IsFiniteMeasure ν]
+    {f : ℝ × Z → ℝ} (hfm : Measurable f)
+    {σ τ : Z → ℝ≥0} (hσm : Measurable σ) (hτm : Measurable τ)
+    (T : ℝ≥0) (hτ : ∀ z, τ z ≤ T)
+    (K : ℝ) (hK : 0 ≤ K) (hb : ∀ t z, |f (t,z)| ≤ K) :
+    (∫ p : Z × Z, ((∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
+        f (t,p.1) * f (t,p.2)) - ∫ t in (0 : ℝ)..
+          (min (min (τ p.1) (τ p.2)) (max (σ p.1) (σ p.2)) : ℝ≥0), f (t,p.1) * f (t,p.2)) ∂(ν.prod ν)) =
+      ∫ t in (0 : ℝ)..(T : ℝ),
+        (∫ z, if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
+  let g (p : ℝ × (Z × Z)) := if (σ p.2.1 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2.1 : ℝ) ∧
+      (σ p.2.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2.2 : ℝ) then f (p.1,p.2.1) * f (p.1,p.2.2) else 0
+  have htm (v : Z → ℝ≥0) (hv : Measurable v) : Measurable (fun z => (v z : ℝ)) :=
+    NNReal.continuous_coe.measurable.comp hv
+  have hgm : Measurable g :=
+    ((hfm.comp (measurable_fst.prodMk (measurable_fst.comp measurable_snd))).mul
+      (hfm.comp (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))).ite
+        ((measurableSet_le ((htm σ hσm).comp (measurable_fst.comp measurable_snd)) measurable_fst).inter
+          ((measurableSet_lt measurable_fst ((htm τ hτm).comp (measurable_fst.comp measurable_snd))).inter
+            ((measurableSet_le ((htm σ hσm).comp (measurable_snd.comp measurable_snd)) measurable_fst).inter
+              (measurableSet_lt measurable_fst ((htm τ hτm).comp (measurable_snd.comp measurable_snd)))))) measurable_const
+  have hgb (p : ℝ × (Z × Z)) : ‖g p‖ ≤ K^2 := by
+    rw [Real.norm_eq_abs]
+    dsimp only [g]
+    split_ifs
+    · rw [abs_mul, pow_two]
+      exact mul_le_mul (hb _ _) (hb _ _) (abs_nonneg _) hK
+    · simpa using sq_nonneg K
+  have hgi : Integrable g ((volume.restrict (Ioc 0 (T : ℝ))).prod (ν.prod ν)) :=
+    Integrable.of_bound hgm.aestronglyMeasurable (K^2) (Eventually.of_forall hgb)
+  have hp (p : Z × Z) : ((∫ t in (0 : ℝ)..(min (τ p.1) (τ p.2) : ℝ≥0),
+      f (t,p.1) * f (t,p.2)) - ∫ t in (0 : ℝ)..
+        (min (min (τ p.1) (τ p.2)) (max (σ p.1) (σ p.2)) : ℝ≥0), f (t,p.1) * f (t,p.2)) =
+        ∫ t in Ioc (0 : ℝ) T, g (t,p) := by
+    let H := min (τ p.1) (τ p.2)
+    let L := min H (max (σ p.1) (σ p.2))
+    have hHT : H ≤ T := (min_le_left _ _).trans (hτ p.1)
+    have hLT : L ≤ T := (min_le_left _ _).trans hHT
+    have hi (S : ℝ≥0) : IntervalIntegrable
+        (fun t : ℝ => if t < (S : ℝ) then f (t,p.1) * f (t,p.2) else 0) volume 0 T := by
+      apply (intervalIntegrable_iff_integrableOn_Icc_of_le T.coe_nonneg).mpr
+      apply Integrable.of_bound (μ := volume.restrict (Icc (0 : ℝ) T)) ?_ (K^2) ?_
+      · exact (((hfm.comp (measurable_id.prodMk measurable_const)).mul
+          (hfm.comp (measurable_id.prodMk measurable_const))).ite
+            (measurableSet_lt measurable_id measurable_const) measurable_const).aestronglyMeasurable
+      · apply Eventually.of_forall
+        intro t
+        rw [Real.norm_eq_abs]
+        split_ifs
+        · rw [abs_mul, pow_two]
+          exact mul_le_mul (hb _ _) (hb _ _) (abs_nonneg _) hK
+        · simpa using sq_nonneg K
+    rw [← interval_integral_prefix_lt hHT, ← interval_integral_prefix_lt hLT,
+      ← intervalIntegral.integral_sub (hi H) (hi L), ← intervalIntegral.integral_of_le T.coe_nonneg]
+    apply intervalIntegral.integral_congr
+    intro t ht
+    dsimp only [g, H, L]
+    simpa only [NNReal.coe_min, NNReal.coe_max] using
+      (real_window_pair_eq (fun t => f (t,p.1) * f (t,p.2)) (σ p.1) (τ p.1) (σ p.2) (τ p.2) t).symm
+  have hg (t : ℝ) : (∫ p : Z × Z, g (t,p) ∂(ν.prod ν)) =
+      (∫ z, if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0 ∂ν)^2 := by
+    calc
+      _ = ∫ p : Z × Z,
+          (if (σ p.1 : ℝ) ≤ t ∧ t < (τ p.1 : ℝ) then f (t,p.1) else 0) *
+          (if (σ p.2 : ℝ) ≤ t ∧ t < (τ p.2 : ℝ) then f (t,p.2) else 0) ∂(ν.prod ν) := by
+        apply integral_congr_ae
+        apply Eventually.of_forall
+        intro p
+        dsimp only [g]
+        split_ifs <;> simp_all
+      _ = _ := by
+        simpa only [pow_two] using (integral_prod_mul (μ := ν) (ν := ν)
+          (fun z => if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0)
+          (fun z => if (σ z : ℝ) ≤ t ∧ t < (τ z : ℝ) then f (t,z) else 0))
+  simp_rw [hp]
+  rw [← integral_integral_swap hgi]
+  simp only [hg, intervalIntegral.integral_of_le T.coe_nonneg]
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField
+
+ 
+
+end SLEExactGauge
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/FiniteMass.lean b/lean/OAI/Probability/SLE/FiniteMass.lean
index b7a3c25cf0e2b0e4cabea2448da8ccd88ba7772d..db5c520435c76b0b167996d0fba5160947aa2d39 100644
--- a/lean/OAI/Probability/SLE/FiniteMass.lean
+++ b/lean/OAI/Probability/SLE/FiniteMass.lean
@@ -1,993 +1,993 @@
-import Mathlib
-import OAI.Probability.SLE.MeasureSupport
-import OAI.Probability.SLE.SpatialMeans
-
-namespace OAI
-
-/-! Finite Mass for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
-lemma continuous_radius_density_crossing
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hz : z ∈ upperHalfPlane)
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (w U : ℝ≥0) (hwU : w ≤ U) (δ : ℝ) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w,
-        limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω < δ} ≤
-      ENNReal.ofReal (δ / initialGreen κ z) := by
-  apply continuous_crossing_from_monotone _ _ _ 0 w U bot_le hwU
-  · intro ω hω
-    exact (terminalRadiusKernel_continuousOn a hε hBc _ _ U ω hω).mono
-      (Icc_subset_Icc le_rfl hwU)
-  · intro ω hω s hs
-    exact terminalRadiusKernel_pos a hε _ _ U s (hs.2.trans hwU) ω hω
-  · intro s hs hvs hsU N
-    apply (finite_radius_density_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz
-      a hε hεlim 0 U s hs hvs hsU δ N).trans
-    simpa only [mul_one] using mul_le_mul'
-      (le_refl (ENNReal.ofReal (δ / initialGreen κ z)))
-      (show P {ω | localizedRadiusStop (a 0) ω < ⊤} ≤ 1 from prob_le_one)
-
-noncomputable def initialAngleWindowConstant (κ : ℝ) (z : ℂ) : ℝ :=
-  Real.exp (codimension κ * (1 - Real.log 2 - Real.log z.im)) / initialGreen κ z
-
-lemma initial_inverse_angle_window_bound
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hz : z ∈ upperHalfPlane)
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (w U : ℝ≥0) (hw : w ≤ 1) (hwU : w ≤ U) (L : ℝ) (hL : 0 < L) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w, L < radiusInverseAngle a s ω} ≤
-      ENNReal.ofReal (initialAngleWindowConstant κ z / L) := by
-  have hs : {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w, L < radiusInverseAngle a s ω} ⊆
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w,
-        limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
-          Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L} := by
-    rintro ω ⟨hω, s, hs, hV⟩
-    exact ⟨hω, s, hs, radiusInverseAngle_large_imp_crossing a hε hκ8 s w U hs.2 hwU L hL ω hω hV⟩
-  apply (measure_mono hs).trans
-  apply (continuous_radius_density_crossing hB hBm hBc hB0 hκ hκ8 γ G F ‹∀ᵐ ω ∂P, _› hz
-    a hε hεlim w U hwU _).trans
-  apply ENNReal.ofReal_le_ofReal
-  unfold initialAngleWindowConstant
-  rw [div_right_comm _ L]
-  apply div_le_div_of_nonneg_right ?_ hL.le
-  apply div_le_div_of_nonneg_right ?_ (initialGreen_pos κ z hz).le
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (codimension_pos κ hκ8).le
-  exact sub_le_sub_right (sub_le_sub_right (show (w : ℝ) ≤ 1 from hw) _) _
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma sum_exp_decay (η : ℝ) (hη : 0 < η) (C : ℝ) (hC : 0 ≤ C) :
-    (∑' n : ℕ, ENNReal.ofReal (C * Real.exp (-η * n))) =
-      ENNReal.ofReal (C / (1 - Real.exp (-η))) := by
-  have hr : Real.exp (-η) < 1 := Real.exp_lt_one_iff.mpr (neg_neg_of_pos hη)
-  have hn : ∀ n : ℕ, Real.exp (-η * n) = Real.exp (-η) ^ n := fun n => by
-    rw [mul_comm, Real.exp_nat_mul]
-  simp_rw [hn]
-  rw [← ENNReal.ofReal_tsum_of_nonneg (fun n => mul_nonneg hC (pow_nonneg (Real.exp_pos _).le _))
-    ((summable_geometric_of_lt_one (Real.exp_pos _).le hr).mul_left C),
-    tsum_mul_left, tsum_geometric_of_lt_one (Real.exp_pos _).le hr, div_eq_mul_inv]
-
-lemma exponential_envelope_bound
-    {Ω : Type*} [MeasurableSpace Ω] (Q : Measure Ω)
-    (V : ℝ≥0 → Ω → ℝ) (E : Set Ω) (U : ℝ≥0)
-    (η M C : ℝ) (hη : 0 < η) (hM : 0 < M) (hC : 0 ≤ C)
-    (hwindow : ∀ n : ℕ, Q {ω | ω ∈ E ∧ ∃ s ∈ Icc (n : ℝ≥0) (min (n + 1) U),
-      M * Real.exp (η * n) < V s ω} ≤ ENNReal.ofReal (C / (M * Real.exp (η * n)))) :
-    Q {ω | ω ∈ E ∧ ∃ s ∈ Icc 0 U, M * Real.exp (η * s) < V s ω} ≤
-      ENNReal.ofReal (C / M / (1 - Real.exp (-η))) := by
-  let W (n : ℕ) : Set Ω := {ω | ω ∈ E ∧ ∃ s ∈ Icc (n : ℝ≥0) (min (n + 1) U),
-      M * Real.exp (η * n) < V s ω}
-  have hsub : {ω | ω ∈ E ∧ ∃ s ∈ Icc 0 U, M * Real.exp (η * s) < V s ω} ⊆ ⋃ n, W n := by
-    rintro ω ⟨hω, s, hs, hv⟩
-    let n : ℕ := ⌊(s : ℝ)⌋₊
-    have hn : (n : ℝ) ≤ s := Nat.floor_le s.coe_nonneg
-    have hu : (s : ℝ) ≤ (n : ℝ) + 1 := (Nat.lt_floor_add_one (s : ℝ)).le
-    apply mem_iUnion.mpr
-    refine ⟨n, hω, s, ?_, ?_⟩
-    · exact ⟨by exact_mod_cast hn, le_min (by exact_mod_cast hu) hs.2⟩
-    · apply lt_of_le_of_lt _ hv
-      apply mul_le_mul_of_nonneg_left _ hM.le
-      exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hn hη.le)
-  calc
-    _ ≤ Q (⋃ n, W n) := measure_mono hsub
-    _ ≤ ∑' n, Q (W n) := measure_iUnion_le W
-    _ ≤ ∑' n : ℕ, ENNReal.ofReal (C / (M * Real.exp (η * n))) := ENNReal.tsum_le_tsum hwindow
-    _ = _ := by
-      have he (n : ℕ) : C / (M * Real.exp (η * n)) = (C / M) * Real.exp (-η * n) := by
-        rw [neg_mul, Real.exp_neg]
-        ring
-      simp_rw [he]
-      exact sum_exp_decay η hη (C / M) (div_nonneg hC hM.le)
-
-lemma integral_exp_neg_half_le_two (u : ℝ) :
-    (∫ s in (0 : ℝ)..u, Real.exp (-s / 2)) ≤ 2 := by
-  have hd (s : ℝ) : HasDerivAt (fun s => -2 * Real.exp (-s / 2)) (Real.exp (-s / 2)) s := by
-    convert (((Real.hasDerivAt_exp (-s / 2)).comp s ((hasDerivAt_id s).neg.div_const 2)).const_mul (-2)) using 1
-    all_goals first | rfl | ring
-  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt
-    (fun s _ => hd s) ((Real.continuous_exp.comp (continuous_id.neg.div_const 2)).intervalIntegrable 0 u)]
-  norm_num
-  nlinarith [Real.exp_pos (-u / 2)]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def angleEnvelopeConstant (κ : ℝ) (z : ℂ) : ℝ :=
-  max (initialAngleWindowConstant κ z) (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ))
-
-lemma angleEnvelopeConstant_pos (κ : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    0 < angleEnvelopeConstant κ z :=
-  (div_pos (Real.exp_pos _) (initialGreen_pos κ z hz)).trans_le (le_max_left _ _)
-
-lemma finite_inverse_angle_envelope
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 U,
-        M * Real.exp (greenExponent κ / 8 * s) < radiusInverseAngle a s ω} ≤
-      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
-  have hrate : 0 < greenExponent κ / 8 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
-  have hC := angleEnvelopeConstant_pos κ z hz
-  have he : -(greenExponent κ / 8) = -greenExponent κ / 8 := by ring
-  rw [← he]
-  apply exponential_envelope_bound (finiteCenteredLaw P κ B γ G z U) (radiusInverseAngle a)
-    {ω | localizedRadiusStop (a U) ω < ⊤} U (greenExponent κ / 8) M
-    (angleEnvelopeConstant κ z) hrate hM hC.le
-  intro n
-  let L := M * Real.exp (greenExponent κ / 8 * n)
-  have hL : 0 < L := mul_pos hM (Real.exp_pos _)
-  change finiteCenteredLaw P κ B γ G z U
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc (n : ℝ≥0) (min ((n : ℝ≥0) + 1) U),
-        L < radiusInverseAngle a s ω} ≤ ENNReal.ofReal (angleEnvelopeConstant κ z / L)
-  by_cases hnU : (n : ℝ≥0) ≤ U
-  · by_cases hn : n = 0
-    · subst n
-      have hh := initial_inverse_angle_window_bound hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim
-        (min 1 U) U (min_le_left _ _) (min_le_right _ _) M hM
-      simpa only [L, Nat.cast_zero, zero_add, mul_zero, Real.exp_zero, mul_one, angleEnvelopeConstant] using
-        hh.trans (ENNReal.ofReal_le_ofReal
-          (div_le_div_of_nonneg_right (le_max_left _ _) hM.le))
-    · have hn1 : 1 ≤ (n : ℝ) := by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hn
-      have hvw : (n : ℝ≥0) ≤ min ((n : ℝ≥0) + 1) U := le_min (by simp) hnU
-      have hh := inverse_angle_window_bound hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim
-        n (min ((n : ℝ≥0) + 1) U) U hn1 hvw (min_le_right _ _) L hL
-      apply hh.trans
-      apply ENNReal.ofReal_le_ofReal
-      apply div_le_div_of_nonneg_right _ hL.le
-      apply le_trans _ (le_max_right _ _)
-      apply mul_le_mul_of_nonneg_left _ (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)).le
-      apply Real.exp_le_exp.mpr
-      have hm : ((min (n + 1) U : ℝ≥0) : ℝ) ≤ (n : ℝ) + 1 := by exact_mod_cast min_le_left ((n : ℝ≥0) + 1) U
-      have hd : (↑(min ((n : ℝ≥0) + 1) U) : ℝ) - (n : ℝ) ≤ 1 := by linarith
-      simpa only [mul_one, NNReal.coe_natCast] using
-        mul_le_mul_of_nonneg_left hd (codimension_pos κ hκ8).le
-  · have hempty : {ω | localizedRadiusStop (a U) ω < ⊤ ∧
-        ∃ s ∈ Icc (n : ℝ≥0) (min ((n : ℝ≥0) + 1) U), L < radiusInverseAngle a s ω} = ∅ := by
-      apply eq_empty_iff_forall_notMem.mpr
-      rintro ω ⟨_, s, hs, _⟩
-      exact hnU (hs.1.trans (hs.2.trans (min_le_right _ _)))
-    change finiteCenteredLaw P κ B γ G z U _ ≤ _
-    rw [hempty, measure_empty]
-    exact bot_le
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma log_height_clock_derivative_nonpos
-    (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    -2 / greenLogNormSq κ q b + logClockRate κ q b / 2 ≤ 0 := by
-  have hh := mul_le_mul_of_nonneg_left (logAngleSq_le_one κ q b)
-    (show (0 : ℝ) ≤ 2 / greenLogNormSq κ q b from
-      div_nonneg (by norm_num) (greenLogNormSq_pos κ q b).le)
-  have he : logClockRate κ q b / 2 = (2 / greenLogNormSq κ q b) * logAngleSq κ q b := by
-    unfold logClockRate logAngleSq
-    field_simp
-    ring
-  rw [he, neg_div]
-  linarith
-
-lemma log_loewner_radius_height_decay
-    {Q : ℝ → ℂ × ℝ} {B : ℝ → ℝ} {κ T : ℝ}
-    (hT : 0 ≤ T) (hQ0 : (Q 0).2 = 0)
-    (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q
-      (physicalLogLoewnerField (Real.sqrt κ * B t) (Q t)) (Icc 0 T) t)
-    (t : ℝ) (ht : t ∈ Icc 0 T) :
-    (Q t).1.im ≤ (Q 0).1.im - logRadiusClock (Q 0).1.im (Q t) / 2 := by
-  let η₀ := (Q 0).1.im
-  have hQc : ContinuousOn Q (Icc 0 T) := fun t ht => (hQ t ht).continuousWithinAt
-  have hSc : ContinuousOn (fun t => logRadiusClock η₀ (Q t)) (Icc 0 T) :=
-    (continuous_logRadiusClock η₀).comp_continuousOn hQc
-  have hanti : AntitoneOn (fun t => (Q t).1.im + logRadiusClock η₀ (Q t) / 2) (Icc 0 T) := by
-    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
-      ((Complex.continuous_im.comp_continuousOn (continuous_fst.comp_continuousOn hQc)).add
-        (hSc.div_const _))
-      (fun t ht => ((log_ode_im_derivative (hQ t (interior_subset ht))).add
-        ((log_ode_clock_derivative η₀ (hQ t (interior_subset ht))).div_const _)).mono interior_subset)
-    intro t ht
-    exact log_height_clock_derivative_nonpos κ (Q t) (B t)
-  have hh := hanti (show (0 : ℝ) ∈ Icc 0 T from ⟨le_rfl, hT⟩) ht ht.1
-  have hS0 : logRadiusClock η₀ (Q 0) = 0 := by simp [logRadiusClock, η₀, hQ0]
-  dsimp only at hh
-  rw [hS0, zero_div, add_zero] at hh
-  change (Q t).1.im ≤ η₀ - _
-  linarith
-
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ} (a : AngularLocalization B hm κ z u ε)
-
-lemma physical_derivative (ω : Ω) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
-    HasDerivWithinAt (fun t => a.solution.Q t ω)
-      (physicalLogLoewnerField (Real.sqrt κ * B t.toNNReal ω) (a.solution.Q t ω))
-        (Icc (0 : ℝ) (a.τ ω : ℝ)) t := by
-  have hd := (a.solution.derivative ω t ht.1).mono
-    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-  have hb := (a.inside ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)).2.2
-  rw [Real.coe_toNNReal t ht.1] at hb
-  rwa [clippedLogLoewnerField_eq_physical _ _ _ _ _ hb] at hd
-
-noncomputable def radiusPhysicalRate (s : ℝ) (ω : Ω) : ℝ :=
-  Real.exp (2 * (a.solution.Q (a.clock.inverse s ω) ω).1.im) /
-    (4 * logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
-      (B (a.clock.inverse s ω).toNNReal ω) ^ 2)
-
-lemma radiusPhysicalRate_continuous
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
-    Continuous (fun s => a.radiusPhysicalRate s ω) := by
-  have hQ := (a.solution.lipschitz ω).continuous.comp (a.clock.continuous_inverse ω)
-  have hB := (hBc ω).comp (continuous_real_toNNReal.comp (a.clock.continuous_inverse ω))
-  have hA := (continuous_logAngleSq κ).comp (hQ.prodMk hB)
-  exact (Real.continuous_exp.comp (continuous_const.mul
-    (Complex.continuous_im.comp hQ.fst))).div (continuous_const.mul (hA.pow 2))
-    (fun s => mul_ne_zero (by norm_num) (pow_ne_zero 2 (logAngleSq_pos _ _ _).ne'))
-
-lemma radiusPhysicalRate_at_clock (ω : Ω) (t : ℝ)
-    (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
-    a.radiusPhysicalRate (a.clock.S t ω) ω *
-      logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω) = 1 := by
-  unfold radiusPhysicalRate
-  rw [a.clock.inverse_clock ω t ht, logClockRate_angle]
-  have hA := (logAngleSq_pos κ (a.solution.Q t ω) (B t.toNNReal ω)).ne'
-  have hE := Real.exp_ne_zero (2 * (a.solution.Q t ω).1.im)
-  field_simp
-
-lemma radiusPhysicalRate_integral
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
-    (∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), a.radiusPhysicalRate s ω) = (a.τ ω : ℝ) := by
-  have hh := intervalIntegral.integral_comp_mul_deriv''
-    (a.clock.continuous ω).continuousOn
-    (f' := fun t => logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω))
-    (a := (0 : ℝ)) (b := (a.τ ω : ℝ)) (g := fun s => a.radiusPhysicalRate s ω)
-    (fun t ht => by
-      have ht' : t ∈ Ioo (0 : ℝ) (a.τ ω : ℝ) := by
-        simpa only [min_eq_left (a.τ ω).coe_nonneg, max_eq_right (a.τ ω).coe_nonneg] using ht
-      exact ((a.radius_clock_derivative ω t ⟨ht'.1.le, ht'.2.le⟩).hasDerivAt
-        (Icc_mem_nhds ht'.1 ht'.2)).hasDerivWithinAt)
-    ((continuous_logClockRate κ).comp
-      ((a.solution.lipschitz ω).continuous.prodMk
-        ((hBc ω).comp continuous_real_toNNReal))).continuousOn
-    (a.radiusPhysicalRate_continuous hBc ω).continuousOn
-  rw [a.clock.zero ω] at hh
-  rw [← hh]
-  trans ∫ t in (0 : ℝ)..(a.τ ω : ℝ), (1 : ℝ)
-  · apply intervalIntegral.integral_congr
-    intro t ht
-    exact a.radiusPhysicalRate_at_clock ω t (by simpa only [uIcc_of_le (a.τ ω).coe_nonneg] using ht)
-  · simp
-
-lemma radius_height_decay (ω : Ω) (s : ℝ) (hs : 0 ≤ s)
-    (hsl : s ≤ a.clock.S (a.τ ω) ω) :
-    (a.solution.Q (a.clock.inverse s ω) ω).1.im ≤ Real.log z.im - s / 2 := by
-  have hh := log_loewner_radius_height_decay (a.τ ω).coe_nonneg
-    (show (a.solution.Q 0 ω).2 = 0 by rw [a.solution.initial]; rfl)
-    (a.physical_derivative ω) (a.clock.inverse s ω) (a.clock.inverse_mem s ω)
-  have he := a.clock.clock_inverse s ω
-  change logRadiusClock (Real.log z.im) (a.solution.Q (a.clock.inverse s ω) ω) =
-    min (max s 0) (a.clock.S (a.τ ω) ω) at he
-  rw [max_eq_left hs, min_eq_left hsl] at he
-  have hinit : (a.solution.Q 0 ω).1.im = Real.log z.im := by
-    rw [a.solution.initial, initialLogCoordinates_im]
-  rwa [hinit, he] at hh
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma inverse_angle_control_rate {A a M s : ℝ}
-    (hA : 0 < A) (ha : 0 < a) (hM : 0 < M)
-    (hV : A ^ (-a / 2) ≤ M * Real.exp (a * s / 8)) :
-    1 / A ^ 2 ≤ M ^ (4 / a) * Real.exp (s / 2) := by
-  have hp : 0 ≤ 4 / a := by positivity
-  have hh := Real.rpow_le_rpow (Real.rpow_pos_of_pos hA _).le hV hp
-  rw [← Real.rpow_mul hA.le] at hh
-  have he : (-a / 2) * (4 / a) = -2 := by field_simp; ring
-  rw [he, Real.rpow_neg hA.le, Real.rpow_two,
-    Real.mul_rpow hM.le (Real.exp_pos _).le, ← Real.exp_mul] at hh
-  have he' : a * s / 8 * (4 / a) = s / 2 := by field_simp; ring
-  rw [he'] at hh
-  simpa only [one_div] using hh
-
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ} (a : AngularLocalization B hm κ z u ε)
-
-lemma physical_rate_of_envelope (hκ : 0 < κ) (hκ8 : κ < 8)
-    (M : ℝ) (hM : 0 < M) (ω : Ω) (s : ℝ) (hs : 0 ≤ s)
-    (hsl : s ≤ a.clock.S (a.τ ω) ω)
-    (hV : (logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
-      (B (a.clock.inverse s ω).toNNReal ω)) ^ (-greenExponent κ / 2) ≤
-        M * Real.exp (greenExponent κ * s / 8)) :
-    a.radiusPhysicalRate s ω ≤
-      (Real.exp (2 * Real.log z.im) / 4 * M ^ (4 / greenExponent κ)) * Real.exp (-s / 2) := by
-  let A := logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
-    (B (a.clock.inverse s ω).toNNReal ω)
-  have hA : 0 < A := logAngleSq_pos _ _ _
-  have hp := inverse_angle_control_rate hA (greenExponent_pos κ hκ hκ8) hM hV
-  have he : Real.exp (2 * (a.solution.Q (a.clock.inverse s ω) ω).1.im) ≤
-      Real.exp (2 * Real.log z.im) * Real.exp (-s) := by
-    rw [← Real.exp_add]
-    apply Real.exp_le_exp.mpr
-    linarith [a.radius_height_decay ω s hs hsl]
-  unfold radiusPhysicalRate
-  change Real.exp _ / (4 * A ^ 2) ≤ _
-  calc
-    _ = (Real.exp (2 * (a.solution.Q (a.clock.inverse s ω) ω).1.im) / 4) * (1 / A ^ 2) := by ring
-    _ ≤ (Real.exp (2 * Real.log z.im) * Real.exp (-s) / 4) *
-        (M ^ (4 / greenExponent κ) * Real.exp (s / 2)) :=
-      mul_le_mul (div_le_div_of_nonneg_right he (by norm_num)) hp (by positivity) (by positivity)
-    _ = _ := by
-      rw [show Real.exp (2 * Real.log z.im) * Real.exp (-s) / 4 *
-          (M ^ (4 / greenExponent κ) * Real.exp (s / 2)) =
-          (Real.exp (2 * Real.log z.im) / 4 * M ^ (4 / greenExponent κ)) *
-            (Real.exp (-s) * Real.exp (s / 2)) by ring, ← Real.exp_add]
-      congr 2
-      ring
-
-end AngularLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def physicalTimeThreshold (κ : ℝ) (z : ℂ) (M : ℝ) : ℝ :=
-  Real.exp (2 * Real.log z.im) / 2 * M ^ (4 / greenExponent κ)
-
-lemma physicalTimeThreshold_pos (κ : ℝ) (z : ℂ) {M : ℝ} (hM : 0 < M) :
-    0 < physicalTimeThreshold κ z M :=
-  mul_pos (div_pos (Real.exp_pos _) (by norm_num)) (Real.rpow_pos_of_pos hM _)
-
-namespace AngularLocalization
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u ε : ℝ} (a : AngularLocalization B hm κ z u ε)
-
-lemma physical_time_of_envelope (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (M : ℝ) (hM : 0 < M) (ω : Ω)
-    (hV : ∀ s ∈ Icc 0 (a.clock.S (a.τ ω) ω),
-      (logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
-        (B (a.clock.inverse s ω).toNNReal ω)) ^ (-greenExponent κ / 2) ≤
-          M * Real.exp (greenExponent κ * s / 8)) :
-    (a.τ ω : ℝ) ≤ physicalTimeThreshold κ z M := by
-  let C := Real.exp (2 * Real.log z.im) / 4 * M ^ (4 / greenExponent κ)
-  have hC : 0 ≤ C := mul_nonneg (div_nonneg (Real.exp_pos _).le (by norm_num))
-    (Real.rpow_pos_of_pos hM _).le
-  rw [← a.radiusPhysicalRate_integral hBc ω]
-  calc
-    _ ≤ ∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), C * Real.exp (-s / 2) := by
-      apply intervalIntegral.integral_mono_on (a.clock.terminal_nonneg ω)
-        ((a.radiusPhysicalRate_continuous hBc ω).intervalIntegrable _ _)
-        ((continuous_const.mul (Real.continuous_exp.comp (continuous_id.neg.div_const 2))).intervalIntegrable _ _)
-      intro s hs
-      exact a.physical_rate_of_envelope hκ hκ8 M hM ω s hs.1 hs.2 (hV s hs)
-    _ = C * (∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), Real.exp (-s / 2)) :=
-      intervalIntegral.integral_const_mul _ _
-    _ ≤ C * 2 := mul_le_mul_of_nonneg_left (integral_exp_neg_half_le_two _) hC
-    _ = _ := by dsimp [C, physicalTimeThreshold]; ring
-
-end AngularLocalization
-
-lemma localized_physical_time_of_envelope
-    {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (U : ℝ≥0) (M : ℝ) (hM : 0 < M)
-    (ω : Ω) (hω : localizedRadiusStop (a U) ω < ⊤)
-    (hV : ∀ s ∈ Icc 0 U, radiusInverseAngle a s ω ≤ M * Real.exp (greenExponent κ / 8 * s)) :
-    localizedRadiusStop (a U) ω ≤ ((physicalTimeThreshold κ z M).toNNReal : WithTop ℝ≥0) := by
-  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top (a U) ω hω
-  rw [localizedRadiusStop_eq_hit (a U) ω n hn, WithTop.coe_le_coe, ← NNReal.coe_le_coe,
-    Real.coe_toNNReal _ (physicalTimeThreshold_pos κ z hM).le]
-  apply (a U n).physical_time_of_envelope hBc hκ hκ8 M hM ω
-  intro s hs
-  have hsu : s ≤ (U : ℝ) := hs.2.trans_eq hn
-  have hh := hV s.toNNReal ⟨bot_le, Real.toNNReal_le_iff_le_coe.mpr hsu⟩
-  unfold radiusInverseAngle at hh
-  rw [limitRadiusKernel_eq_below (a s.toNNReal) (a U) hε
-    (by simpa only [Real.coe_toNNReal s hs.1] using hsu) 0 2 s.toNNReal
-    s.toNNReal.coe_nonneg le_rfl ω, Real.coe_toNNReal s hs.1,
-    limitRadiusKernel_eq_of_surviving_cutoff (a U) hε 0 2 s hs.1 ω n hs.2] at hh
-  unfold AngularLocalization.radiusValue at hh
-  rw [powerLogDensity_radius_angle κ 0 2 (Real.log z.im)] at hh
-  norm_num only [zero_mul, Real.exp_zero, one_mul, div_self (by norm_num : (2 : ℝ) ≠ 0), Real.rpow_one] at hh
-  convert hh using 1
-  congr 2
-  ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma finite_physical_tail_of_localizations
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤
-      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
-  have hwQ : ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z U, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω) :=
-    (withDensity_absolutelyContinuous P _).ae_le hw
-  have hs : {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤ᵐ[finiteCenteredLaw P κ B γ G z U]
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 U,
-        M * Real.exp (greenExponent κ / 8 * s) < radiusInverseAngle a s ω} := by
-    filter_upwards [hwQ, finiteCenteredLaw_live_compact hB κ hκ hκ8 γ G F hw z hz U U.coe_nonneg] with ω hω hlive
-    intro hlate
-    have he := localizedRadiusStop_eq_source (a U) hεlim hBc hz ω (γ ω) (G ω) (F ω) hω
-    obtain ⟨T, _, _, hT, _⟩ := hlive
-    have hl : localizedRadiusStop (a U) ω < ⊤ := by rw [he, hT]; exact ENNReal.coe_lt_top
-    refine ⟨hl, ?_⟩
-    by_contra h
-    push Not at h
-    have hb := localized_physical_time_of_envelope a hε hBc hκ hκ8 U M hM ω hl h
-    rw [he] at hb
-    exact not_lt_of_ge hb hlate
-  exact (measure_mono_ae hs).trans
-    (finite_inverse_angle_envelope hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim U M hM)
-
-lemma finite_physical_tail_regular
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤
-      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
-  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
-  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
-  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
-  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
-  have hεA (n : ℕ) : ε n ≤ A₀ := by
-    dsimp [ε]
-    exact div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
-  have hεlim : Tendsto ε atTop (𝓝 0) := by
-    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
-    simpa only [mul_one_div, mul_zero] using hh
-  have hεanti : Antitone ε := by
-    intro n m hnm
-    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
-  let a (u : ℝ≥0) (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
-    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
-      u.coe_nonneg (hε0 n) (hεA n))
-  exact finite_physical_tail_of_localizations hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hεanti hεlim U M hM
-
- 
-
-theorem finite_physical_tail
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤
-      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
-  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
-  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
-    filter_upwards [hw, hCB] with ω hω he
-    simpa only [he] using hω
-  have hh := finite_physical_tail_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz U M hM
-  rwa [finiteCenteredLaw_congr_driver hCB κ γ G z U] at hh
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem finite_physical_clock_tight
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (δ : ℝ) (hδ : 0 < δ) :
-    ∃ T : ℝ≥0, 0 < T ∧ ∀ U : ℝ≥0,
-      finiteCenteredLaw P κ B γ G z U
-        {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U} ≤ ENNReal.ofReal δ := by
-  let D := 1 - Real.exp (-greenExponent κ / 8)
-  have hD : 0 < D := by
-    apply sub_pos.mpr
-    apply Real.exp_lt_one_iff.mpr
-    exact div_neg_of_neg_of_pos (neg_neg_of_pos (greenExponent_pos κ hκ hκ8)) (by norm_num)
-  let M := angleEnvelopeConstant κ z / (δ * D)
-  have hM : 0 < M := div_pos (angleEnvelopeConstant_pos κ z hz) (mul_pos hδ hD)
-  refine ⟨(physicalTimeThreshold κ z M).toNNReal, Real.toNNReal_pos.mpr (physicalTimeThreshold_pos κ z hM), ?_⟩
-  intro U
-  have hh := finite_physical_tail hB κ hκ hκ8 γ G F hw z hz U M hM
-  have he : angleEnvelopeConstant κ z / M / D = δ := by
-    dsimp [M]
-    field_simp [(angleEnvelopeConstant_pos κ z hz).ne', hD.ne', hδ.ne']
-  change _ ≤ ENNReal.ofReal (angleEnvelopeConstant κ z / M / D) at hh
-  rw [he] at hh
-  exact hh
-
- 
-
-theorem finite_physical_clock_positive
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    ∃ T : ℝ≥0, 0 < T ∧ ∀ U : ℝ≥0, (1 / 2 : ℝ≥0∞) ≤
-      finiteCenteredLaw P κ B γ G z U
-        {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T} := by
-  obtain ⟨T, hT, ht⟩ := finite_physical_clock_tight hB κ hκ hκ8 γ G F hw z hz (1/2) (by norm_num)
-  refine ⟨T, hT, fun U => ?_⟩
-  let Q := finiteCenteredLaw P κ B γ G z U
-  let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F hw z hz U U.coe_nonneg
-  have hu : {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T} ∪
-      {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U} = univ := by
-    ext ω
-    simp only [mem_union, mem_ofPred_eq, mem_univ, iff_true]
-    exact le_or_gt _ _
-  have hbad : Q {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U} ≤ (1 / 2 : ℝ≥0∞) := by
-    have he : ENNReal.ofReal ((1 : ℝ) / 2) = (1 / 2 : ℝ≥0∞) := by
-      rw [ENNReal.ofReal_div_of_pos (by norm_num)]
-      norm_num
-    rw [← he]
-    exact ht U
-  have hh : (1 : ℝ≥0∞) ≤ Q {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T} + 1 / 2 := by
-    have hsum := measure_union_le (μ := Q)
-      {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T}
-      {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U}
-    rw [hu, measure_univ] at hsum
-    exact hsum.trans (add_le_add le_rfl hbad)
-  have hsub := tsub_le_iff_right.mpr hh
-  norm_num at hsub
-  simpa only [one_div] using hsub
-
-end SLEExactGauge
-end
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-noncomputable def finiteSpatialGreenMass (κ : ℝ) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ) (r : ℝ) (T : ℝ≥0) (K : Set ℂ) : ℝ≥0∞ :=
-  ∫⁻ z in K, ENNReal.ofReal (if spatialHitTime γ z r ≤ T then
-    spatialGreenWeight κ U γ G z r else 0)
-def SourceFiniteTimeFirstMomentTarget : Prop :=
-  ∀ (Ω : Type*) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ γ : Ω → ℝ≥0 → ℂ, ∀ a : RegularSLEPresentation P κ γ,
-  ∀ z₀ : ℂ, ∀ R : ℝ, 0 < R → closedBall z₀ R ⊆ upperHalfPlane →
-  ∃ (T : ℝ≥0) (e₀ m₀ : ℝ), 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧
-    ∀ r : ℝ, 0 < r → r ≤ e₀ →
-      ENNReal.ofReal m₀ ≤ ∫⁻ ω : a.samples,
-        finiteSpatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
-          (γ ω) (a.G ω) r T (closedBall z₀ R) ∂(P.comap Subtype.val)
-
-lemma initialGreen_continuousOn (κ : ℝ) : ContinuousOn (initialGreen κ) upperHalfPlane := by
-  have hn (z : ℂ) (hz : z ∈ upperHalfPlane) : ‖z‖ ≠ 0 := by
-    apply norm_ne_zero_iff.mpr
-    intro he
-    subst z
-    exact (lt_irrefl 0) (show (0 : ℝ) < 0 from hz)
-  have hy : ContinuousOn (fun z : ℂ => z.im) upperHalfPlane :=
-    Complex.continuous_im.continuousOn
-  have h1 := (hy.const_mul 2).rpow_const (p := -codimension κ)
-    (fun z hz => Or.inl (mul_ne_zero (by norm_num) (ne_of_gt hz)))
-  have h2 := (hy.div continuous_norm.continuousOn hn).rpow_const (p := greenExponent κ)
-    (fun z hz => Or.inl (div_ne_zero (ne_of_gt hz) (hn z hz)))
-  convert h1.mul h2 using 1
-  funext z
-  simp only [initialGreen, Complex.sin_arg, Pi.mul_apply, Pi.div_apply]
-
-lemma angleEnvelopeConstant_continuousOn (κ : ℝ) :
-    ContinuousOn (angleEnvelopeConstant κ) upperHalfPlane := by
-  have hl : ContinuousOn (fun z : ℂ => Real.log z.im) upperHalfPlane :=
-    Complex.continuous_im.continuousOn.log (fun _ hz => ne_of_gt hz)
-  have he : ContinuousOn (fun z : ℂ => Real.exp
-      (codimension κ * (1 - Real.log 2 - Real.log z.im))) upperHalfPlane :=
-    Real.continuous_exp.comp_continuousOn
-      ((continuousOn_const.sub hl).const_mul (codimension κ))
-  exact (he.div (initialGreen_continuousOn κ)
-    (fun z hz => (initialGreen_pos κ z hz).ne')).sup continuousOn_const
-
-noncomputable def halfClockThreshold (κ : ℝ) (z : ℂ) : ℝ :=
-  physicalTimeThreshold κ z
-    (angleEnvelopeConstant κ z / ((1 / 2) * (1 - Real.exp (-greenExponent κ / 8))))
-
-lemma halfClockThreshold_pos {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {z : ℂ} (hz : z ∈ upperHalfPlane) : 0 < halfClockThreshold κ z := by
-  apply physicalTimeThreshold_pos
-  apply div_pos (angleEnvelopeConstant_pos κ z hz)
-  apply mul_pos (by norm_num)
-  exact sub_pos.mpr (Real.exp_lt_one_iff.mpr (by
-    have := greenExponent_pos κ hκ hκ8
-    linarith))
-
-lemma halfClockThreshold_continuousOn {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ContinuousOn (halfClockThreshold κ) upperHalfPlane := by
-  have hD : 0 < (1 / 2 : ℝ) * (1 - Real.exp (-greenExponent κ / 8)) := by
-    apply mul_pos (by norm_num)
-    exact sub_pos.mpr (Real.exp_lt_one_iff.mpr (by
-      have := greenExponent_pos κ hκ hκ8
-      linarith))
-  have hM := (angleEnvelopeConstant_continuousOn κ).div_const
-    ((1 / 2) * (1 - Real.exp (-greenExponent κ / 8)))
-  have hp := hM.rpow_const (p := 4 / greenExponent κ) (fun z hz =>
-    Or.inl (div_pos (angleEnvelopeConstant_pos κ z hz) hD).ne')
-  have hl := Complex.continuous_im.continuousOn.log
-    (s := upperHalfPlane) (fun _ hz => ne_of_gt hz)
-  exact ((Real.continuous_exp.comp_continuousOn (hl.const_mul 2)).div_const 2).mul hp
-namespace RegularSLEPresentation
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
-lemma spatialHitTime_stopping (z : ℂ) (r : ℝ) :
-    IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
-  a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
-
- 
-
-lemma finite_spatial_density_identity [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
-    (u : ℝ) (hu : 0 ≤ u) (hlevel : (2 * z.im) * Real.exp (-u) < r) (T : ℝ≥0) :
-    (∫ ω : a.samples in {ω | spatialHitTime (γ ω) z r ≤ T},
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
-        (γ ω) (a.G ω) z r ∂(P.comap Subtype.val)) =
-    ∫ ω : a.samples in {ω | spatialHitTime (γ ω) z r ≤ T},
-      stoppedGreen κ (fun t => Real.sqrt κ * a.B t ω)
-        (γ ω) (a.G ω) z u ∂(P.comap Subtype.val) := by
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
-  obtain ⟨M, Z, hM, hZ, hMb, hZb, hClosed, hStop, heq, hNoHit⟩ :=
-    exists_physical_green_stopping_regular a.brownian hBm a.continuous_B a.zero_B
-      κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
-  let σ (ω : a.samples) : ℝ≥0 := boundedSpatialHit (γ ω) z r T
-  have hσ : IsStoppingTime a.filtration (fun ω => (σ ω : WithTop ℝ≥0)) :=
-    a.boundedSpatialHit_stopping z r T
-  let A := {ω : a.samples | spatialHitTime (γ ω) z r ≤ T}
-  have hA : MeasurableSet[hσ.measurableSpace] A := by
-    have hh := (a.spatialHitTime_stopping z r).measurableSet_stopping_time_le_min
-      (isStoppingTime_const a.filtration T)
-    rw [hσ.measurableSet]
-    rw [IsStoppingTime.measurableSet] at hh
-    simpa only [σ, A, boundedSpatialHit_coe] using hh
-  have hAm : MeasurableSet A := hσ.measurableSpace_le _ hA
-  have hWA : (fun ω : a.samples => spatialGreenWeight κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) =ᵐ[μ.restrict A]
-      fun ω => M (σ ω) ω := by
-    filter_upwards [ae_restrict_of_ae heq, ae_restrict_mem hAm] with ω he hω
-    change spatialHitTime (γ ω) z r ≤ (T : WithTop ℝ≥0) at hω
-    rw [he.1]
-    simp only [Function.comp_apply]
-    rw [(a.witness ω).physicalGreen_eq_before_spatial
-      ((a.continuous_B ω).const_mul _) κ z r hr hy u hlevel (σ ω)
-      (boundedSpatialHit_le (γ ω) z r T)]
-    have hn : spatialHitTime (γ ω) z r ≠ ⊤ :=
-      ne_top_of_le_ne_top WithTop.coe_ne_top hω
-    simp only [spatialGreenWeight, ite_eq_right hn, σ, boundedSpatialHit,
-      min_eq_left hω]
-  calc
-    _ = ∫ ω in A, M (σ ω) ω ∂μ := integral_congr_ae hWA
-    _ = ∫ ω in A, μ[Z | hσ.measurableSpace] ω ∂μ :=
-      integral_congr_ae (ae_restrict_of_ae (hStop σ hσ))
-    _ = ∫ ω in A, Z ω ∂μ := setIntegral_condExp hσ.measurableSpace_le hZ hA
-    _ = _ := integral_congr_ae (ae_restrict_of_ae (heq.mono fun ω hω => hω.2))
-
-lemma finite_spatial_lintegral_identity [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
-    (u : ℝ) (hu : 0 ≤ u) (hlevel : (2 * z.im) * Real.exp (-u) < r) (T : ℝ≥0) :
-    (∫⁻ ω : a.samples, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
-        (γ ω) (a.G ω) z r else 0) ∂(P.comap Subtype.val)) =
-    ENNReal.ofReal (initialGreen κ z) *
-      finiteCenteredLaw (P.comap Subtype.val) κ a.B (γ ∘ Subtype.val) a.G z u
-        {ω | spatialHitTime (γ ω) z r ≤ T} := by
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  let A := {ω : a.samples | spatialHitTime (γ ω) z r ≤ T}
-  have hA : MeasurableSet A :=
-    (a.spatialHitTime_stopping z r).measurableSpace_le _
-      ((a.spatialHitTime_stopping z r).measurableSet_le' T)
-  have hi := (a.spatialGreen_expectation hκ hκ8 z r hr hy).1.integrableOn (s := A)
-  have hn : 0 ≤ᵐ[μ.restrict A] (fun ω : a.samples => spatialGreenWeight κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) :=
-    Eventually.of_forall fun ω => by
-      dsimp only [Pi.zero_apply, spatialGreenWeight]
-      split_ifs
-      · exact le_rfl
-      · exact greenDensity_nonneg (a.witness ω) κ _ z
-  have hind (ω : a.samples) : ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r else 0) =
-      A.indicator (fun ω : a.samples => ENNReal.ofReal (spatialGreenWeight κ
-        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r)) ω := by
-    dsimp only [A, Set.indicator, mem_ofPred_eq]
-    split_ifs <;> simp_all only [ENNReal.ofReal_zero]
-  simp_rw [hind]
-  rw [lintegral_indicator hA, ← ofReal_integral_eq_lintegral_ofReal hi hn,
-    a.finite_spatial_density_identity hκ hκ8 z hz r hr hy u hu hlevel T,
-    finiteCenteredLaw_apply a.brownian κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F
-      (Eventually.of_forall a.witness) z hz u hu A hA,
-    ← ENNReal.ofReal_mul (initialGreen_pos κ z hz).le]
-  congr 1
-  simp only [Function.comp_apply]
-  field_simp [(initialGreen_pos κ z hz).ne']
-  rfl
-
-lemma finite_spatial_lintegral_lower [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
-    (T : ℝ≥0) (hT : halfClockThreshold κ z ≤ T) :
-    ENNReal.ofReal (initialGreen κ z / 2) ≤
-      ∫⁻ ω : a.samples, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
-        spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
-          (γ ω) (a.G ω) z r else 0) ∂(P.comap Subtype.val) := by
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  let u := Real.log (4 * z.im / r)
-  have hratio : 0 < 4 * z.im / r := div_pos (mul_pos (by norm_num) hz) hr
-  have hu : 0 ≤ u := Real.log_nonneg ((le_div_iff₀ hr).mpr (by linarith))
-  have hlev : (2 * z.im) * Real.exp (-u) < r := by
-    have he : (2 * z.im) * Real.exp (-u) = r / 2 := by
-      dsimp only [u]
-      rw [Real.exp_neg, Real.exp_log hratio]
-      field_simp [(show 0 < z.im from hz).ne']
-      ring
-    rw [he]
-    linarith
-  let Q := finiteCenteredLaw μ κ a.B (γ ∘ Subtype.val) a.G z u
-  let : IsProbabilityMeasure Q := finiteCenteredLaw_probability a.brownian κ hκ hκ8
-    (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
-  let D := 1 - Real.exp (-greenExponent κ / 8)
-  have hD : 0 < D := by
-    apply sub_pos.mpr
-    apply Real.exp_lt_one_iff.mpr
-    have := greenExponent_pos κ hκ hκ8
-    linarith
-  let M := angleEnvelopeConstant κ z / ((1 / 2) * D)
-  have hM : 0 < M := div_pos (angleEnvelopeConstant_pos κ z hz) (mul_pos (by norm_num) hD)
-  have hbad : Q {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (a.G ω) z u} ≤ 1 / 2 := by
-    have hh := finite_physical_tail a.brownian κ hκ hκ8 (γ ∘ Subtype.val)
-      a.G a.F (Eventually.of_forall a.witness) z hz ⟨u, hu⟩ M hM
-    have he : angleEnvelopeConstant κ z / M / D = 1 / 2 := by
-      dsimp only [M]
-      field_simp [(angleEnvelopeConstant_pos κ z hz).ne', hD.ne']
-    change Q {ω | ENNReal.ofReal (halfClockThreshold κ z) <
-      radiusLevelTime (γ ω) (a.G ω) z u} ≤ ENNReal.ofReal (angleEnvelopeConstant κ z / M / D) at hh
-    rw [he] at hh
-    have hTT : ENNReal.ofReal (halfClockThreshold κ z) ≤ (T : ℝ≥0∞) := by
-      simpa only [ENNReal.ofReal_coe_nnreal] using ENNReal.ofReal_le_ofReal hT
-    have hhalf : ENNReal.ofReal ((1 : ℝ) / 2) = (1 / 2 : ℝ≥0∞) := by
-      rw [ENNReal.ofReal_div_of_pos (by norm_num)]
-      norm_num
-    rw [hhalf] at hh
-    exact (measure_mono (fun ω hω => hTT.trans_lt hω)).trans hh
-  have hgood : (1 / 2 : ℝ≥0∞) ≤ Q {ω | radiusLevelTime (γ ω) (a.G ω) z u ≤ T} := by
-    have he : {ω : a.samples | radiusLevelTime (γ ω) (a.G ω) z u ≤ T} ∪
-        {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (a.G ω) z u} = univ := by
-      ext ω
-      simp only [mem_union, mem_ofPred_eq, mem_univ, iff_true]
-      exact le_or_gt _ _
-    have hh : (1 : ℝ≥0∞) ≤ Q {ω | radiusLevelTime (γ ω) (a.G ω) z u ≤ T} + 1 / 2 := by
-      have hs := measure_union_le (μ := Q)
-        {ω : a.samples | radiusLevelTime (γ ω) (a.G ω) z u ≤ T}
-        {ω : a.samples | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (a.G ω) z u}
-      rw [he, measure_univ] at hs
-      exact hs.trans (add_le_add le_rfl hbad)
-    have hh' := tsub_le_iff_right.mpr hh
-    norm_num at hh'
-    simpa only [one_div] using hh'
-  have hhit : (1 / 2 : ℝ≥0∞) ≤ Q {ω | spatialHitTime (γ ω) z r ≤ T} := by
-    apply hgood.trans (measure_mono _)
-    intro ω hω
-    exact ((a.witness ω).radius_hit_forces_spatial_before
-      ((a.continuous_B ω).const_mul _) z r hr hy u hlev T hω).le
-  rw [a.finite_spatial_lintegral_identity hκ hκ8 z hz r hr hy u hu hlev T]
-  calc
-    _ = ENNReal.ofReal (initialGreen κ z) * (1 / 2) := by
-      rw [ENNReal.ofReal_div_of_pos (by norm_num)]
-      norm_num [div_eq_mul_inv]
-    _ ≤ _ := mul_le_mul' le_rfl hhit
-
-lemma finiteSpatialGreenMass_expectation_swap [IsProbabilityMeasure P] (r : ℝ) (T : ℝ≥0) (K : Set ℂ) :
-    (∫⁻ ω : a.samples, finiteSpatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
-      (γ ω) (a.G ω) r T K ∂(P.comap Subtype.val)) =
-    ∫⁻ z in K, ∫⁻ ω : a.samples, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
-        (γ ω) (a.G ω) z r else 0) ∂(P.comap Subtype.val) := by
-  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
-  have hm (t : ℝ≥0) : Measurable (fun ω : a.samples => γ ω t) :=
-    (a.trace_adapted t).mono (a.filtration.le t) le_rfl
-  have ht := spatialHitTime_measurable (fun ω : a.samples => γ ω) a.trace_continuous hm r
-  have he : MeasurableSet {p : ℂ × a.samples | spatialHitTime (γ p.2) p.1 r ≤ T} :=
-    measurableSet_le ht measurable_const
-  classical
-  have hW := (@Measurable.ite _ _ _ _ _ _ _ (fun _ => Classical.propDecidable _)
-    he (a.spatialGreenWeight_measurable r) (measurable_const (a := (0 : ℝ)))).ennreal_ofReal
-  symm
-  apply lintegral_lintegral_swap
-  convert hW.aemeasurable using 1
-  ext ⟨z, ω⟩
-  simp only [Function.uncurry_apply_pair]
-
- 
-
-theorem source_finite_time_first_moment [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8) (z₀ : ℂ) (R : ℝ)
-    (hR : 0 < R) (hK : closedBall z₀ R ⊆ upperHalfPlane) :
-    ∃ (T : ℝ≥0) (e₀ m₀ : ℝ), 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧
-      ∀ r : ℝ, 0 < r → r ≤ e₀ →
-        ENNReal.ofReal m₀ ≤ ∫⁻ ω : a.samples,
-          finiteSpatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
-            (γ ω) (a.G ω) r T (closedBall z₀ R) ∂(P.comap Subtype.val) := by
-  have hne : (closedBall z₀ R).Nonempty := ⟨z₀, mem_closedBall_self hR.le⟩
-  obtain ⟨zy, hzy, hy⟩ := (isCompact_closedBall z₀ R).exists_isMinOn hne
-    Complex.continuous_im.continuousOn
-  obtain ⟨zg, hzg, hg⟩ := (isCompact_closedBall z₀ R).exists_isMinOn hne
-    ((initialGreen_continuousOn κ).mono hK)
-  obtain ⟨zt, hzt, ht⟩ := (isCompact_closedBall z₀ R).exists_isMaxOn hne
-    ((halfClockThreshold_continuousOn hκ hκ8).mono hK)
-  let T := (halfClockThreshold κ zt).toNNReal
-  have hT : 0 < T := Real.toNNReal_pos.mpr (halfClockThreshold_pos hκ hκ8 (hK hzt))
-  let m := initialGreen κ zg / 2
-  have hm : 0 < m := div_pos (initialGreen_pos κ zg (hK hzg)) (by norm_num)
-  have hvol : 0 < volume (closedBall z₀ R) := measure_closedBall_pos volume z₀ hR
-  have hvolfin : volume (closedBall z₀ R) ≠ ⊤ := (isCompact_closedBall z₀ R).measure_ne_top
-  refine ⟨T, zy.im, m * (volume (closedBall z₀ R)).toReal, hT, hK hzy,
-    mul_pos hm (ENNReal.toReal_pos hvol.ne' hvolfin), ?_⟩
-  intro r hr hre
-  rw [a.finiteSpatialGreenMass_expectation_swap]
-  calc
-    _ = ENNReal.ofReal m * volume (closedBall z₀ R) := by
-      rw [ENNReal.ofReal_mul hm.le, ENNReal.ofReal_toReal hvolfin]
-    _ = ∫⁻ _z in closedBall z₀ R, ENNReal.ofReal m := by
-      rw [lintegral_const, Measure.restrict_apply_univ]
-    _ ≤ _ := by
-      apply setLIntegral_mono' isClosed_closedBall.measurableSet
-      intro z hz
-      apply (ENNReal.ofReal_le_ofReal (div_le_div_of_nonneg_right (hg hz) (by norm_num))).trans
-      apply a.finite_spatial_lintegral_lower hκ hκ8 z (hK hz) r hr (hre.trans (hy hz)) T
-      exact (ht hz).trans (Real.le_coe_toNNReal _)
-
-end RegularSLEPresentation
-
-theorem sourceFiniteTimeFirstMoment : SourceFiniteTimeFirstMomentTarget := by
-  intro Ω mΩ P hP κ hκ hκ8 γ a z₀ R hR hK
-  let : IsProbabilityMeasure P := hP
-  exact a.source_finite_time_first_moment hκ hκ8 z₀ R hR hK
-
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MeasureSupport
+import OAI.Probability.SLE.SpatialMeans
+
+namespace OAI
+
+/-! Finite Mass for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
+lemma continuous_radius_density_crossing
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hz : z ∈ upperHalfPlane)
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (w U : ℝ≥0) (hwU : w ≤ U) (δ : ℝ) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w,
+        limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω < δ} ≤
+      ENNReal.ofReal (δ / initialGreen κ z) := by
+  apply continuous_crossing_from_monotone _ _ _ 0 w U bot_le hwU
+  · intro ω hω
+    exact (terminalRadiusKernel_continuousOn a hε hBc _ _ U ω hω).mono
+      (Icc_subset_Icc le_rfl hwU)
+  · intro ω hω s hs
+    exact terminalRadiusKernel_pos a hε _ _ U s (hs.2.trans hwU) ω hω
+  · intro s hs hvs hsU N
+    apply (finite_radius_density_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz
+      a hε hεlim 0 U s hs hvs hsU δ N).trans
+    simpa only [mul_one] using mul_le_mul'
+      (le_refl (ENNReal.ofReal (δ / initialGreen κ z)))
+      (show P {ω | localizedRadiusStop (a 0) ω < ⊤} ≤ 1 from prob_le_one)
+
+noncomputable def initialAngleWindowConstant (κ : ℝ) (z : ℂ) : ℝ :=
+  Real.exp (codimension κ * (1 - Real.log 2 - Real.log z.im)) / initialGreen κ z
+
+lemma initial_inverse_angle_window_bound
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hz : z ∈ upperHalfPlane)
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (w U : ℝ≥0) (hw : w ≤ 1) (hwU : w ≤ U) (L : ℝ) (hL : 0 < L) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w, L < radiusInverseAngle a s ω} ≤
+      ENNReal.ofReal (initialAngleWindowConstant κ z / L) := by
+  have hs : {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w, L < radiusInverseAngle a s ω} ⊆
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 w,
+        limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
+          Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L} := by
+    rintro ω ⟨hω, s, hs, hV⟩
+    exact ⟨hω, s, hs, radiusInverseAngle_large_imp_crossing a hε hκ8 s w U hs.2 hwU L hL ω hω hV⟩
+  apply (measure_mono hs).trans
+  apply (continuous_radius_density_crossing hB hBm hBc hB0 hκ hκ8 γ G F ‹∀ᵐ ω ∂P, _› hz
+    a hε hεlim w U hwU _).trans
+  apply ENNReal.ofReal_le_ofReal
+  unfold initialAngleWindowConstant
+  rw [div_right_comm _ L]
+  apply div_le_div_of_nonneg_right ?_ hL.le
+  apply div_le_div_of_nonneg_right ?_ (initialGreen_pos κ z hz).le
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (codimension_pos κ hκ8).le
+  exact sub_le_sub_right (sub_le_sub_right (show (w : ℝ) ≤ 1 from hw) _) _
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma sum_exp_decay (η : ℝ) (hη : 0 < η) (C : ℝ) (hC : 0 ≤ C) :
+    (∑' n : ℕ, ENNReal.ofReal (C * Real.exp (-η * n))) =
+      ENNReal.ofReal (C / (1 - Real.exp (-η))) := by
+  have hr : Real.exp (-η) < 1 := Real.exp_lt_one_iff.mpr (neg_neg_of_pos hη)
+  have hn : ∀ n : ℕ, Real.exp (-η * n) = Real.exp (-η) ^ n := fun n => by
+    rw [mul_comm, Real.exp_nat_mul]
+  simp_rw [hn]
+  rw [← ENNReal.ofReal_tsum_of_nonneg (fun n => mul_nonneg hC (pow_nonneg (Real.exp_pos _).le _))
+    ((summable_geometric_of_lt_one (Real.exp_pos _).le hr).mul_left C),
+    tsum_mul_left, tsum_geometric_of_lt_one (Real.exp_pos _).le hr, div_eq_mul_inv]
+
+lemma exponential_envelope_bound
+    {Ω : Type*} [MeasurableSpace Ω] (Q : Measure Ω)
+    (V : ℝ≥0 → Ω → ℝ) (E : Set Ω) (U : ℝ≥0)
+    (η M C : ℝ) (hη : 0 < η) (hM : 0 < M) (hC : 0 ≤ C)
+    (hwindow : ∀ n : ℕ, Q {ω | ω ∈ E ∧ ∃ s ∈ Icc (n : ℝ≥0) (min (n + 1) U),
+      M * Real.exp (η * n) < V s ω} ≤ ENNReal.ofReal (C / (M * Real.exp (η * n)))) :
+    Q {ω | ω ∈ E ∧ ∃ s ∈ Icc 0 U, M * Real.exp (η * s) < V s ω} ≤
+      ENNReal.ofReal (C / M / (1 - Real.exp (-η))) := by
+  let W (n : ℕ) : Set Ω := {ω | ω ∈ E ∧ ∃ s ∈ Icc (n : ℝ≥0) (min (n + 1) U),
+      M * Real.exp (η * n) < V s ω}
+  have hsub : {ω | ω ∈ E ∧ ∃ s ∈ Icc 0 U, M * Real.exp (η * s) < V s ω} ⊆ ⋃ n, W n := by
+    rintro ω ⟨hω, s, hs, hv⟩
+    let n : ℕ := ⌊(s : ℝ)⌋₊
+    have hn : (n : ℝ) ≤ s := Nat.floor_le s.coe_nonneg
+    have hu : (s : ℝ) ≤ (n : ℝ) + 1 := (Nat.lt_floor_add_one (s : ℝ)).le
+    apply mem_iUnion.mpr
+    refine ⟨n, hω, s, ?_, ?_⟩
+    · exact ⟨by exact_mod_cast hn, le_min (by exact_mod_cast hu) hs.2⟩
+    · apply lt_of_le_of_lt _ hv
+      apply mul_le_mul_of_nonneg_left _ hM.le
+      exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hn hη.le)
+  calc
+    _ ≤ Q (⋃ n, W n) := measure_mono hsub
+    _ ≤ ∑' n, Q (W n) := measure_iUnion_le W
+    _ ≤ ∑' n : ℕ, ENNReal.ofReal (C / (M * Real.exp (η * n))) := ENNReal.tsum_le_tsum hwindow
+    _ = _ := by
+      have he (n : ℕ) : C / (M * Real.exp (η * n)) = (C / M) * Real.exp (-η * n) := by
+        rw [neg_mul, Real.exp_neg]
+        ring
+      simp_rw [he]
+      exact sum_exp_decay η hη (C / M) (div_nonneg hC hM.le)
+
+lemma integral_exp_neg_half_le_two (u : ℝ) :
+    (∫ s in (0 : ℝ)..u, Real.exp (-s / 2)) ≤ 2 := by
+  have hd (s : ℝ) : HasDerivAt (fun s => -2 * Real.exp (-s / 2)) (Real.exp (-s / 2)) s := by
+    convert (((Real.hasDerivAt_exp (-s / 2)).comp s ((hasDerivAt_id s).neg.div_const 2)).const_mul (-2)) using 1
+    all_goals first | rfl | ring
+  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (fun s _ => hd s) ((Real.continuous_exp.comp (continuous_id.neg.div_const 2)).intervalIntegrable 0 u)]
+  norm_num
+  nlinarith [Real.exp_pos (-u / 2)]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def angleEnvelopeConstant (κ : ℝ) (z : ℂ) : ℝ :=
+  max (initialAngleWindowConstant κ z) (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ))
+
+lemma angleEnvelopeConstant_pos (κ : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    0 < angleEnvelopeConstant κ z :=
+  (div_pos (Real.exp_pos _) (initialGreen_pos κ z hz)).trans_le (le_max_left _ _)
+
+lemma finite_inverse_angle_envelope
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 U,
+        M * Real.exp (greenExponent κ / 8 * s) < radiusInverseAngle a s ω} ≤
+      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
+  have hrate : 0 < greenExponent κ / 8 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
+  have hC := angleEnvelopeConstant_pos κ z hz
+  have he : -(greenExponent κ / 8) = -greenExponent κ / 8 := by ring
+  rw [← he]
+  apply exponential_envelope_bound (finiteCenteredLaw P κ B γ G z U) (radiusInverseAngle a)
+    {ω | localizedRadiusStop (a U) ω < ⊤} U (greenExponent κ / 8) M
+    (angleEnvelopeConstant κ z) hrate hM hC.le
+  intro n
+  let L := M * Real.exp (greenExponent κ / 8 * n)
+  have hL : 0 < L := mul_pos hM (Real.exp_pos _)
+  change finiteCenteredLaw P κ B γ G z U
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc (n : ℝ≥0) (min ((n : ℝ≥0) + 1) U),
+        L < radiusInverseAngle a s ω} ≤ ENNReal.ofReal (angleEnvelopeConstant κ z / L)
+  by_cases hnU : (n : ℝ≥0) ≤ U
+  · by_cases hn : n = 0
+    · subst n
+      have hh := initial_inverse_angle_window_bound hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim
+        (min 1 U) U (min_le_left _ _) (min_le_right _ _) M hM
+      simpa only [L, Nat.cast_zero, zero_add, mul_zero, Real.exp_zero, mul_one, angleEnvelopeConstant] using
+        hh.trans (ENNReal.ofReal_le_ofReal
+          (div_le_div_of_nonneg_right (le_max_left _ _) hM.le))
+    · have hn1 : 1 ≤ (n : ℝ) := by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hn
+      have hvw : (n : ℝ≥0) ≤ min ((n : ℝ≥0) + 1) U := le_min (by simp) hnU
+      have hh := inverse_angle_window_bound hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim
+        n (min ((n : ℝ≥0) + 1) U) U hn1 hvw (min_le_right _ _) L hL
+      apply hh.trans
+      apply ENNReal.ofReal_le_ofReal
+      apply div_le_div_of_nonneg_right _ hL.le
+      apply le_trans _ (le_max_right _ _)
+      apply mul_le_mul_of_nonneg_left _ (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)).le
+      apply Real.exp_le_exp.mpr
+      have hm : ((min (n + 1) U : ℝ≥0) : ℝ) ≤ (n : ℝ) + 1 := by exact_mod_cast min_le_left ((n : ℝ≥0) + 1) U
+      have hd : (↑(min ((n : ℝ≥0) + 1) U) : ℝ) - (n : ℝ) ≤ 1 := by linarith
+      simpa only [mul_one, NNReal.coe_natCast] using
+        mul_le_mul_of_nonneg_left hd (codimension_pos κ hκ8).le
+  · have hempty : {ω | localizedRadiusStop (a U) ω < ⊤ ∧
+        ∃ s ∈ Icc (n : ℝ≥0) (min ((n : ℝ≥0) + 1) U), L < radiusInverseAngle a s ω} = ∅ := by
+      apply eq_empty_iff_forall_notMem.mpr
+      rintro ω ⟨_, s, hs, _⟩
+      exact hnU (hs.1.trans (hs.2.trans (min_le_right _ _)))
+    change finiteCenteredLaw P κ B γ G z U _ ≤ _
+    rw [hempty, measure_empty]
+    exact bot_le
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma log_height_clock_derivative_nonpos
+    (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    -2 / greenLogNormSq κ q b + logClockRate κ q b / 2 ≤ 0 := by
+  have hh := mul_le_mul_of_nonneg_left (logAngleSq_le_one κ q b)
+    (show (0 : ℝ) ≤ 2 / greenLogNormSq κ q b from
+      div_nonneg (by norm_num) (greenLogNormSq_pos κ q b).le)
+  have he : logClockRate κ q b / 2 = (2 / greenLogNormSq κ q b) * logAngleSq κ q b := by
+    unfold logClockRate logAngleSq
+    field_simp
+    ring
+  rw [he, neg_div]
+  linarith
+
+lemma log_loewner_radius_height_decay
+    {Q : ℝ → ℂ × ℝ} {B : ℝ → ℝ} {κ T : ℝ}
+    (hT : 0 ≤ T) (hQ0 : (Q 0).2 = 0)
+    (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q
+      (physicalLogLoewnerField (Real.sqrt κ * B t) (Q t)) (Icc 0 T) t)
+    (t : ℝ) (ht : t ∈ Icc 0 T) :
+    (Q t).1.im ≤ (Q 0).1.im - logRadiusClock (Q 0).1.im (Q t) / 2 := by
+  let η₀ := (Q 0).1.im
+  have hQc : ContinuousOn Q (Icc 0 T) := fun t ht => (hQ t ht).continuousWithinAt
+  have hSc : ContinuousOn (fun t => logRadiusClock η₀ (Q t)) (Icc 0 T) :=
+    (continuous_logRadiusClock η₀).comp_continuousOn hQc
+  have hanti : AntitoneOn (fun t => (Q t).1.im + logRadiusClock η₀ (Q t) / 2) (Icc 0 T) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
+      ((Complex.continuous_im.comp_continuousOn (continuous_fst.comp_continuousOn hQc)).add
+        (hSc.div_const _))
+      (fun t ht => ((log_ode_im_derivative (hQ t (interior_subset ht))).add
+        ((log_ode_clock_derivative η₀ (hQ t (interior_subset ht))).div_const _)).mono interior_subset)
+    intro t ht
+    exact log_height_clock_derivative_nonpos κ (Q t) (B t)
+  have hh := hanti (show (0 : ℝ) ∈ Icc 0 T from ⟨le_rfl, hT⟩) ht ht.1
+  have hS0 : logRadiusClock η₀ (Q 0) = 0 := by simp [logRadiusClock, η₀, hQ0]
+  dsimp only at hh
+  rw [hS0, zero_div, add_zero] at hh
+  change (Q t).1.im ≤ η₀ - _
+  linarith
+
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ} (a : AngularLocalization B hm κ z u ε)
+
+lemma physical_derivative (ω : Ω) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
+    HasDerivWithinAt (fun t => a.solution.Q t ω)
+      (physicalLogLoewnerField (Real.sqrt κ * B t.toNNReal ω) (a.solution.Q t ω))
+        (Icc (0 : ℝ) (a.τ ω : ℝ)) t := by
+  have hd := (a.solution.derivative ω t ht.1).mono
+    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+  have hb := (a.inside ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)).2.2
+  rw [Real.coe_toNNReal t ht.1] at hb
+  rwa [clippedLogLoewnerField_eq_physical _ _ _ _ _ hb] at hd
+
+noncomputable def radiusPhysicalRate (s : ℝ) (ω : Ω) : ℝ :=
+  Real.exp (2 * (a.solution.Q (a.clock.inverse s ω) ω).1.im) /
+    (4 * logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
+      (B (a.clock.inverse s ω).toNNReal ω) ^ 2)
+
+lemma radiusPhysicalRate_continuous
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
+    Continuous (fun s => a.radiusPhysicalRate s ω) := by
+  have hQ := (a.solution.lipschitz ω).continuous.comp (a.clock.continuous_inverse ω)
+  have hB := (hBc ω).comp (continuous_real_toNNReal.comp (a.clock.continuous_inverse ω))
+  have hA := (continuous_logAngleSq κ).comp (hQ.prodMk hB)
+  exact (Real.continuous_exp.comp (continuous_const.mul
+    (Complex.continuous_im.comp hQ.fst))).div (continuous_const.mul (hA.pow 2))
+    (fun s => mul_ne_zero (by norm_num) (pow_ne_zero 2 (logAngleSq_pos _ _ _).ne'))
+
+lemma radiusPhysicalRate_at_clock (ω : Ω) (t : ℝ)
+    (ht : t ∈ Icc (0 : ℝ) (a.τ ω : ℝ)) :
+    a.radiusPhysicalRate (a.clock.S t ω) ω *
+      logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω) = 1 := by
+  unfold radiusPhysicalRate
+  rw [a.clock.inverse_clock ω t ht, logClockRate_angle]
+  have hA := (logAngleSq_pos κ (a.solution.Q t ω) (B t.toNNReal ω)).ne'
+  have hE := Real.exp_ne_zero (2 * (a.solution.Q t ω).1.im)
+  field_simp
+
+lemma radiusPhysicalRate_integral
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
+    (∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), a.radiusPhysicalRate s ω) = (a.τ ω : ℝ) := by
+  have hh := intervalIntegral.integral_comp_mul_deriv''
+    (a.clock.continuous ω).continuousOn
+    (f' := fun t => logClockRate κ (a.solution.Q t ω) (B t.toNNReal ω))
+    (a := (0 : ℝ)) (b := (a.τ ω : ℝ)) (g := fun s => a.radiusPhysicalRate s ω)
+    (fun t ht => by
+      have ht' : t ∈ Ioo (0 : ℝ) (a.τ ω : ℝ) := by
+        simpa only [min_eq_left (a.τ ω).coe_nonneg, max_eq_right (a.τ ω).coe_nonneg] using ht
+      exact ((a.radius_clock_derivative ω t ⟨ht'.1.le, ht'.2.le⟩).hasDerivAt
+        (Icc_mem_nhds ht'.1 ht'.2)).hasDerivWithinAt)
+    ((continuous_logClockRate κ).comp
+      ((a.solution.lipschitz ω).continuous.prodMk
+        ((hBc ω).comp continuous_real_toNNReal))).continuousOn
+    (a.radiusPhysicalRate_continuous hBc ω).continuousOn
+  rw [a.clock.zero ω] at hh
+  rw [← hh]
+  trans ∫ t in (0 : ℝ)..(a.τ ω : ℝ), (1 : ℝ)
+  · apply intervalIntegral.integral_congr
+    intro t ht
+    exact a.radiusPhysicalRate_at_clock ω t (by simpa only [uIcc_of_le (a.τ ω).coe_nonneg] using ht)
+  · simp
+
+lemma radius_height_decay (ω : Ω) (s : ℝ) (hs : 0 ≤ s)
+    (hsl : s ≤ a.clock.S (a.τ ω) ω) :
+    (a.solution.Q (a.clock.inverse s ω) ω).1.im ≤ Real.log z.im - s / 2 := by
+  have hh := log_loewner_radius_height_decay (a.τ ω).coe_nonneg
+    (show (a.solution.Q 0 ω).2 = 0 by rw [a.solution.initial]; rfl)
+    (a.physical_derivative ω) (a.clock.inverse s ω) (a.clock.inverse_mem s ω)
+  have he := a.clock.clock_inverse s ω
+  change logRadiusClock (Real.log z.im) (a.solution.Q (a.clock.inverse s ω) ω) =
+    min (max s 0) (a.clock.S (a.τ ω) ω) at he
+  rw [max_eq_left hs, min_eq_left hsl] at he
+  have hinit : (a.solution.Q 0 ω).1.im = Real.log z.im := by
+    rw [a.solution.initial, initialLogCoordinates_im]
+  rwa [hinit, he] at hh
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma inverse_angle_control_rate {A a M s : ℝ}
+    (hA : 0 < A) (ha : 0 < a) (hM : 0 < M)
+    (hV : A ^ (-a / 2) ≤ M * Real.exp (a * s / 8)) :
+    1 / A ^ 2 ≤ M ^ (4 / a) * Real.exp (s / 2) := by
+  have hp : 0 ≤ 4 / a := by positivity
+  have hh := Real.rpow_le_rpow (Real.rpow_pos_of_pos hA _).le hV hp
+  rw [← Real.rpow_mul hA.le] at hh
+  have he : (-a / 2) * (4 / a) = -2 := by field_simp; ring
+  rw [he, Real.rpow_neg hA.le, Real.rpow_two,
+    Real.mul_rpow hM.le (Real.exp_pos _).le, ← Real.exp_mul] at hh
+  have he' : a * s / 8 * (4 / a) = s / 2 := by field_simp; ring
+  rw [he'] at hh
+  simpa only [one_div] using hh
+
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ} (a : AngularLocalization B hm κ z u ε)
+
+lemma physical_rate_of_envelope (hκ : 0 < κ) (hκ8 : κ < 8)
+    (M : ℝ) (hM : 0 < M) (ω : Ω) (s : ℝ) (hs : 0 ≤ s)
+    (hsl : s ≤ a.clock.S (a.τ ω) ω)
+    (hV : (logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
+      (B (a.clock.inverse s ω).toNNReal ω)) ^ (-greenExponent κ / 2) ≤
+        M * Real.exp (greenExponent κ * s / 8)) :
+    a.radiusPhysicalRate s ω ≤
+      (Real.exp (2 * Real.log z.im) / 4 * M ^ (4 / greenExponent κ)) * Real.exp (-s / 2) := by
+  let A := logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
+    (B (a.clock.inverse s ω).toNNReal ω)
+  have hA : 0 < A := logAngleSq_pos _ _ _
+  have hp := inverse_angle_control_rate hA (greenExponent_pos κ hκ hκ8) hM hV
+  have he : Real.exp (2 * (a.solution.Q (a.clock.inverse s ω) ω).1.im) ≤
+      Real.exp (2 * Real.log z.im) * Real.exp (-s) := by
+    rw [← Real.exp_add]
+    apply Real.exp_le_exp.mpr
+    linarith [a.radius_height_decay ω s hs hsl]
+  unfold radiusPhysicalRate
+  change Real.exp _ / (4 * A ^ 2) ≤ _
+  calc
+    _ = (Real.exp (2 * (a.solution.Q (a.clock.inverse s ω) ω).1.im) / 4) * (1 / A ^ 2) := by ring
+    _ ≤ (Real.exp (2 * Real.log z.im) * Real.exp (-s) / 4) *
+        (M ^ (4 / greenExponent κ) * Real.exp (s / 2)) :=
+      mul_le_mul (div_le_div_of_nonneg_right he (by norm_num)) hp (by positivity) (by positivity)
+    _ = _ := by
+      rw [show Real.exp (2 * Real.log z.im) * Real.exp (-s) / 4 *
+          (M ^ (4 / greenExponent κ) * Real.exp (s / 2)) =
+          (Real.exp (2 * Real.log z.im) / 4 * M ^ (4 / greenExponent κ)) *
+            (Real.exp (-s) * Real.exp (s / 2)) by ring, ← Real.exp_add]
+      congr 2
+      ring
+
+end AngularLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def physicalTimeThreshold (κ : ℝ) (z : ℂ) (M : ℝ) : ℝ :=
+  Real.exp (2 * Real.log z.im) / 2 * M ^ (4 / greenExponent κ)
+
+lemma physicalTimeThreshold_pos (κ : ℝ) (z : ℂ) {M : ℝ} (hM : 0 < M) :
+    0 < physicalTimeThreshold κ z M :=
+  mul_pos (div_pos (Real.exp_pos _) (by norm_num)) (Real.rpow_pos_of_pos hM _)
+
+namespace AngularLocalization
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u ε : ℝ} (a : AngularLocalization B hm κ z u ε)
+
+lemma physical_time_of_envelope (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (M : ℝ) (hM : 0 < M) (ω : Ω)
+    (hV : ∀ s ∈ Icc 0 (a.clock.S (a.τ ω) ω),
+      (logAngleSq κ (a.solution.Q (a.clock.inverse s ω) ω)
+        (B (a.clock.inverse s ω).toNNReal ω)) ^ (-greenExponent κ / 2) ≤
+          M * Real.exp (greenExponent κ * s / 8)) :
+    (a.τ ω : ℝ) ≤ physicalTimeThreshold κ z M := by
+  let C := Real.exp (2 * Real.log z.im) / 4 * M ^ (4 / greenExponent κ)
+  have hC : 0 ≤ C := mul_nonneg (div_nonneg (Real.exp_pos _).le (by norm_num))
+    (Real.rpow_pos_of_pos hM _).le
+  rw [← a.radiusPhysicalRate_integral hBc ω]
+  calc
+    _ ≤ ∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), C * Real.exp (-s / 2) := by
+      apply intervalIntegral.integral_mono_on (a.clock.terminal_nonneg ω)
+        ((a.radiusPhysicalRate_continuous hBc ω).intervalIntegrable _ _)
+        ((continuous_const.mul (Real.continuous_exp.comp (continuous_id.neg.div_const 2))).intervalIntegrable _ _)
+      intro s hs
+      exact a.physical_rate_of_envelope hκ hκ8 M hM ω s hs.1 hs.2 (hV s hs)
+    _ = C * (∫ s in (0 : ℝ)..(a.clock.S (a.τ ω) ω), Real.exp (-s / 2)) :=
+      intervalIntegral.integral_const_mul _ _
+    _ ≤ C * 2 := mul_le_mul_of_nonneg_left (integral_exp_neg_half_le_two _) hC
+    _ = _ := by dsimp [C, physicalTimeThreshold]; ring
+
+end AngularLocalization
+
+lemma localized_physical_time_of_envelope
+    {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (U : ℝ≥0) (M : ℝ) (hM : 0 < M)
+    (ω : Ω) (hω : localizedRadiusStop (a U) ω < ⊤)
+    (hV : ∀ s ∈ Icc 0 U, radiusInverseAngle a s ω ≤ M * Real.exp (greenExponent κ / 8 * s)) :
+    localizedRadiusStop (a U) ω ≤ ((physicalTimeThreshold κ z M).toNNReal : WithTop ℝ≥0) := by
+  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top (a U) ω hω
+  rw [localizedRadiusStop_eq_hit (a U) ω n hn, WithTop.coe_le_coe, ← NNReal.coe_le_coe,
+    Real.coe_toNNReal _ (physicalTimeThreshold_pos κ z hM).le]
+  apply (a U n).physical_time_of_envelope hBc hκ hκ8 M hM ω
+  intro s hs
+  have hsu : s ≤ (U : ℝ) := hs.2.trans_eq hn
+  have hh := hV s.toNNReal ⟨bot_le, Real.toNNReal_le_iff_le_coe.mpr hsu⟩
+  unfold radiusInverseAngle at hh
+  rw [limitRadiusKernel_eq_below (a s.toNNReal) (a U) hε
+    (by simpa only [Real.coe_toNNReal s hs.1] using hsu) 0 2 s.toNNReal
+    s.toNNReal.coe_nonneg le_rfl ω, Real.coe_toNNReal s hs.1,
+    limitRadiusKernel_eq_of_surviving_cutoff (a U) hε 0 2 s hs.1 ω n hs.2] at hh
+  unfold AngularLocalization.radiusValue at hh
+  rw [powerLogDensity_radius_angle κ 0 2 (Real.log z.im)] at hh
+  norm_num only [zero_mul, Real.exp_zero, one_mul, div_self (by norm_num : (2 : ℝ) ≠ 0), Real.rpow_one] at hh
+  convert hh using 1
+  congr 2
+  ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma finite_physical_tail_of_localizations
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤
+      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
+  have hwQ : ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z U, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω) :=
+    (withDensity_absolutelyContinuous P _).ae_le hw
+  have hs : {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤ᵐ[finiteCenteredLaw P κ B γ G z U]
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc 0 U,
+        M * Real.exp (greenExponent κ / 8 * s) < radiusInverseAngle a s ω} := by
+    filter_upwards [hwQ, finiteCenteredLaw_live_compact hB κ hκ hκ8 γ G F hw z hz U U.coe_nonneg] with ω hω hlive
+    intro hlate
+    have he := localizedRadiusStop_eq_source (a U) hεlim hBc hz ω (γ ω) (G ω) (F ω) hω
+    obtain ⟨T, _, _, hT, _⟩ := hlive
+    have hl : localizedRadiusStop (a U) ω < ⊤ := by rw [he, hT]; exact ENNReal.coe_lt_top
+    refine ⟨hl, ?_⟩
+    by_contra h
+    push Not at h
+    have hb := localized_physical_time_of_envelope a hε hBc hκ hκ8 U M hM ω hl h
+    rw [he] at hb
+    exact not_lt_of_ge hb hlate
+  exact (measure_mono_ae hs).trans
+    (finite_inverse_angle_envelope hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim U M hM)
+
+lemma finite_physical_tail_regular
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤
+      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
+  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
+  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
+  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
+  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
+  have hεA (n : ℕ) : ε n ≤ A₀ := by
+    dsimp [ε]
+    exact div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
+  have hεlim : Tendsto ε atTop (𝓝 0) := by
+    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
+    simpa only [mul_one_div, mul_zero] using hh
+  have hεanti : Antitone ε := by
+    intro n m hnm
+    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
+  let a (u : ℝ≥0) (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
+    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
+      u.coe_nonneg (hε0 n) (hεA n))
+  exact finite_physical_tail_of_localizations hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hεanti hεlim U M hM
+
+ 
+
+theorem finite_physical_tail
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (U : ℝ≥0) (M : ℝ) (hM : 0 < M) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | ENNReal.ofReal (physicalTimeThreshold κ z M) < radiusLevelTime (γ ω) (G ω) z U} ≤
+      ENNReal.ofReal (angleEnvelopeConstant κ z / M / (1 - Real.exp (-greenExponent κ / 8))) := by
+  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
+  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
+    filter_upwards [hw, hCB] with ω hω he
+    simpa only [he] using hω
+  have hh := finite_physical_tail_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz U M hM
+  rwa [finiteCenteredLaw_congr_driver hCB κ γ G z U] at hh
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem finite_physical_clock_tight
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (δ : ℝ) (hδ : 0 < δ) :
+    ∃ T : ℝ≥0, 0 < T ∧ ∀ U : ℝ≥0,
+      finiteCenteredLaw P κ B γ G z U
+        {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U} ≤ ENNReal.ofReal δ := by
+  let D := 1 - Real.exp (-greenExponent κ / 8)
+  have hD : 0 < D := by
+    apply sub_pos.mpr
+    apply Real.exp_lt_one_iff.mpr
+    exact div_neg_of_neg_of_pos (neg_neg_of_pos (greenExponent_pos κ hκ hκ8)) (by norm_num)
+  let M := angleEnvelopeConstant κ z / (δ * D)
+  have hM : 0 < M := div_pos (angleEnvelopeConstant_pos κ z hz) (mul_pos hδ hD)
+  refine ⟨(physicalTimeThreshold κ z M).toNNReal, Real.toNNReal_pos.mpr (physicalTimeThreshold_pos κ z hM), ?_⟩
+  intro U
+  have hh := finite_physical_tail hB κ hκ hκ8 γ G F hw z hz U M hM
+  have he : angleEnvelopeConstant κ z / M / D = δ := by
+    dsimp [M]
+    field_simp [(angleEnvelopeConstant_pos κ z hz).ne', hD.ne', hδ.ne']
+  change _ ≤ ENNReal.ofReal (angleEnvelopeConstant κ z / M / D) at hh
+  rw [he] at hh
+  exact hh
+
+ 
+
+theorem finite_physical_clock_positive
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    ∃ T : ℝ≥0, 0 < T ∧ ∀ U : ℝ≥0, (1 / 2 : ℝ≥0∞) ≤
+      finiteCenteredLaw P κ B γ G z U
+        {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T} := by
+  obtain ⟨T, hT, ht⟩ := finite_physical_clock_tight hB κ hκ hκ8 γ G F hw z hz (1/2) (by norm_num)
+  refine ⟨T, hT, fun U => ?_⟩
+  let Q := finiteCenteredLaw P κ B γ G z U
+  let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F hw z hz U U.coe_nonneg
+  have hu : {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T} ∪
+      {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U} = univ := by
+    ext ω
+    simp only [mem_union, mem_ofPred_eq, mem_univ, iff_true]
+    exact le_or_gt _ _
+  have hbad : Q {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U} ≤ (1 / 2 : ℝ≥0∞) := by
+    have he : ENNReal.ofReal ((1 : ℝ) / 2) = (1 / 2 : ℝ≥0∞) := by
+      rw [ENNReal.ofReal_div_of_pos (by norm_num)]
+      norm_num
+    rw [← he]
+    exact ht U
+  have hh : (1 : ℝ≥0∞) ≤ Q {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T} + 1 / 2 := by
+    have hsum := measure_union_le (μ := Q)
+      {ω | radiusLevelTime (γ ω) (G ω) z U ≤ T}
+      {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z U}
+    rw [hu, measure_univ] at hsum
+    exact hsum.trans (add_le_add le_rfl hbad)
+  have hsub := tsub_le_iff_right.mpr hh
+  norm_num at hsub
+  simpa only [one_div] using hsub
+
+end SLEExactGauge
+end
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+noncomputable def finiteSpatialGreenMass (κ : ℝ) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ) (r : ℝ) (T : ℝ≥0) (K : Set ℂ) : ℝ≥0∞ :=
+  ∫⁻ z in K, ENNReal.ofReal (if spatialHitTime γ z r ≤ T then
+    spatialGreenWeight κ U γ G z r else 0)
+def SourceFiniteTimeFirstMomentTarget : Prop :=
+  ∀ (Ω : Type*) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ γ : Ω → ℝ≥0 → ℂ, ∀ a : RegularSLEPresentation P κ γ,
+  ∀ z₀ : ℂ, ∀ R : ℝ, 0 < R → closedBall z₀ R ⊆ upperHalfPlane →
+  ∃ (T : ℝ≥0) (e₀ m₀ : ℝ), 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧
+    ∀ r : ℝ, 0 < r → r ≤ e₀ →
+      ENNReal.ofReal m₀ ≤ ∫⁻ ω : a.samples,
+        finiteSpatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
+          (γ ω) (a.G ω) r T (closedBall z₀ R) ∂(P.comap Subtype.val)
+
+lemma initialGreen_continuousOn (κ : ℝ) : ContinuousOn (initialGreen κ) upperHalfPlane := by
+  have hn (z : ℂ) (hz : z ∈ upperHalfPlane) : ‖z‖ ≠ 0 := by
+    apply norm_ne_zero_iff.mpr
+    intro he
+    subst z
+    exact (lt_irrefl 0) (show (0 : ℝ) < 0 from hz)
+  have hy : ContinuousOn (fun z : ℂ => z.im) upperHalfPlane :=
+    Complex.continuous_im.continuousOn
+  have h1 := (hy.const_mul 2).rpow_const (p := -codimension κ)
+    (fun z hz => Or.inl (mul_ne_zero (by norm_num) (ne_of_gt hz)))
+  have h2 := (hy.div continuous_norm.continuousOn hn).rpow_const (p := greenExponent κ)
+    (fun z hz => Or.inl (div_ne_zero (ne_of_gt hz) (hn z hz)))
+  convert h1.mul h2 using 1
+  funext z
+  simp only [initialGreen, Complex.sin_arg, Pi.mul_apply, Pi.div_apply]
+
+lemma angleEnvelopeConstant_continuousOn (κ : ℝ) :
+    ContinuousOn (angleEnvelopeConstant κ) upperHalfPlane := by
+  have hl : ContinuousOn (fun z : ℂ => Real.log z.im) upperHalfPlane :=
+    Complex.continuous_im.continuousOn.log (fun _ hz => ne_of_gt hz)
+  have he : ContinuousOn (fun z : ℂ => Real.exp
+      (codimension κ * (1 - Real.log 2 - Real.log z.im))) upperHalfPlane :=
+    Real.continuous_exp.comp_continuousOn
+      ((continuousOn_const.sub hl).const_mul (codimension κ))
+  exact (he.div (initialGreen_continuousOn κ)
+    (fun z hz => (initialGreen_pos κ z hz).ne')).sup continuousOn_const
+
+noncomputable def halfClockThreshold (κ : ℝ) (z : ℂ) : ℝ :=
+  physicalTimeThreshold κ z
+    (angleEnvelopeConstant κ z / ((1 / 2) * (1 - Real.exp (-greenExponent κ / 8))))
+
+lemma halfClockThreshold_pos {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {z : ℂ} (hz : z ∈ upperHalfPlane) : 0 < halfClockThreshold κ z := by
+  apply physicalTimeThreshold_pos
+  apply div_pos (angleEnvelopeConstant_pos κ z hz)
+  apply mul_pos (by norm_num)
+  exact sub_pos.mpr (Real.exp_lt_one_iff.mpr (by
+    have := greenExponent_pos κ hκ hκ8
+    linarith))
+
+lemma halfClockThreshold_continuousOn {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ContinuousOn (halfClockThreshold κ) upperHalfPlane := by
+  have hD : 0 < (1 / 2 : ℝ) * (1 - Real.exp (-greenExponent κ / 8)) := by
+    apply mul_pos (by norm_num)
+    exact sub_pos.mpr (Real.exp_lt_one_iff.mpr (by
+      have := greenExponent_pos κ hκ hκ8
+      linarith))
+  have hM := (angleEnvelopeConstant_continuousOn κ).div_const
+    ((1 / 2) * (1 - Real.exp (-greenExponent κ / 8)))
+  have hp := hM.rpow_const (p := 4 / greenExponent κ) (fun z hz =>
+    Or.inl (div_pos (angleEnvelopeConstant_pos κ z hz) hD).ne')
+  have hl := Complex.continuous_im.continuousOn.log
+    (s := upperHalfPlane) (fun _ hz => ne_of_gt hz)
+  exact ((Real.continuous_exp.comp_continuousOn (hl.const_mul 2)).div_const 2).mul hp
+namespace RegularSLEPresentation
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+lemma spatialHitTime_stopping (z : ℂ) (r : ℝ) :
+    IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
+  a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
+
+ 
+
+lemma finite_spatial_density_identity [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
+    (u : ℝ) (hu : 0 ≤ u) (hlevel : (2 * z.im) * Real.exp (-u) < r) (T : ℝ≥0) :
+    (∫ ω : a.samples in {ω | spatialHitTime (γ ω) z r ≤ T},
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
+        (γ ω) (a.G ω) z r ∂(P.comap Subtype.val)) =
+    ∫ ω : a.samples in {ω | spatialHitTime (γ ω) z r ≤ T},
+      stoppedGreen κ (fun t => Real.sqrt κ * a.B t ω)
+        (γ ω) (a.G ω) z u ∂(P.comap Subtype.val) := by
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
+  obtain ⟨M, Z, hM, hZ, hMb, hZb, hClosed, hStop, heq, hNoHit⟩ :=
+    exists_physical_green_stopping_regular a.brownian hBm a.continuous_B a.zero_B
+      κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
+  let σ (ω : a.samples) : ℝ≥0 := boundedSpatialHit (γ ω) z r T
+  have hσ : IsStoppingTime a.filtration (fun ω => (σ ω : WithTop ℝ≥0)) :=
+    a.boundedSpatialHit_stopping z r T
+  let A := {ω : a.samples | spatialHitTime (γ ω) z r ≤ T}
+  have hA : MeasurableSet[hσ.measurableSpace] A := by
+    have hh := (a.spatialHitTime_stopping z r).measurableSet_stopping_time_le_min
+      (isStoppingTime_const a.filtration T)
+    rw [hσ.measurableSet]
+    rw [IsStoppingTime.measurableSet] at hh
+    simpa only [σ, A, boundedSpatialHit_coe] using hh
+  have hAm : MeasurableSet A := hσ.measurableSpace_le _ hA
+  have hWA : (fun ω : a.samples => spatialGreenWeight κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) =ᵐ[μ.restrict A]
+      fun ω => M (σ ω) ω := by
+    filter_upwards [ae_restrict_of_ae heq, ae_restrict_mem hAm] with ω he hω
+    change spatialHitTime (γ ω) z r ≤ (T : WithTop ℝ≥0) at hω
+    rw [he.1]
+    simp only [Function.comp_apply]
+    rw [(a.witness ω).physicalGreen_eq_before_spatial
+      ((a.continuous_B ω).const_mul _) κ z r hr hy u hlevel (σ ω)
+      (boundedSpatialHit_le (γ ω) z r T)]
+    have hn : spatialHitTime (γ ω) z r ≠ ⊤ :=
+      ne_top_of_le_ne_top WithTop.coe_ne_top hω
+    simp only [spatialGreenWeight, ite_eq_right hn, σ, boundedSpatialHit,
+      min_eq_left hω]
+  calc
+    _ = ∫ ω in A, M (σ ω) ω ∂μ := integral_congr_ae hWA
+    _ = ∫ ω in A, μ[Z | hσ.measurableSpace] ω ∂μ :=
+      integral_congr_ae (ae_restrict_of_ae (hStop σ hσ))
+    _ = ∫ ω in A, Z ω ∂μ := setIntegral_condExp hσ.measurableSpace_le hZ hA
+    _ = _ := integral_congr_ae (ae_restrict_of_ae (heq.mono fun ω hω => hω.2))
+
+lemma finite_spatial_lintegral_identity [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
+    (u : ℝ) (hu : 0 ≤ u) (hlevel : (2 * z.im) * Real.exp (-u) < r) (T : ℝ≥0) :
+    (∫⁻ ω : a.samples, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
+        (γ ω) (a.G ω) z r else 0) ∂(P.comap Subtype.val)) =
+    ENNReal.ofReal (initialGreen κ z) *
+      finiteCenteredLaw (P.comap Subtype.val) κ a.B (γ ∘ Subtype.val) a.G z u
+        {ω | spatialHitTime (γ ω) z r ≤ T} := by
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  let A := {ω : a.samples | spatialHitTime (γ ω) z r ≤ T}
+  have hA : MeasurableSet A :=
+    (a.spatialHitTime_stopping z r).measurableSpace_le _
+      ((a.spatialHitTime_stopping z r).measurableSet_le' T)
+  have hi := (a.spatialGreen_expectation hκ hκ8 z r hr hy).1.integrableOn (s := A)
+  have hn : 0 ≤ᵐ[μ.restrict A] (fun ω : a.samples => spatialGreenWeight κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) :=
+    Eventually.of_forall fun ω => by
+      dsimp only [Pi.zero_apply, spatialGreenWeight]
+      split_ifs
+      · exact le_rfl
+      · exact greenDensity_nonneg (a.witness ω) κ _ z
+  have hind (ω : a.samples) : ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r else 0) =
+      A.indicator (fun ω : a.samples => ENNReal.ofReal (spatialGreenWeight κ
+        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r)) ω := by
+    dsimp only [A, Set.indicator, mem_ofPred_eq]
+    split_ifs <;> simp_all only [ENNReal.ofReal_zero]
+  simp_rw [hind]
+  rw [lintegral_indicator hA, ← ofReal_integral_eq_lintegral_ofReal hi hn,
+    a.finite_spatial_density_identity hκ hκ8 z hz r hr hy u hu hlevel T,
+    finiteCenteredLaw_apply a.brownian κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F
+      (Eventually.of_forall a.witness) z hz u hu A hA,
+    ← ENNReal.ofReal_mul (initialGreen_pos κ z hz).le]
+  congr 1
+  simp only [Function.comp_apply]
+  field_simp [(initialGreen_pos κ z hz).ne']
+  rfl
+
+lemma finite_spatial_lintegral_lower [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
+    (T : ℝ≥0) (hT : halfClockThreshold κ z ≤ T) :
+    ENNReal.ofReal (initialGreen κ z / 2) ≤
+      ∫⁻ ω : a.samples, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
+        spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
+          (γ ω) (a.G ω) z r else 0) ∂(P.comap Subtype.val) := by
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  let u := Real.log (4 * z.im / r)
+  have hratio : 0 < 4 * z.im / r := div_pos (mul_pos (by norm_num) hz) hr
+  have hu : 0 ≤ u := Real.log_nonneg ((le_div_iff₀ hr).mpr (by linarith))
+  have hlev : (2 * z.im) * Real.exp (-u) < r := by
+    have he : (2 * z.im) * Real.exp (-u) = r / 2 := by
+      dsimp only [u]
+      rw [Real.exp_neg, Real.exp_log hratio]
+      field_simp [(show 0 < z.im from hz).ne']
+      ring
+    rw [he]
+    linarith
+  let Q := finiteCenteredLaw μ κ a.B (γ ∘ Subtype.val) a.G z u
+  let : IsProbabilityMeasure Q := finiteCenteredLaw_probability a.brownian κ hκ hκ8
+    (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
+  let D := 1 - Real.exp (-greenExponent κ / 8)
+  have hD : 0 < D := by
+    apply sub_pos.mpr
+    apply Real.exp_lt_one_iff.mpr
+    have := greenExponent_pos κ hκ hκ8
+    linarith
+  let M := angleEnvelopeConstant κ z / ((1 / 2) * D)
+  have hM : 0 < M := div_pos (angleEnvelopeConstant_pos κ z hz) (mul_pos (by norm_num) hD)
+  have hbad : Q {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (a.G ω) z u} ≤ 1 / 2 := by
+    have hh := finite_physical_tail a.brownian κ hκ hκ8 (γ ∘ Subtype.val)
+      a.G a.F (Eventually.of_forall a.witness) z hz ⟨u, hu⟩ M hM
+    have he : angleEnvelopeConstant κ z / M / D = 1 / 2 := by
+      dsimp only [M]
+      field_simp [(angleEnvelopeConstant_pos κ z hz).ne', hD.ne']
+    change Q {ω | ENNReal.ofReal (halfClockThreshold κ z) <
+      radiusLevelTime (γ ω) (a.G ω) z u} ≤ ENNReal.ofReal (angleEnvelopeConstant κ z / M / D) at hh
+    rw [he] at hh
+    have hTT : ENNReal.ofReal (halfClockThreshold κ z) ≤ (T : ℝ≥0∞) := by
+      simpa only [ENNReal.ofReal_coe_nnreal] using ENNReal.ofReal_le_ofReal hT
+    have hhalf : ENNReal.ofReal ((1 : ℝ) / 2) = (1 / 2 : ℝ≥0∞) := by
+      rw [ENNReal.ofReal_div_of_pos (by norm_num)]
+      norm_num
+    rw [hhalf] at hh
+    exact (measure_mono (fun ω hω => hTT.trans_lt hω)).trans hh
+  have hgood : (1 / 2 : ℝ≥0∞) ≤ Q {ω | radiusLevelTime (γ ω) (a.G ω) z u ≤ T} := by
+    have he : {ω : a.samples | radiusLevelTime (γ ω) (a.G ω) z u ≤ T} ∪
+        {ω | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (a.G ω) z u} = univ := by
+      ext ω
+      simp only [mem_union, mem_ofPred_eq, mem_univ, iff_true]
+      exact le_or_gt _ _
+    have hh : (1 : ℝ≥0∞) ≤ Q {ω | radiusLevelTime (γ ω) (a.G ω) z u ≤ T} + 1 / 2 := by
+      have hs := measure_union_le (μ := Q)
+        {ω : a.samples | radiusLevelTime (γ ω) (a.G ω) z u ≤ T}
+        {ω : a.samples | (T : ℝ≥0∞) < radiusLevelTime (γ ω) (a.G ω) z u}
+      rw [he, measure_univ] at hs
+      exact hs.trans (add_le_add le_rfl hbad)
+    have hh' := tsub_le_iff_right.mpr hh
+    norm_num at hh'
+    simpa only [one_div] using hh'
+  have hhit : (1 / 2 : ℝ≥0∞) ≤ Q {ω | spatialHitTime (γ ω) z r ≤ T} := by
+    apply hgood.trans (measure_mono _)
+    intro ω hω
+    exact ((a.witness ω).radius_hit_forces_spatial_before
+      ((a.continuous_B ω).const_mul _) z r hr hy u hlev T hω).le
+  rw [a.finite_spatial_lintegral_identity hκ hκ8 z hz r hr hy u hu hlev T]
+  calc
+    _ = ENNReal.ofReal (initialGreen κ z) * (1 / 2) := by
+      rw [ENNReal.ofReal_div_of_pos (by norm_num)]
+      norm_num [div_eq_mul_inv]
+    _ ≤ _ := mul_le_mul' le_rfl hhit
+
+lemma finiteSpatialGreenMass_expectation_swap [IsProbabilityMeasure P] (r : ℝ) (T : ℝ≥0) (K : Set ℂ) :
+    (∫⁻ ω : a.samples, finiteSpatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
+      (γ ω) (a.G ω) r T K ∂(P.comap Subtype.val)) =
+    ∫⁻ z in K, ∫⁻ ω : a.samples, ENNReal.ofReal (if spatialHitTime (γ ω) z r ≤ T then
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
+        (γ ω) (a.G ω) z r else 0) ∂(P.comap Subtype.val) := by
+  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
+  have hm (t : ℝ≥0) : Measurable (fun ω : a.samples => γ ω t) :=
+    (a.trace_adapted t).mono (a.filtration.le t) le_rfl
+  have ht := spatialHitTime_measurable (fun ω : a.samples => γ ω) a.trace_continuous hm r
+  have he : MeasurableSet {p : ℂ × a.samples | spatialHitTime (γ p.2) p.1 r ≤ T} :=
+    measurableSet_le ht measurable_const
+  classical
+  have hW := (@Measurable.ite _ _ _ _ _ _ _ (fun _ => Classical.propDecidable _)
+    he (a.spatialGreenWeight_measurable r) (measurable_const (a := (0 : ℝ)))).ennreal_ofReal
+  symm
+  apply lintegral_lintegral_swap
+  convert hW.aemeasurable using 1
+  ext ⟨z, ω⟩
+  simp only [Function.uncurry_apply_pair]
+
+ 
+
+theorem source_finite_time_first_moment [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8) (z₀ : ℂ) (R : ℝ)
+    (hR : 0 < R) (hK : closedBall z₀ R ⊆ upperHalfPlane) :
+    ∃ (T : ℝ≥0) (e₀ m₀ : ℝ), 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧
+      ∀ r : ℝ, 0 < r → r ≤ e₀ →
+        ENNReal.ofReal m₀ ≤ ∫⁻ ω : a.samples,
+          finiteSpatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
+            (γ ω) (a.G ω) r T (closedBall z₀ R) ∂(P.comap Subtype.val) := by
+  have hne : (closedBall z₀ R).Nonempty := ⟨z₀, mem_closedBall_self hR.le⟩
+  obtain ⟨zy, hzy, hy⟩ := (isCompact_closedBall z₀ R).exists_isMinOn hne
+    Complex.continuous_im.continuousOn
+  obtain ⟨zg, hzg, hg⟩ := (isCompact_closedBall z₀ R).exists_isMinOn hne
+    ((initialGreen_continuousOn κ).mono hK)
+  obtain ⟨zt, hzt, ht⟩ := (isCompact_closedBall z₀ R).exists_isMaxOn hne
+    ((halfClockThreshold_continuousOn hκ hκ8).mono hK)
+  let T := (halfClockThreshold κ zt).toNNReal
+  have hT : 0 < T := Real.toNNReal_pos.mpr (halfClockThreshold_pos hκ hκ8 (hK hzt))
+  let m := initialGreen κ zg / 2
+  have hm : 0 < m := div_pos (initialGreen_pos κ zg (hK hzg)) (by norm_num)
+  have hvol : 0 < volume (closedBall z₀ R) := measure_closedBall_pos volume z₀ hR
+  have hvolfin : volume (closedBall z₀ R) ≠ ⊤ := (isCompact_closedBall z₀ R).measure_ne_top
+  refine ⟨T, zy.im, m * (volume (closedBall z₀ R)).toReal, hT, hK hzy,
+    mul_pos hm (ENNReal.toReal_pos hvol.ne' hvolfin), ?_⟩
+  intro r hr hre
+  rw [a.finiteSpatialGreenMass_expectation_swap]
+  calc
+    _ = ENNReal.ofReal m * volume (closedBall z₀ R) := by
+      rw [ENNReal.ofReal_mul hm.le, ENNReal.ofReal_toReal hvolfin]
+    _ = ∫⁻ _z in closedBall z₀ R, ENNReal.ofReal m := by
+      rw [lintegral_const, Measure.restrict_apply_univ]
+    _ ≤ _ := by
+      apply setLIntegral_mono' isClosed_closedBall.measurableSet
+      intro z hz
+      apply (ENNReal.ofReal_le_ofReal (div_le_div_of_nonneg_right (hg hz) (by norm_num))).trans
+      apply a.finite_spatial_lintegral_lower hκ hκ8 z (hK hz) r hr (hre.trans (hy hz)) T
+      exact (ht hz).trans (Real.le_coe_toNNReal _)
+
+end RegularSLEPresentation
+
+theorem sourceFiniteTimeFirstMoment : SourceFiniteTimeFirstMomentTarget := by
+  intro Ω mΩ P hP κ hκ hκ8 γ a z₀ R hR hK
+  let : IsProbabilityMeasure P := hP
+  exact a.source_finite_time_first_moment hκ hκ8 z₀ R hR hK
+
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/FiniteRadiusStop.lean b/lean/OAI/Probability/SLE/FiniteRadiusStop.lean
index da4232c86c012e208a9d14827c4e680c20edca31..6ab1c55698fff189a6023643cd70d945f14b2567 100644
--- a/lean/OAI/Probability/SLE/FiniteRadiusStop.lean
+++ b/lean/OAI/Probability/SLE/FiniteRadiusStop.lean
@@ -1,822 +1,822 @@
-import Mathlib
-import OAI.Probability.SLE.BoundaryDensity
-
-namespace OAI
-
-/-! Finite Radius Stop for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma continuous_nonneg_at_boundary (D : ℝ≥0 → ℝ) (hD : Continuous D)
-    (h0 : 0 ≤ D 0) (τ : ℝ≥0) (hb : ∀ t, t < τ → 0 < D t) :
-    ∀ t, t ≤ τ → 0 ≤ D t := by
-  intro t ht
-  by_cases hτ : τ = 0
-  · have : t = 0 := le_antisymm (ht.trans hτ.le) zero_le
-    simpa only [this] using h0
-  · have hτpos : 0 < τ := pos_iff_ne_zero.mpr hτ
-    apply le_on_closure (s := Iio τ) (f := fun _ : ℝ≥0 => (0 : ℝ)) (g := D)
-      (fun s hs => (hb s hs).le) continuous_const.continuousOn hD.continuousOn
-    rw [closure_Iio' (show (Iio τ).Nonempty from ⟨0, hτpos⟩)]
-    exact ht
-
- 
-
-theorem first_margin_stopping
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
-    (D : ℝ≥0 → Ω → ℝ) (hDa : Adapted f D)
-    (hDc : ∀ ω, Continuous (fun t => D t ω)) (hD0 : ∀ ω, 0 ≤ D 0 ω)
-    (T : ℝ≥0) :
-    ∃ τ : Ω → ℝ≥0,
-      IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0)) ∧
-      (∀ ω, τ ω ≤ T) ∧
-      (∀ ω t, t ≤ τ ω → 0 ≤ D t ω) ∧
-      (∀ ω t, t < τ ω → 0 < D t ω) ∧
-      (∀ ω, τ ω < T → D (τ ω) ω = 0) := by
-  let H (t : ℝ≥0) (ω : Ω) := max 0 (D t ω)
-  have hHa : Adapted f H := fun t => measurable_const.max (hDa t)
-  have hHc (ω : Ω) : Continuous (fun t => H t ω) := continuous_const.max (hDc ω)
-  have hHn (t : ℝ≥0) (ω : Ω) : 0 ≤ H t ω := le_max_left _ _
-  let τ (ω : Ω) := hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω
-  obtain ⟨hτ, hb, hpre, hat⟩ := continuous_nonnegative_first_zero_stopping f H hHa hHc hHn T
-  have hDpre (ω : Ω) (t : ℝ≥0) (ht : t < τ ω) : 0 < D t ω := by
-    have hh := hpre ω t ht
-    simpa only [H, lt_max_iff, lt_self_iff_false, false_or] using hh
-  have hDle (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) : 0 ≤ D t ω :=
-    continuous_nonneg_at_boundary (fun s => D s ω) (hDc ω) (hD0 ω) (τ ω) (hDpre ω) t ht
-  refine ⟨τ, hτ, hb, hDle, hDpre, ?_⟩
-  intro ω ht
-  apply le_antisymm _ (hDle ω (τ ω) le_rfl)
-  have hh := le_max_right 0 (D (τ ω) ω)
-  exact hh.trans (hat ω ht).le
-
- 
-noncomputable def logStopMargin (κ η₀ u lo hi ε N : ℝ) (q : ℂ × ℝ) (b : ℝ) : ℝ :=
-  min (u - logRadiusClock η₀ q)
-    (min (logAngleSq κ q b - ε)
-      (min (q.1.im - lo) (min (hi - q.1.im) (N - |b|))))
-
-lemma continuous_logStopMargin (κ η₀ u lo hi ε N : ℝ) :
-    Continuous (fun p : (ℂ × ℝ) × ℝ => logStopMargin κ η₀ u lo hi ε N p.1 p.2) := by
-  unfold logStopMargin
-  exact (continuous_const.sub ((continuous_logRadiusClock η₀).comp continuous_fst)).min
-    (((continuous_logAngleSq κ).sub continuous_const).min
-      (((Complex.continuous_im.comp continuous_fst.fst).sub continuous_const).min
-        ((continuous_const.sub (Complex.continuous_im.comp continuous_fst.fst)).min
-          (continuous_const.sub continuous_snd.abs))))
-
-lemma logStopMargin_nonneg_iff (κ η₀ u lo hi ε N : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    0 ≤ logStopMargin κ η₀ u lo hi ε N q b ↔
-      logRadiusClock η₀ q ≤ u ∧ ε ≤ logAngleSq κ q b ∧
-        lo ≤ q.1.im ∧ q.1.im ≤ hi ∧ |b| ≤ N := by
-  simp only [logStopMargin, le_min_iff, sub_nonneg]
-
-lemma logStopMargin_pos_iff (κ η₀ u lo hi ε N : ℝ) (q : ℂ × ℝ) (b : ℝ) :
-    0 < logStopMargin κ η₀ u lo hi ε N q b ↔
-      logRadiusClock η₀ q < u ∧ ε < logAngleSq κ q b ∧
-        lo < q.1.im ∧ q.1.im < hi ∧ |b| < N := by
-  simp only [logStopMargin, lt_min_iff, sub_pos]
-
-lemma ClippedLogSolution.norm_le
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (ω : Ω) (t T : ℝ≥0) (ht : t ≤ T) :
-    ‖c.Q t ω‖ ≤ ‖q₀‖ + c.L * T := by
-  have hh := (c.lipschitz ω).norm_sub_le (t : ℝ) 0
-  rw [c.initial ω, sub_zero, Real.norm_eq_abs, abs_of_nonneg t.coe_nonneg] at hh
-  have hn := norm_add_le (c.Q t ω - q₀) q₀
-  rw [sub_add_cancel] at hn
-  have hmul : (c.L : ℝ) * t ≤ c.L * T :=
-    mul_le_mul_of_nonneg_left (by exact_mod_cast ht) c.L.coe_nonneg
-  linarith
-
- 
-
-theorem ClippedLogSolution.finite_radius_cutoff
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hq0 : q₀.2 = 0)
-    (u ε N : ℝ) (hu : 0 ≤ u) (hε : 0 < ε) (hN : 0 ≤ N)
-    (hε0 : ε ≤ logAngleSq κ q₀ 0)
-    (hlo : lo < q₀.1.im - u / (2 * ε)) (hhi : q₀.1.im < hi)
-    (T : ℝ≥0) (hT : u < (4 * ε ^ 2 / Real.exp (2 * q₀.1.im)) * T) :
-    ∃ τ : Ω → ℝ≥0,
-      IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (τ ω : WithTop ℝ≥0)) ∧
-      (∀ ω, τ ω < T) ∧
-      (∀ ω t, t ≤ τ ω →
-        logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
-        ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
-        lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi ∧ |B t ω| ≤ N) ∧
-      (∀ ω t, t < τ ω →
-        logRadiusClock q₀.1.im (c.Q t ω) < u ∧
-        ε < logAngleSq κ (c.Q t ω) (B t ω) ∧
-        lo < (c.Q t ω).1.im ∧ (c.Q t ω).1.im < hi ∧ |B t ω| < N) ∧
-      (∀ ω, logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
-        logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε ∨ |B (τ ω) ω| = N) ∧
-      Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
-      (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) = greenLogDensity κ q₀ 0 := by
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let D (t : ℝ≥0) (ω : Ω) := logStopMargin κ q₀.1.im u lo hi ε N (c.Q t ω) (B t ω)
-  have hBa : Adapted f B :=
-    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
-  have hDa : Adapted f D := by
-    intro t
-    change Measurable[f t] ((fun p : (ℂ × ℝ) × ℝ => logStopMargin κ q₀.1.im u lo hi ε N p.1 p.2) ∘
-      (fun ω => (c.Q t ω, B t ω)))
-    exact (continuous_logStopMargin κ q₀.1.im u lo hi ε N).measurable.comp
-      ((c.adapted t).prodMk (hBa t))
-  have hDc (ω : Ω) : Continuous (fun t => D t ω) := by
-    change Continuous ((fun p : (ℂ × ℝ) × ℝ => logStopMargin κ q₀.1.im u lo hi ε N p.1 p.2) ∘
-      (fun t : ℝ≥0 => (c.Q t ω, B t ω)))
-    exact (continuous_logStopMargin κ q₀.1.im u lo hi ε N).comp
-      (((c.lipschitz ω).continuous.comp NNReal.continuous_coe).prodMk (hBc ω))
-  have hD0 (ω : Ω) : 0 ≤ D 0 ω := by
-    change 0 ≤ logStopMargin κ q₀.1.im u lo hi ε N (c.Q (0 : ℝ≥0) ω) (B 0 ω)
-    rw [NNReal.coe_zero, c.initial ω, hB0 ω, logStopMargin_nonneg_iff]
-    refine ⟨?_, hε0, ?_, hhi.le, ?_⟩
-    · simpa only [logRadiusClock, sub_self, zero_add, hq0] using hu
-    · have hd : 0 ≤ u / (2 * ε) := div_nonneg hu (by positivity)
-      linarith
-    · simpa only [abs_zero] using hN
-  obtain ⟨τ, hτ, hb, hleD, hltD, hzero⟩ := first_margin_stopping f D hDa hDc hD0 T
-  have hin (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) :
-      logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
-      ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
-      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi ∧ |B t ω| ≤ N :=
-    (logStopMargin_nonneg_iff κ q₀.1.im u lo hi ε N _ _).mp (hleD ω t ht)
-  have hg (ω : Ω) :
-      (c.Q (τ ω) ω).1.im ≤ q₀.1.im ∧
-      q₀.1.im - logRadiusClock q₀.1.im (c.Q (τ ω) ω) / (2 * ε) ≤ (c.Q (τ ω) ω).1.im ∧
-      (4 * ε ^ 2 / Real.exp (2 * q₀.1.im)) * τ ω ≤ logRadiusClock q₀.1.im (c.Q (τ ω) ω) := by
-    have hd (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) :
-        HasDerivWithinAt (fun t => c.Q t ω)
-          (physicalLogLoewnerField (Real.sqrt κ * B s.toNNReal ω) (c.Q s ω))
-          (Icc 0 (τ ω : ℝ)) s := by
-      have hst : s.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr hs.2
-      have hbounds := (hin ω s.toNNReal hst).2.2
-      rw [Real.coe_toNNReal _ hs.1] at hbounds
-      have hh := (c.derivative ω s hs.1).mono (show Icc (0 : ℝ) (τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-      rwa [clippedLogLoewnerField_eq_physical lo hi hle _ _ ⟨hbounds.1, hbounds.2.1⟩] at hh
-    have ha (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) :
-        ε ≤ logAngleSq κ (c.Q s ω) (B s.toNNReal ω) := by
-      have hh := (hin ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.1
-      simpa only [Real.coe_toNNReal _ hs.1] using hh
-    have hq : (c.Q 0 ω).2 = 0 := by rw [c.initial ω, hq0]
-    have hh := log_loewner_height_clock_bounds (τ ω).coe_nonneg hε hq hd ha
-    simpa only [c.initial ω] using hh
-  have hτlt (ω : Ω) : τ ω < T := by
-    by_contra hn
-    have he : τ ω = T := le_antisymm (hb ω) (not_lt.mp hn)
-    have hh := (hg ω).2.2.trans (hin ω (τ ω) le_rfl).1
-    rw [he] at hh
-    exact (not_le_of_gt hT) hh
-  have he (ω : Ω) :
-      logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
-      logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε ∨ |B (τ ω) ω| = N := by
-    obtain ⟨hr, ha, hl, hh, hbn⟩ := hin ω (τ ω) le_rfl
-    have hstrictlo : lo < (c.Q (τ ω) ω).1.im := by
-      have hdiv := div_le_div_of_nonneg_right hr (show 0 ≤ 2 * ε by positivity)
-      linarith [(hg ω).2.1]
-    have hstricthi : (c.Q (τ ω) ω).1.im < hi := (hg ω).1.trans_lt hhi
-    by_contra hn
-    push Not at hn
-    have hp : 0 < D (τ ω) ω :=
-      (logStopMargin_pos_iff κ q₀.1.im u lo hi ε N _ _).mpr
-        ⟨lt_of_le_of_ne hr hn.1, lt_of_le_of_ne ha (Ne.symm hn.2.1),
-          hstrictlo, hstricthi, lt_of_le_of_ne hbn hn.2.2⟩
-    rw [hzero ω (hτlt ω)] at hp
-    exact lt_irrefl 0 hp
-  let R := max (‖q₀‖ + c.L * T) N
-  have hR : 0 ≤ R := hN.trans (le_max_right _ _)
-  have hinside (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) :
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi := by
-    refine ⟨?_, (hin ω t ht).2.2.1, (hin ω t ht).2.2.2.1⟩
-    rw [Prod.norm_def, max_le_iff, Real.norm_eq_abs]
-    exact ⟨(c.norm_le ω t T (ht.trans (hb ω))).trans (le_max_left _ _),
-      (hin ω t ht).2.2.2.2.trans (le_max_right _ _)⟩
-  obtain ⟨hint, hmean⟩ := c.stopped_green hB hBm hBc hκ hκ8 R hR τ hτ T hb hinside
-  refine ⟨τ, hτ, hτlt, hin, ?_, he, hint, hmean⟩
-  intro ω t ht
-  exact (logStopMargin_pos_iff κ q₀.1.im u lo hi ε N _ _).mp (hltD ω t ht)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma continuous_path_bounded_by_nat (B : ℝ≥0 → ℝ) (hB : Continuous B) (T : ℝ≥0) :
-    ∃ n : ℕ, ∀ t : ℝ≥0, t ≤ T → |B t| < (n : ℝ) + 1 := by
-  obtain ⟨C, hC⟩ := (isCompact_Icc : IsCompact (Icc (0 : ℝ≥0) T)).exists_bound_of_continuousOn hB.continuousOn
-  obtain ⟨n, hn⟩ := exists_nat_gt C
-  refine ⟨n, fun t ht => ?_⟩
-  have hh : |B t| ≤ C := by simpa only [Real.norm_eq_abs] using hC t ⟨zero_le, ht⟩
-  linarith
-
-lemma first_radius_angle_le (S A : ℝ≥0 → ℝ) (u ε : ℝ) (s t : ℝ≥0)
-    (hpre : ∀ v, v < s → S v < u ∧ ε < A v)
-    (hat : S t = u ∨ A t = ε) : s ≤ t := by
-  by_contra hn
-  obtain ⟨hr, ha⟩ := hpre t (not_le.mp hn)
-  rcases hat with h | h
-  · exact (ne_of_lt hr) h
-  · exact (ne_of_gt ha) h
-
-lemma bounded_iSup_stopping
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
-    (τ : ℕ → Ω → ℝ≥0)
-    (hτ : ∀ n, IsStoppingTime f (fun ω => (τ n ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ n ω, τ n ω ≤ T) :
-    IsStoppingTime f (fun ω => ((⨆ n, τ n ω : ℝ≥0) : WithTop ℝ≥0)) := by
-  intro t
-  have he : {ω | ((⨆ n, τ n ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
-      ⋂ n : ℕ, {ω | (τ n ω : WithTop ℝ≥0) ≤ t} := by
-    ext ω
-    simp only [mem_ofPred_eq, WithTop.coe_le_coe, mem_iInter]
-    exact ciSup_le_iff ⟨T, by rintro v ⟨n, rfl⟩; exact hb n ω⟩
-  rw [he]
-  exact MeasurableSet.iInter (fun n => hτ n t)
-
-lemma greenLogDensity_nonneg (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) : 0 ≤ greenLogDensity κ q b :=
-  (Real.exp_pos _).le
-
-lemma greenLogDensity_le_radius
-    (κ η₀ u : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (q : ℂ × ℝ) (b : ℝ)
-    (hclock : logRadiusClock η₀ q ≤ u) :
-    greenLogDensity κ q b ≤ Real.exp (codimension κ * (u - Real.log 2 - η₀)) := by
-  simpa only [Real.one_rpow, mul_one] using
-    greenLogDensity_le_at_angular_exit κ η₀ u 1 hκ hκ8 q b hclock (logAngleSq_le_one κ q b)
-
- 
-
-theorem ClippedLogSolution.finite_angular_law
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hq0 : q₀.2 = 0)
-    (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
-    (hε0 : ε ≤ logAngleSq κ q₀ 0)
-    (hlo : lo < q₀.1.im - u / (2 * ε)) (hhi : q₀.1.im < hi)
-    (T : ℝ≥0) (hT : u < (4 * ε ^ 2 / Real.exp (2 * q₀.1.im)) * T) :
-    ∃ τ : Ω → ℝ≥0,
-      IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (τ ω : WithTop ℝ≥0)) ∧
-      (∀ ω, τ ω < T) ∧
-      (∀ ω t, t ≤ τ ω →
-        logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
-        ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
-        lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) ∧
-      (∀ ω t, t < τ ω →
-        logRadiusClock q₀.1.im (c.Q t ω) < u ∧ ε < logAngleSq κ (c.Q t ω) (B t ω)) ∧
-      (∀ ω, logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
-        logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε) ∧
-      Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
-      (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) = greenLogDensity κ q₀ 0 := by
-  classical
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hall (n : ℕ) := c.finite_radius_cutoff hB hBm hBc hB0 hκ hκ8 hq0
-    u ε ((n : ℝ) + 1) hu hε (by positivity) hε0 hlo hhi T hT
-  choose τn hstop htime hin hpre hboundary hint hmean using hall
-  let τ (ω : Ω) : ℝ≥0 := ⨆ n, τn n ω
-  have hb (n : ℕ) (ω : Ω) : τn n ω ≤ T := (htime n ω).le
-  have hbd (ω : Ω) : BddAbove (range (fun n => τn n ω)) :=
-    ⟨T, by rintro v ⟨n, rfl⟩; exact hb n ω⟩
-  have hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)) := bounded_iSup_stopping _ τn hstop T hb
-  have hstable (ω : Ω) : ∃ n : ℕ, τ ω = τn n ω ∧
-      (∀ m, n ≤ m → τn m ω = τ ω) ∧
-      (logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
-        logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε) := by
-    obtain ⟨n, hnb⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
-    have hhit (m : ℕ) (hnm : n ≤ m) :
-        logRadiusClock q₀.1.im (c.Q (τn m ω) ω) = u ∨
-          logAngleSq κ (c.Q (τn m ω) ω) (B (τn m ω) ω) = ε := by
-      rcases hboundary m ω with h | h | h
-      · exact Or.inl h
-      · exact Or.inr h
-      · have hh := hnb (τn m ω) (hb m ω)
-        have hnm' : (n : ℝ) ≤ m := by exact_mod_cast hnm
-        linarith
-    have hle (m : ℕ) : τn m ω ≤ τn n ω :=
-      first_radius_angle_le _ _ u ε _ _ (fun t ht => ⟨(hpre m ω t ht).1, (hpre m ω t ht).2.1⟩)
-        (hhit n le_rfl)
-    have he : τ ω = τn n ω := le_antisymm (ciSup_le hle) (le_ciSup (hbd ω) n)
-    refine ⟨n, he, ?_, ?_⟩
-    · intro m hnm
-      rw [he]
-      apply le_antisymm (hle m)
-      exact first_radius_angle_le _ _ u ε _ _
-        (fun t ht => ⟨(hpre n ω t ht).1, (hpre n ω t ht).2.1⟩) (hhit m hnm)
-    · rw [he]
-      exact hhit n le_rfl
-  have hτtime (ω : Ω) : τ ω < T := by
-    obtain ⟨n, he, _⟩ := hstable ω
-    rw [he]
-    exact htime n ω
-  have hτin (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) :
-      logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
-      ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
-      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi := by
-    obtain ⟨n, he, _⟩ := hstable ω
-    obtain ⟨h1, h2, h3, h4, _⟩ := hin n ω t (by simpa only [he] using ht)
-    exact ⟨h1, h2, h3, h4⟩
-  have hτpre (ω : Ω) (t : ℝ≥0) (ht : t < τ ω) :
-      logRadiusClock q₀.1.im (c.Q t ω) < u ∧ ε < logAngleSq κ (c.Q t ω) (B t ω) := by
-    obtain ⟨n, he, _⟩ := hstable ω
-    have hh := hpre n ω t (by simpa only [he] using ht)
-    exact ⟨hh.1, hh.2.1⟩
-  let K := Real.exp (codimension κ * (u - Real.log 2 - q₀.1.im))
-  have hbound (n : ℕ) (ω : Ω) :
-      ‖greenLogDensity κ (c.Q (τn n ω) ω) (B (τn n ω) ω)‖ ≤ K := by
-    rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
-    exact greenLogDensity_le_radius κ q₀.1.im u hκ hκ8 _ _ (hin n ω (τn n ω) le_rfl).1
-  have hlim (ω : Ω) : Tendsto
-      (fun n => greenLogDensity κ (c.Q (τn n ω) ω) (B (τn n ω) ω)) atTop
-      (𝓝 (greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω))) := by
-    obtain ⟨n, _, he, _⟩ := hstable ω
-    apply tendsto_const_nhds.congr'
-    exact (eventually_ge_atTop n).mono (fun m hm => by dsimp only; rw [he m hm])
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hQm : Measurable (fun ω => c.Q (τ ω) ω) :=
-    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
-  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
-  have hfm : Measurable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) :=
-    (greenLogDensity_contDiff κ).continuous.measurable.comp (hQm.prodMk hBmτ)
-  have hfi : Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P := by
-    apply Integrable.of_bound hfm.aestronglyMeasurable K
-    exact Eventually.of_forall fun ω => by
-      rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
-      exact greenLogDensity_le_radius κ q₀.1.im u hκ hκ8 _ _ (hτin ω (τ ω) le_rfl).1
-  have hconv := tendsto_integral_of_dominated_convergence (fun _ : Ω => K)
-    (fun n => (hint n).aestronglyMeasurable) (integrable_const K)
-    (fun n => Eventually.of_forall (hbound n)) (Eventually.of_forall hlim)
-  have hmean' : (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
-      greenLogDensity κ q₀ 0 := by
-    apply tendsto_nhds_unique hconv
-    simp only [hmean]
-    exact tendsto_const_nhds
-  exact ⟨τ, hτ, hτtime, hτin, hτpre, fun ω => (hstable ω).choose_spec.2.2, hfi, hmean'⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def initialLogCoordinates (z : ℂ) : ℂ × ℝ :=
-  ((z.re : ℂ) + (Real.log z.im : ℂ) * Complex.I, 0)
-
-noncomputable def restoreLogPoint (q : ℂ × ℝ) : ℂ :=
-  (q.1.re : ℂ) + (Real.exp q.1.im : ℂ) * Complex.I
-
-lemma restoreLogPoint_im (q : ℂ × ℝ) : (restoreLogPoint q).im = Real.exp q.1.im := by
-  simp only [restoreLogPoint, Complex.add_im, Complex.ofReal_im, Complex.mul_im,
-    Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero, zero_add]
-
-lemma restoreLogPoint_initial (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    restoreLogPoint (initialLogCoordinates z) = z := by
-  apply Complex.ext <;> simp [restoreLogPoint, initialLogCoordinates, Real.exp_log hz]
-
-lemma restoreLogPoint_derivative {Q : ℝ → ℂ × ℝ} {D : Set ℝ} {t u : ℝ}
-    (hQ : HasDerivWithinAt Q (physicalLogLoewnerField u (Q t)) D t) :
-    HasDerivWithinAt (fun s => restoreLogPoint (Q s))
-      (2 / (restoreLogPoint (Q t) - (u : ℂ))) D t := by
-  have hr := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hQ.fst
-  have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hQ.fst
-  have hd := (Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t hr).add
-    ((Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t hi.exp).mul_const Complex.I)
-  convert hd using 1 <;> try rfl
-  · let Z := restoreLogPoint (Q t) - (u : ℂ)
-    have he : (((Q t).1.re - u : ℝ) : ℂ) + (Real.exp (Q t).1.im : ℂ) * Complex.I = Z := by
-      simp only [Z, restoreLogPoint, Complex.ofReal_sub]; ring
-    have hZi : Z.im = Real.exp (Q t).1.im := by
-      dsimp only [Z]
-      rw [Complex.sub_im, Complex.ofReal_im, sub_zero, restoreLogPoint_im]
-    simp only [Function.comp_def, Complex.reCLM_apply, Complex.ofRealCLM_apply,
-      Complex.imCLM_apply, physicalLogLoewnerField, he, Complex.add_re, Complex.add_im,
-      Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re, Complex.mul_im,
-      Complex.I_re, Complex.I_im, mul_zero, sub_self, add_zero, mul_one, zero_add]
-    change 2 / Z = ((2 * Z⁻¹.re : ℝ) : ℂ) +
-      ((Real.exp (Q t).1.im * (-2 * Complex.normSq Z⁻¹) : ℝ) : ℂ) * Complex.I
-    apply Complex.ext
-    · simp only [div_eq_mul_inv, Complex.mul_re, Complex.re_ofNat, Complex.im_ofNat,
-        zero_mul, sub_zero, Complex.add_re, Complex.ofReal_re, Complex.ofReal_im,
-        Complex.I_re, Complex.I_im, mul_zero, sub_self, add_zero]
-    · simp only [div_eq_mul_inv, Complex.mul_im, Complex.re_ofNat, Complex.im_ofNat,
-        zero_mul, add_zero, Complex.add_im, Complex.ofReal_re, Complex.ofReal_im,
-        Complex.I_re, Complex.I_im, mul_zero, mul_one, zero_add, Complex.inv_im,
-        Complex.normSq_inv, hZi]
-      ring
-
- 
-
-theorem clipped_log_identification
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (T : ℝ≥0)
-    (lo hi : ℝ) (hle : lo ≤ hi) (Q : ℝ → ℂ × ℝ)
-    (hQ0 : Q 0 = initialLogCoordinates z)
-    (hQd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q
-      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t)
-    (hb : ∀ t ∈ Icc 0 (T : ℝ), (Q t).1.im ∈ Icc lo hi) :
-    z ∈ survivingDomain γ T ∧ EqOn Q (logLoewnerCoordinates G z) (Icc 0 (T : ℝ)) := by
-  let W (t : ℝ) := restoreLogPoint (Q t)
-  have hW0 : W 0 = z := by
-    change restoreLogPoint (Q 0) = z
-    rw [hQ0, restoreLogPoint_initial z hz]
-  have hp (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : 0 < (W t).im := by
-    change 0 < (restoreLogPoint (Q t)).im
-    rw [restoreLogPoint_im]; exact Real.exp_pos _
-  have hd (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt W
-      (2 / (W t - (U t.toNNReal : ℂ))) (Icc 0 (T : ℝ)) t := by
-    apply restoreLogPoint_derivative
-    have hh := hQd t ht
-    rwa [clippedLogLoewnerField_eq_physical lo hi hle _ _ (hb t ht)] at hh
-  obtain ⟨hlive, he⟩ := loewner_survival_of_positive_solution hU h T W hp hd
-  rw [hW0] at hlive he
-  have hη : Real.log (G T z).im = (Q T).1.im := by
-    rw [he]
-    change Real.log (restoreLogPoint (Q T)).im = _
-    rw [restoreLogPoint_im, Real.log_exp]
-  have hh₀ := hb 0 ⟨le_rfl, T.coe_nonneg⟩
-  rw [hQ0] at hh₀
-  have hhi : Real.log z.im ≤ hi := by simpa [initialLogCoordinates] using hh₀.2
-  exact ⟨hlive, clipped_log_loewner_agrees hU h T z hlive lo hi hle
-    (by rw [hη]; exact (hb T ⟨T.coe_nonneg, le_rfl⟩).1) hhi Q hQ0 hQd⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem exists_joint_clipped_flow
-    {Ω : Type*} [MeasurableSpace Ω]
-    (U : ℝ≥0 → Ω → ℝ)
-    (hUm : Measurable (fun p : ℝ≥0 × Ω => U p.1 p.2))
-    (hUc : ∀ ω, Continuous (fun t => U t ω))
-    (q₀ : Ω → ℂ × ℝ) (hq₀ : Measurable q₀)
-    (lo hi : ℝ) (hle : lo ≤ hi) :
-    ∃ Q : ℝ → Ω → ℂ × ℝ,
-      Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧
-      (∀ ω, Q 0 ω = q₀ ω) ∧
-      (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω)
-        (clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (Q t ω)) (Ici 0) t) := by
-  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) (x : ℂ × ℝ) :=
-    clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (q₀ ω + x)
-  have hUr : Measurable (fun p : ℝ × Ω => U p.1.toNNReal p.2) :=
-    hUm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hV : Measurable (fun p : (ℝ × Ω) × (ℂ × ℝ) => V p.1.1 p.1.2 p.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun p : (ℝ × Ω) × (ℂ × ℝ) => (U p.1.1.toNNReal p.1.2, q₀ p.1.2 + p.2)))
-    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp
-      ((hUr.comp measurable_fst).prodMk
-        ((hq₀.comp (measurable_snd.comp measurable_fst)).add measurable_snd))
-  have hVc (ω : Ω) (x : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω x) := by
-    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun t : ℝ => (U t.toNNReal ω, q₀ ω + x)))
-    exact (continuous_clippedLogLoewnerField lo hi hle).comp
-      (((hUc ω).comp continuous_real_toNNReal).prodMk continuous_const)
-  have hVl (ω : Ω) (t : ℝ) : LipschitzWith K (V t ω) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro x y
-    simpa only [V, dist_add_left] using (hlip (U t.toNNReal ω)).dist_le_mul (q₀ ω + x) (q₀ ω + y)
-  obtain ⟨X, hXm, _, hX0, hXd, _⟩ := exists_measurable_global_bounded_ode_picard
-    V 0 L K hV hVc hVl (fun ω t x => hbound _ _)
-  refine ⟨fun t ω => q₀ ω + X t ω, (hq₀.comp measurable_snd).add hXm, ?_, ?_⟩
-  · intro ω; change q₀ ω + X 0 ω = q₀ ω; rw [hX0, add_zero]
-  · intro ω t ht
-    simpa only [V] using (hXd ω t ht).const_add (q₀ ω)
-
-lemma measurable_initialLogCoordinates : Measurable initialLogCoordinates := by
-  exact ((Complex.measurable_ofReal.comp Complex.measurable_re).add
-    ((Complex.measurable_ofReal.comp (Real.measurable_log.comp Complex.measurable_im)).mul_const
-      Complex.I)).prodMk measurable_const
-
- 
-theorem exists_spatial_clipped_flow
-    {Ω : Type*} [MeasurableSpace Ω]
-    (U : ℝ≥0 → Ω → ℝ)
-    (hUm : Measurable (fun p : ℝ≥0 × Ω => U p.1 p.2))
-    (hUc : ∀ ω, Continuous (fun t => U t ω)) (n : ℕ) :
-    ∃ Q : ℝ → ℂ × Ω → ℂ × ℝ,
-      Measurable (fun p : ℝ × (ℂ × Ω) => Q p.1 p.2) ∧
-      (∀ z ω, Q 0 (z, ω) = initialLogCoordinates z) ∧
-      (∀ z ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s (z, ω))
-        (clippedLogLoewnerField (-(n : ℝ)) n (by linarith [Nat.cast_nonneg (α := ℝ) n])
-          (U t.toNNReal ω) (Q t (z, ω))) (Ici 0) t) := by
-  obtain ⟨Q, hQm, hQ0, hQd⟩ := exists_joint_clipped_flow (fun t (p : ℂ × Ω) => U t p.2)
-    (hUm.comp (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))
-    (fun p => hUc p.2) (fun p => initialLogCoordinates p.1)
-    (measurable_initialLogCoordinates.comp measurable_fst) (-(n : ℝ)) n (by linarith [Nat.cast_nonneg (α := ℝ) n])
-  exact ⟨Q, hQm, fun z ω => hQ0 (z, ω), fun z ω t ht => hQd (z, ω) t ht⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma clipped_log_height_derivative
-    {U : ℝ≥0 → ℝ} {lo hi : ℝ} {hle : lo ≤ hi} {Q : ℝ → ℂ × ℝ}
-    {t : ℝ} {S : Set ℝ}
-    (hd : HasDerivWithinAt Q
-      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) S t) :
-    HasDerivWithinAt (fun s => (Q s).1.im)
-      (-2 * Complex.normSq (clippedCenter lo hi hle (U t.toNNReal) (Q t))⁻¹) S t := by
-  simpa only [Function.comp_def, Complex.imCLM_apply, clippedLogLoewnerField,
-    Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re,
-    Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero, zero_add]
-    using Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hd.fst
-
-lemma clipped_log_height_antitone
-    {U : ℝ≥0 → ℝ} {lo hi : ℝ} {hle : lo ≤ hi} {Q : ℝ → ℂ × ℝ}
-    (T : ℝ≥0)
-    (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q
-      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t) :
-    AntitoneOn (fun t => (Q t).1.im) (Icc 0 (T : ℝ)) := by
-  have he (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := clipped_log_height_derivative (hd t ht)
-  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
-    (fun t ht => (he t ht).continuousWithinAt)
-    (fun t ht => (he t (interior_subset ht)).mono interior_subset)
-  intro t ht
-  exact mul_nonpos_of_nonpos_of_nonneg (by norm_num) (Complex.normSq_nonneg _)
-
- 
-
-lemma clipped_log_identification_of_endpoint
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (T : ℝ≥0) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (lo hi : ℝ) (hle : lo ≤ hi) (Q : ℝ → ℂ × ℝ)
-    (hQ0 : Q 0 = initialLogCoordinates z)
-    (hQd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q
-      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t)
-    (hlo : lo ≤ (Q T).1.im) (hhi : Real.log z.im ≤ hi) :
-    z ∈ survivingDomain γ T ∧ EqOn Q (logLoewnerCoordinates G z) (Icc 0 (T : ℝ)) := by
-  apply clipped_log_identification hU h z hz T lo hi hle Q hQ0 hQd
-  have hm := clipped_log_height_antitone T hQd
-  intro t ht
-  refine ⟨hlo.trans (hm ht ⟨T.coe_nonneg, le_rfl⟩ ht.2), ?_⟩
-  have hh := hm (show (0 : ℝ) ∈ Icc 0 (T : ℝ) from ⟨le_rfl, T.coe_nonneg⟩) ht ht.1
-  change (Q t).1.im ≤ (Q 0).1.im at hh
-  rw [hQ0] at hh
-  have hinit : (initialLogCoordinates z).1.im = Real.log z.im := by
-    simp [initialLogCoordinates]
-  rw [hinit] at hh
-  exact hh.trans hhi
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def powerLogDensity (κ p a : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  Real.exp (-p * Real.log 2 + p * q.2 +
-    (a - p) * q.1.im -
-      (a / 2) * Real.log (greenLogNormSq κ q x))
-lemma powerLogDensity_contDiff (κ p a : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => powerLogDensity κ p a zz.1 zz.2) := by
-  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.re) :=
-    Complex.reCLM.contDiff.comp contDiff_fst.fst
-  have him : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.im) :=
-    Complex.imCLM.contDiff.comp contDiff_fst.fst
-  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => greenLogNormSq κ zz.1 zz.2) := by
-    exact ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add
-      ((contDiff_const.mul him).exp)
-  have hlog := hN.log (fun zz => (greenLogNormSq_pos κ zz.1 zz.2).ne')
-  unfold powerLogDensity
-  apply ContDiff.exp
-  exact (((contDiff_const.add (contDiff_const.mul contDiff_fst.snd)).add
-    (contDiff_const.mul him)).sub (contDiff_const.mul hlog))
-
-lemma powerLogDensity_spatial_derivative (κ p a : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    HasDerivAt (powerLogDensity κ p a q)
-      (powerLogDensity κ p a q x * (a * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
-        greenLogNormSq κ q x)) x := by
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
-    simp
-  have hlog := hN.log (greenLogNormSq_pos κ q x).ne'
-  have hd := ((hasDerivAt_const x (-p * Real.log 2 + p * q.2 +
-    (a - p) * q.1.im)).sub
-      (hlog.const_mul (a / 2))).exp
-  convert hd using 1 <;> try rfl
-  dsimp only [powerLogDensity, Pi.sub_apply, Pi.mul_apply, Function.comp_def]
-  ring
-
-lemma powerLogDensity_spatial_second (κ p a : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
-    iteratedDeriv 2 (powerLogDensity κ p a q) x =
-      powerLogDensity κ p a q x * κ * a *
-        ((a + 1) * (q.1.re - Real.sqrt κ * x) ^ 2 - Real.exp (2 * q.1.im)) /
-        (greenLogNormSq κ q x) ^ 2 := by
-  have hd : deriv (powerLogDensity κ p a q) = fun x =>
-      powerLogDensity κ p a q x * (a * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
-        greenLogNormSq κ q x) := funext fun x => (powerLogDensity_spatial_derivative κ p a q x).deriv
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
-    simp
-  have hcoef := (hX.const_mul (a * Real.sqrt κ)).div hN
-    (greenLogNormSq_pos κ q x).ne'
-  have hsecond := (powerLogDensity_spatial_derivative κ p a q x).mul hcoef
-  rw [show (2 : ℕ) = 1 + 1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
-  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, id_eq] at hsecond
-  change HasDerivAt (fun y => powerLogDensity κ p a q y *
-    (a * Real.sqrt κ * (q.1.re - Real.sqrt κ * y) / greenLogNormSq κ q y)) _ x at hsecond
-  rw [hsecond.deriv]
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  dsimp only [greenLogNormSq]
-  generalize powerLogDensity κ p a q x = f
-  generalize q.1.re - Real.sqrt κ * x = X
-  generalize Real.exp (2 * q.1.im) = Y
-  have hs := Real.sq_sqrt hκ
-  generalize Real.sqrt κ = r at *
-  rw [← hs]
-  field_simp
-  ring
-
-lemma powerLogDensity_directional (κ p a : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q) v =
-      powerLogDensity κ p a q x * (p * v.2 +
-        (a - p) * v.1.im - (a / 2) *
-          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
-            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x)) := by
-  have hzre : HasDerivAt (fun t : ℝ => (q + t • v).1.re) v.1.re 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
-  have hzim : HasDerivAt (fun t : ℝ => (q + t • v).1.im) v.1.im 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
-  have hj : HasDerivAt (fun t : ℝ => (q + t • v).2) v.2 0 := by
-    simpa only [Prod.snd_add, Prod.smul_snd, smul_eq_mul, id_eq, one_mul]
-      using ((hasDerivAt_id (0 : ℝ)).mul_const v.2).const_add q.2
-  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q + t • v) x)
-      (2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
-        2 * Real.exp (2 * q.1.im) * v.1.im) 0 := by
-    convert ((hzre.sub_const (Real.sqrt κ * x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
-    simp only [zero_smul, add_zero, Nat.cast_ofNat]
-    ring
-  have hlog := hN.log (by simpa only [zero_smul, add_zero] using (greenLogNormSq_pos κ q x).ne')
-  have hcalc := (((hj.const_mul (p)).const_add (-p * Real.log 2)).add
-    (hzim.const_mul (a - p))).sub
-      (hlog.const_mul (a / 2))
-  have hline : HasDerivAt (fun t : ℝ => powerLogDensity κ p a (q + t • v) x)
-      (powerLogDensity κ p a q x * (p * v.2 +
-        (a - p) * v.1.im - (a / 2) *
-          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
-            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x))) 0 := by
-    convert hcalc.exp using 1 <;> try rfl
-    simp only [Pi.sub_apply, Pi.add_apply, zero_smul, add_zero]
-    simp only [powerLogDensity]
-  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => powerLogDensity κ p a r x) :=
-    (powerLogDensity_contDiff κ p a).comp (contDiff_id.prodMk contDiff_const)
-  have haff : HasDerivAt (fun t : ℝ => q + t • v) v 0 := by
-    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
-  have hfd : HasFDerivAt (fun r : ℂ × ℝ => powerLogDensity κ p a r x)
-      (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q) (q + (0 : ℝ) • v) := by
-    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
-  have hchain := hfd.comp_hasDerivAt 0 haff
-  simpa only [Function.comp_def] using hchain.unique hline
-
-lemma powerLogDensity_pair_directional (κ p a : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => powerLogDensity κ p a zz.1 zz.2) (q, x) (v, 0) =
-      fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q v := by
-  have hd := ((powerLogDensity_contDiff κ p a).differentiable (by simp)) (q, x)
-  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
-  have hc := (hd.hasFDerivAt.comp q hp).fderiv
-  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) hc
-  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
-    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
-
- 
-
-lemma exists_compact_power_field (κ p a R : ℝ) (hR : 0 ≤ R) :
-    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
-      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = powerLogDensity κ p a q x) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R →
-        fderiv ℝ φ (q, x) = fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => powerLogDensity κ p a zz.1 zz.2) (q, x)) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
-        iteratedDeriv 2 (powerLogDensity κ p a q) x) := by
-  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
-  let φ (zz : (ℂ × ℝ) × ℝ) : ℝ := b zz * powerLogDensity κ p a zz.1 zz.2
-  have hnear (zz : (ℂ × ℝ) × ℝ) (hp : ‖zz‖ ≤ R) :
-      φ =ᶠ[𝓝 zz] (fun r : (ℂ × ℝ) × ℝ => powerLogDensity κ p a r.1 r.2) := by
-    have hball : zz ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
-      simp only [Metric.mem_ball, dist_zero_right]
-      change ‖zz‖ < R + 1
-      linarith
-    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with r hr
-    simp only [φ, hr, Pi.one_apply, one_mul]
-  refine ⟨φ, b.contDiff.mul (powerLogDensity_contDiff κ p a), b.hasCompactSupport.mul_right,
-    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
-  · intro q x h
-    exact (hnear (q, x) h).fderiv_eq
-  · intro q x h
-    have he := (hnear (q, x) h).comp_tendsto
-      ((continuous_const.prodMk continuous_id).tendsto x)
-    exact he.iteratedDeriv_eq 2
-
- 
-lemma power_physical_generator (κ p a : ℝ) (hκ : 0 ≤ κ)
-    (q : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q)
-      (physicalLogLoewnerField (Real.sqrt κ * x) q) +
-      (1 / 2 : ℝ) * iteratedDeriv 2 (powerLogDensity κ p a q) x =
-      powerLogDensity κ p a q x *
-        ((-4 * a + κ * a * (a + 1) / 2) * (q.1.re - Real.sqrt κ * x) ^ 2 +
-          (4 * p - κ * a / 2) * Real.exp (2 * q.1.im)) /
-        (greenLogNormSq κ q x) ^ 2 := by
-  rw [powerLogDensity_directional, powerLogDensity_spatial_second κ p a hκ]
-  obtain ⟨hr, hi, hj⟩ := physicalLogLoewnerField_components κ q x
-  rw [hr, hi, hj]
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  field_simp
-  dsimp only [greenLogNormSq]
-  ring
-
-lemma powerLogDensity_radius_angle (κ p a η₀ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    powerLogDensity κ p a q x =
-      Real.exp (p * (logRadiusClock η₀ q - Real.log 2 - η₀)) *
-        (logAngleSq κ q x) ^ (a / 2) := by
-  rw [Real.rpow_def_of_pos (logAngleSq_pos κ q x), ← Real.exp_add]
-  unfold logAngleSq
-  rw [Real.log_div (Real.exp_ne_zero _) (greenLogNormSq_pos κ q x).ne', Real.log_exp]
-  unfold powerLogDensity logRadiusClock
-  congr 1
-  ring
-
- 
-
-lemma green_weighted_power_generator (κ b : ℝ) (hκ : 0 < κ)
-    (q : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ (codimension κ)
-        (greenExponent κ - b) r x) q)
-      (physicalLogLoewnerField (Real.sqrt κ * x) q) +
-      (1 / 2 : ℝ) * iteratedDeriv 2
-        (powerLogDensity κ (codimension κ) (greenExponent κ - b) q) x =
-      powerLogDensity κ (codimension κ) (greenExponent κ - b) q x *
-        logClockRate κ q x * (κ * b / 8) *
-          (1 - (greenExponent κ - b) * ((logAngleSq κ q x)⁻¹ - 1)) := by
-  rw [power_physical_generator κ _ _ hκ.le]
-  unfold logClockRate logAngleSq
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  have hy := Real.exp_ne_zero (2 * q.1.im)
-  dsimp only [greenExponent, codimension, dimension]
-  field_simp
-  dsimp only [greenLogNormSq]
-  ring
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.BoundaryDensity
+
+namespace OAI
+
+/-! Finite Radius Stop for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma continuous_nonneg_at_boundary (D : ℝ≥0 → ℝ) (hD : Continuous D)
+    (h0 : 0 ≤ D 0) (τ : ℝ≥0) (hb : ∀ t, t < τ → 0 < D t) :
+    ∀ t, t ≤ τ → 0 ≤ D t := by
+  intro t ht
+  by_cases hτ : τ = 0
+  · have : t = 0 := le_antisymm (ht.trans hτ.le) zero_le
+    simpa only [this] using h0
+  · have hτpos : 0 < τ := pos_iff_ne_zero.mpr hτ
+    apply le_on_closure (s := Iio τ) (f := fun _ : ℝ≥0 => (0 : ℝ)) (g := D)
+      (fun s hs => (hb s hs).le) continuous_const.continuousOn hD.continuousOn
+    rw [closure_Iio' (show (Iio τ).Nonempty from ⟨0, hτpos⟩)]
+    exact ht
+
+ 
+
+theorem first_margin_stopping
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
+    (D : ℝ≥0 → Ω → ℝ) (hDa : Adapted f D)
+    (hDc : ∀ ω, Continuous (fun t => D t ω)) (hD0 : ∀ ω, 0 ≤ D 0 ω)
+    (T : ℝ≥0) :
+    ∃ τ : Ω → ℝ≥0,
+      IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0)) ∧
+      (∀ ω, τ ω ≤ T) ∧
+      (∀ ω t, t ≤ τ ω → 0 ≤ D t ω) ∧
+      (∀ ω t, t < τ ω → 0 < D t ω) ∧
+      (∀ ω, τ ω < T → D (τ ω) ω = 0) := by
+  let H (t : ℝ≥0) (ω : Ω) := max 0 (D t ω)
+  have hHa : Adapted f H := fun t => measurable_const.max (hDa t)
+  have hHc (ω : Ω) : Continuous (fun t => H t ω) := continuous_const.max (hDc ω)
+  have hHn (t : ℝ≥0) (ω : Ω) : 0 ≤ H t ω := le_max_left _ _
+  let τ (ω : Ω) := hittingBtwn H ({0} : Set ℝ) (0 : ℝ≥0) T ω
+  obtain ⟨hτ, hb, hpre, hat⟩ := continuous_nonnegative_first_zero_stopping f H hHa hHc hHn T
+  have hDpre (ω : Ω) (t : ℝ≥0) (ht : t < τ ω) : 0 < D t ω := by
+    have hh := hpre ω t ht
+    simpa only [H, lt_max_iff, lt_self_iff_false, false_or] using hh
+  have hDle (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) : 0 ≤ D t ω :=
+    continuous_nonneg_at_boundary (fun s => D s ω) (hDc ω) (hD0 ω) (τ ω) (hDpre ω) t ht
+  refine ⟨τ, hτ, hb, hDle, hDpre, ?_⟩
+  intro ω ht
+  apply le_antisymm _ (hDle ω (τ ω) le_rfl)
+  have hh := le_max_right 0 (D (τ ω) ω)
+  exact hh.trans (hat ω ht).le
+
+ 
+noncomputable def logStopMargin (κ η₀ u lo hi ε N : ℝ) (q : ℂ × ℝ) (b : ℝ) : ℝ :=
+  min (u - logRadiusClock η₀ q)
+    (min (logAngleSq κ q b - ε)
+      (min (q.1.im - lo) (min (hi - q.1.im) (N - |b|))))
+
+lemma continuous_logStopMargin (κ η₀ u lo hi ε N : ℝ) :
+    Continuous (fun p : (ℂ × ℝ) × ℝ => logStopMargin κ η₀ u lo hi ε N p.1 p.2) := by
+  unfold logStopMargin
+  exact (continuous_const.sub ((continuous_logRadiusClock η₀).comp continuous_fst)).min
+    (((continuous_logAngleSq κ).sub continuous_const).min
+      (((Complex.continuous_im.comp continuous_fst.fst).sub continuous_const).min
+        ((continuous_const.sub (Complex.continuous_im.comp continuous_fst.fst)).min
+          (continuous_const.sub continuous_snd.abs))))
+
+lemma logStopMargin_nonneg_iff (κ η₀ u lo hi ε N : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    0 ≤ logStopMargin κ η₀ u lo hi ε N q b ↔
+      logRadiusClock η₀ q ≤ u ∧ ε ≤ logAngleSq κ q b ∧
+        lo ≤ q.1.im ∧ q.1.im ≤ hi ∧ |b| ≤ N := by
+  simp only [logStopMargin, le_min_iff, sub_nonneg]
+
+lemma logStopMargin_pos_iff (κ η₀ u lo hi ε N : ℝ) (q : ℂ × ℝ) (b : ℝ) :
+    0 < logStopMargin κ η₀ u lo hi ε N q b ↔
+      logRadiusClock η₀ q < u ∧ ε < logAngleSq κ q b ∧
+        lo < q.1.im ∧ q.1.im < hi ∧ |b| < N := by
+  simp only [logStopMargin, lt_min_iff, sub_pos]
+
+lemma ClippedLogSolution.norm_le
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (ω : Ω) (t T : ℝ≥0) (ht : t ≤ T) :
+    ‖c.Q t ω‖ ≤ ‖q₀‖ + c.L * T := by
+  have hh := (c.lipschitz ω).norm_sub_le (t : ℝ) 0
+  rw [c.initial ω, sub_zero, Real.norm_eq_abs, abs_of_nonneg t.coe_nonneg] at hh
+  have hn := norm_add_le (c.Q t ω - q₀) q₀
+  rw [sub_add_cancel] at hn
+  have hmul : (c.L : ℝ) * t ≤ c.L * T :=
+    mul_le_mul_of_nonneg_left (by exact_mod_cast ht) c.L.coe_nonneg
+  linarith
+
+ 
+
+theorem ClippedLogSolution.finite_radius_cutoff
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hq0 : q₀.2 = 0)
+    (u ε N : ℝ) (hu : 0 ≤ u) (hε : 0 < ε) (hN : 0 ≤ N)
+    (hε0 : ε ≤ logAngleSq κ q₀ 0)
+    (hlo : lo < q₀.1.im - u / (2 * ε)) (hhi : q₀.1.im < hi)
+    (T : ℝ≥0) (hT : u < (4 * ε ^ 2 / Real.exp (2 * q₀.1.im)) * T) :
+    ∃ τ : Ω → ℝ≥0,
+      IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (τ ω : WithTop ℝ≥0)) ∧
+      (∀ ω, τ ω < T) ∧
+      (∀ ω t, t ≤ τ ω →
+        logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
+        ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
+        lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi ∧ |B t ω| ≤ N) ∧
+      (∀ ω t, t < τ ω →
+        logRadiusClock q₀.1.im (c.Q t ω) < u ∧
+        ε < logAngleSq κ (c.Q t ω) (B t ω) ∧
+        lo < (c.Q t ω).1.im ∧ (c.Q t ω).1.im < hi ∧ |B t ω| < N) ∧
+      (∀ ω, logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
+        logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε ∨ |B (τ ω) ω| = N) ∧
+      Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
+      (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) = greenLogDensity κ q₀ 0 := by
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let D (t : ℝ≥0) (ω : Ω) := logStopMargin κ q₀.1.im u lo hi ε N (c.Q t ω) (B t ω)
+  have hBa : Adapted f B :=
+    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
+  have hDa : Adapted f D := by
+    intro t
+    change Measurable[f t] ((fun p : (ℂ × ℝ) × ℝ => logStopMargin κ q₀.1.im u lo hi ε N p.1 p.2) ∘
+      (fun ω => (c.Q t ω, B t ω)))
+    exact (continuous_logStopMargin κ q₀.1.im u lo hi ε N).measurable.comp
+      ((c.adapted t).prodMk (hBa t))
+  have hDc (ω : Ω) : Continuous (fun t => D t ω) := by
+    change Continuous ((fun p : (ℂ × ℝ) × ℝ => logStopMargin κ q₀.1.im u lo hi ε N p.1 p.2) ∘
+      (fun t : ℝ≥0 => (c.Q t ω, B t ω)))
+    exact (continuous_logStopMargin κ q₀.1.im u lo hi ε N).comp
+      (((c.lipschitz ω).continuous.comp NNReal.continuous_coe).prodMk (hBc ω))
+  have hD0 (ω : Ω) : 0 ≤ D 0 ω := by
+    change 0 ≤ logStopMargin κ q₀.1.im u lo hi ε N (c.Q (0 : ℝ≥0) ω) (B 0 ω)
+    rw [NNReal.coe_zero, c.initial ω, hB0 ω, logStopMargin_nonneg_iff]
+    refine ⟨?_, hε0, ?_, hhi.le, ?_⟩
+    · simpa only [logRadiusClock, sub_self, zero_add, hq0] using hu
+    · have hd : 0 ≤ u / (2 * ε) := div_nonneg hu (by positivity)
+      linarith
+    · simpa only [abs_zero] using hN
+  obtain ⟨τ, hτ, hb, hleD, hltD, hzero⟩ := first_margin_stopping f D hDa hDc hD0 T
+  have hin (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) :
+      logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
+      ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
+      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi ∧ |B t ω| ≤ N :=
+    (logStopMargin_nonneg_iff κ q₀.1.im u lo hi ε N _ _).mp (hleD ω t ht)
+  have hg (ω : Ω) :
+      (c.Q (τ ω) ω).1.im ≤ q₀.1.im ∧
+      q₀.1.im - logRadiusClock q₀.1.im (c.Q (τ ω) ω) / (2 * ε) ≤ (c.Q (τ ω) ω).1.im ∧
+      (4 * ε ^ 2 / Real.exp (2 * q₀.1.im)) * τ ω ≤ logRadiusClock q₀.1.im (c.Q (τ ω) ω) := by
+    have hd (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) :
+        HasDerivWithinAt (fun t => c.Q t ω)
+          (physicalLogLoewnerField (Real.sqrt κ * B s.toNNReal ω) (c.Q s ω))
+          (Icc 0 (τ ω : ℝ)) s := by
+      have hst : s.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr hs.2
+      have hbounds := (hin ω s.toNNReal hst).2.2
+      rw [Real.coe_toNNReal _ hs.1] at hbounds
+      have hh := (c.derivative ω s hs.1).mono (show Icc (0 : ℝ) (τ ω : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+      rwa [clippedLogLoewnerField_eq_physical lo hi hle _ _ ⟨hbounds.1, hbounds.2.1⟩] at hh
+    have ha (s : ℝ) (hs : s ∈ Icc 0 (τ ω : ℝ)) :
+        ε ≤ logAngleSq κ (c.Q s ω) (B s.toNNReal ω) := by
+      have hh := (hin ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.1
+      simpa only [Real.coe_toNNReal _ hs.1] using hh
+    have hq : (c.Q 0 ω).2 = 0 := by rw [c.initial ω, hq0]
+    have hh := log_loewner_height_clock_bounds (τ ω).coe_nonneg hε hq hd ha
+    simpa only [c.initial ω] using hh
+  have hτlt (ω : Ω) : τ ω < T := by
+    by_contra hn
+    have he : τ ω = T := le_antisymm (hb ω) (not_lt.mp hn)
+    have hh := (hg ω).2.2.trans (hin ω (τ ω) le_rfl).1
+    rw [he] at hh
+    exact (not_le_of_gt hT) hh
+  have he (ω : Ω) :
+      logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
+      logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε ∨ |B (τ ω) ω| = N := by
+    obtain ⟨hr, ha, hl, hh, hbn⟩ := hin ω (τ ω) le_rfl
+    have hstrictlo : lo < (c.Q (τ ω) ω).1.im := by
+      have hdiv := div_le_div_of_nonneg_right hr (show 0 ≤ 2 * ε by positivity)
+      linarith [(hg ω).2.1]
+    have hstricthi : (c.Q (τ ω) ω).1.im < hi := (hg ω).1.trans_lt hhi
+    by_contra hn
+    push Not at hn
+    have hp : 0 < D (τ ω) ω :=
+      (logStopMargin_pos_iff κ q₀.1.im u lo hi ε N _ _).mpr
+        ⟨lt_of_le_of_ne hr hn.1, lt_of_le_of_ne ha (Ne.symm hn.2.1),
+          hstrictlo, hstricthi, lt_of_le_of_ne hbn hn.2.2⟩
+    rw [hzero ω (hτlt ω)] at hp
+    exact lt_irrefl 0 hp
+  let R := max (‖q₀‖ + c.L * T) N
+  have hR : 0 ≤ R := hN.trans (le_max_right _ _)
+  have hinside (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) :
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi := by
+    refine ⟨?_, (hin ω t ht).2.2.1, (hin ω t ht).2.2.2.1⟩
+    rw [Prod.norm_def, max_le_iff, Real.norm_eq_abs]
+    exact ⟨(c.norm_le ω t T (ht.trans (hb ω))).trans (le_max_left _ _),
+      (hin ω t ht).2.2.2.2.trans (le_max_right _ _)⟩
+  obtain ⟨hint, hmean⟩ := c.stopped_green hB hBm hBc hκ hκ8 R hR τ hτ T hb hinside
+  refine ⟨τ, hτ, hτlt, hin, ?_, he, hint, hmean⟩
+  intro ω t ht
+  exact (logStopMargin_pos_iff κ q₀.1.im u lo hi ε N _ _).mp (hltD ω t ht)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma continuous_path_bounded_by_nat (B : ℝ≥0 → ℝ) (hB : Continuous B) (T : ℝ≥0) :
+    ∃ n : ℕ, ∀ t : ℝ≥0, t ≤ T → |B t| < (n : ℝ) + 1 := by
+  obtain ⟨C, hC⟩ := (isCompact_Icc : IsCompact (Icc (0 : ℝ≥0) T)).exists_bound_of_continuousOn hB.continuousOn
+  obtain ⟨n, hn⟩ := exists_nat_gt C
+  refine ⟨n, fun t ht => ?_⟩
+  have hh : |B t| ≤ C := by simpa only [Real.norm_eq_abs] using hC t ⟨zero_le, ht⟩
+  linarith
+
+lemma first_radius_angle_le (S A : ℝ≥0 → ℝ) (u ε : ℝ) (s t : ℝ≥0)
+    (hpre : ∀ v, v < s → S v < u ∧ ε < A v)
+    (hat : S t = u ∨ A t = ε) : s ≤ t := by
+  by_contra hn
+  obtain ⟨hr, ha⟩ := hpre t (not_le.mp hn)
+  rcases hat with h | h
+  · exact (ne_of_lt hr) h
+  · exact (ne_of_gt ha) h
+
+lemma bounded_iSup_stopping
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
+    (τ : ℕ → Ω → ℝ≥0)
+    (hτ : ∀ n, IsStoppingTime f (fun ω => (τ n ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ n ω, τ n ω ≤ T) :
+    IsStoppingTime f (fun ω => ((⨆ n, τ n ω : ℝ≥0) : WithTop ℝ≥0)) := by
+  intro t
+  have he : {ω | ((⨆ n, τ n ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
+      ⋂ n : ℕ, {ω | (τ n ω : WithTop ℝ≥0) ≤ t} := by
+    ext ω
+    simp only [mem_ofPred_eq, WithTop.coe_le_coe, mem_iInter]
+    exact ciSup_le_iff ⟨T, by rintro v ⟨n, rfl⟩; exact hb n ω⟩
+  rw [he]
+  exact MeasurableSet.iInter (fun n => hτ n t)
+
+lemma greenLogDensity_nonneg (κ : ℝ) (q : ℂ × ℝ) (b : ℝ) : 0 ≤ greenLogDensity κ q b :=
+  (Real.exp_pos _).le
+
+lemma greenLogDensity_le_radius
+    (κ η₀ u : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (q : ℂ × ℝ) (b : ℝ)
+    (hclock : logRadiusClock η₀ q ≤ u) :
+    greenLogDensity κ q b ≤ Real.exp (codimension κ * (u - Real.log 2 - η₀)) := by
+  simpa only [Real.one_rpow, mul_one] using
+    greenLogDensity_le_at_angular_exit κ η₀ u 1 hκ hκ8 q b hclock (logAngleSq_le_one κ q b)
+
+ 
+
+theorem ClippedLogSolution.finite_angular_law
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hq0 : q₀.2 = 0)
+    (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
+    (hε0 : ε ≤ logAngleSq κ q₀ 0)
+    (hlo : lo < q₀.1.im - u / (2 * ε)) (hhi : q₀.1.im < hi)
+    (T : ℝ≥0) (hT : u < (4 * ε ^ 2 / Real.exp (2 * q₀.1.im)) * T) :
+    ∃ τ : Ω → ℝ≥0,
+      IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (τ ω : WithTop ℝ≥0)) ∧
+      (∀ ω, τ ω < T) ∧
+      (∀ ω t, t ≤ τ ω →
+        logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
+        ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
+        lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) ∧
+      (∀ ω t, t < τ ω →
+        logRadiusClock q₀.1.im (c.Q t ω) < u ∧ ε < logAngleSq κ (c.Q t ω) (B t ω)) ∧
+      (∀ ω, logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
+        logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε) ∧
+      Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
+      (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) = greenLogDensity κ q₀ 0 := by
+  classical
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hall (n : ℕ) := c.finite_radius_cutoff hB hBm hBc hB0 hκ hκ8 hq0
+    u ε ((n : ℝ) + 1) hu hε (by positivity) hε0 hlo hhi T hT
+  choose τn hstop htime hin hpre hboundary hint hmean using hall
+  let τ (ω : Ω) : ℝ≥0 := ⨆ n, τn n ω
+  have hb (n : ℕ) (ω : Ω) : τn n ω ≤ T := (htime n ω).le
+  have hbd (ω : Ω) : BddAbove (range (fun n => τn n ω)) :=
+    ⟨T, by rintro v ⟨n, rfl⟩; exact hb n ω⟩
+  have hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)) := bounded_iSup_stopping _ τn hstop T hb
+  have hstable (ω : Ω) : ∃ n : ℕ, τ ω = τn n ω ∧
+      (∀ m, n ≤ m → τn m ω = τ ω) ∧
+      (logRadiusClock q₀.1.im (c.Q (τ ω) ω) = u ∨
+        logAngleSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = ε) := by
+    obtain ⟨n, hnb⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
+    have hhit (m : ℕ) (hnm : n ≤ m) :
+        logRadiusClock q₀.1.im (c.Q (τn m ω) ω) = u ∨
+          logAngleSq κ (c.Q (τn m ω) ω) (B (τn m ω) ω) = ε := by
+      rcases hboundary m ω with h | h | h
+      · exact Or.inl h
+      · exact Or.inr h
+      · have hh := hnb (τn m ω) (hb m ω)
+        have hnm' : (n : ℝ) ≤ m := by exact_mod_cast hnm
+        linarith
+    have hle (m : ℕ) : τn m ω ≤ τn n ω :=
+      first_radius_angle_le _ _ u ε _ _ (fun t ht => ⟨(hpre m ω t ht).1, (hpre m ω t ht).2.1⟩)
+        (hhit n le_rfl)
+    have he : τ ω = τn n ω := le_antisymm (ciSup_le hle) (le_ciSup (hbd ω) n)
+    refine ⟨n, he, ?_, ?_⟩
+    · intro m hnm
+      rw [he]
+      apply le_antisymm (hle m)
+      exact first_radius_angle_le _ _ u ε _ _
+        (fun t ht => ⟨(hpre n ω t ht).1, (hpre n ω t ht).2.1⟩) (hhit m hnm)
+    · rw [he]
+      exact hhit n le_rfl
+  have hτtime (ω : Ω) : τ ω < T := by
+    obtain ⟨n, he, _⟩ := hstable ω
+    rw [he]
+    exact htime n ω
+  have hτin (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) :
+      logRadiusClock q₀.1.im (c.Q t ω) ≤ u ∧
+      ε ≤ logAngleSq κ (c.Q t ω) (B t ω) ∧
+      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi := by
+    obtain ⟨n, he, _⟩ := hstable ω
+    obtain ⟨h1, h2, h3, h4, _⟩ := hin n ω t (by simpa only [he] using ht)
+    exact ⟨h1, h2, h3, h4⟩
+  have hτpre (ω : Ω) (t : ℝ≥0) (ht : t < τ ω) :
+      logRadiusClock q₀.1.im (c.Q t ω) < u ∧ ε < logAngleSq κ (c.Q t ω) (B t ω) := by
+    obtain ⟨n, he, _⟩ := hstable ω
+    have hh := hpre n ω t (by simpa only [he] using ht)
+    exact ⟨hh.1, hh.2.1⟩
+  let K := Real.exp (codimension κ * (u - Real.log 2 - q₀.1.im))
+  have hbound (n : ℕ) (ω : Ω) :
+      ‖greenLogDensity κ (c.Q (τn n ω) ω) (B (τn n ω) ω)‖ ≤ K := by
+    rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
+    exact greenLogDensity_le_radius κ q₀.1.im u hκ hκ8 _ _ (hin n ω (τn n ω) le_rfl).1
+  have hlim (ω : Ω) : Tendsto
+      (fun n => greenLogDensity κ (c.Q (τn n ω) ω) (B (τn n ω) ω)) atTop
+      (𝓝 (greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω))) := by
+    obtain ⟨n, _, he, _⟩ := hstable ω
+    apply tendsto_const_nhds.congr'
+    exact (eventually_ge_atTop n).mono (fun m hm => by dsimp only; rw [he m hm])
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hQm : Measurable (fun ω => c.Q (τ ω) ω) :=
+    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
+  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
+  have hfm : Measurable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) :=
+    (greenLogDensity_contDiff κ).continuous.measurable.comp (hQm.prodMk hBmτ)
+  have hfi : Integrable (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω)) P := by
+    apply Integrable.of_bound hfm.aestronglyMeasurable K
+    exact Eventually.of_forall fun ω => by
+      rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
+      exact greenLogDensity_le_radius κ q₀.1.im u hκ hκ8 _ _ (hτin ω (τ ω) le_rfl).1
+  have hconv := tendsto_integral_of_dominated_convergence (fun _ : Ω => K)
+    (fun n => (hint n).aestronglyMeasurable) (integrable_const K)
+    (fun n => Eventually.of_forall (hbound n)) (Eventually.of_forall hlim)
+  have hmean' : (∫ ω, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
+      greenLogDensity κ q₀ 0 := by
+    apply tendsto_nhds_unique hconv
+    simp only [hmean]
+    exact tendsto_const_nhds
+  exact ⟨τ, hτ, hτtime, hτin, hτpre, fun ω => (hstable ω).choose_spec.2.2, hfi, hmean'⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def initialLogCoordinates (z : ℂ) : ℂ × ℝ :=
+  ((z.re : ℂ) + (Real.log z.im : ℂ) * Complex.I, 0)
+
+noncomputable def restoreLogPoint (q : ℂ × ℝ) : ℂ :=
+  (q.1.re : ℂ) + (Real.exp q.1.im : ℂ) * Complex.I
+
+lemma restoreLogPoint_im (q : ℂ × ℝ) : (restoreLogPoint q).im = Real.exp q.1.im := by
+  simp only [restoreLogPoint, Complex.add_im, Complex.ofReal_im, Complex.mul_im,
+    Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero, zero_add]
+
+lemma restoreLogPoint_initial (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    restoreLogPoint (initialLogCoordinates z) = z := by
+  apply Complex.ext <;> simp [restoreLogPoint, initialLogCoordinates, Real.exp_log hz]
+
+lemma restoreLogPoint_derivative {Q : ℝ → ℂ × ℝ} {D : Set ℝ} {t u : ℝ}
+    (hQ : HasDerivWithinAt Q (physicalLogLoewnerField u (Q t)) D t) :
+    HasDerivWithinAt (fun s => restoreLogPoint (Q s))
+      (2 / (restoreLogPoint (Q t) - (u : ℂ))) D t := by
+  have hr := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hQ.fst
+  have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hQ.fst
+  have hd := (Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t hr).add
+    ((Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t hi.exp).mul_const Complex.I)
+  convert hd using 1 <;> try rfl
+  · let Z := restoreLogPoint (Q t) - (u : ℂ)
+    have he : (((Q t).1.re - u : ℝ) : ℂ) + (Real.exp (Q t).1.im : ℂ) * Complex.I = Z := by
+      simp only [Z, restoreLogPoint, Complex.ofReal_sub]; ring
+    have hZi : Z.im = Real.exp (Q t).1.im := by
+      dsimp only [Z]
+      rw [Complex.sub_im, Complex.ofReal_im, sub_zero, restoreLogPoint_im]
+    simp only [Function.comp_def, Complex.reCLM_apply, Complex.ofRealCLM_apply,
+      Complex.imCLM_apply, physicalLogLoewnerField, he, Complex.add_re, Complex.add_im,
+      Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re, Complex.mul_im,
+      Complex.I_re, Complex.I_im, mul_zero, sub_self, add_zero, mul_one, zero_add]
+    change 2 / Z = ((2 * Z⁻¹.re : ℝ) : ℂ) +
+      ((Real.exp (Q t).1.im * (-2 * Complex.normSq Z⁻¹) : ℝ) : ℂ) * Complex.I
+    apply Complex.ext
+    · simp only [div_eq_mul_inv, Complex.mul_re, Complex.re_ofNat, Complex.im_ofNat,
+        zero_mul, sub_zero, Complex.add_re, Complex.ofReal_re, Complex.ofReal_im,
+        Complex.I_re, Complex.I_im, mul_zero, sub_self, add_zero]
+    · simp only [div_eq_mul_inv, Complex.mul_im, Complex.re_ofNat, Complex.im_ofNat,
+        zero_mul, add_zero, Complex.add_im, Complex.ofReal_re, Complex.ofReal_im,
+        Complex.I_re, Complex.I_im, mul_zero, mul_one, zero_add, Complex.inv_im,
+        Complex.normSq_inv, hZi]
+      ring
+
+ 
+
+theorem clipped_log_identification
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (T : ℝ≥0)
+    (lo hi : ℝ) (hle : lo ≤ hi) (Q : ℝ → ℂ × ℝ)
+    (hQ0 : Q 0 = initialLogCoordinates z)
+    (hQd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q
+      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t)
+    (hb : ∀ t ∈ Icc 0 (T : ℝ), (Q t).1.im ∈ Icc lo hi) :
+    z ∈ survivingDomain γ T ∧ EqOn Q (logLoewnerCoordinates G z) (Icc 0 (T : ℝ)) := by
+  let W (t : ℝ) := restoreLogPoint (Q t)
+  have hW0 : W 0 = z := by
+    change restoreLogPoint (Q 0) = z
+    rw [hQ0, restoreLogPoint_initial z hz]
+  have hp (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : 0 < (W t).im := by
+    change 0 < (restoreLogPoint (Q t)).im
+    rw [restoreLogPoint_im]; exact Real.exp_pos _
+  have hd (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt W
+      (2 / (W t - (U t.toNNReal : ℂ))) (Icc 0 (T : ℝ)) t := by
+    apply restoreLogPoint_derivative
+    have hh := hQd t ht
+    rwa [clippedLogLoewnerField_eq_physical lo hi hle _ _ (hb t ht)] at hh
+  obtain ⟨hlive, he⟩ := loewner_survival_of_positive_solution hU h T W hp hd
+  rw [hW0] at hlive he
+  have hη : Real.log (G T z).im = (Q T).1.im := by
+    rw [he]
+    change Real.log (restoreLogPoint (Q T)).im = _
+    rw [restoreLogPoint_im, Real.log_exp]
+  have hh₀ := hb 0 ⟨le_rfl, T.coe_nonneg⟩
+  rw [hQ0] at hh₀
+  have hhi : Real.log z.im ≤ hi := by simpa [initialLogCoordinates] using hh₀.2
+  exact ⟨hlive, clipped_log_loewner_agrees hU h T z hlive lo hi hle
+    (by rw [hη]; exact (hb T ⟨T.coe_nonneg, le_rfl⟩).1) hhi Q hQ0 hQd⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem exists_joint_clipped_flow
+    {Ω : Type*} [MeasurableSpace Ω]
+    (U : ℝ≥0 → Ω → ℝ)
+    (hUm : Measurable (fun p : ℝ≥0 × Ω => U p.1 p.2))
+    (hUc : ∀ ω, Continuous (fun t => U t ω))
+    (q₀ : Ω → ℂ × ℝ) (hq₀ : Measurable q₀)
+    (lo hi : ℝ) (hle : lo ≤ hi) :
+    ∃ Q : ℝ → Ω → ℂ × ℝ,
+      Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧
+      (∀ ω, Q 0 ω = q₀ ω) ∧
+      (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω)
+        (clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (Q t ω)) (Ici 0) t) := by
+  obtain ⟨L, K, hbound, hlip⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) (x : ℂ × ℝ) :=
+    clippedLogLoewnerField lo hi hle (U t.toNNReal ω) (q₀ ω + x)
+  have hUr : Measurable (fun p : ℝ × Ω => U p.1.toNNReal p.2) :=
+    hUm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hV : Measurable (fun p : (ℝ × Ω) × (ℂ × ℝ) => V p.1.1 p.1.2 p.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun p : (ℝ × Ω) × (ℂ × ℝ) => (U p.1.1.toNNReal p.1.2, q₀ p.1.2 + p.2)))
+    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp
+      ((hUr.comp measurable_fst).prodMk
+        ((hq₀.comp (measurable_snd.comp measurable_fst)).add measurable_snd))
+  have hVc (ω : Ω) (x : ℂ × ℝ) : Continuous (fun t : ℝ => V t ω x) := by
+    change Continuous ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun t : ℝ => (U t.toNNReal ω, q₀ ω + x)))
+    exact (continuous_clippedLogLoewnerField lo hi hle).comp
+      (((hUc ω).comp continuous_real_toNNReal).prodMk continuous_const)
+  have hVl (ω : Ω) (t : ℝ) : LipschitzWith K (V t ω) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro x y
+    simpa only [V, dist_add_left] using (hlip (U t.toNNReal ω)).dist_le_mul (q₀ ω + x) (q₀ ω + y)
+  obtain ⟨X, hXm, _, hX0, hXd, _⟩ := exists_measurable_global_bounded_ode_picard
+    V 0 L K hV hVc hVl (fun ω t x => hbound _ _)
+  refine ⟨fun t ω => q₀ ω + X t ω, (hq₀.comp measurable_snd).add hXm, ?_, ?_⟩
+  · intro ω; change q₀ ω + X 0 ω = q₀ ω; rw [hX0, add_zero]
+  · intro ω t ht
+    simpa only [V] using (hXd ω t ht).const_add (q₀ ω)
+
+lemma measurable_initialLogCoordinates : Measurable initialLogCoordinates := by
+  exact ((Complex.measurable_ofReal.comp Complex.measurable_re).add
+    ((Complex.measurable_ofReal.comp (Real.measurable_log.comp Complex.measurable_im)).mul_const
+      Complex.I)).prodMk measurable_const
+
+ 
+theorem exists_spatial_clipped_flow
+    {Ω : Type*} [MeasurableSpace Ω]
+    (U : ℝ≥0 → Ω → ℝ)
+    (hUm : Measurable (fun p : ℝ≥0 × Ω => U p.1 p.2))
+    (hUc : ∀ ω, Continuous (fun t => U t ω)) (n : ℕ) :
+    ∃ Q : ℝ → ℂ × Ω → ℂ × ℝ,
+      Measurable (fun p : ℝ × (ℂ × Ω) => Q p.1 p.2) ∧
+      (∀ z ω, Q 0 (z, ω) = initialLogCoordinates z) ∧
+      (∀ z ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s (z, ω))
+        (clippedLogLoewnerField (-(n : ℝ)) n (by linarith [Nat.cast_nonneg (α := ℝ) n])
+          (U t.toNNReal ω) (Q t (z, ω))) (Ici 0) t) := by
+  obtain ⟨Q, hQm, hQ0, hQd⟩ := exists_joint_clipped_flow (fun t (p : ℂ × Ω) => U t p.2)
+    (hUm.comp (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))
+    (fun p => hUc p.2) (fun p => initialLogCoordinates p.1)
+    (measurable_initialLogCoordinates.comp measurable_fst) (-(n : ℝ)) n (by linarith [Nat.cast_nonneg (α := ℝ) n])
+  exact ⟨Q, hQm, fun z ω => hQ0 (z, ω), fun z ω t ht => hQd (z, ω) t ht⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma clipped_log_height_derivative
+    {U : ℝ≥0 → ℝ} {lo hi : ℝ} {hle : lo ≤ hi} {Q : ℝ → ℂ × ℝ}
+    {t : ℝ} {S : Set ℝ}
+    (hd : HasDerivWithinAt Q
+      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) S t) :
+    HasDerivWithinAt (fun s => (Q s).1.im)
+      (-2 * Complex.normSq (clippedCenter lo hi hle (U t.toNNReal) (Q t))⁻¹) S t := by
+  simpa only [Function.comp_def, Complex.imCLM_apply, clippedLogLoewnerField,
+    Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re,
+    Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero, zero_add]
+    using Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hd.fst
+
+lemma clipped_log_height_antitone
+    {U : ℝ≥0 → ℝ} {lo hi : ℝ} {hle : lo ≤ hi} {Q : ℝ → ℂ × ℝ}
+    (T : ℝ≥0)
+    (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q
+      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t) :
+    AntitoneOn (fun t => (Q t).1.im) (Icc 0 (T : ℝ)) := by
+  have he (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := clipped_log_height_derivative (hd t ht)
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
+    (fun t ht => (he t ht).continuousWithinAt)
+    (fun t ht => (he t (interior_subset ht)).mono interior_subset)
+  intro t ht
+  exact mul_nonpos_of_nonpos_of_nonneg (by norm_num) (Complex.normSq_nonneg _)
+
+ 
+
+lemma clipped_log_identification_of_endpoint
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (T : ℝ≥0) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (lo hi : ℝ) (hle : lo ≤ hi) (Q : ℝ → ℂ × ℝ)
+    (hQ0 : Q 0 = initialLogCoordinates z)
+    (hQd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q
+      (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t)
+    (hlo : lo ≤ (Q T).1.im) (hhi : Real.log z.im ≤ hi) :
+    z ∈ survivingDomain γ T ∧ EqOn Q (logLoewnerCoordinates G z) (Icc 0 (T : ℝ)) := by
+  apply clipped_log_identification hU h z hz T lo hi hle Q hQ0 hQd
+  have hm := clipped_log_height_antitone T hQd
+  intro t ht
+  refine ⟨hlo.trans (hm ht ⟨T.coe_nonneg, le_rfl⟩ ht.2), ?_⟩
+  have hh := hm (show (0 : ℝ) ∈ Icc 0 (T : ℝ) from ⟨le_rfl, T.coe_nonneg⟩) ht ht.1
+  change (Q t).1.im ≤ (Q 0).1.im at hh
+  rw [hQ0] at hh
+  have hinit : (initialLogCoordinates z).1.im = Real.log z.im := by
+    simp [initialLogCoordinates]
+  rw [hinit] at hh
+  exact hh.trans hhi
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def powerLogDensity (κ p a : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  Real.exp (-p * Real.log 2 + p * q.2 +
+    (a - p) * q.1.im -
+      (a / 2) * Real.log (greenLogNormSq κ q x))
+lemma powerLogDensity_contDiff (κ p a : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => powerLogDensity κ p a zz.1 zz.2) := by
+  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.re) :=
+    Complex.reCLM.contDiff.comp contDiff_fst.fst
+  have him : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.im) :=
+    Complex.imCLM.contDiff.comp contDiff_fst.fst
+  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => greenLogNormSq κ zz.1 zz.2) := by
+    exact ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add
+      ((contDiff_const.mul him).exp)
+  have hlog := hN.log (fun zz => (greenLogNormSq_pos κ zz.1 zz.2).ne')
+  unfold powerLogDensity
+  apply ContDiff.exp
+  exact (((contDiff_const.add (contDiff_const.mul contDiff_fst.snd)).add
+    (contDiff_const.mul him)).sub (contDiff_const.mul hlog))
+
+lemma powerLogDensity_spatial_derivative (κ p a : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    HasDerivAt (powerLogDensity κ p a q)
+      (powerLogDensity κ p a q x * (a * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
+        greenLogNormSq κ q x)) x := by
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
+    simp
+  have hlog := hN.log (greenLogNormSq_pos κ q x).ne'
+  have hd := ((hasDerivAt_const x (-p * Real.log 2 + p * q.2 +
+    (a - p) * q.1.im)).sub
+      (hlog.const_mul (a / 2))).exp
+  convert hd using 1 <;> try rfl
+  dsimp only [powerLogDensity, Pi.sub_apply, Pi.mul_apply, Function.comp_def]
+  ring
+
+lemma powerLogDensity_spatial_second (κ p a : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
+    iteratedDeriv 2 (powerLogDensity κ p a q) x =
+      powerLogDensity κ p a q x * κ * a *
+        ((a + 1) * (q.1.re - Real.sqrt κ * x) ^ 2 - Real.exp (2 * q.1.im)) /
+        (greenLogNormSq κ q x) ^ 2 := by
+  have hd : deriv (powerLogDensity κ p a q) = fun x =>
+      powerLogDensity κ p a q x * (a * Real.sqrt κ * (q.1.re - Real.sqrt κ * x) /
+        greenLogNormSq κ q x) := funext fun x => (powerLogDensity_spatial_derivative κ p a q x).deriv
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2 * (q.1.re - Real.sqrt κ * x) * (-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2 * q.1.im)) using 1 <;> try rfl
+    simp
+  have hcoef := (hX.const_mul (a * Real.sqrt κ)).div hN
+    (greenLogNormSq_pos κ q x).ne'
+  have hsecond := (powerLogDensity_spatial_derivative κ p a q x).mul hcoef
+  rw [show (2 : ℕ) = 1 + 1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
+  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, id_eq] at hsecond
+  change HasDerivAt (fun y => powerLogDensity κ p a q y *
+    (a * Real.sqrt κ * (q.1.re - Real.sqrt κ * y) / greenLogNormSq κ q y)) _ x at hsecond
+  rw [hsecond.deriv]
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  dsimp only [greenLogNormSq]
+  generalize powerLogDensity κ p a q x = f
+  generalize q.1.re - Real.sqrt κ * x = X
+  generalize Real.exp (2 * q.1.im) = Y
+  have hs := Real.sq_sqrt hκ
+  generalize Real.sqrt κ = r at *
+  rw [← hs]
+  field_simp
+  ring
+
+lemma powerLogDensity_directional (κ p a : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q) v =
+      powerLogDensity κ p a q x * (p * v.2 +
+        (a - p) * v.1.im - (a / 2) *
+          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
+            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x)) := by
+  have hzre : HasDerivAt (fun t : ℝ => (q + t • v).1.re) v.1.re 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
+  have hzim : HasDerivAt (fun t : ℝ => (q + t • v).1.im) v.1.im 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
+  have hj : HasDerivAt (fun t : ℝ => (q + t • v).2) v.2 0 := by
+    simpa only [Prod.snd_add, Prod.smul_snd, smul_eq_mul, id_eq, one_mul]
+      using ((hasDerivAt_id (0 : ℝ)).mul_const v.2).const_add q.2
+  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q + t • v) x)
+      (2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
+        2 * Real.exp (2 * q.1.im) * v.1.im) 0 := by
+    convert ((hzre.sub_const (Real.sqrt κ * x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
+    simp only [zero_smul, add_zero, Nat.cast_ofNat]
+    ring
+  have hlog := hN.log (by simpa only [zero_smul, add_zero] using (greenLogNormSq_pos κ q x).ne')
+  have hcalc := (((hj.const_mul (p)).const_add (-p * Real.log 2)).add
+    (hzim.const_mul (a - p))).sub
+      (hlog.const_mul (a / 2))
+  have hline : HasDerivAt (fun t : ℝ => powerLogDensity κ p a (q + t • v) x)
+      (powerLogDensity κ p a q x * (p * v.2 +
+        (a - p) * v.1.im - (a / 2) *
+          ((2 * (q.1.re - Real.sqrt κ * x) * v.1.re +
+            2 * Real.exp (2 * q.1.im) * v.1.im) / greenLogNormSq κ q x))) 0 := by
+    convert hcalc.exp using 1 <;> try rfl
+    simp only [Pi.sub_apply, Pi.add_apply, zero_smul, add_zero]
+    simp only [powerLogDensity]
+  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => powerLogDensity κ p a r x) :=
+    (powerLogDensity_contDiff κ p a).comp (contDiff_id.prodMk contDiff_const)
+  have haff : HasDerivAt (fun t : ℝ => q + t • v) v 0 := by
+    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
+  have hfd : HasFDerivAt (fun r : ℂ × ℝ => powerLogDensity κ p a r x)
+      (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q) (q + (0 : ℝ) • v) := by
+    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
+  have hchain := hfd.comp_hasDerivAt 0 haff
+  simpa only [Function.comp_def] using hchain.unique hline
+
+lemma powerLogDensity_pair_directional (κ p a : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => powerLogDensity κ p a zz.1 zz.2) (q, x) (v, 0) =
+      fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q v := by
+  have hd := ((powerLogDensity_contDiff κ p a).differentiable (by simp)) (q, x)
+  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
+  have hc := (hd.hasFDerivAt.comp q hp).fderiv
+  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) hc
+  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
+    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
+
+ 
+
+lemma exists_compact_power_field (κ p a R : ℝ) (hR : 0 ≤ R) :
+    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
+      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = powerLogDensity κ p a q x) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R →
+        fderiv ℝ φ (q, x) = fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => powerLogDensity κ p a zz.1 zz.2) (q, x)) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
+        iteratedDeriv 2 (powerLogDensity κ p a q) x) := by
+  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let φ (zz : (ℂ × ℝ) × ℝ) : ℝ := b zz * powerLogDensity κ p a zz.1 zz.2
+  have hnear (zz : (ℂ × ℝ) × ℝ) (hp : ‖zz‖ ≤ R) :
+      φ =ᶠ[𝓝 zz] (fun r : (ℂ × ℝ) × ℝ => powerLogDensity κ p a r.1 r.2) := by
+    have hball : zz ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
+      simp only [Metric.mem_ball, dist_zero_right]
+      change ‖zz‖ < R + 1
+      linarith
+    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with r hr
+    simp only [φ, hr, Pi.one_apply, one_mul]
+  refine ⟨φ, b.contDiff.mul (powerLogDensity_contDiff κ p a), b.hasCompactSupport.mul_right,
+    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
+  · intro q x h
+    exact (hnear (q, x) h).fderiv_eq
+  · intro q x h
+    have he := (hnear (q, x) h).comp_tendsto
+      ((continuous_const.prodMk continuous_id).tendsto x)
+    exact he.iteratedDeriv_eq 2
+
+ 
+lemma power_physical_generator (κ p a : ℝ) (hκ : 0 ≤ κ)
+    (q : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ p a r x) q)
+      (physicalLogLoewnerField (Real.sqrt κ * x) q) +
+      (1 / 2 : ℝ) * iteratedDeriv 2 (powerLogDensity κ p a q) x =
+      powerLogDensity κ p a q x *
+        ((-4 * a + κ * a * (a + 1) / 2) * (q.1.re - Real.sqrt κ * x) ^ 2 +
+          (4 * p - κ * a / 2) * Real.exp (2 * q.1.im)) /
+        (greenLogNormSq κ q x) ^ 2 := by
+  rw [powerLogDensity_directional, powerLogDensity_spatial_second κ p a hκ]
+  obtain ⟨hr, hi, hj⟩ := physicalLogLoewnerField_components κ q x
+  rw [hr, hi, hj]
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  field_simp
+  dsimp only [greenLogNormSq]
+  ring
+
+lemma powerLogDensity_radius_angle (κ p a η₀ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    powerLogDensity κ p a q x =
+      Real.exp (p * (logRadiusClock η₀ q - Real.log 2 - η₀)) *
+        (logAngleSq κ q x) ^ (a / 2) := by
+  rw [Real.rpow_def_of_pos (logAngleSq_pos κ q x), ← Real.exp_add]
+  unfold logAngleSq
+  rw [Real.log_div (Real.exp_ne_zero _) (greenLogNormSq_pos κ q x).ne', Real.log_exp]
+  unfold powerLogDensity logRadiusClock
+  congr 1
+  ring
+
+ 
+
+lemma green_weighted_power_generator (κ b : ℝ) (hκ : 0 < κ)
+    (q : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun r : ℂ × ℝ => powerLogDensity κ (codimension κ)
+        (greenExponent κ - b) r x) q)
+      (physicalLogLoewnerField (Real.sqrt κ * x) q) +
+      (1 / 2 : ℝ) * iteratedDeriv 2
+        (powerLogDensity κ (codimension κ) (greenExponent κ - b) q) x =
+      powerLogDensity κ (codimension κ) (greenExponent κ - b) q x *
+        logClockRate κ q x * (κ * b / 8) *
+          (1 - (greenExponent κ - b) * ((logAngleSq κ q x)⁻¹ - 1)) := by
+  rw [power_physical_generator κ _ _ hκ.le]
+  unfold logClockRate logAngleSq
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  have hy := Real.exp_ne_zero (2 * q.1.im)
+  dsimp only [greenExponent, codimension, dimension]
+  field_simp
+  dsimp only [greenLogNormSq]
+  ring
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/FiniteStoppedLaw.lean b/lean/OAI/Probability/SLE/FiniteStoppedLaw.lean
index 803c523a9ea98fb30e46eec7ce5473e752f11684..837565de13814200c63ccfb573a248b08d1771cc 100644
--- a/lean/OAI/Probability/SLE/FiniteStoppedLaw.lean
+++ b/lean/OAI/Probability/SLE/FiniteStoppedLaw.lean
@@ -1,806 +1,806 @@
-import Mathlib
-import OAI.Probability.SLE.KernelExpectation
-
-namespace OAI
-
-/-! Finite Stopped Law for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem finite_stopped_green_regular
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
-    Integrable (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z u) P ∧
-    (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z u ∂P) = initialGreen κ z := by
-  let A := logAngleSq κ (initialLogCoordinates z) 0
-  have hA : 0 < A := logAngleSq_pos _ _ _
-  let ε (n : ℕ) := A / ((n : ℝ) + 1)
-  have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
-  have hεA (n : ℕ) : ε n ≤ A := div_le_self hA.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
-  have hεlim : Tendsto ε atTop (𝓝 0) := by
-    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A
-    simpa only [mul_one_div, mul_zero] using hh
-  choose f hfi hfm hfb hfw using fun n => exists_finite_angular_kernel
-    hB hBm hBc hB0 κ hκ hκ8 z hz u hu (ε n) (hε n) (hεA n)
-  let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
-  let g (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
-  have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
-  have hlim : ∀ᵐ ω ∂P, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) := by
-    filter_upwards [hw] with ω hω
-    have hU : Continuous (fun t => Real.sqrt κ * B t ω) := (hBc ω).const_mul _
-    by_cases hlevel : ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
-        fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u)
-    · obtain ⟨T, hT, hTR⟩ := hlevel
-      obtain ⟨δ, hδ, hδb⟩ := live_angleSq_lower_bound hU hω z T hT
-      have hev : ∀ᶠ n in atTop, ε n < δ := (tendsto_order.mp hεlim).2 δ hδ
-      apply tendsto_const_nhds.congr'
-      filter_upwards [hev] with n hn
-      obtain ⟨t, ht, htu, hb, he, _⟩ := hfw n ω (γ ω) (G ω) (F ω) hω
-      have hTc : Real.log ((2 * z.im) / fullRadius (G ω) T z) = u := by
-        rw [← logRadiusClock_logCoordinates hU hω z T hT]
-        exact (fullRadius_level_iff_logClock hU hω z T hT u).mp hTR
-      have htT : t ≤ T := live_clock_le_time hU hω z t T ht hT (by simpa only [hTc] using htu)
-      have htb : δ ≤ (Real.sin (Complex.arg (G ω t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 :=
-        hδb t htT
-      have htc : Real.log ((2 * z.im) / fullRadius (G ω) t z) = u := hb.resolve_right (by linarith)
-      have htR : fullRadius (G ω) t z = (2 * z.im) * Real.exp (-u) := by
-        apply (fullRadius_level_iff_logClock hU hω z t ht u).mpr
-        rwa [logRadiusClock_logCoordinates hU hω z t ht]
-      exact (stoppedGreen_eq_of_live_level hU hω κ z u t ht htR).trans he.symm
-    · have hg : g ω = 0 := stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hlevel
-      rw [hg]
-      have hb (n : ℕ) : f n ω ≤ K * (ε n) ^ (greenExponent κ / 2) := by
-        obtain ⟨t, ht, _, htb, _, hbe⟩ := hfw n ω (γ ω) (G ω) (F ω) hω
-        apply hbe
-        apply htb.resolve_left
-        intro he
-        apply hlevel
-        refine ⟨t, ht, (fullRadius_level_iff_logClock hU hω z t ht u).mpr ?_⟩
-        rwa [logRadiusClock_logCoordinates hU hω z t ht]
-      have hupp : Tendsto (fun n => K * (ε n) ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
-        have hh := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
-        simpa only [Function.comp_apply, Real.zero_rpow hp.ne', mul_zero] using hh
-      exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hupp
-        (fun n => (hfb n ω).1) hb
-  have hbound (n : ℕ) : ∀ᵐ ω ∂P, ‖f n ω‖ ≤ K := by
-    apply Eventually.of_forall
-    intro ω
-    simpa only [Real.norm_eq_abs, abs_of_nonneg (hfb n ω).1] using (hfb n ω).2
-  have hgm : AEStronglyMeasurable g P :=
-    aestronglyMeasurable_of_tendsto_ae atTop (fun n => (hfi n).aestronglyMeasurable) hlim
-  have hgi : Integrable g P := ⟨hgm,
-    hasFiniteIntegral_of_dominated_convergence (integrable_const K).hasFiniteIntegral hbound hlim⟩
-  have hc := tendsto_integral_of_dominated_convergence (fun _ : Ω => K)
-    (fun n => (hfi n).aestronglyMeasurable) (integrable_const K) hbound hlim
-  refine ⟨hgi, ?_⟩
-  apply tendsto_nhds_unique hc
-  simpa only [hfm] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => initialGreen κ z) atTop (𝓝 (initialGreen κ z)))
-
- 
-
-theorem finite_stopped_green_of_witness
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
-    Integrable (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z u) P ∧
-    (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z u ∂P) = initialGreen κ z := by
-  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
-  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
-    filter_upwards [hw, hCB] with ω hω hω'
-    simpa only [hω'] using hω
-  have heq : (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * C t ω)
-      (γ ω) (G ω) z u) =ᵐ[P] (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z u) := by
-    filter_upwards [hCB] with ω hω
-    simp only [hω]
-  obtain ⟨hfi, hfm⟩ := finite_stopped_green_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz u hu
-  exact ⟨hfi.congr heq, (integral_congr_ae heq).symm.trans hfm⟩
-
-universe uLevel
-def FiniteStoppedGreenTarget : Prop :=
-  ∀ (Ω : Type uLevel) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (κ : ℝ), 0 < κ → κ < 8 →
-    ∀ (B : ℝ≥0 → Ω → ℝ), ProbabilityTheory.IsBrownianReal B P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ),
-      (∀ t, Measurable (fun ω => γ ω t)) →
-      (∀ᵐ ω ∂P, IsCapacityTwoWitness
-        (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) →
-      ∀ (z : ℂ), z ∈ upperHalfPlane → ∀ (u : ℝ), 0 ≤ u →
-        Integrable (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-          (γ ω) (G ω) z u) P ∧
-        (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-          (γ ω) (G ω) z u ∂P) = initialGreen κ z
-
- 
-
-theorem finite_stopped_green : FiniteStoppedGreenTarget := by
-  intro Ω mΩ P hP κ hκ hκ8 B hB γ G F _hγ hw z hz u hu
-  let := hP
-  exact finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma initialGreen_pos (κ : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    0 < initialGreen κ z := by
-  have hz0 : z ≠ 0 := by intro he; simp only [he, upperHalfPlane, Set.mem_ofPred_eq, Complex.zero_im, lt_self_iff_false] at hz
-  have hsin : 0 < Real.sin (Complex.arg z) := by
-    rw [Complex.sin_arg]
-    exact div_pos hz (norm_pos_iff.mpr hz0)
-  exact mul_pos (Real.rpow_pos_of_pos (mul_pos (by norm_num) hz) _) (Real.rpow_pos_of_pos hsin _)
-
-lemma greenDensity_nonneg
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (κ : ℝ) (t : ℝ≥0) (z : ℂ) :
-    0 ≤ greenDensity κ U γ G t z := by
-  classical
-  by_cases hz : z ∈ survivingDomain γ t
-  · rw [greenDensity, ite_eq_left hz]
-    have hy := h.imag_positive t z hz
-    apply mul_nonneg
-    · exact Real.rpow_nonneg (div_nonneg (mul_nonneg (by norm_num) hy.le) (norm_nonneg _)) _
-    · apply Real.rpow_nonneg
-      rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero]
-      exact div_nonneg hy.le (norm_nonneg _)
-  · simp only [greenDensity, ite_eq_right hz, le_refl]
-
-lemma stoppedGreen_nonneg
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (κ : ℝ) (z : ℂ) (u : ℝ) :
-    0 ≤ stoppedGreen κ U γ G z u := by
-  unfold stoppedGreen
-  split_ifs
-  · exact greenDensity_nonneg h κ _ z
-  · exact le_rfl
-
- 
-
-noncomputable def finiteCenteredLaw
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
-    (z : ℂ) (u : ℝ) : Measure Ω :=
-  P.withDensity (fun ω => ENNReal.ofReal
-    (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u / initialGreen κ z))
-
-theorem finiteCenteredLaw_probability
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
-    IsProbabilityMeasure (finiteCenteredLaw P κ B γ G z u) := by
-  obtain ⟨hi, he⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
-  have hp := initialGreen_pos κ z hz
-  apply isProbabilityMeasure_iff.mpr
-  rw [finiteCenteredLaw, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ,
-    ← ofReal_integral_eq_lintegral_ofReal (hi.div_const _) ?_, integral_div, he, div_self hp.ne', ENNReal.ofReal_one]
-  filter_upwards [hw] with ω hω
-  exact div_nonneg (stoppedGreen_nonneg hω κ z u) hp.le
-
- 
-
-theorem finiteCenteredLaw_live_compact
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
-    ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z u,
-      ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
-        fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u) ∧
-        radiusLevelTime (γ ω) (G ω) z u = (T : ℝ≥0∞) ∧
-        ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ≥0, t ≤ T →
-          δ ≤ (Real.sin (Complex.arg (G ω t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 := by
-  obtain ⟨hi, _⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
-  apply (ae_withDensity_iff' ((hi.div_const _).aestronglyMeasurable.aemeasurable.ennreal_ofReal)).mpr
-  filter_upwards [hw, hB.cont] with ω hω hcont
-  intro hweight
-  have hex : ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
-      fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u) := by
-    by_contra hn
-    apply hweight
-    rw [stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hn, zero_div, ENNReal.ofReal_zero]
-  obtain ⟨T, hT, hTR⟩ := hex
-  have hU : Continuous (fun t => Real.sqrt κ * B t ω) := hcont.const_mul _
-  exact ⟨T, hT, hTR, radiusLevelTime_eq_of_live_level hU hω z u T hT hTR,
-    live_angleSq_lower_bound hU hω z T hT⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge
-
-lemma continuousOn_of_integral_balance {F H : ℝ → ℝ} {v A C L : ℝ}
-    (hv : 0 ≤ v) (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
-    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ s in (0 : ℝ)..u, H s) =
-      L + A * (∫ s in (0 : ℝ)..u, F s)) : ContinuousOn F (Icc 0 v) := by
-  have hcF : ContinuousOn (fun u => ∫ s in (0 : ℝ)..u, F s) (Icc 0 v) := by
-    simpa only [uIcc_of_le hv] using intervalIntegral.continuousOn_primitive_interval (f := F) (a := 0) (b := v) (μ := volume)
-      (by simpa only [uIcc_of_le hv] using (intervalIntegrable_iff_integrableOn_Icc_of_le hv).mp hFi)
-  have hcH : ContinuousOn (fun u => ∫ s in (0 : ℝ)..u, H s) (Icc 0 v) := by
-    simpa only [uIcc_of_le hv] using intervalIntegral.continuousOn_primitive_interval (f := H) (a := 0) (b := v) (μ := volume)
-      (by simpa only [uIcc_of_le hv] using (intervalIntegrable_iff_integrableOn_Icc_of_le hv).mp hHi)
-  have hc : ContinuousOn (fun u => L + A * (∫ s in (0 : ℝ)..u, F s) - C * (∫ s in (0 : ℝ)..u, H s)) (Icc 0 v) :=
-    (continuousOn_const.add (continuousOn_const.mul hcF)).sub (continuousOn_const.mul hcH)
-  apply hc.congr
-  intro u hu
-  dsimp only
-  linarith [hbal u hu]
-
-lemma integral_drift_of_balance {F H : ℝ → ℝ} {v A C L q : ℝ}
-    (hv : 0 ≤ v) (hC : 0 ≤ C) (hq : 0 ≤ q)
-    (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
-    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ s in (0 : ℝ)..u, H s) =
-      L + A * (∫ s in (0 : ℝ)..u, F s))
-    (hpow : ∀ᵐ s ∂volume.restrict (Ioc 0 v), (F s) ^ q ≤ H s) :
-    ContinuousOn F (Icc 0 v) ∧
-      ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
-        F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ q := by
-  have hc := continuousOn_of_integral_balance hv hFi hHi hbal
-  refine ⟨hc, ?_⟩
-  intro s hs t ht hst
-  have hsub : uIcc s t ⊆ uIcc 0 v := by
-    simp only [uIcc_of_le hst, uIcc_of_le hv]
-    exact Icc_subset_Icc hs.1 ht.2
-  have hsub0 (x : ℝ) (hx : x ∈ Icc 0 v) : uIcc 0 x ⊆ uIcc 0 v := by
-    simp only [uIcc_of_le hx.1, uIcc_of_le hv]
-    exact Icc_subset_Icc le_rfl hx.2
-  have hFi' := hFi.mono_set hsub
-  have hHi' := hHi.mono_set hsub
-  have hpI : IntervalIntegrable (fun u => F u ^ q) volume s t := by
-    apply ContinuousOn.intervalIntegrable
-    apply (Real.continuous_rpow_const hq).comp_continuousOn
-    exact hc.mono (by simpa only [uIcc_of_le hst] using Icc_subset_Icc hs.1 ht.2)
-  have hpw : (∫ u in s..t, (F u) ^ q) ≤ ∫ u in s..t, H u := by
-    apply intervalIntegral.integral_mono_ae_restrict hst hpI hHi'
-    rw [restrict_Ioc_eq_restrict_Icc] at hpow
-    exact ae_restrict_of_ae_restrict_of_subset (Icc_subset_Icc hs.1 ht.2) hpow
-  have hdiff : F t - F s = A * (∫ u in s..t, F u) - C * (∫ u in s..t, H u) := by
-    rw [← intervalIntegral.integral_interval_sub_left (hFi.mono_set (hsub0 t ht)) (hFi.mono_set (hsub0 s hs)),
-      ← intervalIntegral.integral_interval_sub_left (hHi.mono_set (hsub0 t ht)) (hHi.mono_set (hsub0 s hs))]
-    linarith [hbal s hs, hbal t ht]
-  rw [intervalIntegral.integral_sub (hFi'.const_mul A) (hpI.const_mul C),
-    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, hdiff]
-  exact sub_le_sub_left (mul_le_mul_of_nonneg_left hpw hC) _
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
-
-noncomputable def radiusMoment (P : Measure Ω)
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (b s : ℝ) : ℝ :=
-  (∫ ω, limitRadiusKernel c (codimension κ) (greenExponent κ - b) s ω ∂P) / initialGreen κ z
-
-lemma radiusMoment_nonneg {P : Measure Ω}
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hI : 0 ≤ initialGreen κ z) (b s : ℝ) :
-    0 ≤ radiusMoment P c b s :=
-  div_nonneg (integral_nonneg fun ω => limitRadiusKernel_nonneg c _ _ s ω) hI
-
-lemma radiusMoment_intervalIntegrable {P : Measure Ω} [IsFiniteMeasure P]
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) {b u : ℝ} (hu : 0 ≤ u)
-    (hi : Integrable (Function.uncurry (limitRadiusKernel c (codimension κ) (greenExponent κ - b)))
-      ((volume.restrict (Ioc 0 u)).prod P)) : IntervalIntegrable (radiusMoment P c b) volume 0 u := by
-  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hu]
-  exact hi.integral_prod_left.div_const _
-
-lemma radiusMoment_intervalIntegral {P : Measure Ω} [IsFiniteMeasure P]
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) {b u : ℝ} (hu : 0 ≤ u)
-    (hi : Integrable (Function.uncurry (limitRadiusKernel c (codimension κ) (greenExponent κ - b)))
-      ((volume.restrict (Ioc 0 u)).prod P)) :
-    (∫ s in (0 : ℝ)..u, radiusMoment P c b s) =
-      (∫ q : ℝ × Ω, limitRadiusKernel c (codimension κ) (greenExponent κ - b) q.1 q.2
-        ∂((volume.restrict (Ioc 0 u)).prod P)) / initialGreen κ z := by
-  simp only [intervalIntegral.integral_of_le hu, radiusMoment, integral_div]
-  congr 1
-  exact (integral_prod _ hi).symm
-
-lemma balance_divide {X H Y F A C : ℝ} (he : X + C * H = Y + A * F) (I : ℝ) :
-    X / I + C * (H / I) = Y / I + A * (F / I) := by
-  linear_combination (1 / I) * he
-
-lemma radiusMoment_balance
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ)
-    (u : ℝ) (hu : u ∈ Icc 0 (v : ℝ)) :
-    radiusMoment P (a v) b u + (κ * b / 8) * (greenExponent κ - b) *
-      (∫ s in (0 : ℝ)..u, radiusMoment P (a v) (b + 2) s) =
-    powerLogDensity κ (codimension κ) (greenExponent κ - b) (initialLogCoordinates z) 0 / initialGreen κ z +
-      (κ * b / 8) * (1 + (greenExponent κ - b)) *
-        (∫ s in (0 : ℝ)..u, radiusMoment P (a v) b s) := by
-  let := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨hi, he⟩ := limit_power_balance (a ⟨u, hu.1⟩) (a v) hε hεlim hε0 hB hBm hBc hB0
-    hκ hκ8 b hb0 hb hu.1 hu.2
-  dsimp only [NNReal.toReal] at hi he
-  have hi' : Integrable (Function.uncurry (limitRadiusKernel (a v) (codimension κ) (greenExponent κ - (b + 2))))
-      ((volume.restrict (Ioc 0 u)).prod P) := by
-    rw [show greenExponent κ - (b + 2) = greenExponent κ - b - 2 by ring]
-    convert hi using 1; rfl
-  have hl := limitRadiusKernel_region_integrable (P := P) (a v) hε hBm (codimension κ)
-    (greenExponent κ - b) (codimension_pos κ hκ8).le (sub_pos.mpr hb).le u
-  rw [radiusMoment_intervalIntegral _ hu.1 hi', radiusMoment_intervalIntegral _ hu.1 hl]
-  dsimp only [radiusMoment]
-  rw [sub_add_eq_sub_sub]
-  convert balance_divide he (initialGreen κ z) using 1 <;> rfl
-
- 
-
-theorem radiusMoment_drift
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
-    ContinuousOn (radiusMoment P (a v) b) (Icc 0 v) ∧
-      ∀ s ∈ Icc 0 (v : ℝ), ∀ t ∈ Icc 0 (v : ℝ), s ≤ t →
-        radiusMoment P (a v) b t - radiusMoment P (a v) b s ≤
-          ∫ u in s..t, (κ * b / 8) * (1 + (greenExponent κ - b)) * radiusMoment P (a v) b u -
-            (κ * b / 8) * (greenExponent κ - b) * (radiusMoment P (a v) b u) ^ (1 + 2 / b) := by
-  let := hB.isGaussianProcess.isProbabilityMeasure
-  have hI : 0 < initialGreen κ z := initialGreen_pos κ z hz
-  have hh := (limit_power_balance (a v) (a v) hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb v.coe_nonneg le_rfl).1
-  have hhigh : IntervalIntegrable (radiusMoment P (a v) (b + 2)) volume 0 v := by
-    apply radiusMoment_intervalIntegrable _ v.coe_nonneg
-    simpa only [sub_add_eq_sub_sub] using hh
-  have hlow := radiusMoment_intervalIntegrable (P := P) (a v) v.coe_nonneg
-    (limitRadiusKernel_region_integrable (a v) hε hBm (codimension κ) (greenExponent κ - b)
-      (codimension_pos κ hκ8).le (sub_pos.mpr hb).le v)
-  apply integral_drift_of_balance v.coe_nonneg (by positivity) (by positivity) hlow hhigh
-    (radiusMoment_balance a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb)
-  filter_upwards [hh.prod_right_ae, ae_restrict_mem measurableSet_Ioc] with s hsi hs
-  have hm := limitRadiusKernel_green_mean (a ⟨s, hs.1.le⟩) (a v) hε hεlim (fun n => (hε0 n).le)
-    hB hBm hBc hB0 hκ hκ8 hz hs.1.le hs.2
-  have hj := limitRadiusKernel_inverse_moment_jensen (a v) hε hBm (codimension κ)
-    (greenExponent κ) b s (initialGreen κ z) (codimension_pos κ hκ8).le
-    (greenExponent_pos κ hκ hκ8).le hb0 hb.le hI hm hsi
-  simpa only [radiusMoment, sub_add_eq_sub_sub] using hj
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-noncomputable def stoppedInverseAngle (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
-    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (u b : ℝ) : ℝ :=
-  (stoppedAngleSq κ B γ G z u) ^ (-b / 2)
-
-lemma finiteCenteredLaw_inverseAngle_moment
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ≥0) (hu : 0 ≤ (u : ℝ))
-    {ε : ℕ → ℝ} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (b : ℝ) (hb : b ≤ greenExponent κ) :
-    Integrable (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z u b)
-      (finiteCenteredLaw P κ B γ G z u) ∧
-    (∫ ω, stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z u b
-      ∂finiteCenteredLaw P κ B γ G z u) = radiusMoment P a b u := by
-  let w (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
-  let f (ω : Ω) := stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z u b
-  let d (ω : Ω) := ENNReal.ofReal (w ω / initialGreen κ z)
-  have hwi := (finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu).1
-  have hd : AEMeasurable d P := (hwi.div_const _).aestronglyMeasurable.aemeasurable.ennreal_ofReal
-  have hid : Integrable (limitRadiusKernel a (codimension κ) (greenExponent κ - b) u) P :=
-    limitRadiusKernel_integrable a hε hBm _ _ (codimension_pos κ hκ8).le (sub_nonneg.mpr hb) u
-  have he : (fun ω => (d ω).toReal • f ω) =ᵐ[P]
-      (fun ω => limitRadiusKernel a (codimension κ) (greenExponent κ - b) u ω / initialGreen κ z) := by
-    filter_upwards [hw] with ω hω
-    have hw0 := stoppedGreen_nonneg hω κ z u
-    rw [show (d ω).toReal = w ω / initialGreen κ z from
-      ENNReal.toReal_ofReal (div_nonneg hw0 (initialGreen_pos κ z hz).le), smul_eq_mul]
-    rw [limitRadiusKernel_inverse_factor,
-      limitRadiusKernel_eq_stoppedGreen a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω,
-      limitRadiusKernel_eq_stoppedAngleSq a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω]
-    · change w ω / initialGreen κ z * f ω = w ω * f ω / initialGreen κ z
-      ring
-    · exact hε
-  constructor
-  · apply (integrable_withDensity_iff_integrable_smul₀' hd
-      (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)).mpr
-    exact (hid.div_const _).congr he.symm
-  · change (∫ ω, f ω ∂P.withDensity d) = _
-    rw [integral_withDensity_eq_integral_toReal_smul₀ hd
-      (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top), integral_congr_ae he, integral_div]
-    rfl
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
-lemma limitRadiusKernel_zero_eq_indicator_level
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (hεlim : Tendsto ε atTop (𝓝 0))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
-    (hu : 0 ≤ u) (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F) (p : ℝ) :
-    limitRadiusKernel a p 0 u ω =
-      if radiusLevelTime γ G z u < ∞ then Real.exp (p * (u - Real.log 2 - Real.log z.im)) else 0 := by
-  by_cases hh : ∃ T : ℝ≥0, z ∈ survivingDomain γ T ∧ fullRadius G T z = (2 * z.im) * Real.exp (-u)
-  · obtain ⟨T, hT, hR⟩ := hh
-    rw [limitRadiusKernel_eq_stopped_of_live_level a hε hεlim hBc hz hu ω γ G F hw T hT hR,
-      radiusLevelTime_eq_of_live_level ((hBc ω).const_mul _) hw z u T hT hR, ite_eq_left ENNReal.coe_lt_top,
-      powerLogDensity_radius_angle κ p 0 (Real.log z.im), zero_div, Real.rpow_zero, mul_one,
-      (fullRadius_level_iff_logClock ((hBc ω).const_mul _) hw z T hT u).mp hR]
-  · rw [limitRadiusKernel_eq_zero_of_no_level a hBc hz ω γ G F hw _ _ hh,
-      radiusLevelTime_eq_top_of_no_level γ G z u hh, ite_eq_right (lt_irrefl _)]
-
-lemma integral_limitRadiusKernel_zero_eq_probability
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (hεlim : Tendsto ε atTop (𝓝 0))
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
-    (hu : 0 ≤ u) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) (p : ℝ) :
-    (∫ ω, limitRadiusKernel a p 0 u ω ∂P) =
-      Real.exp (p * (u - Real.log 2 - Real.log z.im)) *
-        (P {ω | radiusLevelTime (γ ω) (G ω) z u < ∞}).toReal := by
-  let K := Real.exp (p * (u - Real.log 2 - Real.log z.im))
-  let A := {ω | limitRadiusKernel a p 0 u ω ≠ 0}
-  have hAm : MeasurableSet A := (measurableSet_eq_fun
-    ((limitRadiusKernel_measurable a hε hBm p 0).comp (measurable_const.prodMk measurable_id)) measurable_const).compl
-  have hA : A =ᵐ[P] {ω | radiusLevelTime (γ ω) (G ω) z u < ∞} := by
-    filter_upwards [hw] with ω hω
-    change (limitRadiusKernel a p 0 u ω ≠ 0) = (radiusLevelTime (γ ω) (G ω) z u < ∞)
-    rw [limitRadiusKernel_zero_eq_indicator_level a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω p]
-    split_ifs with h <;> simp [h, Real.exp_ne_zero]
-  have he : limitRadiusKernel a p 0 u = A.indicator (fun _ => K) := by
-    funext ω
-    rw [Set.indicator_apply]
-    change limitRadiusKernel a p 0 u ω = if limitRadiusKernel a p 0 u ω ≠ 0 then K else 0
-    simp only [limitRadiusKernel_zero, max_eq_left hu]
-    split_ifs <;> simp_all [Real.exp_ne_zero, K]
-  rw [he, integral_indicator hAm, setIntegral_const, smul_eq_mul, mul_comm]
-  change K * (P A).toReal = _
-  rw [measure_congr hA]
-
-lemma initialGreen_div_zeroKernel (κ u : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    initialGreen κ z / Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) =
-      Real.exp (-codimension κ * u) * (Real.sin (Complex.arg z)) ^ greenExponent κ := by
-  have hr : 0 < 2 * z.im := mul_pos (by norm_num) hz
-  rw [initialGreen, Real.rpow_def_of_pos hr, Real.log_mul (by norm_num) (ne_of_gt hz)]
-  rw [mul_comm (Real.exp _) ((Real.sin (Complex.arg z)) ^ greenExponent κ),
-    mul_div_assoc, ← Real.exp_sub]
-  rw [show (Real.log 2 + Real.log z.im) * -codimension κ -
-    codimension κ * (u - Real.log 2 - Real.log z.im) = -codimension κ * u by ring]
-  ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge
-
- 
-
-lemma liminf_slope_of_integral_bound {F G : ℝ → ℝ} {a b : ℝ}
-    (hab : a ≤ b) (hG : ContinuousOn G (Icc a b))
-    (hinc : ∀ s ∈ Icc a b, ∀ t ∈ Icc a b, s ≤ t →
-      F t - F s ≤ ∫ u in s..t, G u) :
-    ∀ x ∈ Ico a b, ∀ r, G x < r → ∃ᶠ z in 𝓝[>] x, slope F x z < r := by
-  let g (t : ℝ) := G (projIcc a b hab t)
-  have hgc : Continuous g := hG.comp_continuous
-    (continuous_subtype_val.comp continuous_projIcc) (fun t => (projIcc a b hab t).property)
-  have he (t : ℝ) (ht : t ∈ Icc a b) : g t = G t := by
-    simp only [g, projIcc_of_mem hab ht]
-  intro x hx r hr
-  have hd : HasDerivAt (fun z => ∫ u in a..z, g u) (g x) x :=
-    intervalIntegral.integral_hasDerivAt_right (hgc.intervalIntegrable _ _)
-      hgc.stronglyMeasurable.stronglyMeasurableAtFilter hgc.continuousAt
-  have hs := (hasDerivWithinAt_iff_tendsto_slope' (lt_irrefl x)).1
-    (hd.hasDerivWithinAt : HasDerivWithinAt (fun z => ∫ u in a..z, g u) (g x) (Ioi x) x)
-  have hsl : ∀ᶠ z in 𝓝[>] x, slope (fun y => ∫ u in a..y, g u) x z < r :=
-    hs (Iio_mem_nhds (by rw [he x ⟨hx.1, hx.2.le⟩]; exact hr))
-  apply Filter.Eventually.frequently
-  filter_upwards [hsl, Ioc_mem_nhdsGT hx.2] with z hz hzI
-  apply lt_of_le_of_lt _ hz
-  rw [slope_def_field, slope_def_field,
-    intervalIntegral.integral_interval_sub_left (hgc.intervalIntegrable a z) (hgc.intervalIntegrable a x)]
-  apply div_le_div_of_nonneg_right _ (sub_pos.mpr hzI.1).le
-  rw [intervalIntegral.integral_congr (fun t ht => he t (by
-    have ht' : t ∈ Icc x z := by simpa only [uIcc_of_le hzI.1.le] using ht
-    exact ⟨hx.1.trans ht'.1, ht'.2.trans hzI.2⟩))]
-  exact hinc x ⟨hx.1, hx.2.le⟩ z ⟨hx.1.trans hzI.1.le, hzI.2⟩ hzI.1.le
-
-lemma image_le_of_integral_drift {F G H H' : ℝ → ℝ} {a b : ℝ}
-    (hab : a ≤ b) (hF : ContinuousOn F (Icc a b)) (hG : ContinuousOn G (Icc a b))
-    (hinc : ∀ s ∈ Icc a b, ∀ t ∈ Icc a b, s ≤ t → F t - F s ≤ ∫ u in s..t, G u)
-    (ha : F a ≤ H a) (hH : ContinuousOn H (Icc a b))
-    (hH' : ∀ x ∈ Ico a b, HasDerivWithinAt H (H' x) (Ici x) x)
-    (hbound : ∀ x ∈ Ico a b, F x = H x → G x < H' x) :
-    ∀ x ∈ Icc a b, F x ≤ H x := by
-  exact fun x hx => image_le_of_liminf_slope_right_lt_deriv_boundary' hF
-    (liminf_slope_of_integral_bound hab hG hinc) ha hH hH' hbound hx
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge
-
-noncomputable def lyapunovBarrier (r K δ t : ℝ) : ℝ := K ^ r * (t + δ) ^ (-r)
-
-lemma lyapunovBarrier_pos {r K δ t : ℝ} (hK : 0 < K) (ht : 0 < t + δ) :
-    0 < lyapunovBarrier r K δ t := mul_pos (Real.rpow_pos_of_pos hK _) (Real.rpow_pos_of_pos ht _)
-
-lemma lyapunovBarrier_deriv {r K δ t : ℝ} (ht : 0 < t + δ) :
-    HasDerivAt (lyapunovBarrier r K δ) (-r * lyapunovBarrier r K δ t / (t + δ)) t := by
-  have hd := (((hasDerivAt_id t).add_const δ).rpow_const (p := -r) (Or.inl ht.ne')).const_mul (K ^ r)
-  convert hd using 1 <;> try rfl
-  dsimp only [lyapunovBarrier, id_eq]
-  rw [Real.rpow_sub ht, Real.rpow_one]
-  ring
-
-lemma lyapunovBarrier_power {r K δ t : ℝ} (hr : 0 < r) (hK : 0 < K) (ht : 0 < t + δ) :
-    (lyapunovBarrier r K δ t) ^ (1 + 1 / r) =
-      lyapunovBarrier r K δ t * K / (t + δ) := by
-  have he : lyapunovBarrier r K δ t = (K / (t + δ)) ^ r := by
-    rw [Real.div_rpow hK.le ht.le, lyapunovBarrier, Real.rpow_neg ht.le]
-    rfl
-  rw [he, ← Real.rpow_mul (div_pos hK ht).le]
-  have hf : r * (1 + 1 / r) = r + 1 := by field_simp
-  rw [hf, Real.rpow_add (div_pos hK ht), Real.rpow_one]
-  ring
-
-lemma exists_small_barrier {r K C : ℝ} (hr : 0 < r) (hK : 0 < K) :
-    ∃ δ : ℝ, 0 < δ ∧ δ ≤ 1 ∧ C ≤ lyapunovBarrier r K δ 0 := by
-  have ht : Tendsto (fun δ : ℝ => K ^ r * δ ^ (-r)) (𝓝[>] (0 : ℝ)) atTop :=
-    (tendsto_rpow_neg_nhdsGT_zero (neg_neg_of_pos hr)).const_mul_atTop (Real.rpow_pos_of_pos hK r)
-  have he := (ht.eventually_ge_atTop C).and (Ioc_mem_nhdsGT (by norm_num : (0 : ℝ) < 1))
-  obtain ⟨δ, hd, hδ⟩ := he.exists
-  exact ⟨δ, hδ.1, hδ.2, by simpa only [lyapunovBarrier, zero_add] using hd⟩
-
- 
-
-theorem comingDown_one_window {F : ℝ → ℝ} {v A C r : ℝ}
-    (hv : 0 ≤ v) (hv1 : v ≤ 1) (hA : 0 ≤ A) (hC : 0 < C) (hr : 0 < r)
-    (hF : ContinuousOn F (Icc 0 v))
-    (hinc : ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
-      F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ (1 + 1 / r)) :
-    ∀ t ∈ Ioc 0 v, F t ≤ ((2 * A + r + 1) / C / t) ^ r := by
-  let K := (2 * A + r + 1) / C
-  have hK : 0 < K := by dsimp [K]; positivity
-  obtain ⟨δ, hδ, hδ1, hinit⟩ := exists_small_barrier (C := F 0) hr hK
-  have hcont : ContinuousOn (lyapunovBarrier r K δ) (Icc 0 v) := by
-    intro t ht
-    exact (lyapunovBarrier_deriv (r := r) (K := K) (by linarith [ht.1] : 0 < t + δ)).continuousAt.continuousWithinAt
-  have hq : 0 ≤ 1 + 1 / r := by positivity
-  have hgc : ContinuousOn (fun u => A * F u - C * (F u) ^ (1 + 1 / r)) (Icc 0 v) :=
-    (continuousOn_const.mul hF).sub (continuousOn_const.mul ((Real.continuous_rpow_const hq).comp_continuousOn hF))
-  have hfence : ∀ t ∈ Icc 0 v, F t ≤ lyapunovBarrier r K δ t := by
-    apply image_le_of_integral_drift hv hF hgc hinc hinit hcont
-      (fun t ht => (lyapunovBarrier_deriv (r := r) (K := K) (by linarith [ht.1] : 0 < t + δ)).hasDerivWithinAt)
-    intro t ht he
-    have htδ : 0 < t + δ := by linarith [ht.1]
-    have hH : 0 < lyapunovBarrier r K δ t := lyapunovBarrier_pos hK htδ
-    rw [he, lyapunovBarrier_power hr hK htδ]
-    apply (lt_div_iff₀ htδ).mpr
-    have ht2 : t + δ ≤ 2 := by linarith [ht.2]
-    have hk : C * K = 2 * A + r + 1 := by dsimp [K]; field_simp
-    have htA : A * (t + δ) ≤ 2 * A := by nlinarith
-    have hx : A * (t + δ) - C * K < -r := by linarith
-    have hm := mul_lt_mul_of_pos_left hx hH
-    field_simp at *
-    nlinarith
-  intro t ht
-  apply (hfence t ⟨ht.1.le, ht.2⟩).trans
-  have htδ : 0 < t + δ := by linarith [ht.1]
-  have he : lyapunovBarrier r K δ t = (K / (t + δ)) ^ r := by
-    rw [Real.div_rpow hK.le htδ.le, lyapunovBarrier, Real.rpow_neg htδ.le]
-    rfl
-  rw [he]
-  exact Real.rpow_le_rpow (div_nonneg hK.le htδ.le)
-    (div_le_div_of_nonneg_left hK.le ht.1 (by linarith)) hr.le
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge
-
-lemma comingDown_shifted {F : ℝ → ℝ} {v A C r : ℝ}
-    (_hv : 0 ≤ v) (hA : 0 ≤ A) (hC : 0 < C) (hr : 0 < r)
-    (hF : ContinuousOn F (Icc 0 v))
-    (hinc : ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
-      F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ (1 + 1 / r))
-    {x ℓ : ℝ} (hx : 0 ≤ x) (hℓ : 0 < ℓ) (hℓ1 : ℓ ≤ 1) (hxℓ : x + ℓ ≤ v) :
-    F (x + ℓ) ≤ ((2 * A + r + 1) / C / ℓ) ^ r := by
-  have hmap : MapsTo (fun s : ℝ => x + s) (Icc 0 ℓ) (Icc 0 v) := by
-    intro s hs
-    constructor <;> linarith [hs.1, hs.2]
-  have hcont : ContinuousOn (fun s => F (x + s)) (Icc 0 ℓ) :=
-    hF.comp (continuous_const.add continuous_id).continuousOn hmap
-  apply comingDown_one_window hℓ.le hℓ1 hA hC hr hcont ?_ ℓ ⟨hℓ, le_rfl⟩
-  intro s hs t ht hst
-  rw [intervalIntegral.integral_comp_add_left (fun u => A * F u - C * F u ^ (1 + 1 / r)) x]
-  exact hinc _ (hmap hs) _ (hmap ht) (by linarith)
-
- 
-
-theorem comingDown_uniform {F : ℝ → ℝ} {v A C r η : ℝ}
-    (hv : 0 ≤ v) (hA : 0 ≤ A) (hC : 0 < C) (hr : 0 < r)
-    (hF : ContinuousOn F (Icc 0 v))
-    (hinc : ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
-      F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ (1 + 1 / r))
-    (hη : 0 < η) (hη1 : η ≤ 1) :
-    ∀ t ∈ Icc η v, F t ≤ ((2 * A + r + 1) / C / η) ^ r := by
-  intro t ht
-  have hh := comingDown_shifted hv hA hC hr hF hinc (x := t - η) (sub_nonneg.mpr ht.1) hη hη1
-    (by linarith [ht.2] : t - η + η ≤ v)
-  simpa only [sub_add_cancel] using hh
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def lyapunovA (κ b : ℝ) : ℝ := (κ * b / 8) * (1 + (greenExponent κ - b))
-noncomputable def lyapunovC (κ b : ℝ) : ℝ := (κ * b / 8) * (greenExponent κ - b)
-noncomputable def angularMomentBound (κ b η : ℝ) : ℝ :=
-  ((2 * lyapunovA κ b + b / 2 + 1) / lyapunovC κ b / η) ^ (b / 2)
-
-lemma lyapunovA_nonneg {κ b : ℝ} (hκ : 0 < κ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
-    0 ≤ lyapunovA κ b := by dsimp [lyapunovA]; positivity
-lemma lyapunovC_pos {κ b : ℝ} (hκ : 0 < κ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
-    0 < lyapunovC κ b := by dsimp [lyapunovC]; positivity
-lemma angularMomentBound_pos {κ b η : ℝ} (hκ : 0 < κ) (hb0 : 0 < b)
-    (hb : b < greenExponent κ) (hη : 0 < η) : 0 < angularMomentBound κ b η := by
-  have hA := lyapunovA_nonneg hκ hb0 hb
-  have hC := lyapunovC_pos hκ hb0 hb
-  dsimp only [angularMomentBound]
-  apply Real.rpow_pos_of_pos
-  positivity
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
-
- 
-
-theorem radiusMoment_bound
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ)
-    (η : ℝ) (hη : 0 < η) (hη1 : η ≤ 1) :
-    ∀ t ∈ Icc η (v : ℝ), radiusMoment P (a v) b t ≤ angularMomentBound κ b η := by
-  obtain ⟨hc, hd⟩ := radiusMoment_drift a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz b hb0 hb (v := v)
-  have he : 1 + 1 / (b / 2) = 1 + 2 / b := by field_simp
-  apply comingDown_uniform v.coe_nonneg (lyapunovA_nonneg hκ hb0 hb) (lyapunovC_pos hκ hb0 hb)
-    (by positivity : 0 < b / 2) hc ?_ hη hη1
-  simpa only [he, lyapunovA, lyapunovC] using hd
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {v : ℝ} {ε : ℕ → ℝ}
-
-lemma limitRadiusKernel_antitone_power
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
-    (p α β s : ℝ) (hαβ : α ≤ β) (ω : Ω) :
-    limitRadiusKernel c p β s ω ≤ limitRadiusKernel c p α s ω := by
-  have he := limitRadiusKernel_power_factor c hε p α (β - α) s ω
-  rw [show α + (β - α) = β by ring] at he
-  rw [he]
-  apply mul_le_of_le_one_right (limitRadiusKernel_nonneg c p α s ω)
-  exact Real.rpow_le_one (limitRadiusKernel_nonneg c 0 2 s ω)
-    (limitRadiusKernel_angle_le_one c s ω) (by linarith)
-
-lemma integral_limitRadiusKernel_antitone_power
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
-    {P : Measure Ω} (p α β s : ℝ) (hαβ : α ≤ β)
-    (hiα : Integrable (limitRadiusKernel c p α s) P)
-    (hiβ : Integrable (limitRadiusKernel c p β s) P) :
-    (∫ ω, limitRadiusKernel c p β s ω ∂P) ≤ ∫ ω, limitRadiusKernel c p α s ω ∂P :=
-  integral_mono hiβ hiα (limitRadiusKernel_antitone_power c hε p α β s hαβ)
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.KernelExpectation
+
+namespace OAI
+
+/-! Finite Stopped Law for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem finite_stopped_green_regular
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
+    Integrable (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z u) P ∧
+    (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z u ∂P) = initialGreen κ z := by
+  let A := logAngleSq κ (initialLogCoordinates z) 0
+  have hA : 0 < A := logAngleSq_pos _ _ _
+  let ε (n : ℕ) := A / ((n : ℝ) + 1)
+  have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
+  have hεA (n : ℕ) : ε n ≤ A := div_le_self hA.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
+  have hεlim : Tendsto ε atTop (𝓝 0) := by
+    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A
+    simpa only [mul_one_div, mul_zero] using hh
+  choose f hfi hfm hfb hfw using fun n => exists_finite_angular_kernel
+    hB hBm hBc hB0 κ hκ hκ8 z hz u hu (ε n) (hε n) (hεA n)
+  let K := Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im))
+  let g (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
+  have hp : 0 < greenExponent κ / 2 := div_pos (greenExponent_pos κ hκ hκ8) (by norm_num)
+  have hlim : ∀ᵐ ω ∂P, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) := by
+    filter_upwards [hw] with ω hω
+    have hU : Continuous (fun t => Real.sqrt κ * B t ω) := (hBc ω).const_mul _
+    by_cases hlevel : ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
+        fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u)
+    · obtain ⟨T, hT, hTR⟩ := hlevel
+      obtain ⟨δ, hδ, hδb⟩ := live_angleSq_lower_bound hU hω z T hT
+      have hev : ∀ᶠ n in atTop, ε n < δ := (tendsto_order.mp hεlim).2 δ hδ
+      apply tendsto_const_nhds.congr'
+      filter_upwards [hev] with n hn
+      obtain ⟨t, ht, htu, hb, he, _⟩ := hfw n ω (γ ω) (G ω) (F ω) hω
+      have hTc : Real.log ((2 * z.im) / fullRadius (G ω) T z) = u := by
+        rw [← logRadiusClock_logCoordinates hU hω z T hT]
+        exact (fullRadius_level_iff_logClock hU hω z T hT u).mp hTR
+      have htT : t ≤ T := live_clock_le_time hU hω z t T ht hT (by simpa only [hTc] using htu)
+      have htb : δ ≤ (Real.sin (Complex.arg (G ω t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 :=
+        hδb t htT
+      have htc : Real.log ((2 * z.im) / fullRadius (G ω) t z) = u := hb.resolve_right (by linarith)
+      have htR : fullRadius (G ω) t z = (2 * z.im) * Real.exp (-u) := by
+        apply (fullRadius_level_iff_logClock hU hω z t ht u).mpr
+        rwa [logRadiusClock_logCoordinates hU hω z t ht]
+      exact (stoppedGreen_eq_of_live_level hU hω κ z u t ht htR).trans he.symm
+    · have hg : g ω = 0 := stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hlevel
+      rw [hg]
+      have hb (n : ℕ) : f n ω ≤ K * (ε n) ^ (greenExponent κ / 2) := by
+        obtain ⟨t, ht, _, htb, _, hbe⟩ := hfw n ω (γ ω) (G ω) (F ω) hω
+        apply hbe
+        apply htb.resolve_left
+        intro he
+        apply hlevel
+        refine ⟨t, ht, (fullRadius_level_iff_logClock hU hω z t ht u).mpr ?_⟩
+        rwa [logRadiusClock_logCoordinates hU hω z t ht]
+      have hupp : Tendsto (fun n => K * (ε n) ^ (greenExponent κ / 2)) atTop (𝓝 0) := by
+        have hh := ((Real.continuousAt_rpow_const 0 (greenExponent κ / 2) (Or.inr hp.le)).tendsto.comp hεlim).const_mul K
+        simpa only [Function.comp_apply, Real.zero_rpow hp.ne', mul_zero] using hh
+      exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hupp
+        (fun n => (hfb n ω).1) hb
+  have hbound (n : ℕ) : ∀ᵐ ω ∂P, ‖f n ω‖ ≤ K := by
+    apply Eventually.of_forall
+    intro ω
+    simpa only [Real.norm_eq_abs, abs_of_nonneg (hfb n ω).1] using (hfb n ω).2
+  have hgm : AEStronglyMeasurable g P :=
+    aestronglyMeasurable_of_tendsto_ae atTop (fun n => (hfi n).aestronglyMeasurable) hlim
+  have hgi : Integrable g P := ⟨hgm,
+    hasFiniteIntegral_of_dominated_convergence (integrable_const K).hasFiniteIntegral hbound hlim⟩
+  have hc := tendsto_integral_of_dominated_convergence (fun _ : Ω => K)
+    (fun n => (hfi n).aestronglyMeasurable) (integrable_const K) hbound hlim
+  refine ⟨hgi, ?_⟩
+  apply tendsto_nhds_unique hc
+  simpa only [hfm] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => initialGreen κ z) atTop (𝓝 (initialGreen κ z)))
+
+ 
+
+theorem finite_stopped_green_of_witness
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
+    Integrable (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z u) P ∧
+    (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z u ∂P) = initialGreen κ z := by
+  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
+  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
+    filter_upwards [hw, hCB] with ω hω hω'
+    simpa only [hω'] using hω
+  have heq : (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * C t ω)
+      (γ ω) (G ω) z u) =ᵐ[P] (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z u) := by
+    filter_upwards [hCB] with ω hω
+    simp only [hω]
+  obtain ⟨hfi, hfm⟩ := finite_stopped_green_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz u hu
+  exact ⟨hfi.congr heq, (integral_congr_ae heq).symm.trans hfm⟩
+
+universe uLevel
+def FiniteStoppedGreenTarget : Prop :=
+  ∀ (Ω : Type uLevel) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (κ : ℝ), 0 < κ → κ < 8 →
+    ∀ (B : ℝ≥0 → Ω → ℝ), ProbabilityTheory.IsBrownianReal B P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ),
+      (∀ t, Measurable (fun ω => γ ω t)) →
+      (∀ᵐ ω ∂P, IsCapacityTwoWitness
+        (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) →
+      ∀ (z : ℂ), z ∈ upperHalfPlane → ∀ (u : ℝ), 0 ≤ u →
+        Integrable (fun ω => stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+          (γ ω) (G ω) z u) P ∧
+        (∫ ω, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+          (γ ω) (G ω) z u ∂P) = initialGreen κ z
+
+ 
+
+theorem finite_stopped_green : FiniteStoppedGreenTarget := by
+  intro Ω mΩ P hP κ hκ hκ8 B hB γ G F _hγ hw z hz u hu
+  let := hP
+  exact finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma initialGreen_pos (κ : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    0 < initialGreen κ z := by
+  have hz0 : z ≠ 0 := by intro he; simp only [he, upperHalfPlane, Set.mem_ofPred_eq, Complex.zero_im, lt_self_iff_false] at hz
+  have hsin : 0 < Real.sin (Complex.arg z) := by
+    rw [Complex.sin_arg]
+    exact div_pos hz (norm_pos_iff.mpr hz0)
+  exact mul_pos (Real.rpow_pos_of_pos (mul_pos (by norm_num) hz) _) (Real.rpow_pos_of_pos hsin _)
+
+lemma greenDensity_nonneg
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (κ : ℝ) (t : ℝ≥0) (z : ℂ) :
+    0 ≤ greenDensity κ U γ G t z := by
+  classical
+  by_cases hz : z ∈ survivingDomain γ t
+  · rw [greenDensity, ite_eq_left hz]
+    have hy := h.imag_positive t z hz
+    apply mul_nonneg
+    · exact Real.rpow_nonneg (div_nonneg (mul_nonneg (by norm_num) hy.le) (norm_nonneg _)) _
+    · apply Real.rpow_nonneg
+      rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero]
+      exact div_nonneg hy.le (norm_nonneg _)
+  · simp only [greenDensity, ite_eq_right hz, le_refl]
+
+lemma stoppedGreen_nonneg
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (κ : ℝ) (z : ℂ) (u : ℝ) :
+    0 ≤ stoppedGreen κ U γ G z u := by
+  unfold stoppedGreen
+  split_ifs
+  · exact greenDensity_nonneg h κ _ z
+  · exact le_rfl
+
+ 
+
+noncomputable def finiteCenteredLaw
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
+    (z : ℂ) (u : ℝ) : Measure Ω :=
+  P.withDensity (fun ω => ENNReal.ofReal
+    (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u / initialGreen κ z))
+
+theorem finiteCenteredLaw_probability
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
+    IsProbabilityMeasure (finiteCenteredLaw P κ B γ G z u) := by
+  obtain ⟨hi, he⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
+  have hp := initialGreen_pos κ z hz
+  apply isProbabilityMeasure_iff.mpr
+  rw [finiteCenteredLaw, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ,
+    ← ofReal_integral_eq_lintegral_ofReal (hi.div_const _) ?_, integral_div, he, div_self hp.ne', ENNReal.ofReal_one]
+  filter_upwards [hw] with ω hω
+  exact div_nonneg (stoppedGreen_nonneg hω κ z u) hp.le
+
+ 
+
+theorem finiteCenteredLaw_live_compact
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) :
+    ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z u,
+      ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
+        fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u) ∧
+        radiusLevelTime (γ ω) (G ω) z u = (T : ℝ≥0∞) ∧
+        ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ≥0, t ≤ T →
+          δ ≤ (Real.sin (Complex.arg (G ω t z - ((Real.sqrt κ * B t ω : ℝ) : ℂ)))) ^ 2 := by
+  obtain ⟨hi, _⟩ := finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu
+  apply (ae_withDensity_iff' ((hi.div_const _).aestronglyMeasurable.aemeasurable.ennreal_ofReal)).mpr
+  filter_upwards [hw, hB.cont] with ω hω hcont
+  intro hweight
+  have hex : ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
+      fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u) := by
+    by_contra hn
+    apply hweight
+    rw [stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hn, zero_div, ENNReal.ofReal_zero]
+  obtain ⟨T, hT, hTR⟩ := hex
+  have hU : Continuous (fun t => Real.sqrt κ * B t ω) := hcont.const_mul _
+  exact ⟨T, hT, hTR, radiusLevelTime_eq_of_live_level hU hω z u T hT hTR,
+    live_angleSq_lower_bound hU hω z T hT⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge
+
+lemma continuousOn_of_integral_balance {F H : ℝ → ℝ} {v A C L : ℝ}
+    (hv : 0 ≤ v) (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
+    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ s in (0 : ℝ)..u, H s) =
+      L + A * (∫ s in (0 : ℝ)..u, F s)) : ContinuousOn F (Icc 0 v) := by
+  have hcF : ContinuousOn (fun u => ∫ s in (0 : ℝ)..u, F s) (Icc 0 v) := by
+    simpa only [uIcc_of_le hv] using intervalIntegral.continuousOn_primitive_interval (f := F) (a := 0) (b := v) (μ := volume)
+      (by simpa only [uIcc_of_le hv] using (intervalIntegrable_iff_integrableOn_Icc_of_le hv).mp hFi)
+  have hcH : ContinuousOn (fun u => ∫ s in (0 : ℝ)..u, H s) (Icc 0 v) := by
+    simpa only [uIcc_of_le hv] using intervalIntegral.continuousOn_primitive_interval (f := H) (a := 0) (b := v) (μ := volume)
+      (by simpa only [uIcc_of_le hv] using (intervalIntegrable_iff_integrableOn_Icc_of_le hv).mp hHi)
+  have hc : ContinuousOn (fun u => L + A * (∫ s in (0 : ℝ)..u, F s) - C * (∫ s in (0 : ℝ)..u, H s)) (Icc 0 v) :=
+    (continuousOn_const.add (continuousOn_const.mul hcF)).sub (continuousOn_const.mul hcH)
+  apply hc.congr
+  intro u hu
+  dsimp only
+  linarith [hbal u hu]
+
+lemma integral_drift_of_balance {F H : ℝ → ℝ} {v A C L q : ℝ}
+    (hv : 0 ≤ v) (hC : 0 ≤ C) (hq : 0 ≤ q)
+    (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
+    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ s in (0 : ℝ)..u, H s) =
+      L + A * (∫ s in (0 : ℝ)..u, F s))
+    (hpow : ∀ᵐ s ∂volume.restrict (Ioc 0 v), (F s) ^ q ≤ H s) :
+    ContinuousOn F (Icc 0 v) ∧
+      ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
+        F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ q := by
+  have hc := continuousOn_of_integral_balance hv hFi hHi hbal
+  refine ⟨hc, ?_⟩
+  intro s hs t ht hst
+  have hsub : uIcc s t ⊆ uIcc 0 v := by
+    simp only [uIcc_of_le hst, uIcc_of_le hv]
+    exact Icc_subset_Icc hs.1 ht.2
+  have hsub0 (x : ℝ) (hx : x ∈ Icc 0 v) : uIcc 0 x ⊆ uIcc 0 v := by
+    simp only [uIcc_of_le hx.1, uIcc_of_le hv]
+    exact Icc_subset_Icc le_rfl hx.2
+  have hFi' := hFi.mono_set hsub
+  have hHi' := hHi.mono_set hsub
+  have hpI : IntervalIntegrable (fun u => F u ^ q) volume s t := by
+    apply ContinuousOn.intervalIntegrable
+    apply (Real.continuous_rpow_const hq).comp_continuousOn
+    exact hc.mono (by simpa only [uIcc_of_le hst] using Icc_subset_Icc hs.1 ht.2)
+  have hpw : (∫ u in s..t, (F u) ^ q) ≤ ∫ u in s..t, H u := by
+    apply intervalIntegral.integral_mono_ae_restrict hst hpI hHi'
+    rw [restrict_Ioc_eq_restrict_Icc] at hpow
+    exact ae_restrict_of_ae_restrict_of_subset (Icc_subset_Icc hs.1 ht.2) hpow
+  have hdiff : F t - F s = A * (∫ u in s..t, F u) - C * (∫ u in s..t, H u) := by
+    rw [← intervalIntegral.integral_interval_sub_left (hFi.mono_set (hsub0 t ht)) (hFi.mono_set (hsub0 s hs)),
+      ← intervalIntegral.integral_interval_sub_left (hHi.mono_set (hsub0 t ht)) (hHi.mono_set (hsub0 s hs))]
+    linarith [hbal s hs, hbal t ht]
+  rw [intervalIntegral.integral_sub (hFi'.const_mul A) (hpI.const_mul C),
+    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, hdiff]
+  exact sub_le_sub_left (mul_le_mul_of_nonneg_left hpw hC) _
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
+
+noncomputable def radiusMoment (P : Measure Ω)
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (b s : ℝ) : ℝ :=
+  (∫ ω, limitRadiusKernel c (codimension κ) (greenExponent κ - b) s ω ∂P) / initialGreen κ z
+
+lemma radiusMoment_nonneg {P : Measure Ω}
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hI : 0 ≤ initialGreen κ z) (b s : ℝ) :
+    0 ≤ radiusMoment P c b s :=
+  div_nonneg (integral_nonneg fun ω => limitRadiusKernel_nonneg c _ _ s ω) hI
+
+lemma radiusMoment_intervalIntegrable {P : Measure Ω} [IsFiniteMeasure P]
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) {b u : ℝ} (hu : 0 ≤ u)
+    (hi : Integrable (Function.uncurry (limitRadiusKernel c (codimension κ) (greenExponent κ - b)))
+      ((volume.restrict (Ioc 0 u)).prod P)) : IntervalIntegrable (radiusMoment P c b) volume 0 u := by
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hu]
+  exact hi.integral_prod_left.div_const _
+
+lemma radiusMoment_intervalIntegral {P : Measure Ω} [IsFiniteMeasure P]
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) {b u : ℝ} (hu : 0 ≤ u)
+    (hi : Integrable (Function.uncurry (limitRadiusKernel c (codimension κ) (greenExponent κ - b)))
+      ((volume.restrict (Ioc 0 u)).prod P)) :
+    (∫ s in (0 : ℝ)..u, radiusMoment P c b s) =
+      (∫ q : ℝ × Ω, limitRadiusKernel c (codimension κ) (greenExponent κ - b) q.1 q.2
+        ∂((volume.restrict (Ioc 0 u)).prod P)) / initialGreen κ z := by
+  simp only [intervalIntegral.integral_of_le hu, radiusMoment, integral_div]
+  congr 1
+  exact (integral_prod _ hi).symm
+
+lemma balance_divide {X H Y F A C : ℝ} (he : X + C * H = Y + A * F) (I : ℝ) :
+    X / I + C * (H / I) = Y / I + A * (F / I) := by
+  linear_combination (1 / I) * he
+
+lemma radiusMoment_balance
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ)
+    (u : ℝ) (hu : u ∈ Icc 0 (v : ℝ)) :
+    radiusMoment P (a v) b u + (κ * b / 8) * (greenExponent κ - b) *
+      (∫ s in (0 : ℝ)..u, radiusMoment P (a v) (b + 2) s) =
+    powerLogDensity κ (codimension κ) (greenExponent κ - b) (initialLogCoordinates z) 0 / initialGreen κ z +
+      (κ * b / 8) * (1 + (greenExponent κ - b)) *
+        (∫ s in (0 : ℝ)..u, radiusMoment P (a v) b s) := by
+  let := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨hi, he⟩ := limit_power_balance (a ⟨u, hu.1⟩) (a v) hε hεlim hε0 hB hBm hBc hB0
+    hκ hκ8 b hb0 hb hu.1 hu.2
+  dsimp only [NNReal.toReal] at hi he
+  have hi' : Integrable (Function.uncurry (limitRadiusKernel (a v) (codimension κ) (greenExponent κ - (b + 2))))
+      ((volume.restrict (Ioc 0 u)).prod P) := by
+    rw [show greenExponent κ - (b + 2) = greenExponent κ - b - 2 by ring]
+    convert hi using 1; rfl
+  have hl := limitRadiusKernel_region_integrable (P := P) (a v) hε hBm (codimension κ)
+    (greenExponent κ - b) (codimension_pos κ hκ8).le (sub_pos.mpr hb).le u
+  rw [radiusMoment_intervalIntegral _ hu.1 hi', radiusMoment_intervalIntegral _ hu.1 hl]
+  dsimp only [radiusMoment]
+  rw [sub_add_eq_sub_sub]
+  convert balance_divide he (initialGreen κ z) using 1 <;> rfl
+
+ 
+
+theorem radiusMoment_drift
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
+    ContinuousOn (radiusMoment P (a v) b) (Icc 0 v) ∧
+      ∀ s ∈ Icc 0 (v : ℝ), ∀ t ∈ Icc 0 (v : ℝ), s ≤ t →
+        radiusMoment P (a v) b t - radiusMoment P (a v) b s ≤
+          ∫ u in s..t, (κ * b / 8) * (1 + (greenExponent κ - b)) * radiusMoment P (a v) b u -
+            (κ * b / 8) * (greenExponent κ - b) * (radiusMoment P (a v) b u) ^ (1 + 2 / b) := by
+  let := hB.isGaussianProcess.isProbabilityMeasure
+  have hI : 0 < initialGreen κ z := initialGreen_pos κ z hz
+  have hh := (limit_power_balance (a v) (a v) hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb v.coe_nonneg le_rfl).1
+  have hhigh : IntervalIntegrable (radiusMoment P (a v) (b + 2)) volume 0 v := by
+    apply radiusMoment_intervalIntegrable _ v.coe_nonneg
+    simpa only [sub_add_eq_sub_sub] using hh
+  have hlow := radiusMoment_intervalIntegrable (P := P) (a v) v.coe_nonneg
+    (limitRadiusKernel_region_integrable (a v) hε hBm (codimension κ) (greenExponent κ - b)
+      (codimension_pos κ hκ8).le (sub_pos.mpr hb).le v)
+  apply integral_drift_of_balance v.coe_nonneg (by positivity) (by positivity) hlow hhigh
+    (radiusMoment_balance a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb)
+  filter_upwards [hh.prod_right_ae, ae_restrict_mem measurableSet_Ioc] with s hsi hs
+  have hm := limitRadiusKernel_green_mean (a ⟨s, hs.1.le⟩) (a v) hε hεlim (fun n => (hε0 n).le)
+    hB hBm hBc hB0 hκ hκ8 hz hs.1.le hs.2
+  have hj := limitRadiusKernel_inverse_moment_jensen (a v) hε hBm (codimension κ)
+    (greenExponent κ) b s (initialGreen κ z) (codimension_pos κ hκ8).le
+    (greenExponent_pos κ hκ hκ8).le hb0 hb.le hI hm hsi
+  simpa only [radiusMoment, sub_add_eq_sub_sub] using hj
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+noncomputable def stoppedInverseAngle (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
+    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (u b : ℝ) : ℝ :=
+  (stoppedAngleSq κ B γ G z u) ^ (-b / 2)
+
+lemma finiteCenteredLaw_inverseAngle_moment
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ≥0) (hu : 0 ≤ (u : ℝ))
+    {ε : ℕ → ℝ} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (b : ℝ) (hb : b ≤ greenExponent κ) :
+    Integrable (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z u b)
+      (finiteCenteredLaw P κ B γ G z u) ∧
+    (∫ ω, stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z u b
+      ∂finiteCenteredLaw P κ B γ G z u) = radiusMoment P a b u := by
+  let w (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
+  let f (ω : Ω) := stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z u b
+  let d (ω : Ω) := ENNReal.ofReal (w ω / initialGreen κ z)
+  have hwi := (finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu).1
+  have hd : AEMeasurable d P := (hwi.div_const _).aestronglyMeasurable.aemeasurable.ennreal_ofReal
+  have hid : Integrable (limitRadiusKernel a (codimension κ) (greenExponent κ - b) u) P :=
+    limitRadiusKernel_integrable a hε hBm _ _ (codimension_pos κ hκ8).le (sub_nonneg.mpr hb) u
+  have he : (fun ω => (d ω).toReal • f ω) =ᵐ[P]
+      (fun ω => limitRadiusKernel a (codimension κ) (greenExponent κ - b) u ω / initialGreen κ z) := by
+    filter_upwards [hw] with ω hω
+    have hw0 := stoppedGreen_nonneg hω κ z u
+    rw [show (d ω).toReal = w ω / initialGreen κ z from
+      ENNReal.toReal_ofReal (div_nonneg hw0 (initialGreen_pos κ z hz).le), smul_eq_mul]
+    rw [limitRadiusKernel_inverse_factor,
+      limitRadiusKernel_eq_stoppedGreen a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω,
+      limitRadiusKernel_eq_stoppedAngleSq a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω]
+    · change w ω / initialGreen κ z * f ω = w ω * f ω / initialGreen κ z
+      ring
+    · exact hε
+  constructor
+  · apply (integrable_withDensity_iff_integrable_smul₀' hd
+      (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)).mpr
+    exact (hid.div_const _).congr he.symm
+  · change (∫ ω, f ω ∂P.withDensity d) = _
+    rw [integral_withDensity_eq_integral_toReal_smul₀ hd
+      (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top), integral_congr_ae he, integral_div]
+    rfl
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+lemma limitRadiusKernel_zero_eq_indicator_level
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (hεlim : Tendsto ε atTop (𝓝 0))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
+    (hu : 0 ≤ u) (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F) (p : ℝ) :
+    limitRadiusKernel a p 0 u ω =
+      if radiusLevelTime γ G z u < ∞ then Real.exp (p * (u - Real.log 2 - Real.log z.im)) else 0 := by
+  by_cases hh : ∃ T : ℝ≥0, z ∈ survivingDomain γ T ∧ fullRadius G T z = (2 * z.im) * Real.exp (-u)
+  · obtain ⟨T, hT, hR⟩ := hh
+    rw [limitRadiusKernel_eq_stopped_of_live_level a hε hεlim hBc hz hu ω γ G F hw T hT hR,
+      radiusLevelTime_eq_of_live_level ((hBc ω).const_mul _) hw z u T hT hR, ite_eq_left ENNReal.coe_lt_top,
+      powerLogDensity_radius_angle κ p 0 (Real.log z.im), zero_div, Real.rpow_zero, mul_one,
+      (fullRadius_level_iff_logClock ((hBc ω).const_mul _) hw z T hT u).mp hR]
+  · rw [limitRadiusKernel_eq_zero_of_no_level a hBc hz ω γ G F hw _ _ hh,
+      radiusLevelTime_eq_top_of_no_level γ G z u hh, ite_eq_right (lt_irrefl _)]
+
+lemma integral_limitRadiusKernel_zero_eq_probability
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (hεlim : Tendsto ε atTop (𝓝 0))
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane)
+    (hu : 0 ≤ u) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) (p : ℝ) :
+    (∫ ω, limitRadiusKernel a p 0 u ω ∂P) =
+      Real.exp (p * (u - Real.log 2 - Real.log z.im)) *
+        (P {ω | radiusLevelTime (γ ω) (G ω) z u < ∞}).toReal := by
+  let K := Real.exp (p * (u - Real.log 2 - Real.log z.im))
+  let A := {ω | limitRadiusKernel a p 0 u ω ≠ 0}
+  have hAm : MeasurableSet A := (measurableSet_eq_fun
+    ((limitRadiusKernel_measurable a hε hBm p 0).comp (measurable_const.prodMk measurable_id)) measurable_const).compl
+  have hA : A =ᵐ[P] {ω | radiusLevelTime (γ ω) (G ω) z u < ∞} := by
+    filter_upwards [hw] with ω hω
+    change (limitRadiusKernel a p 0 u ω ≠ 0) = (radiusLevelTime (γ ω) (G ω) z u < ∞)
+    rw [limitRadiusKernel_zero_eq_indicator_level a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω p]
+    split_ifs with h <;> simp [h, Real.exp_ne_zero]
+  have he : limitRadiusKernel a p 0 u = A.indicator (fun _ => K) := by
+    funext ω
+    rw [Set.indicator_apply]
+    change limitRadiusKernel a p 0 u ω = if limitRadiusKernel a p 0 u ω ≠ 0 then K else 0
+    simp only [limitRadiusKernel_zero, max_eq_left hu]
+    split_ifs <;> simp_all [Real.exp_ne_zero, K]
+  rw [he, integral_indicator hAm, setIntegral_const, smul_eq_mul, mul_comm]
+  change K * (P A).toReal = _
+  rw [measure_congr hA]
+
+lemma initialGreen_div_zeroKernel (κ u : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    initialGreen κ z / Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) =
+      Real.exp (-codimension κ * u) * (Real.sin (Complex.arg z)) ^ greenExponent κ := by
+  have hr : 0 < 2 * z.im := mul_pos (by norm_num) hz
+  rw [initialGreen, Real.rpow_def_of_pos hr, Real.log_mul (by norm_num) (ne_of_gt hz)]
+  rw [mul_comm (Real.exp _) ((Real.sin (Complex.arg z)) ^ greenExponent κ),
+    mul_div_assoc, ← Real.exp_sub]
+  rw [show (Real.log 2 + Real.log z.im) * -codimension κ -
+    codimension κ * (u - Real.log 2 - Real.log z.im) = -codimension κ * u by ring]
+  ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge
+
+ 
+
+lemma liminf_slope_of_integral_bound {F G : ℝ → ℝ} {a b : ℝ}
+    (hab : a ≤ b) (hG : ContinuousOn G (Icc a b))
+    (hinc : ∀ s ∈ Icc a b, ∀ t ∈ Icc a b, s ≤ t →
+      F t - F s ≤ ∫ u in s..t, G u) :
+    ∀ x ∈ Ico a b, ∀ r, G x < r → ∃ᶠ z in 𝓝[>] x, slope F x z < r := by
+  let g (t : ℝ) := G (projIcc a b hab t)
+  have hgc : Continuous g := hG.comp_continuous
+    (continuous_subtype_val.comp continuous_projIcc) (fun t => (projIcc a b hab t).property)
+  have he (t : ℝ) (ht : t ∈ Icc a b) : g t = G t := by
+    simp only [g, projIcc_of_mem hab ht]
+  intro x hx r hr
+  have hd : HasDerivAt (fun z => ∫ u in a..z, g u) (g x) x :=
+    intervalIntegral.integral_hasDerivAt_right (hgc.intervalIntegrable _ _)
+      hgc.stronglyMeasurable.stronglyMeasurableAtFilter hgc.continuousAt
+  have hs := (hasDerivWithinAt_iff_tendsto_slope' (lt_irrefl x)).1
+    (hd.hasDerivWithinAt : HasDerivWithinAt (fun z => ∫ u in a..z, g u) (g x) (Ioi x) x)
+  have hsl : ∀ᶠ z in 𝓝[>] x, slope (fun y => ∫ u in a..y, g u) x z < r :=
+    hs (Iio_mem_nhds (by rw [he x ⟨hx.1, hx.2.le⟩]; exact hr))
+  apply Filter.Eventually.frequently
+  filter_upwards [hsl, Ioc_mem_nhdsGT hx.2] with z hz hzI
+  apply lt_of_le_of_lt _ hz
+  rw [slope_def_field, slope_def_field,
+    intervalIntegral.integral_interval_sub_left (hgc.intervalIntegrable a z) (hgc.intervalIntegrable a x)]
+  apply div_le_div_of_nonneg_right _ (sub_pos.mpr hzI.1).le
+  rw [intervalIntegral.integral_congr (fun t ht => he t (by
+    have ht' : t ∈ Icc x z := by simpa only [uIcc_of_le hzI.1.le] using ht
+    exact ⟨hx.1.trans ht'.1, ht'.2.trans hzI.2⟩))]
+  exact hinc x ⟨hx.1, hx.2.le⟩ z ⟨hx.1.trans hzI.1.le, hzI.2⟩ hzI.1.le
+
+lemma image_le_of_integral_drift {F G H H' : ℝ → ℝ} {a b : ℝ}
+    (hab : a ≤ b) (hF : ContinuousOn F (Icc a b)) (hG : ContinuousOn G (Icc a b))
+    (hinc : ∀ s ∈ Icc a b, ∀ t ∈ Icc a b, s ≤ t → F t - F s ≤ ∫ u in s..t, G u)
+    (ha : F a ≤ H a) (hH : ContinuousOn H (Icc a b))
+    (hH' : ∀ x ∈ Ico a b, HasDerivWithinAt H (H' x) (Ici x) x)
+    (hbound : ∀ x ∈ Ico a b, F x = H x → G x < H' x) :
+    ∀ x ∈ Icc a b, F x ≤ H x := by
+  exact fun x hx => image_le_of_liminf_slope_right_lt_deriv_boundary' hF
+    (liminf_slope_of_integral_bound hab hG hinc) ha hH hH' hbound hx
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge
+
+noncomputable def lyapunovBarrier (r K δ t : ℝ) : ℝ := K ^ r * (t + δ) ^ (-r)
+
+lemma lyapunovBarrier_pos {r K δ t : ℝ} (hK : 0 < K) (ht : 0 < t + δ) :
+    0 < lyapunovBarrier r K δ t := mul_pos (Real.rpow_pos_of_pos hK _) (Real.rpow_pos_of_pos ht _)
+
+lemma lyapunovBarrier_deriv {r K δ t : ℝ} (ht : 0 < t + δ) :
+    HasDerivAt (lyapunovBarrier r K δ) (-r * lyapunovBarrier r K δ t / (t + δ)) t := by
+  have hd := (((hasDerivAt_id t).add_const δ).rpow_const (p := -r) (Or.inl ht.ne')).const_mul (K ^ r)
+  convert hd using 1 <;> try rfl
+  dsimp only [lyapunovBarrier, id_eq]
+  rw [Real.rpow_sub ht, Real.rpow_one]
+  ring
+
+lemma lyapunovBarrier_power {r K δ t : ℝ} (hr : 0 < r) (hK : 0 < K) (ht : 0 < t + δ) :
+    (lyapunovBarrier r K δ t) ^ (1 + 1 / r) =
+      lyapunovBarrier r K δ t * K / (t + δ) := by
+  have he : lyapunovBarrier r K δ t = (K / (t + δ)) ^ r := by
+    rw [Real.div_rpow hK.le ht.le, lyapunovBarrier, Real.rpow_neg ht.le]
+    rfl
+  rw [he, ← Real.rpow_mul (div_pos hK ht).le]
+  have hf : r * (1 + 1 / r) = r + 1 := by field_simp
+  rw [hf, Real.rpow_add (div_pos hK ht), Real.rpow_one]
+  ring
+
+lemma exists_small_barrier {r K C : ℝ} (hr : 0 < r) (hK : 0 < K) :
+    ∃ δ : ℝ, 0 < δ ∧ δ ≤ 1 ∧ C ≤ lyapunovBarrier r K δ 0 := by
+  have ht : Tendsto (fun δ : ℝ => K ^ r * δ ^ (-r)) (𝓝[>] (0 : ℝ)) atTop :=
+    (tendsto_rpow_neg_nhdsGT_zero (neg_neg_of_pos hr)).const_mul_atTop (Real.rpow_pos_of_pos hK r)
+  have he := (ht.eventually_ge_atTop C).and (Ioc_mem_nhdsGT (by norm_num : (0 : ℝ) < 1))
+  obtain ⟨δ, hd, hδ⟩ := he.exists
+  exact ⟨δ, hδ.1, hδ.2, by simpa only [lyapunovBarrier, zero_add] using hd⟩
+
+ 
+
+theorem comingDown_one_window {F : ℝ → ℝ} {v A C r : ℝ}
+    (hv : 0 ≤ v) (hv1 : v ≤ 1) (hA : 0 ≤ A) (hC : 0 < C) (hr : 0 < r)
+    (hF : ContinuousOn F (Icc 0 v))
+    (hinc : ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
+      F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ (1 + 1 / r)) :
+    ∀ t ∈ Ioc 0 v, F t ≤ ((2 * A + r + 1) / C / t) ^ r := by
+  let K := (2 * A + r + 1) / C
+  have hK : 0 < K := by dsimp [K]; positivity
+  obtain ⟨δ, hδ, hδ1, hinit⟩ := exists_small_barrier (C := F 0) hr hK
+  have hcont : ContinuousOn (lyapunovBarrier r K δ) (Icc 0 v) := by
+    intro t ht
+    exact (lyapunovBarrier_deriv (r := r) (K := K) (by linarith [ht.1] : 0 < t + δ)).continuousAt.continuousWithinAt
+  have hq : 0 ≤ 1 + 1 / r := by positivity
+  have hgc : ContinuousOn (fun u => A * F u - C * (F u) ^ (1 + 1 / r)) (Icc 0 v) :=
+    (continuousOn_const.mul hF).sub (continuousOn_const.mul ((Real.continuous_rpow_const hq).comp_continuousOn hF))
+  have hfence : ∀ t ∈ Icc 0 v, F t ≤ lyapunovBarrier r K δ t := by
+    apply image_le_of_integral_drift hv hF hgc hinc hinit hcont
+      (fun t ht => (lyapunovBarrier_deriv (r := r) (K := K) (by linarith [ht.1] : 0 < t + δ)).hasDerivWithinAt)
+    intro t ht he
+    have htδ : 0 < t + δ := by linarith [ht.1]
+    have hH : 0 < lyapunovBarrier r K δ t := lyapunovBarrier_pos hK htδ
+    rw [he, lyapunovBarrier_power hr hK htδ]
+    apply (lt_div_iff₀ htδ).mpr
+    have ht2 : t + δ ≤ 2 := by linarith [ht.2]
+    have hk : C * K = 2 * A + r + 1 := by dsimp [K]; field_simp
+    have htA : A * (t + δ) ≤ 2 * A := by nlinarith
+    have hx : A * (t + δ) - C * K < -r := by linarith
+    have hm := mul_lt_mul_of_pos_left hx hH
+    field_simp at *
+    nlinarith
+  intro t ht
+  apply (hfence t ⟨ht.1.le, ht.2⟩).trans
+  have htδ : 0 < t + δ := by linarith [ht.1]
+  have he : lyapunovBarrier r K δ t = (K / (t + δ)) ^ r := by
+    rw [Real.div_rpow hK.le htδ.le, lyapunovBarrier, Real.rpow_neg htδ.le]
+    rfl
+  rw [he]
+  exact Real.rpow_le_rpow (div_nonneg hK.le htδ.le)
+    (div_le_div_of_nonneg_left hK.le ht.1 (by linarith)) hr.le
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge
+
+lemma comingDown_shifted {F : ℝ → ℝ} {v A C r : ℝ}
+    (_hv : 0 ≤ v) (hA : 0 ≤ A) (hC : 0 < C) (hr : 0 < r)
+    (hF : ContinuousOn F (Icc 0 v))
+    (hinc : ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
+      F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ (1 + 1 / r))
+    {x ℓ : ℝ} (hx : 0 ≤ x) (hℓ : 0 < ℓ) (hℓ1 : ℓ ≤ 1) (hxℓ : x + ℓ ≤ v) :
+    F (x + ℓ) ≤ ((2 * A + r + 1) / C / ℓ) ^ r := by
+  have hmap : MapsTo (fun s : ℝ => x + s) (Icc 0 ℓ) (Icc 0 v) := by
+    intro s hs
+    constructor <;> linarith [hs.1, hs.2]
+  have hcont : ContinuousOn (fun s => F (x + s)) (Icc 0 ℓ) :=
+    hF.comp (continuous_const.add continuous_id).continuousOn hmap
+  apply comingDown_one_window hℓ.le hℓ1 hA hC hr hcont ?_ ℓ ⟨hℓ, le_rfl⟩
+  intro s hs t ht hst
+  rw [intervalIntegral.integral_comp_add_left (fun u => A * F u - C * F u ^ (1 + 1 / r)) x]
+  exact hinc _ (hmap hs) _ (hmap ht) (by linarith)
+
+ 
+
+theorem comingDown_uniform {F : ℝ → ℝ} {v A C r η : ℝ}
+    (hv : 0 ≤ v) (hA : 0 ≤ A) (hC : 0 < C) (hr : 0 < r)
+    (hF : ContinuousOn F (Icc 0 v))
+    (hinc : ∀ s ∈ Icc 0 v, ∀ t ∈ Icc 0 v, s ≤ t →
+      F t - F s ≤ ∫ u in s..t, A * F u - C * (F u) ^ (1 + 1 / r))
+    (hη : 0 < η) (hη1 : η ≤ 1) :
+    ∀ t ∈ Icc η v, F t ≤ ((2 * A + r + 1) / C / η) ^ r := by
+  intro t ht
+  have hh := comingDown_shifted hv hA hC hr hF hinc (x := t - η) (sub_nonneg.mpr ht.1) hη hη1
+    (by linarith [ht.2] : t - η + η ≤ v)
+  simpa only [sub_add_cancel] using hh
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def lyapunovA (κ b : ℝ) : ℝ := (κ * b / 8) * (1 + (greenExponent κ - b))
+noncomputable def lyapunovC (κ b : ℝ) : ℝ := (κ * b / 8) * (greenExponent κ - b)
+noncomputable def angularMomentBound (κ b η : ℝ) : ℝ :=
+  ((2 * lyapunovA κ b + b / 2 + 1) / lyapunovC κ b / η) ^ (b / 2)
+
+lemma lyapunovA_nonneg {κ b : ℝ} (hκ : 0 < κ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
+    0 ≤ lyapunovA κ b := by dsimp [lyapunovA]; positivity
+lemma lyapunovC_pos {κ b : ℝ} (hκ : 0 < κ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
+    0 < lyapunovC κ b := by dsimp [lyapunovC]; positivity
+lemma angularMomentBound_pos {κ b η : ℝ} (hκ : 0 < κ) (hb0 : 0 < b)
+    (hb : b < greenExponent κ) (hη : 0 < η) : 0 < angularMomentBound κ b η := by
+  have hA := lyapunovA_nonneg hκ hb0 hb
+  have hC := lyapunovC_pos hκ hb0 hb
+  dsimp only [angularMomentBound]
+  apply Real.rpow_pos_of_pos
+  positivity
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
+
+ 
+
+theorem radiusMoment_bound
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ)
+    (η : ℝ) (hη : 0 < η) (hη1 : η ≤ 1) :
+    ∀ t ∈ Icc η (v : ℝ), radiusMoment P (a v) b t ≤ angularMomentBound κ b η := by
+  obtain ⟨hc, hd⟩ := radiusMoment_drift a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz b hb0 hb (v := v)
+  have he : 1 + 1 / (b / 2) = 1 + 2 / b := by field_simp
+  apply comingDown_uniform v.coe_nonneg (lyapunovA_nonneg hκ hb0 hb) (lyapunovC_pos hκ hb0 hb)
+    (by positivity : 0 < b / 2) hc ?_ hη hη1
+  simpa only [he, lyapunovA, lyapunovC] using hd
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {v : ℝ} {ε : ℕ → ℝ}
+
+lemma limitRadiusKernel_antitone_power
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
+    (p α β s : ℝ) (hαβ : α ≤ β) (ω : Ω) :
+    limitRadiusKernel c p β s ω ≤ limitRadiusKernel c p α s ω := by
+  have he := limitRadiusKernel_power_factor c hε p α (β - α) s ω
+  rw [show α + (β - α) = β by ring] at he
+  rw [he]
+  apply mul_le_of_le_one_right (limitRadiusKernel_nonneg c p α s ω)
+  exact Real.rpow_le_one (limitRadiusKernel_nonneg c 0 2 s ω)
+    (limitRadiusKernel_angle_le_one c s ω) (by linarith)
+
+lemma integral_limitRadiusKernel_antitone_power
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
+    {P : Measure Ω} (p α β s : ℝ) (hαβ : α ≤ β)
+    (hiα : Integrable (limitRadiusKernel c p α s) P)
+    (hiβ : Integrable (limitRadiusKernel c p β s) P) :
+    (∫ ω, limitRadiusKernel c p β s ω ∂P) ≤ ∫ ω, limitRadiusKernel c p α s ω ∂P :=
+  integral_mono hiβ hiα (limitRadiusKernel_antitone_power c hε p α β s hαβ)
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualDenseConditional.lean b/lean/OAI/Probability/SLE/Gauge/ActualDenseConditional.lean
index 238f6c5824c5791eab7a0e0df354a6dab1f82a6a..67b477ed3af70fd27e3d3c35640ae18828791081 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualDenseConditional.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualDenseConditional.lean
@@ -1,132 +1,132 @@
-import OAI.Probability.SLE.Gauge.ActualDenseExtended
-import OAI.Probability.SLE.Angle.ActualAngleWarmupExtended
-import OAI.Probability.SLE.Geometry.ExtendedMarkedAngle
-import OAI.Probability.SLE.Geometry.FiniteCenteredConsistency
-
-namespace OAI
-
-/-! The complete conditional dense test: one unit of angle warmup followed
-by the dense experiment, under any common later finite centered law. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-local instance conditionalDenseDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance conditionalDenseDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance conditionalDensePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance conditionalDensePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem uniform_actual_conditional_dense_test
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hγm : ∀ t, Measurable (fun ω => γ ω t))
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W) :
-    ∃ h b C : ℝ, 0 < h ∧ h ≤ 1 / 2 ∧ 0 < b ∧ 0 < C ∧
-      ∃ δ : ℕ → ℝ, (∀ N, 0 < N → 0 < δ N) ∧
-      ∀ (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u),
-        ∀ N : ℕ, 0 < N → ∀ v : ℝ, 0 < v →
-          v < δ N * ((2 * z.im) * Real.exp (-(u + 1))) →
-          ∀ V : ℝ, u + 1 + Serpentine.experimentRadiusLevel h ≤ V →
-          ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB
-            (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
-            κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A →
-            ENNReal.ofReal ((1 / 2) * Real.exp (-C * (N : ℝ) ^ 2)) *
-              finiteCenteredLaw P κ B γ G z V A ≤
-            finiteCenteredLaw P κ B γ G z V
-              (A ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-                (fun ω => radiusLevelTime (γ ω) (G ω) z (u + 1 + Serpentine.experimentRadiusLevel h))
-                (closedBall z ((2 * z.im) * Real.exp (-(u + 1))))
-                (b * ((2 * z.im) * Real.exp (-(u + 1))) ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
-  classical
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  obtain ⟨Φ, Gc, Fc, _, hWc⟩ := exists_measurable_canonical_witness
-    (P.map (sourceDriverPath B hc)) κ htrace
-  obtain ⟨s, hs, hs1, hwarm⟩ :=
-    actual_radius_angle_warmup P κ hκ hκ8 B hB hBm hc hB0 γ G F hw Φ Gc Fc hWc
-  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hdense⟩ :=
-    uniform_actual_dense_at_extended_good_radius_state P κ hκ hκ8 B hB hBm hc hB0 γ G F
-      hγm hw htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW hs hs1
-  have hhalf : h ≤ 1 / 2 := by linarith
-  have hq : 0 ≤ Serpentine.experimentRadiusLevel h :=
-    Serpentine.experimentRadiusLevel_nonneg hh (by linarith)
-  refine ⟨h, b, C, hh, hhalf, hb, hC, δ, hδ, ?_⟩
-  intro z hz u hu N hN v hv hvcut V hV A hA
-  let U := u + 1 + Serpentine.experimentRadiusLevel h
-  have hU : 0 ≤ U := by dsimp [U]; linarith
-  have hu1 : 0 ≤ u + 1 := by linarith
-  let E : Set Ω := {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
-    s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}
-  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u + 1)
-  have hσ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz (u + 1) hu1
-  have hE : MeasurableSet[hσ.measurableSpace] E :=
-    canonical_good_angle_measurable_at_radius hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz (u + 1) s hu1
-  have hA1 : MeasurableSet[hσ.measurableSpace] A :=
-    (actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u (u + 1)
-      hu (by linarith)) A hA
-  have hEext (ω : Ω) (hω : ω ∈ E) :
-      ω ∈ extendedMarkedGood κ B γ G σ z s := by
-    rwa [← canonical_good_angle_eq_extendedMarkedGood κ B hc γ G F hw z (u + 1) s hu1]
-  have hbase (ω : Ω) (hω : ω ∈ E) :
-      fullRadius (G ω) ((σ ω).untopD 0) z = (2 * z.im) * Real.exp (-(u + 1)) := by
-    exact (actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z (u + 1) (hEext ω hω).1).2
-  let S := stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-    (fun ω => radiusLevelTime (γ ω) (G ω) z U)
-    (closedBall z ((2 * z.im) * Real.exp (-(u + 1))))
-    (b * ((2 * z.im) * Real.exp (-(u + 1))) ^ dimension κ * (N : ℝ) ^ codimension κ)
-  have hd := hdense σ hσ z hz (u + 1) hu1 E hE (fun ω hω => (hEext ω hω).1)
-    (fun ω hω => (hEext ω hω).2) hbase N hN v hv hvcut A hA1
-  have hσU := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz U hU
-  have hpast := actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
-    z hz (u + 1) U hu1 (by dsimp [U]; linarith)
-  have hAU : MeasurableSet[hσU.measurableSpace] A := hpast A hA1
-  have hEU : MeasurableSet[hσU.measurableSpace] E := hpast E hE
-  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t ω => γ ω t) := by
-    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ _ γ G F hw
-    · intro t
-      exact measurable_const.mul
-        ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
-    · intro ω
-      exact (hc ω).const_mul _
-  have hS : MeasurableSet[hσU.measurableSpace] S :=
-    stoppedMassTest_measurable_at_stop _ γ hγa (fun ω => (hw ω).1) _ hσU κ v _ _
-  have hcons (D : Set Ω) (hD : MeasurableSet[hσU.measurableSpace] D) :
-      finiteCenteredLaw P κ B γ G z V D = finiteCenteredLaw P κ B γ G z U D :=
-    finiteCenteredLaw_consistent_at_radius_stop hB hBm hc hB0 κ hκ hκ8 γ G F
-      (Eventually.of_forall hw) z hz U V hU hV _ hσU Filter.EventuallyEq.rfl D hD
-  have hd' : ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
-      finiteCenteredLaw P κ B γ G z V (A ∩ E) ≤ finiteCenteredLaw P κ B γ G z V (A ∩ S) := by
-    rw [hcons _ (hAU.inter hEU), hcons _ (hAU.inter hS)]
-    exact hd.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩))
-  have hwarm' := hwarm z hz u hu V (by linarith) A hA
-  calc
-    _ = ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
-        (ENNReal.ofReal (1 / 2 : ℝ) * finiteCenteredLaw P κ B γ G z V A) := by
-      rw [ENNReal.ofReal_mul (by norm_num)]
-      ac_rfl
-    _ ≤ ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
-        finiteCenteredLaw P κ B γ G z V (A ∩ E) := mul_le_mul' le_rfl hwarm'
-    _ ≤ _ := hd'
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualDenseExtended
+import OAI.Probability.SLE.Angle.ActualAngleWarmupExtended
+import OAI.Probability.SLE.Geometry.ExtendedMarkedAngle
+import OAI.Probability.SLE.Geometry.FiniteCenteredConsistency
+
+namespace OAI
+
+/-! The complete conditional dense test: one unit of angle warmup followed
+by the dense experiment, under any common later finite centered law. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+local instance conditionalDenseDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance conditionalDenseDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance conditionalDensePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance conditionalDensePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem uniform_actual_conditional_dense_test
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hγm : ∀ t, Measurable (fun ω => γ ω t))
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W) :
+    ∃ h b C : ℝ, 0 < h ∧ h ≤ 1 / 2 ∧ 0 < b ∧ 0 < C ∧
+      ∃ δ : ℕ → ℝ, (∀ N, 0 < N → 0 < δ N) ∧
+      ∀ (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u),
+        ∀ N : ℕ, 0 < N → ∀ v : ℝ, 0 < v →
+          v < δ N * ((2 * z.im) * Real.exp (-(u + 1))) →
+          ∀ V : ℝ, u + 1 + Serpentine.experimentRadiusLevel h ≤ V →
+          ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB
+            (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
+            κ hκ hκ8 γ G F hw z hz u hu).measurableSpace] A →
+            ENNReal.ofReal ((1 / 2) * Real.exp (-C * (N : ℝ) ^ 2)) *
+              finiteCenteredLaw P κ B γ G z V A ≤
+            finiteCenteredLaw P κ B γ G z V
+              (A ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+                (fun ω => radiusLevelTime (γ ω) (G ω) z (u + 1 + Serpentine.experimentRadiusLevel h))
+                (closedBall z ((2 * z.im) * Real.exp (-(u + 1))))
+                (b * ((2 * z.im) * Real.exp (-(u + 1))) ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
+  classical
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  obtain ⟨Φ, Gc, Fc, _, hWc⟩ := exists_measurable_canonical_witness
+    (P.map (sourceDriverPath B hc)) κ htrace
+  obtain ⟨s, hs, hs1, hwarm⟩ :=
+    actual_radius_angle_warmup P κ hκ hκ8 B hB hBm hc hB0 γ G F hw Φ Gc Fc hWc
+  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hdense⟩ :=
+    uniform_actual_dense_at_extended_good_radius_state P κ hκ hκ8 B hB hBm hc hB0 γ G F
+      hγm hw htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW hs hs1
+  have hhalf : h ≤ 1 / 2 := by linarith
+  have hq : 0 ≤ Serpentine.experimentRadiusLevel h :=
+    Serpentine.experimentRadiusLevel_nonneg hh (by linarith)
+  refine ⟨h, b, C, hh, hhalf, hb, hC, δ, hδ, ?_⟩
+  intro z hz u hu N hN v hv hvcut V hV A hA
+  let U := u + 1 + Serpentine.experimentRadiusLevel h
+  have hU : 0 ≤ U := by dsimp [U]; linarith
+  have hu1 : 0 ≤ u + 1 := by linarith
+  let E : Set Ω := {ω | canonicalRadiusLevel κ (u + 1, z, sourceDriverPath B hc ω) < ∞ ∧
+    s ≤ canonicalStoppedSine κ (u + 1, z, sourceDriverPath B hc ω)}
+  let σ : Ω → WithTop ℝ≥0 := fun ω => radiusLevelTime (γ ω) (G ω) z (u + 1)
+  have hσ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz (u + 1) hu1
+  have hE : MeasurableSet[hσ.measurableSpace] E :=
+    canonical_good_angle_measurable_at_radius hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz (u + 1) s hu1
+  have hA1 : MeasurableSet[hσ.measurableSpace] A :=
+    (actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u (u + 1)
+      hu (by linarith)) A hA
+  have hEext (ω : Ω) (hω : ω ∈ E) :
+      ω ∈ extendedMarkedGood κ B γ G σ z s := by
+    rwa [← canonical_good_angle_eq_extendedMarkedGood κ B hc γ G F hw z (u + 1) s hu1]
+  have hbase (ω : Ω) (hω : ω ∈ E) :
+      fullRadius (G ω) ((σ ω).untopD 0) z = (2 * z.im) * Real.exp (-(u + 1)) := by
+    exact (actual_radius_level_finite_state (hw ω) ((hc ω).const_mul _) z (u + 1) (hEext ω hω).1).2
+  let S := stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+    (fun ω => radiusLevelTime (γ ω) (G ω) z U)
+    (closedBall z ((2 * z.im) * Real.exp (-(u + 1))))
+    (b * ((2 * z.im) * Real.exp (-(u + 1))) ^ dimension κ * (N : ℝ) ^ codimension κ)
+  have hd := hdense σ hσ z hz (u + 1) hu1 E hE (fun ω hω => (hEext ω hω).1)
+    (fun ω hω => (hEext ω hω).2) hbase N hN v hv hvcut A hA1
+  have hσU := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz U hU
+  have hpast := actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
+    z hz (u + 1) U hu1 (by dsimp [U]; linarith)
+  have hAU : MeasurableSet[hσU.measurableSpace] A := hpast A hA1
+  have hEU : MeasurableSet[hσU.measurableSpace] E := hpast E hE
+  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t ω => γ ω t) := by
+    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ _ γ G F hw
+    · intro t
+      exact measurable_const.mul
+        ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
+    · intro ω
+      exact (hc ω).const_mul _
+  have hS : MeasurableSet[hσU.measurableSpace] S :=
+    stoppedMassTest_measurable_at_stop _ γ hγa (fun ω => (hw ω).1) _ hσU κ v _ _
+  have hcons (D : Set Ω) (hD : MeasurableSet[hσU.measurableSpace] D) :
+      finiteCenteredLaw P κ B γ G z V D = finiteCenteredLaw P κ B γ G z U D :=
+    finiteCenteredLaw_consistent_at_radius_stop hB hBm hc hB0 κ hκ hκ8 γ G F
+      (Eventually.of_forall hw) z hz U V hU hV _ hσU Filter.EventuallyEq.rfl D hD
+  have hd' : ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
+      finiteCenteredLaw P κ B γ G z V (A ∩ E) ≤ finiteCenteredLaw P κ B γ G z V (A ∩ S) := by
+    rw [hcons _ (hAU.inter hEU), hcons _ (hAU.inter hS)]
+    exact hd.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩))
+  have hwarm' := hwarm z hz u hu V (by linarith) A hA
+  calc
+    _ = ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
+        (ENNReal.ofReal (1 / 2 : ℝ) * finiteCenteredLaw P κ B γ G z V A) := by
+      rw [ENNReal.ofReal_mul (by norm_num)]
+      ac_rfl
+    _ ≤ ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
+        finiteCenteredLaw P κ B γ G z V (A ∩ E) := mul_le_mul' le_rfl hwarm'
+    _ ≤ _ := hd'
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualDenseCoverProbability.lean b/lean/OAI/Probability/SLE/Gauge/ActualDenseCoverProbability.lean
index 42cf04ae5010256be10e4e5be3d6de5cfcf4861e..ac1ce72fdb83adb62ac792a2105b5797f9852e5e 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualDenseCoverProbability.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualDenseCoverProbability.lean
@@ -1,138 +1,138 @@
-import OAI.Probability.SLE.Gauge.ActualMissedCell
-import OAI.Probability.SLE.SausagePointBound
-
-namespace OAI
-
-/-! The missed-cell estimate from finite centered laws. Constants and the
-physical mesh are uniform over interior cutoffs and moving grid centers. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-local instance coverProbabilityDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance coverProbabilityDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance coverProbabilityPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance coverProbabilityPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem uniform_actual_dense_cover_probability
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hγm : ∀ t, Measurable (fun ω => γ ω t))
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W) :
-    ∃ T b c : ℝ, 0 < T ∧ 0 < b ∧ 0 < c ∧ ∃ v : ℕ → ℝ,
-      (∀ n, 0 < v n) ∧ Tendsto v atTop (𝓝 0) ∧
-      ∀ η : ℝ, 0 < η → ∃ D : ℝ, 0 < D ∧
-        ∀ (R : ℝ) (K : Set ℂ), K ⊆ closedBall 0 R → (∀ z ∈ K, η ≤ z.im) →
-        ∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-          P {ω | denseMissedCell κ T b c (v n) n ij (γ ω)} ≤
-            ENNReal.ofReal (D * coverRadius T n ^ codimension κ *
-              Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
-  classical
-  obtain ⟨h, b, C, hh, hh1, hb, hC, δ, hδ, hdense⟩ :=
-    uniform_actual_conditional_dense_test P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw
-      htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW
-  let q := Serpentine.experimentRadiusLevel h
-  have hq : 0 ≤ q := Serpentine.experimentRadiusLevel_nonneg hh (by linarith)
-  let T := 2 + q
-  have hT : 0 < T := by dsimp only [T]; linarith
-  have hgap : 1 + q ≤ T := by dsimp only [T]; linarith
-  let c := 1 / (2 * Real.sqrt C)
-  have hcpos : 0 < c := by dsimp only [c]; positivity
-  have hsmall : C * c ^ 2 ≤ 1 / 4 := by
-    dsimp only [c]
-    rw [div_pow, mul_pow, Real.sq_sqrt hC.le]
-    field_simp [hC.ne']; norm_num
-  let δ' (N : ℕ) := if N = 0 then 1 else δ N
-  have hδ' (N : ℕ) : 0 < δ' N := by
-    dsimp only [δ']
-    split_ifs with hN
-    · norm_num
-    · exact hδ N (Nat.pos_of_ne_zero hN)
-  obtain ⟨v, hv, hvlim, _, _, hvmesh⟩ := exists_cover_physical_mesh T c δ' hδ'
-  refine ⟨T, b, c, hT, hb, hcpos, v, hv, hvlim, ?_⟩
-  intro η hη
-  let I := (2 : ℝ) ^ (-codimension κ) * (η / 2) ^ (-codimension κ)
-  let D := uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ * I
-  have hIC : 0 < uniformInverseAngleConstant κ 1 :=
-    uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
-  have hI : 0 < I := by dsimp only [I]; positivity
-  have hD : 0 < D := by dsimp only [D]; positivity
-  refine ⟨D, hD, ?_⟩
-  intro R K hK hheight
-  have hgeom := eventually_retained_grid_cover_levels hT hη hK hheight
-  have hheight' : ∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-      η / 2 ≤ (coverGridCenter (coverRadius T n) ij).im := by
-    filter_upwards [(coverRadius_tendsto_zero hT).eventually
-      (gt_mem_nhds (show 0 < η / 4 by positivity))] with n hn
-    intro ij hij
-    have hz := (retainedCoverGrid_center_bounds hK hheight hij).2
-    linarith
-  filter_upwards [hgeom, hheight', coverMultiplicity_eventually_ge hcpos 1,
-    denseVisit_probability_eventually_lower (by norm_num : (0 : ℝ) < 1 / 2) hC.le hcpos hsmall,
-    eventually_ge_atTop (1 : ℕ)] with n hngeom hnheight hN hnprob hn1
-  intro ij hij
-  let z := coverGridCenter (coverRadius T n) ij
-  obtain ⟨hz, hlevel, _, hvisit, hmeshsmall⟩ := hngeom ij hij
-  have hNpos : 0 < coverMultiplicity c n := lt_of_lt_of_le Nat.zero_lt_one hN
-  have hV := coverVisitLevel_nonneg_of_gap z T n hvisit
-  have hfailed := actual_dense_window_failure_bound P κ hκ hκ8 B hB hBm hc hB0 γ G F hw
-    z hz T q b (v n) C (coverVisitLevel z T n) hT.le hq hgap hV
-    (coverMultiplicity c n) n hn1 hlevel hnprob ?_
-  · have hmiss := actual_missed_cell_bound_of_centered_failure P κ hκ hκ8 B hB hBm hc hB0
-      γ G F hw htrace T q b c (v n) hT.le hq hgap n ij hz hlevel hvisit hmeshsmall hfailed
-    refine hmiss.trans (ENNReal.ofReal_le_ofReal ?_)
-    have hp : 0 ≤ codimension κ := by dsimp only [codimension, dimension]; linarith
-    have hIz : initialGreen κ z ≤ I :=
-      (initialGreen_le_height κ hκ hκ8 z hz).trans
-        (mul_le_mul_of_nonneg_left
-          (Real.rpow_le_rpow_of_nonpos (half_pos hη) (hnheight ij hij) (neg_nonpos.mpr hp))
-          (Real.rpow_nonneg (by norm_num) _))
-    apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
-    apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (coverRadius_pos T n).le _)
-    exact mul_le_mul_of_nonneg_left hIz
-      (mul_nonneg hIC.le (Real.rpow_nonneg (by positivity) _))
-  · intro j hj A hA
-    have hr := radius_target_eq_coverRadius z hz T j (hlevel j hj)
-    have hr' : (2 * z.im) * Real.exp (-(coverLogLevel z T j + 1)) =
-        Real.exp (-1) * coverRadius T j := by
-      rw [neg_add, Real.exp_add, ← mul_assoc, hr]
-      ring
-    have hcut : v n < δ (coverMultiplicity c n) * ((2 * z.im) *
-        Real.exp (-(coverLogLevel z T j + 1))) := by
-      have hhmesh := hvmesh n j hj
-      rw [hr']
-      dsimp only [δ'] at hhmesh
-      rw [ite_eq_right (Nat.ne_of_gt hNpos)] at hhmesh
-      simpa only [mul_comm] using hhmesh
-    have hend : coverLogLevel z T j + 1 + q ≤ coverVisitLevel z T n := by
-      have he := coverLogLevel_next_gap z T j (hlevel j hj)
-      have hjlast : j + 1 ≤ 2 * n / 3 + 1 :=
-        Nat.add_le_add_right (Finset.mem_Icc.mp hj).2 1
-      have hm := coverLogLevel_monotone z hT.le hjlast
-      linarith
-    have hd := hdense z hz (coverLogLevel z T j) (coverLogLevel_nonneg _ _ _)
-      (coverMultiplicity c n) hNpos (v n) (hv n) hcut (coverVisitLevel z T n) hend A hA
-    simpa only [hr', actualDenseScaleTest, q] using hd
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualMissedCell
+import OAI.Probability.SLE.SausagePointBound
+
+namespace OAI
+
+/-! The missed-cell estimate from finite centered laws. Constants and the
+physical mesh are uniform over interior cutoffs and moving grid centers. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+local instance coverProbabilityDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance coverProbabilityDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance coverProbabilityPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance coverProbabilityPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem uniform_actual_dense_cover_probability
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hγm : ∀ t, Measurable (fun ω => γ ω t))
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W) :
+    ∃ T b c : ℝ, 0 < T ∧ 0 < b ∧ 0 < c ∧ ∃ v : ℕ → ℝ,
+      (∀ n, 0 < v n) ∧ Tendsto v atTop (𝓝 0) ∧
+      ∀ η : ℝ, 0 < η → ∃ D : ℝ, 0 < D ∧
+        ∀ (R : ℝ) (K : Set ℂ), K ⊆ closedBall 0 R → (∀ z ∈ K, η ≤ z.im) →
+        ∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+          P {ω | denseMissedCell κ T b c (v n) n ij (γ ω)} ≤
+            ENNReal.ofReal (D * coverRadius T n ^ codimension κ *
+              Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
+  classical
+  obtain ⟨h, b, C, hh, hh1, hb, hC, δ, hδ, hdense⟩ :=
+    uniform_actual_conditional_dense_test P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw
+      htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW
+  let q := Serpentine.experimentRadiusLevel h
+  have hq : 0 ≤ q := Serpentine.experimentRadiusLevel_nonneg hh (by linarith)
+  let T := 2 + q
+  have hT : 0 < T := by dsimp only [T]; linarith
+  have hgap : 1 + q ≤ T := by dsimp only [T]; linarith
+  let c := 1 / (2 * Real.sqrt C)
+  have hcpos : 0 < c := by dsimp only [c]; positivity
+  have hsmall : C * c ^ 2 ≤ 1 / 4 := by
+    dsimp only [c]
+    rw [div_pow, mul_pow, Real.sq_sqrt hC.le]
+    field_simp [hC.ne']; norm_num
+  let δ' (N : ℕ) := if N = 0 then 1 else δ N
+  have hδ' (N : ℕ) : 0 < δ' N := by
+    dsimp only [δ']
+    split_ifs with hN
+    · norm_num
+    · exact hδ N (Nat.pos_of_ne_zero hN)
+  obtain ⟨v, hv, hvlim, _, _, hvmesh⟩ := exists_cover_physical_mesh T c δ' hδ'
+  refine ⟨T, b, c, hT, hb, hcpos, v, hv, hvlim, ?_⟩
+  intro η hη
+  let I := (2 : ℝ) ^ (-codimension κ) * (η / 2) ^ (-codimension κ)
+  let D := uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ * I
+  have hIC : 0 < uniformInverseAngleConstant κ 1 :=
+    uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
+  have hI : 0 < I := by dsimp only [I]; positivity
+  have hD : 0 < D := by dsimp only [D]; positivity
+  refine ⟨D, hD, ?_⟩
+  intro R K hK hheight
+  have hgeom := eventually_retained_grid_cover_levels hT hη hK hheight
+  have hheight' : ∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+      η / 2 ≤ (coverGridCenter (coverRadius T n) ij).im := by
+    filter_upwards [(coverRadius_tendsto_zero hT).eventually
+      (gt_mem_nhds (show 0 < η / 4 by positivity))] with n hn
+    intro ij hij
+    have hz := (retainedCoverGrid_center_bounds hK hheight hij).2
+    linarith
+  filter_upwards [hgeom, hheight', coverMultiplicity_eventually_ge hcpos 1,
+    denseVisit_probability_eventually_lower (by norm_num : (0 : ℝ) < 1 / 2) hC.le hcpos hsmall,
+    eventually_ge_atTop (1 : ℕ)] with n hngeom hnheight hN hnprob hn1
+  intro ij hij
+  let z := coverGridCenter (coverRadius T n) ij
+  obtain ⟨hz, hlevel, _, hvisit, hmeshsmall⟩ := hngeom ij hij
+  have hNpos : 0 < coverMultiplicity c n := lt_of_lt_of_le Nat.zero_lt_one hN
+  have hV := coverVisitLevel_nonneg_of_gap z T n hvisit
+  have hfailed := actual_dense_window_failure_bound P κ hκ hκ8 B hB hBm hc hB0 γ G F hw
+    z hz T q b (v n) C (coverVisitLevel z T n) hT.le hq hgap hV
+    (coverMultiplicity c n) n hn1 hlevel hnprob ?_
+  · have hmiss := actual_missed_cell_bound_of_centered_failure P κ hκ hκ8 B hB hBm hc hB0
+      γ G F hw htrace T q b c (v n) hT.le hq hgap n ij hz hlevel hvisit hmeshsmall hfailed
+    refine hmiss.trans (ENNReal.ofReal_le_ofReal ?_)
+    have hp : 0 ≤ codimension κ := by dsimp only [codimension, dimension]; linarith
+    have hIz : initialGreen κ z ≤ I :=
+      (initialGreen_le_height κ hκ hκ8 z hz).trans
+        (mul_le_mul_of_nonneg_left
+          (Real.rpow_le_rpow_of_nonpos (half_pos hη) (hnheight ij hij) (neg_nonpos.mpr hp))
+          (Real.rpow_nonneg (by norm_num) _))
+    apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
+    apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (coverRadius_pos T n).le _)
+    exact mul_le_mul_of_nonneg_left hIz
+      (mul_nonneg hIC.le (Real.rpow_nonneg (by positivity) _))
+  · intro j hj A hA
+    have hr := radius_target_eq_coverRadius z hz T j (hlevel j hj)
+    have hr' : (2 * z.im) * Real.exp (-(coverLogLevel z T j + 1)) =
+        Real.exp (-1) * coverRadius T j := by
+      rw [neg_add, Real.exp_add, ← mul_assoc, hr]
+      ring
+    have hcut : v n < δ (coverMultiplicity c n) * ((2 * z.im) *
+        Real.exp (-(coverLogLevel z T j + 1))) := by
+      have hhmesh := hvmesh n j hj
+      rw [hr']
+      dsimp only [δ'] at hhmesh
+      rw [ite_eq_right (Nat.ne_of_gt hNpos)] at hhmesh
+      simpa only [mul_comm] using hhmesh
+    have hend : coverLogLevel z T j + 1 + q ≤ coverVisitLevel z T n := by
+      have he := coverLogLevel_next_gap z T j (hlevel j hj)
+      have hjlast : j + 1 ≤ 2 * n / 3 + 1 :=
+        Nat.add_le_add_right (Finset.mem_Icc.mp hj).2 1
+      have hm := coverLogLevel_monotone z hT.le hjlast
+      linarith
+    have hd := hdense z hz (coverLogLevel z T j) (coverLogLevel_nonneg _ _ _)
+      (coverMultiplicity c n) hNpos (v n) (hv n) hcut (coverVisitLevel z T n) hend A hA
+    simpa only [hr', actualDenseScaleTest, q] using hd
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualDenseExtended.lean b/lean/OAI/Probability/SLE/Gauge/ActualDenseExtended.lean
index d86292c9ae696e4367578fb64ddacff20a6097ec..e1d57fb9d0adc0a41cd82f01221c95b4e560dcf6 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualDenseExtended.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualDenseExtended.lean
@@ -1,88 +1,88 @@
-import OAI.Probability.SLE.Gauge.ActualDenseGoodState
-import OAI.Probability.SLE.Geometry.ConditionalStopExtension
-
-namespace OAI
-
-/-! The proved dense experiment at possibly infinite actual radius stops.
-Only the good-state past event is localized through finite horizons. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-local instance extendedDenseDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance extendedDenseDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance extendedDensePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance extendedDensePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem uniform_actual_dense_at_extended_good_radius_state
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hγm : ∀ t, Measurable (fun ω => γ ω t))
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W)
-    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
-    ∃ h b C : ℝ, 0 < h ∧ h ≤ s / 2 ∧ 0 < b ∧ 0 < C ∧
-      ∃ δ : ℕ → ℝ, (∀ N, 0 < N → 0 < δ N) ∧
-      ∀ (σ : Ω → WithTop ℝ≥0)
-        (hσ : IsStoppingTime (Filtration.natural B
-          (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) σ)
-        (z : ℂ), z ∈ upperHalfPlane → ∀ u : ℝ, 0 ≤ u →
-        ∀ E : Set Ω, MeasurableSet[hσ.measurableSpace] E →
-          (∀ ω ∈ E, σ ω ≠ ⊤) →
-          E ⊆ stoppedMarkedGood κ B γ G (fun ω => (σ ω).untopD 0) z s →
-          (∀ ω ∈ E, fullRadius (G ω) ((σ ω).untopD 0) z = (2 * z.im) * Real.exp (-u)) →
-          ∀ N : ℕ, 0 < N → ∀ v : ℝ, 0 < v →
-            v < δ N * ((2 * z.im) * Real.exp (-u)) →
-            ∀ A : Set Ω, MeasurableSet[hσ.measurableSpace] A →
-              ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
-                finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h) (A ∩ E) ≤
-              finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h)
-                (A ∩ E ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-                  (fun ω => radiusLevelTime (γ ω) (G ω) z (u + Serpentine.experimentRadiusLevel h))
-                  (closedBall z ((2 * z.im) * Real.exp (-u)))
-                  (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
-  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hstep⟩ :=
-    uniform_actual_dense_at_good_radius_state P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw
-      htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW hs hs1
-  refine ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, ?_⟩
-  intro σ hσ z hz u hu E hE hfinite hgood hbase N hN v hv hvcut A hA
-  let Q := finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h)
-  let S := stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-    (fun ω => radiusLevelTime (γ ω) (G ω) z (u + Serpentine.experimentRadiusLevel h))
-    (closedBall z ((2 * z.im) * Real.exp (-u)))
-    (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)
-  let state : Ω → ℝ≥0 → Prop := fun ω t =>
-    ω ∈ stoppedMarkedGood κ B γ G (fun _ => t) z s ∧
-      fullRadius (G ω) t z = (2 * z.im) * Real.exp (-u)
-  apply conditional_event_bound_at_extended_stop Q _
-    (ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2))) S state _ σ hσ (A ∩ E) (hA.inter hE)
-    (Eventually.of_forall (fun ω hω => hfinite ω hω.2))
-  · intro ω hω
-    exact ⟨hgood hω.2, hbase ω hω.2⟩
-  · intro τ hτ D hD hstate
-    have hDgood : D ⊆ stoppedMarkedGood κ B γ G τ z s := fun ω hω => (hstate ω hω).1
-    have hDbase : ∀ ω ∈ D, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u) :=
-      fun ω hω => (hstate ω hω).2
-    simpa only [inter_self] using hstep τ hτ z hz u hu D hD hDgood hDbase N hN v hv hvcut D hD
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualDenseGoodState
+import OAI.Probability.SLE.Geometry.ConditionalStopExtension
+
+namespace OAI
+
+/-! The proved dense experiment at possibly infinite actual radius stops.
+Only the good-state past event is localized through finite horizons. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+local instance extendedDenseDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance extendedDenseDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance extendedDensePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance extendedDensePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem uniform_actual_dense_at_extended_good_radius_state
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hγm : ∀ t, Measurable (fun ω => γ ω t))
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W)
+    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
+    ∃ h b C : ℝ, 0 < h ∧ h ≤ s / 2 ∧ 0 < b ∧ 0 < C ∧
+      ∃ δ : ℕ → ℝ, (∀ N, 0 < N → 0 < δ N) ∧
+      ∀ (σ : Ω → WithTop ℝ≥0)
+        (hσ : IsStoppingTime (Filtration.natural B
+          (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) σ)
+        (z : ℂ), z ∈ upperHalfPlane → ∀ u : ℝ, 0 ≤ u →
+        ∀ E : Set Ω, MeasurableSet[hσ.measurableSpace] E →
+          (∀ ω ∈ E, σ ω ≠ ⊤) →
+          E ⊆ stoppedMarkedGood κ B γ G (fun ω => (σ ω).untopD 0) z s →
+          (∀ ω ∈ E, fullRadius (G ω) ((σ ω).untopD 0) z = (2 * z.im) * Real.exp (-u)) →
+          ∀ N : ℕ, 0 < N → ∀ v : ℝ, 0 < v →
+            v < δ N * ((2 * z.im) * Real.exp (-u)) →
+            ∀ A : Set Ω, MeasurableSet[hσ.measurableSpace] A →
+              ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
+                finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h) (A ∩ E) ≤
+              finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h)
+                (A ∩ E ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+                  (fun ω => radiusLevelTime (γ ω) (G ω) z (u + Serpentine.experimentRadiusLevel h))
+                  (closedBall z ((2 * z.im) * Real.exp (-u)))
+                  (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
+  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hstep⟩ :=
+    uniform_actual_dense_at_good_radius_state P κ hκ hκ8 B hB hBm hc hB0 γ G F hγm hw
+      htrace hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW hs hs1
+  refine ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, ?_⟩
+  intro σ hσ z hz u hu E hE hfinite hgood hbase N hN v hv hvcut A hA
+  let Q := finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h)
+  let S := stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+    (fun ω => radiusLevelTime (γ ω) (G ω) z (u + Serpentine.experimentRadiusLevel h))
+    (closedBall z ((2 * z.im) * Real.exp (-u)))
+    (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)
+  let state : Ω → ℝ≥0 → Prop := fun ω t =>
+    ω ∈ stoppedMarkedGood κ B γ G (fun _ => t) z s ∧
+      fullRadius (G ω) t z = (2 * z.im) * Real.exp (-u)
+  apply conditional_event_bound_at_extended_stop Q _
+    (ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2))) S state _ σ hσ (A ∩ E) (hA.inter hE)
+    (Eventually.of_forall (fun ω hω => hfinite ω hω.2))
+  · intro ω hω
+    exact ⟨hgood hω.2, hbase ω hω.2⟩
+  · intro τ hτ D hD hstate
+    have hDgood : D ⊆ stoppedMarkedGood κ B γ G τ z s := fun ω hω => (hstate ω hω).1
+    have hDbase : ∀ ω ∈ D, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u) :=
+      fun ω hω => (hstate ω hω).2
+    simpa only [inter_self] using hstep τ hτ z hz u hu D hD hDgood hDbase N hN v hv hvcut D hD
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualDenseGoodState.lean b/lean/OAI/Probability/SLE/Gauge/ActualDenseGoodState.lean
index 5d6193761f7ae622e910c3a043404d29c683d213..e4a46825172f13cb5fe662a872f5901595405f73 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualDenseGoodState.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualDenseGoodState.lean
@@ -1,168 +1,168 @@
-import OAI.Probability.SLE.Gauge.ActualDenseIntegration
-import OAI.Probability.SLE.Geometry.FiniteDensityAtRadiusState
-import OAI.Probability.SLE.Geometry.StoppedMarkedStateMeasurability
-import OAI.Probability.SLE.Geometry.FixedCanonicalRestart
-
-namespace OAI
-
-/-! The dense experiment at an actual finite stopping time, conditional on
-its live good-angle radius state. Constants are chosen before the stopping
-time, marked point and stopped-past event. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open FrontSuccessMeasurability
-open Target
-
-local instance actualGoodDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance actualGoodDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance actualGoodPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance actualGoodPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-lemma measurable_actual_radius_level
-    {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) :
-    Measurable (fun ω => radiusLevelTime (γ ω) (G ω) z u) := by
-  have hbp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
-  have hk : Measurable (fun ω => canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω)) :=
-    measurable_canonicalRadiusLevel_random_state κ hbp measurable_const measurable_const
-  have he : (fun ω => canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω)) =
-      (fun ω => radiusLevelTime (γ ω) (G ω) z u) :=
-    funext (fun ω => canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) u hu z)
-  rwa [he] at hk
-
-lemma measurable_actual_stopped_mass_test
-    {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hγm : ∀ t, Measurable (fun ω => γ ω t))
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (v r b : ℝ) :
-    MeasurableSet (stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-      (fun ω => radiusLevelTime (γ ω) (G ω) z u) (closedBall z r) b) := by
-  have hlevel := measurable_actual_radius_level κ B hm hc γ G F hw z u hu
-  have hpath : Measurable (sourceTracePath γ (fun ω => (hw ω).1)) :=
-    source_path_measurable _ hγm
-  exact measurableSet_le measurable_const ((wholeSausageCharge_measurable κ v _).comp
-    (measurable_stoppedAtPath _ hpath _ hlevel))
-
-theorem uniform_actual_dense_at_good_radius_state
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hγm : ∀ t, Measurable (fun ω => γ ω t))
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W)
-    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
-    ∃ h b C : ℝ, 0 < h ∧ h ≤ s / 2 ∧ 0 < b ∧ 0 < C ∧
-      ∃ δ : ℕ → ℝ, (∀ N, 0 < N → 0 < δ N) ∧
-      ∀ (τ : Ω → ℝ≥0)
-        (hτ : IsStoppingTime (Filtration.natural B
-          (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable))
-          (fun ω => (τ ω : WithTop ℝ≥0)))
-        (z : ℂ), z ∈ upperHalfPlane → ∀ u : ℝ, 0 ≤ u →
-        ∀ E : Set Ω, MeasurableSet[hτ.measurableSpace] E →
-          E ⊆ stoppedMarkedGood κ B γ G τ z s →
-          (∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u)) →
-          ∀ N : ℕ, 0 < N → ∀ v : ℝ, 0 < v →
-            v < δ N * ((2 * z.im) * Real.exp (-u)) →
-            ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
-              ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
-                finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h) (A ∩ E) ≤
-              finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h)
-                (A ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-                  (fun ω => radiusLevelTime (γ ω) (G ω) z (u + Serpentine.experimentRadiusLevel h))
-                  (closedBall z ((2 * z.im) * Real.exp (-u)))
-                  (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
-  classical
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  obtain ⟨Φ, Gc, Fc, hΦ, hΦW, hΦγ⟩ :=
-    canonical_witness_original_trace P κ B hm hc γ G F hw htrace
-  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hsections⟩ :=
-    Serpentine.uniform_canonical_conformal_dense_sections P κ hκ hκ8 B hB hBm hc hB0 γ G F hw
-      Φ hΦ hΦγ hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW
-      (1 / 16) (by norm_num) (by norm_num) hs hs1
-  refine ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, ?_⟩
-  intro τ hτ z hz u hu E hE hgood hbase N hN v hv hvcut A hA
-  let a := stoppedMarkedScale κ B γ G τ z
-  let w := stoppedMarkedUnit κ B γ G τ z
-  let Y := normalizedRestartDriverPath B hc τ a
-  let q := Serpentine.experimentRadiusLevel h
-  have hq : 0 < q := by
-    dsimp [q, Serpentine.experimentRadiusLevel]
-    have hlog : Real.log (h / 80) < 0 := Real.log_neg (by positivity) (by linarith)
-    linarith
-  have ham : Measurable[hτ.measurableSpace] a :=
-    source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z
-  have ha : ∀ ω, 0 < a ω := stoppedMarkedScale_pos κ B γ G τ z
-  have hwm : Measurable[hτ.measurableSpace] w :=
-    source_stoppedMarkedUnit_measurable B hm hc κ γ G F hw τ hτ z
-  obtain ⟨Fb, hFb, hYm, _, hFbEq, _, hYlaw, hrestart⟩ :=
-    source_normalized_restart_with_factor κ hκ hκ8 B hB hm hc γ G F hw Φ hΦ Gc Fc hΦW
-      hboundary τ hτ a ham ha
-  have hlive (ω : Ω) (hω : ω ∈ E) : z ∈ survivingDomain (γ ω) (τ ω) :=
-    (hgood hω).1
-  have hgeometry (ω : Ω) (hω : ω ∈ E) :=
-    source_stoppedMarkedState_geometry B hc κ γ G F hw τ z ω (hlive ω hω)
-  have hpoint (ω : Ω) (hω : ω ∈ E) :
-      F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ)) = z := by
-    simpa only [normalizedInverse, add_comm] using (hgeometry ω hω).2.2.1
-  have hcanonical (ω : Ω) (hω : ω ∈ E) :
-      ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) ≤
-      ((P.map (sourceDriverPath B hc)).withDensity
-        (fun p => canonicalFiniteDensity κ (q, w ω, p)))
-        (canonicalDenseEvent κ Φ (Fb ω) q (w ω) b
-          ((2 * z.im) * Real.exp (-u)) N v z) := by
-    let f := normalizedInverse (F ω (τ ω)) (Real.sqrt κ * B (τ ω) ω) (a ω : ℝ)
-    have hgeo := hgeometry ω hω
-    have hchart (ζ : closedHalfPlane) (hζ : (ζ : ℂ) ∈ upperHalfPlane) : Fb ω ζ = f ζ := by
-      simpa only [f, normalizedInverse, add_comm] using hFbEq ω ζ hζ
-    have hrad : 2 * (w ω).im * ‖deriv f (w ω)‖ = (2 * z.im) * Real.exp (-u) :=
-      hgeo.2.2.2.2.2.2.trans (hbase ω hω)
-    have hv' : v < δ N * (2 * (w ω).im * ‖deriv f (w ω)‖) := by rwa [hrad]
-    have hh := hsections f (Fb ω) (w ω) hgeo.2.2.2.1 hgeo.2.2.2.2.1 hchart
-      hgeo.1 (hgood hω).2 hgeo.2.2.2.2.2.1 N hN v hv hv'
-    have hfw : f (w ω) = z := hgeo.2.2.1
-    simpa only [q, hfw, hrad] using hh
-  obtain ⟨M, hM, hMfinite, hMactual, hMpast⟩ :=
-    exists_finiteDensity_at_radius_state hB hBm hc hB0 κ hκ hκ8 γ G F hw
-      τ hτ z hz u q hu hq E hE hlive hbase
-  have hS : MeasurableSet (stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-      (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
-      (closedBall z ((2 * z.im) * Real.exp (-u)))
-      (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)) :=
-    measurable_actual_stopped_mass_test κ B hm hc γ G F hγm hw z (u + q) (by linarith) v _ _
-  have h := actual_dense_sections_integrate P (P.map (sourceDriverPath B hc))
-    hτ.measurableSpace hτ.measurableSpace_le κ B hc γ G F hw Y hYm hYlaw Φ hΦ Fb hFb
-    τ a ha (fun ω => Gc (Y ω)) (fun ω => Fc (Y ω))
-    (hrestart.mono (fun _ hω => hω.1))
-    (fun ω => normalizedRestartDriverPath_driver κ B hc τ a ω)
-    (hrestart.mono (fun _ hω t => (hω.2 t).symm)) w hwm E hE
-    (fun ω hω => (hgeometry ω hω).2.1) z u q b ((2 * z.im) * Real.exp (-u)) v hq.le N
-    hpoint hbase M hM hMfinite hMactual
-    (ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2))) ENNReal.ofReal_ne_top hcanonical hS A hA
-  rw [hMpast A hA] at h
-  exact h
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualDenseIntegration
+import OAI.Probability.SLE.Geometry.FiniteDensityAtRadiusState
+import OAI.Probability.SLE.Geometry.StoppedMarkedStateMeasurability
+import OAI.Probability.SLE.Geometry.FixedCanonicalRestart
+
+namespace OAI
+
+/-! The dense experiment at an actual finite stopping time, conditional on
+its live good-angle radius state. Constants are chosen before the stopping
+time, marked point and stopped-past event. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open FrontSuccessMeasurability
+open Target
+
+local instance actualGoodDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance actualGoodDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance actualGoodPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance actualGoodPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+lemma measurable_actual_radius_level
+    {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) :
+    Measurable (fun ω => radiusLevelTime (γ ω) (G ω) z u) := by
+  have hbp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
+  have hk : Measurable (fun ω => canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω)) :=
+    measurable_canonicalRadiusLevel_random_state κ hbp measurable_const measurable_const
+  have he : (fun ω => canonicalRadiusLevel κ (u, z, sourceDriverPath B hc ω)) =
+      (fun ω => radiusLevelTime (γ ω) (G ω) z u) :=
+    funext (fun ω => canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) u hu z)
+  rwa [he] at hk
+
+lemma measurable_actual_stopped_mass_test
+    {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hγm : ∀ t, Measurable (fun ω => γ ω t))
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (v r b : ℝ) :
+    MeasurableSet (stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+      (fun ω => radiusLevelTime (γ ω) (G ω) z u) (closedBall z r) b) := by
+  have hlevel := measurable_actual_radius_level κ B hm hc γ G F hw z u hu
+  have hpath : Measurable (sourceTracePath γ (fun ω => (hw ω).1)) :=
+    source_path_measurable _ hγm
+  exact measurableSet_le measurable_const ((wholeSausageCharge_measurable κ v _).comp
+    (measurable_stoppedAtPath _ hpath _ hlevel))
+
+theorem uniform_actual_dense_at_good_radius_state
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hγm : ∀ t, Measurable (fun ω => γ ω t))
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W)
+    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
+    ∃ h b C : ℝ, 0 < h ∧ h ≤ s / 2 ∧ 0 < b ∧ 0 < C ∧
+      ∃ δ : ℕ → ℝ, (∀ N, 0 < N → 0 < δ N) ∧
+      ∀ (τ : Ω → ℝ≥0)
+        (hτ : IsStoppingTime (Filtration.natural B
+          (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable))
+          (fun ω => (τ ω : WithTop ℝ≥0)))
+        (z : ℂ), z ∈ upperHalfPlane → ∀ u : ℝ, 0 ≤ u →
+        ∀ E : Set Ω, MeasurableSet[hτ.measurableSpace] E →
+          E ⊆ stoppedMarkedGood κ B γ G τ z s →
+          (∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u)) →
+          ∀ N : ℕ, 0 < N → ∀ v : ℝ, 0 < v →
+            v < δ N * ((2 * z.im) * Real.exp (-u)) →
+            ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
+              ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) *
+                finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h) (A ∩ E) ≤
+              finiteCenteredLaw P κ B γ G z (u + Serpentine.experimentRadiusLevel h)
+                (A ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+                  (fun ω => radiusLevelTime (γ ω) (G ω) z (u + Serpentine.experimentRadiusLevel h))
+                  (closedBall z ((2 * z.im) * Real.exp (-u)))
+                  (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
+  classical
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  obtain ⟨Φ, Gc, Fc, hΦ, hΦW, hΦγ⟩ :=
+    canonical_witness_original_trace P κ B hm hc γ G F hw htrace
+  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hsections⟩ :=
+    Serpentine.uniform_canonical_conformal_dense_sections P κ hκ hκ8 B hB hBm hc hB0 γ G F hw
+      Φ hΦ hΦγ hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW
+      (1 / 16) (by norm_num) (by norm_num) hs hs1
+  refine ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, ?_⟩
+  intro τ hτ z hz u hu E hE hgood hbase N hN v hv hvcut A hA
+  let a := stoppedMarkedScale κ B γ G τ z
+  let w := stoppedMarkedUnit κ B γ G τ z
+  let Y := normalizedRestartDriverPath B hc τ a
+  let q := Serpentine.experimentRadiusLevel h
+  have hq : 0 < q := by
+    dsimp [q, Serpentine.experimentRadiusLevel]
+    have hlog : Real.log (h / 80) < 0 := Real.log_neg (by positivity) (by linarith)
+    linarith
+  have ham : Measurable[hτ.measurableSpace] a :=
+    source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z
+  have ha : ∀ ω, 0 < a ω := stoppedMarkedScale_pos κ B γ G τ z
+  have hwm : Measurable[hτ.measurableSpace] w :=
+    source_stoppedMarkedUnit_measurable B hm hc κ γ G F hw τ hτ z
+  obtain ⟨Fb, hFb, hYm, _, hFbEq, _, hYlaw, hrestart⟩ :=
+    source_normalized_restart_with_factor κ hκ hκ8 B hB hm hc γ G F hw Φ hΦ Gc Fc hΦW
+      hboundary τ hτ a ham ha
+  have hlive (ω : Ω) (hω : ω ∈ E) : z ∈ survivingDomain (γ ω) (τ ω) :=
+    (hgood hω).1
+  have hgeometry (ω : Ω) (hω : ω ∈ E) :=
+    source_stoppedMarkedState_geometry B hc κ γ G F hw τ z ω (hlive ω hω)
+  have hpoint (ω : Ω) (hω : ω ∈ E) :
+      F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ)) = z := by
+    simpa only [normalizedInverse, add_comm] using (hgeometry ω hω).2.2.1
+  have hcanonical (ω : Ω) (hω : ω ∈ E) :
+      ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2)) ≤
+      ((P.map (sourceDriverPath B hc)).withDensity
+        (fun p => canonicalFiniteDensity κ (q, w ω, p)))
+        (canonicalDenseEvent κ Φ (Fb ω) q (w ω) b
+          ((2 * z.im) * Real.exp (-u)) N v z) := by
+    let f := normalizedInverse (F ω (τ ω)) (Real.sqrt κ * B (τ ω) ω) (a ω : ℝ)
+    have hgeo := hgeometry ω hω
+    have hchart (ζ : closedHalfPlane) (hζ : (ζ : ℂ) ∈ upperHalfPlane) : Fb ω ζ = f ζ := by
+      simpa only [f, normalizedInverse, add_comm] using hFbEq ω ζ hζ
+    have hrad : 2 * (w ω).im * ‖deriv f (w ω)‖ = (2 * z.im) * Real.exp (-u) :=
+      hgeo.2.2.2.2.2.2.trans (hbase ω hω)
+    have hv' : v < δ N * (2 * (w ω).im * ‖deriv f (w ω)‖) := by rwa [hrad]
+    have hh := hsections f (Fb ω) (w ω) hgeo.2.2.2.1 hgeo.2.2.2.2.1 hchart
+      hgeo.1 (hgood hω).2 hgeo.2.2.2.2.2.1 N hN v hv hv'
+    have hfw : f (w ω) = z := hgeo.2.2.1
+    simpa only [q, hfw, hrad] using hh
+  obtain ⟨M, hM, hMfinite, hMactual, hMpast⟩ :=
+    exists_finiteDensity_at_radius_state hB hBm hc hB0 κ hκ hκ8 γ G F hw
+      τ hτ z hz u q hu hq E hE hlive hbase
+  have hS : MeasurableSet (stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+      (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
+      (closedBall z ((2 * z.im) * Real.exp (-u)))
+      (b * ((2 * z.im) * Real.exp (-u)) ^ dimension κ * (N : ℝ) ^ codimension κ)) :=
+    measurable_actual_stopped_mass_test κ B hm hc γ G F hγm hw z (u + q) (by linarith) v _ _
+  have h := actual_dense_sections_integrate P (P.map (sourceDriverPath B hc))
+    hτ.measurableSpace hτ.measurableSpace_le κ B hc γ G F hw Y hYm hYlaw Φ hΦ Fb hFb
+    τ a ha (fun ω => Gc (Y ω)) (fun ω => Fc (Y ω))
+    (hrestart.mono (fun _ hω => hω.1))
+    (fun ω => normalizedRestartDriverPath_driver κ B hc τ a ω)
+    (hrestart.mono (fun _ hω t => (hω.2 t).symm)) w hwm E hE
+    (fun ω hω => (hgeometry ω hω).2.1) z u q b ((2 * z.im) * Real.exp (-u)) v hq.le N
+    hpoint hbase M hM hMfinite hMactual
+    (ENNReal.ofReal (Real.exp (-C * (N : ℝ) ^ 2))) ENNReal.ofReal_ne_top hcanonical hS A hA
+  rw [hMpast A hA] at h
+  exact h
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualDenseIntegration.lean b/lean/OAI/Probability/SLE/Gauge/ActualDenseIntegration.lean
index bf74a648255e341df5bd3789eb186694ba734c52..cb72ecbb9cf7e04a62f86e64bab338cf7187682a 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualDenseIntegration.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualDenseIntegration.lean
@@ -1,94 +1,94 @@
-import OAI.Probability.SLE.Gauge.CanonicalDenseIntegration
-import OAI.Probability.SLE.Gauge.RestartDenseWeight
-
-namespace OAI
-
-/-! Actual physical stopped-mass probabilities from the proved canonical
-sections and the actual normalized restarted witnesses. All restrictions on
-the old marked state lie in the stopped past. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open FrontSuccessMeasurability
-
-local instance actualDenseIntegrationRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance actualDenseIntegrationRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance actualDenseIntegrationComplexMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance actualDenseIntegrationComplexBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem actual_dense_sections_integrate
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (ν : Measure C(ℝ≥0, ℝ)) [IsFiniteMeasure ν]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (Y : Ω → C(ℝ≥0, ℝ)) (hY : Measurable[mΩ] Y)
-    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
-      @Measure.map Ω C(ℝ≥0, ℝ) mΩ _ Y (P.restrict A) = P A • ν)
-    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
-    (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable[m] Fb)
-    (τ a : Ω → ℝ≥0) (ha : ∀ ω, 0 < a ω)
-    (Gc Fc : Ω → ℝ≥0 → ℂ → ℂ)
-    (hb : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (normalizedRestartDriver (fun t => Real.sqrt κ * B t ω) (τ ω) (a ω))
-      (Φ (Y ω)) (Gc ω) (Fc ω))
-    (hdriver : ∀ ω, (fun t => Real.sqrt κ * Y ω t) =
-      normalizedRestartDriver (fun t => Real.sqrt κ * B t ω) (τ ω) (a ω))
-    (hpath : ∀ᵐ ω ∂P, ∀ t, finiteBoundaryPath (Fb ω) (Φ (Y ω)) t =
-      γ ω (τ ω + (a ω) ^ 2 * t))
-    (w : Ω → ℂ) (hwm : Measurable[m] w)
-    (E : Set Ω) (hE : MeasurableSet[m] E) (hwH : ∀ ω ∈ E, w ω ∈ upperHalfPlane)
-    (z : ℂ) (u q b r v : ℝ) (hq : 0 ≤ q) (N : ℕ)
-    (hpoint : ∀ ω ∈ E,
-      F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ)) = z)
-    (hbase : ∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u))
-    (M : Ω → ℝ≥0∞) (hM : Measurable[m] M) (hMfinite : ∀ ω, M ω ≠ ∞)
-    (hMactual : ∀ᵐ ω ∂P, ω ∈ E → M ω = ENNReal.ofReal
-      (greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (τ ω) z / initialGreen κ z))
-    (p : ℝ≥0∞) (hp : p ≠ ∞)
-    (hsections : ∀ ω ∈ E, p ≤
-      (ν.withDensity (fun y => canonicalFiniteDensity κ (q, w ω, y)))
-        (canonicalDenseEvent κ Φ (Fb ω) q (w ω) b r N v z))
-    (hS : MeasurableSet[mΩ] (stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-      (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
-      (closedBall z r) (b * r ^ dimension κ * (N : ℝ) ^ codimension κ)))
-    (A : Set Ω) (hA : MeasurableSet[m] A) :
-    p * (∫⁻ ω in A ∩ E, M ω ∂P) ≤
-      @finiteCenteredLaw Ω mΩ P κ B γ G z (u + q)
-        (A ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-          (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
-          (closedBall z r) (b * r ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
-  classical
-  let : MeasurableSpace Ω := mΩ
-  let S := stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
-    (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
-    (closedBall z r) (b * r ^ dimension κ * (N : ℝ) ^ codimension κ)
-  let Z : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal
-    (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z (u + q) / initialGreen κ z)
-  apply canonical_dense_sections_integrate P ν m hm Y hY hlaw κ b r v q N z Φ hΦ Fb hFb
-    w hwm E hE M hM hMfinite p hp hsections Z S hS _ A hA
-  filter_upwards [hb, hpath, hMactual] with ω hω hpathω hMω hωE
-  have hpt := hpoint ω hωE
-  have hbaseω : fullRadius (G ω) (τ ω)
-      (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))) =
-      (2 * (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))).im) *
-        Real.exp (-u) := by simpa only [hpt] using hbase ω hωE
-  have hbound := (hw ω).normalized_restart_dense_weight_le ((hc ω).const_mul _) (τ ω) (a ω)
-    (ha ω) κ Φ (Y ω) hω (hdriver ω) (w ω) (hwH ω hωE) u q hq hbaseω
-    (Fb ω) hpathω b r v N
-  rw [hpt, ← hMω hωE] at hbound
-  split_ifs at hbound with hs
-  · rw [indicator_of_mem (show ω ∈ S from hs)]
-    exact hbound
-  · rw [indicator_of_notMem (show ω ∉ S from hs)]
-    exact hbound
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CanonicalDenseIntegration
+import OAI.Probability.SLE.Gauge.RestartDenseWeight
+
+namespace OAI
+
+/-! Actual physical stopped-mass probabilities from the proved canonical
+sections and the actual normalized restarted witnesses. All restrictions on
+the old marked state lie in the stopped past. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open FrontSuccessMeasurability
+
+local instance actualDenseIntegrationRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance actualDenseIntegrationRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance actualDenseIntegrationComplexMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance actualDenseIntegrationComplexBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem actual_dense_sections_integrate
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (ν : Measure C(ℝ≥0, ℝ)) [IsFiniteMeasure ν]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (Y : Ω → C(ℝ≥0, ℝ)) (hY : Measurable[mΩ] Y)
+    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
+      @Measure.map Ω C(ℝ≥0, ℝ) mΩ _ Y (P.restrict A) = P A • ν)
+    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
+    (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable[m] Fb)
+    (τ a : Ω → ℝ≥0) (ha : ∀ ω, 0 < a ω)
+    (Gc Fc : Ω → ℝ≥0 → ℂ → ℂ)
+    (hb : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (normalizedRestartDriver (fun t => Real.sqrt κ * B t ω) (τ ω) (a ω))
+      (Φ (Y ω)) (Gc ω) (Fc ω))
+    (hdriver : ∀ ω, (fun t => Real.sqrt κ * Y ω t) =
+      normalizedRestartDriver (fun t => Real.sqrt κ * B t ω) (τ ω) (a ω))
+    (hpath : ∀ᵐ ω ∂P, ∀ t, finiteBoundaryPath (Fb ω) (Φ (Y ω)) t =
+      γ ω (τ ω + (a ω) ^ 2 * t))
+    (w : Ω → ℂ) (hwm : Measurable[m] w)
+    (E : Set Ω) (hE : MeasurableSet[m] E) (hwH : ∀ ω ∈ E, w ω ∈ upperHalfPlane)
+    (z : ℂ) (u q b r v : ℝ) (hq : 0 ≤ q) (N : ℕ)
+    (hpoint : ∀ ω ∈ E,
+      F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ)) = z)
+    (hbase : ∀ ω ∈ E, fullRadius (G ω) (τ ω) z = (2 * z.im) * Real.exp (-u))
+    (M : Ω → ℝ≥0∞) (hM : Measurable[m] M) (hMfinite : ∀ ω, M ω ≠ ∞)
+    (hMactual : ∀ᵐ ω ∂P, ω ∈ E → M ω = ENNReal.ofReal
+      (greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (τ ω) z / initialGreen κ z))
+    (p : ℝ≥0∞) (hp : p ≠ ∞)
+    (hsections : ∀ ω ∈ E, p ≤
+      (ν.withDensity (fun y => canonicalFiniteDensity κ (q, w ω, y)))
+        (canonicalDenseEvent κ Φ (Fb ω) q (w ω) b r N v z))
+    (hS : MeasurableSet[mΩ] (stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+      (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
+      (closedBall z r) (b * r ^ dimension κ * (N : ℝ) ^ codimension κ)))
+    (A : Set Ω) (hA : MeasurableSet[m] A) :
+    p * (∫⁻ ω in A ∩ E, M ω ∂P) ≤
+      @finiteCenteredLaw Ω mΩ P κ B γ G z (u + q)
+        (A ∩ stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+          (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
+          (closedBall z r) (b * r ^ dimension κ * (N : ℝ) ^ codimension κ)) := by
+  classical
+  let : MeasurableSpace Ω := mΩ
+  let S := stoppedMassTest κ v (sourceTracePath γ (fun ω => (hw ω).1))
+    (fun ω => radiusLevelTime (γ ω) (G ω) z (u + q))
+    (closedBall z r) (b * r ^ dimension κ * (N : ℝ) ^ codimension κ)
+  let Z : Ω → ℝ≥0∞ := fun ω => ENNReal.ofReal
+    (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z (u + q) / initialGreen κ z)
+  apply canonical_dense_sections_integrate P ν m hm Y hY hlaw κ b r v q N z Φ hΦ Fb hFb
+    w hwm E hE M hM hMfinite p hp hsections Z S hS _ A hA
+  filter_upwards [hb, hpath, hMactual] with ω hω hpathω hMω hωE
+  have hpt := hpoint ω hωE
+  have hbaseω : fullRadius (G ω) (τ ω)
+      (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))) =
+      (2 * (F ω (τ ω) ((a ω : ℂ) * w ω + (Real.sqrt κ * B (τ ω) ω : ℝ))).im) *
+        Real.exp (-u) := by simpa only [hpt] using hbase ω hωE
+  have hbound := (hw ω).normalized_restart_dense_weight_le ((hc ω).const_mul _) (τ ω) (a ω)
+    (ha ω) κ Φ (Y ω) hω (hdriver ω) (w ω) (hwH ω hωE) u q hq hbaseω
+    (Fb ω) hpathω b r v N
+  rw [hpt, ← hMω hωE] at hbound
+  split_ifs at hbound with hs
+  · rw [indicator_of_mem (show ω ∈ S from hs)]
+    exact hbound
+  · rw [indicator_of_notMem (show ω ∉ S from hs)]
+    exact hbound
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualDenseWindow.lean b/lean/OAI/Probability/SLE/Gauge/ActualDenseWindow.lean
index a268f5f07583b8983a7cfa278960f6f52a32040c..8c580bcc25e01441e24f8a9e105c69ba25ec923f 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualDenseWindow.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualDenseWindow.lean
@@ -1,144 +1,144 @@
-import OAI.Probability.SLE.Gauge.ActualDenseConditional
-import OAI.Probability.SLE.Gauge.WindowDenseTests
-import OAI.Probability.SLE.Gauge.CoverRadiusLevels
-
-namespace OAI
-
-/-! Repeated actual dense tests on the retained scale window. All tests
-are evaluated under one later finite centered law. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance actualWindowPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance actualWindowPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-def actualDenseScaleTest {Ω : Type*} (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
-    (hc : ∀ ω, Continuous (γ ω)) (G : Ω → ℝ≥0 → ℂ → ℂ)
-    (z : ℂ) (T q b v : ℝ) (N j : ℕ) : Set Ω :=
-  stoppedMassTest κ v (sourceTracePath γ hc)
-    (fun ω => radiusLevelTime (γ ω) (G ω) z (coverLogLevel z T j + 1 + q))
-    (closedBall z (Real.exp (-1) * coverRadius T j))
-    (b * (Real.exp (-1) * coverRadius T j) ^ dimension κ * (N : ℝ) ^ codimension κ)
-
-def actualDenseWindowFailure {Ω : Type*} (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
-    (hc : ∀ ω, Continuous (γ ω)) (G : Ω → ℝ≥0 → ℂ → ℂ)
-    (z : ℂ) (T q b v : ℝ) (N n : ℕ) : Set Ω :=
-  ⋂ j ∈ coverTestWindow n, (actualDenseScaleTest κ γ hc G z T q b v N j)ᶜ
-
-lemma coverLogLevel_next_gap (z : ℂ) (T : ℝ) (j : ℕ)
-    (hj : coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) :
-    coverLogLevel z T j + T ≤ coverLogLevel z T (j + 1) := by
-  rw [hj]
-  apply le_trans _ (le_max_right _ _)
-  simp only [Nat.cast_add, Nat.cast_one]
-  linarith
-
-theorem actual_dense_window_failure_bound
-    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (T q b v C V : ℝ)
-    (hT : 0 ≤ T) (hq : 0 ≤ q) (hgap : 1 + q ≤ T) (hV : 0 ≤ V)
-    (N n : ℕ) (hn : 1 ≤ n)
-    (hlevel : ∀ j ∈ coverTestWindow n,
-      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ))
-    (hprob : 1 / Real.sqrt (n : ℝ) ≤ (1 / 2) * Real.exp (-C * (N : ℝ) ^ 2))
-    (hdense : ∀ j ∈ coverTestWindow n, ∀ A : Set Ω,
-      MeasurableSet[(actual_radiusLevelTime_stopping hB
-        (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
-        κ hκ hκ8 γ G F hw z hz (coverLogLevel z T j) (coverLogLevel_nonneg _ _ _)).measurableSpace] A →
-      ENNReal.ofReal ((1 / 2) * Real.exp (-C * (N : ℝ) ^ 2)) *
-        finiteCenteredLaw P κ B γ G z V A ≤
-      finiteCenteredLaw P κ B γ G z V
-        (A ∩ actualDenseScaleTest κ γ (fun ω => (hw ω).1) G z T q b v N j)) :
-    finiteCenteredLaw P κ B γ G z V
-      (actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G z T q b v N n) ≤
-      ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let τ (j : ℕ) (ω : Ω) : WithTop ℝ≥0 :=
-    radiusLevelTime (γ ω) (G ω) z (coverLogLevel z T j)
-  have hτ (j : ℕ) : IsStoppingTime ℱ (τ j) :=
-    actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz _
-      (coverLogLevel_nonneg _ _ _)
-  have hmono (ω : Ω) : Monotone (fun j => τ j ω) := by
-    intro i j hij
-    exact radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z
-      (coverLogLevel_nonneg _ _ _) (coverLogLevel_monotone z hT hij)
-  let ℱc := clockStopFiltration ℱ τ hτ hmono
-  have hγa : Adapted ℱ (fun t ω => γ ω t) := by
-    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ _ γ G F hw
-    · intro t
-      exact measurable_const.mul
-        ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
-    · intro ω
-      exact (hc ω).const_mul _
-  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw) z hz V hV
-  apply cover_window_failure_bound_on_window (finiteCenteredLaw P κ B γ G z V) ℱc
-    (actualDenseScaleTest κ γ (fun ω => (hw ω).1) G z T q b v N) n hn
-  · intro j hj
-    have hend : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T (j + 1) :=
-      (by linarith : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T j + T).trans
-        (coverLogLevel_next_gap z T j (hlevel j hj))
-    have hstop := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
-      z hz (coverLogLevel z T j + 1 + q) (by linarith [coverLogLevel_nonneg z T j])
-    exact (actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
-      z hz _ _ (by linarith [coverLogLevel_nonneg z T j]) hend)
-      _ (stoppedMassTest_measurable_at_stop ℱ γ hγa (fun ω => (hw ω).1)
-        _ hstop κ v _ _)
-  · intro j hj A hA
-    exact (mul_le_mul' (ENNReal.ofReal_le_ofReal hprob) le_rfl).trans (hdense j hj A hA)
-
-theorem actual_dense_window_failure_measurable_at_end
-    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (T q b v : ℝ)
-    (hT : 0 ≤ T) (hq : 0 ≤ q) (hgap : 1 + q ≤ T) (N n : ℕ)
-    (hlevel : ∀ j ∈ coverTestWindow n,
-      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) :
-    MeasurableSet[(actual_radiusLevelTime_stopping hB
-      (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
-      κ hκ hκ8 γ G F hw z hz (coverLogLevel z T (2 * n / 3 + 1))
-      (coverLogLevel_nonneg _ _ _)).measurableSpace]
-      (actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G z T q b v N n) := by
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have hγa : Adapted ℱ (fun t ω => γ ω t) := by
-    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ _ γ G F hw
-    · intro t
-      exact measurable_const.mul
-        ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
-    · intro ω
-      exact (hc ω).const_mul _
-  apply MeasurableSet.biInter (coverTestWindow n).countable_toSet
-  intro j hj
-  apply MeasurableSet.compl
-  have hend : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T (2 * n / 3 + 1) := by
-    have hj' : j + 1 ≤ 2 * n / 3 + 1 := Nat.add_le_add_right (Finset.mem_Icc.mp hj).2 1
-    exact (by linarith : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T j + T).trans
-      ((coverLogLevel_next_gap z T j (hlevel j hj)).trans (coverLogLevel_monotone z hT hj'))
-  have hstop := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
-    z hz (coverLogLevel z T j + 1 + q) (by linarith [coverLogLevel_nonneg z T j])
-  exact (actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
-    z hz _ _ (by linarith [coverLogLevel_nonneg z T j]) hend)
-    _ (stoppedMassTest_measurable_at_stop ℱ γ hγa (fun ω => (hw ω).1)
-      _ hstop κ v _ _)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualDenseConditional
+import OAI.Probability.SLE.Gauge.WindowDenseTests
+import OAI.Probability.SLE.Gauge.CoverRadiusLevels
+
+namespace OAI
+
+/-! Repeated actual dense tests on the retained scale window. All tests
+are evaluated under one later finite centered law. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance actualWindowPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance actualWindowPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+def actualDenseScaleTest {Ω : Type*} (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
+    (hc : ∀ ω, Continuous (γ ω)) (G : Ω → ℝ≥0 → ℂ → ℂ)
+    (z : ℂ) (T q b v : ℝ) (N j : ℕ) : Set Ω :=
+  stoppedMassTest κ v (sourceTracePath γ hc)
+    (fun ω => radiusLevelTime (γ ω) (G ω) z (coverLogLevel z T j + 1 + q))
+    (closedBall z (Real.exp (-1) * coverRadius T j))
+    (b * (Real.exp (-1) * coverRadius T j) ^ dimension κ * (N : ℝ) ^ codimension κ)
+
+def actualDenseWindowFailure {Ω : Type*} (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
+    (hc : ∀ ω, Continuous (γ ω)) (G : Ω → ℝ≥0 → ℂ → ℂ)
+    (z : ℂ) (T q b v : ℝ) (N n : ℕ) : Set Ω :=
+  ⋂ j ∈ coverTestWindow n, (actualDenseScaleTest κ γ hc G z T q b v N j)ᶜ
+
+lemma coverLogLevel_next_gap (z : ℂ) (T : ℝ) (j : ℕ)
+    (hj : coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) :
+    coverLogLevel z T j + T ≤ coverLogLevel z T (j + 1) := by
+  rw [hj]
+  apply le_trans _ (le_max_right _ _)
+  simp only [Nat.cast_add, Nat.cast_one]
+  linarith
+
+theorem actual_dense_window_failure_bound
+    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (T q b v C V : ℝ)
+    (hT : 0 ≤ T) (hq : 0 ≤ q) (hgap : 1 + q ≤ T) (hV : 0 ≤ V)
+    (N n : ℕ) (hn : 1 ≤ n)
+    (hlevel : ∀ j ∈ coverTestWindow n,
+      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ))
+    (hprob : 1 / Real.sqrt (n : ℝ) ≤ (1 / 2) * Real.exp (-C * (N : ℝ) ^ 2))
+    (hdense : ∀ j ∈ coverTestWindow n, ∀ A : Set Ω,
+      MeasurableSet[(actual_radiusLevelTime_stopping hB
+        (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
+        κ hκ hκ8 γ G F hw z hz (coverLogLevel z T j) (coverLogLevel_nonneg _ _ _)).measurableSpace] A →
+      ENNReal.ofReal ((1 / 2) * Real.exp (-C * (N : ℝ) ^ 2)) *
+        finiteCenteredLaw P κ B γ G z V A ≤
+      finiteCenteredLaw P κ B γ G z V
+        (A ∩ actualDenseScaleTest κ γ (fun ω => (hw ω).1) G z T q b v N j)) :
+    finiteCenteredLaw P κ B γ G z V
+      (actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G z T q b v N n) ≤
+      ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let τ (j : ℕ) (ω : Ω) : WithTop ℝ≥0 :=
+    radiusLevelTime (γ ω) (G ω) z (coverLogLevel z T j)
+  have hτ (j : ℕ) : IsStoppingTime ℱ (τ j) :=
+    actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz _
+      (coverLogLevel_nonneg _ _ _)
+  have hmono (ω : Ω) : Monotone (fun j => τ j ω) := by
+    intro i j hij
+    exact radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z
+      (coverLogLevel_nonneg _ _ _) (coverLogLevel_monotone z hT hij)
+  let ℱc := clockStopFiltration ℱ τ hτ hmono
+  have hγa : Adapted ℱ (fun t ω => γ ω t) := by
+    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ _ γ G F hw
+    · intro t
+      exact measurable_const.mul
+        ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
+    · intro ω
+      exact (hc ω).const_mul _
+  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw) z hz V hV
+  apply cover_window_failure_bound_on_window (finiteCenteredLaw P κ B γ G z V) ℱc
+    (actualDenseScaleTest κ γ (fun ω => (hw ω).1) G z T q b v N) n hn
+  · intro j hj
+    have hend : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T (j + 1) :=
+      (by linarith : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T j + T).trans
+        (coverLogLevel_next_gap z T j (hlevel j hj))
+    have hstop := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
+      z hz (coverLogLevel z T j + 1 + q) (by linarith [coverLogLevel_nonneg z T j])
+    exact (actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
+      z hz _ _ (by linarith [coverLogLevel_nonneg z T j]) hend)
+      _ (stoppedMassTest_measurable_at_stop ℱ γ hγa (fun ω => (hw ω).1)
+        _ hstop κ v _ _)
+  · intro j hj A hA
+    exact (mul_le_mul' (ENNReal.ofReal_le_ofReal hprob) le_rfl).trans (hdense j hj A hA)
+
+theorem actual_dense_window_failure_measurable_at_end
+    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (T q b v : ℝ)
+    (hT : 0 ≤ T) (hq : 0 ≤ q) (hgap : 1 + q ≤ T) (N n : ℕ)
+    (hlevel : ∀ j ∈ coverTestWindow n,
+      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) :
+    MeasurableSet[(actual_radiusLevelTime_stopping hB
+      (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc hB0
+      κ hκ hκ8 γ G F hw z hz (coverLogLevel z T (2 * n / 3 + 1))
+      (coverLogLevel_nonneg _ _ _)).measurableSpace]
+      (actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G z T q b v N n) := by
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have hγa : Adapted ℱ (fun t ω => γ ω t) := by
+    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω) _ _ γ G F hw
+    · intro t
+      exact measurable_const.mul
+        ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
+    · intro ω
+      exact (hc ω).const_mul _
+  apply MeasurableSet.biInter (coverTestWindow n).countable_toSet
+  intro j hj
+  apply MeasurableSet.compl
+  have hend : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T (2 * n / 3 + 1) := by
+    have hj' : j + 1 ≤ 2 * n / 3 + 1 := Nat.add_le_add_right (Finset.mem_Icc.mp hj).2 1
+    exact (by linarith : coverLogLevel z T j + 1 + q ≤ coverLogLevel z T j + T).trans
+      ((coverLogLevel_next_gap z T j (hlevel j hj)).trans (coverLogLevel_monotone z hT hj'))
+  have hstop := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
+    z hz (coverLogLevel z T j + 1 + q) (by linarith [coverLogLevel_nonneg z T j])
+  exact (actual_radiusLevelTime_past_mono hB hm hBm hc hB0 κ hκ hκ8 γ G F hw
+    z hz _ _ (by linarith [coverLogLevel_nonneg z T j]) hend)
+    _ (stoppedMassTest_measurable_at_stop ℱ γ hγa (fun ω => (hw ω).1)
+      _ hstop κ v _ _)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualMissedCell.lean b/lean/OAI/Probability/SLE/Gauge/ActualMissedCell.lean
index 7c95febd42efddd6192ceb5d42bf110751e0da99..d7539e8ab40b09afad36e5b251c30606f2e9de56 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualMissedCell.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualMissedCell.lean
@@ -1,138 +1,138 @@
-import OAI.Probability.SLE.Gauge.ActualDenseWindow
-import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
-import OAI.Probability.SLE.Gauge.DenseCandidateCover
-import OAI.Probability.SLE.Mass.UniformCoverLevels
-
-namespace OAI
-
-/-! Original-law probabilities of visited cells with no terminal sausage
-candidate, obtained from adapted failures under one finite centered law. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance actualMissedDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance actualMissedDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-
-lemma radius_gap_power (a u v p : ℝ) (ha : 0 < a) :
-    Real.exp (-p * v) * (a * Real.exp (-u)) ^ p =
-      (a * Real.exp (-(u + v))) ^ p := by
-  rw [Real.rpow_def_of_pos (mul_pos ha (Real.exp_pos _)),
-    Real.rpow_def_of_pos (mul_pos ha (Real.exp_pos _)),
-    Real.log_mul ha.ne' (Real.exp_ne_zero _),
-    Real.log_mul ha.ne' (Real.exp_ne_zero _), Real.log_exp, Real.log_exp,
-    ← Real.exp_add]
-  congr 1
-  ring
-
-lemma denseMissedCell_subset_actual_failure
-    {Ω : Type*} (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (hc : ∀ ω, Continuous (γ ω))
-    (G : Ω → ℝ≥0 → ℂ → ℂ) (T q b c v : ℝ) (n : ℕ) (ij : ℤ × ℤ) :
-    {ω | denseMissedCell κ T b c v n ij (γ ω)} ⊆
-      actualDenseWindowFailure κ γ hc G (coverGridCenter (coverRadius T n) ij)
-        T q b v (coverMultiplicity c n) n := by
-  intro ω hω
-  apply mem_iInter₂.mpr
-  intro j hj hsuccess
-  exact (not_le_of_gt (hω.2 j hj))
-    (hsuccess.trans (stoppedWholeSausageCharge_le_whole κ (sourceTracePath γ hc ω)
-      _ v _))
-
-theorem actual_missed_cell_bound_of_centered_failure
-    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (T q b c v : ℝ) (hT : 0 ≤ T) (hq : 0 ≤ q) (hgap : 1 + q ≤ T)
-    (n : ℕ) (ij : ℤ × ℤ)
-    (hz : coverGridCenter (coverRadius T n) ij ∈ upperHalfPlane)
-    (hlevel : ∀ j ∈ coverTestWindow n,
-      coverLogLevel (coverGridCenter (coverRadius T n) ij) T j =
-        Real.log (2 * (coverGridCenter (coverRadius T n) ij).im) + T * (j : ℝ))
-    (hvisit : coverLogLevel (coverGridCenter (coverRadius T n) ij) T (2 * n / 3 + 1) + 1 ≤
-      coverVisitLevel (coverGridCenter (coverRadius T n) ij) T n)
-    (hsmall : 4 * coverRadius T n ≤ (coverGridCenter (coverRadius T n) ij).im)
-    (hfailure : finiteCenteredLaw P κ B γ G (coverGridCenter (coverRadius T n) ij)
-      (coverVisitLevel (coverGridCenter (coverRadius T n) ij) T n)
-      (actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G
-        (coverGridCenter (coverRadius T n) ij) T q b v (coverMultiplicity c n) n) ≤
-      ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4))) :
-    P {ω | denseMissedCell κ T b c v n ij (γ ω)} ≤
-      ENNReal.ofReal ((uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ *
-        initialGreen κ (coverGridCenter (coverRadius T n) ij)) *
-        coverRadius T n ^ codimension κ * Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
-  let z := coverGridCenter (coverRadius T n) ij
-  let u := coverLogLevel z T (2 * n / 3 + 1)
-  let V := coverVisitLevel z T n
-  have hgapV : u + 1 ≤ V := hvisit
-  have hzim : 0 < z.im := hz
-  let w : ℝ≥0 := ⟨V - u, by linarith⟩
-  have hw1 : 1 ≤ (w : ℝ) := by change 1 ≤ V-u; linarith
-  have hu : 0 ≤ u := coverLogLevel_nonneg _ _ _
-  have hV : 0 ≤ V := coverVisitLevel_nonneg_of_gap z T n hvisit
-  let A := actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G z T q b v (coverMultiplicity c n) n
-  have hA := actual_dense_window_failure_measurable_at_end P κ hκ hκ8 B hB hBm hc hB0
-    γ G F hw z hz T q b v hT hq hgap (coverMultiplicity c n) n hlevel
-  have happroach := actual_conditional_approach_at_radius_stop P κ hκ hκ8 B hB hBm hc hB0
-    γ G F hw htrace z hz u hu w hw1 A hA
-  have huv : u + (w : ℝ) = V := by change u + (V-u) = V; ring
-  rw [huv] at happroach
-  have hsubset : {ω | denseMissedCell κ T b c v n ij (γ ω)} ⊆
-      A ∩ {ω | radiusLevelTime (γ ω) (G ω) z V < ∞} := by
-    intro ω hω
-    refine ⟨denseMissedCell_subset_actual_failure κ γ (fun ω => (hw ω).1) G T q b c v n ij hω, ?_⟩
-    obtain ⟨x, ⟨t, rfl⟩, ht⟩ := hω.1
-    have hdist : dist (γ ω t) z < 4 * coverRadius T n := by
-      change dist (γ ω t) z ≤ 2 * coverRadius T n at ht
-      linarith [coverRadius_pos T n]
-    obtain ⟨s, _, hslive, hsR⟩ := (hw ω).radius_approach_of_trace_approach
-      ((hc ω).const_mul _) z (4 * coverRadius T n) (by positivity [coverRadius_pos T n]) hsmall t hdist
-    have hscale : (128 * Real.pi) * (4 * coverRadius T n) = (2 * z.im) * Real.exp (-V) := by
-      rw [coverVisitLevel_radius z hz T n]
-      ring
-    exact radiusLevelTime_finite_of_live_radius_lt ((hc ω).const_mul _) (hw ω) z V hV s hslive
-      (by rwa [← hscale])
-  have hK : uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * (w : ℝ)) *
-      ((2 * z.im) * Real.exp (-u)) ^ codimension κ * initialGreen κ z =
-      (uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ * initialGreen κ z) *
-        coverRadius T n ^ codimension κ := by
-    calc
-      _ = uniformInverseAngleConstant κ 1 *
-          (Real.exp (-codimension κ * (w : ℝ)) * ((2 * z.im) * Real.exp (-u)) ^ codimension κ) *
-          initialGreen κ z := by ring
-      _ = uniformInverseAngleConstant κ 1 *
-          ((2 * z.im) * Real.exp (-V)) ^ codimension κ * initialGreen κ z := by
-        rw [radius_gap_power (2 * z.im) u (w : ℝ) (codimension κ) (mul_pos (by norm_num) hzim), huv]
-      _ = _ := by
-        rw [coverVisitLevel_radius z hz T n,
-          Real.mul_rpow (by positivity : 0 ≤ 512 * Real.pi) (coverRadius_pos T n).le]
-        ring
-  rw [hK] at happroach
-  have hKnonneg : 0 ≤ (uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ *
-      initialGreen κ z) * coverRadius T n ^ codimension κ := by
-    have hC := uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
-    have hI := initialGreen_pos κ z hz
-    exact mul_nonneg (mul_nonneg
-      (mul_nonneg hC.le (Real.rpow_nonneg (by positivity) _)) hI.le)
-      (Real.rpow_nonneg (coverRadius_pos T n).le _)
-  calc
-    _ ≤ P (A ∩ {ω | radiusLevelTime (γ ω) (G ω) z V < ∞}) := measure_mono hsubset
-    _ ≤ _ := happroach
-    _ ≤ ENNReal.ofReal ((uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ *
-          initialGreen κ z) * coverRadius T n ^ codimension κ) *
-        ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) :=
-      mul_le_mul' le_rfl hfailure
-    _ = _ := (ENNReal.ofReal_mul hKnonneg).symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualDenseWindow
+import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
+import OAI.Probability.SLE.Gauge.DenseCandidateCover
+import OAI.Probability.SLE.Mass.UniformCoverLevels
+
+namespace OAI
+
+/-! Original-law probabilities of visited cells with no terminal sausage
+candidate, obtained from adapted failures under one finite centered law. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance actualMissedDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance actualMissedDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+
+lemma radius_gap_power (a u v p : ℝ) (ha : 0 < a) :
+    Real.exp (-p * v) * (a * Real.exp (-u)) ^ p =
+      (a * Real.exp (-(u + v))) ^ p := by
+  rw [Real.rpow_def_of_pos (mul_pos ha (Real.exp_pos _)),
+    Real.rpow_def_of_pos (mul_pos ha (Real.exp_pos _)),
+    Real.log_mul ha.ne' (Real.exp_ne_zero _),
+    Real.log_mul ha.ne' (Real.exp_ne_zero _), Real.log_exp, Real.log_exp,
+    ← Real.exp_add]
+  congr 1
+  ring
+
+lemma denseMissedCell_subset_actual_failure
+    {Ω : Type*} (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (hc : ∀ ω, Continuous (γ ω))
+    (G : Ω → ℝ≥0 → ℂ → ℂ) (T q b c v : ℝ) (n : ℕ) (ij : ℤ × ℤ) :
+    {ω | denseMissedCell κ T b c v n ij (γ ω)} ⊆
+      actualDenseWindowFailure κ γ hc G (coverGridCenter (coverRadius T n) ij)
+        T q b v (coverMultiplicity c n) n := by
+  intro ω hω
+  apply mem_iInter₂.mpr
+  intro j hj hsuccess
+  exact (not_le_of_gt (hω.2 j hj))
+    (hsuccess.trans (stoppedWholeSausageCharge_le_whole κ (sourceTracePath γ hc ω)
+      _ v _))
+
+theorem actual_missed_cell_bound_of_centered_failure
+    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (T q b c v : ℝ) (hT : 0 ≤ T) (hq : 0 ≤ q) (hgap : 1 + q ≤ T)
+    (n : ℕ) (ij : ℤ × ℤ)
+    (hz : coverGridCenter (coverRadius T n) ij ∈ upperHalfPlane)
+    (hlevel : ∀ j ∈ coverTestWindow n,
+      coverLogLevel (coverGridCenter (coverRadius T n) ij) T j =
+        Real.log (2 * (coverGridCenter (coverRadius T n) ij).im) + T * (j : ℝ))
+    (hvisit : coverLogLevel (coverGridCenter (coverRadius T n) ij) T (2 * n / 3 + 1) + 1 ≤
+      coverVisitLevel (coverGridCenter (coverRadius T n) ij) T n)
+    (hsmall : 4 * coverRadius T n ≤ (coverGridCenter (coverRadius T n) ij).im)
+    (hfailure : finiteCenteredLaw P κ B γ G (coverGridCenter (coverRadius T n) ij)
+      (coverVisitLevel (coverGridCenter (coverRadius T n) ij) T n)
+      (actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G
+        (coverGridCenter (coverRadius T n) ij) T q b v (coverMultiplicity c n) n) ≤
+      ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4))) :
+    P {ω | denseMissedCell κ T b c v n ij (γ ω)} ≤
+      ENNReal.ofReal ((uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ *
+        initialGreen κ (coverGridCenter (coverRadius T n) ij)) *
+        coverRadius T n ^ codimension κ * Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
+  let z := coverGridCenter (coverRadius T n) ij
+  let u := coverLogLevel z T (2 * n / 3 + 1)
+  let V := coverVisitLevel z T n
+  have hgapV : u + 1 ≤ V := hvisit
+  have hzim : 0 < z.im := hz
+  let w : ℝ≥0 := ⟨V - u, by linarith⟩
+  have hw1 : 1 ≤ (w : ℝ) := by change 1 ≤ V-u; linarith
+  have hu : 0 ≤ u := coverLogLevel_nonneg _ _ _
+  have hV : 0 ≤ V := coverVisitLevel_nonneg_of_gap z T n hvisit
+  let A := actualDenseWindowFailure κ γ (fun ω => (hw ω).1) G z T q b v (coverMultiplicity c n) n
+  have hA := actual_dense_window_failure_measurable_at_end P κ hκ hκ8 B hB hBm hc hB0
+    γ G F hw z hz T q b v hT hq hgap (coverMultiplicity c n) n hlevel
+  have happroach := actual_conditional_approach_at_radius_stop P κ hκ hκ8 B hB hBm hc hB0
+    γ G F hw htrace z hz u hu w hw1 A hA
+  have huv : u + (w : ℝ) = V := by change u + (V-u) = V; ring
+  rw [huv] at happroach
+  have hsubset : {ω | denseMissedCell κ T b c v n ij (γ ω)} ⊆
+      A ∩ {ω | radiusLevelTime (γ ω) (G ω) z V < ∞} := by
+    intro ω hω
+    refine ⟨denseMissedCell_subset_actual_failure κ γ (fun ω => (hw ω).1) G T q b c v n ij hω, ?_⟩
+    obtain ⟨x, ⟨t, rfl⟩, ht⟩ := hω.1
+    have hdist : dist (γ ω t) z < 4 * coverRadius T n := by
+      change dist (γ ω t) z ≤ 2 * coverRadius T n at ht
+      linarith [coverRadius_pos T n]
+    obtain ⟨s, _, hslive, hsR⟩ := (hw ω).radius_approach_of_trace_approach
+      ((hc ω).const_mul _) z (4 * coverRadius T n) (by positivity [coverRadius_pos T n]) hsmall t hdist
+    have hscale : (128 * Real.pi) * (4 * coverRadius T n) = (2 * z.im) * Real.exp (-V) := by
+      rw [coverVisitLevel_radius z hz T n]
+      ring
+    exact radiusLevelTime_finite_of_live_radius_lt ((hc ω).const_mul _) (hw ω) z V hV s hslive
+      (by rwa [← hscale])
+  have hK : uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * (w : ℝ)) *
+      ((2 * z.im) * Real.exp (-u)) ^ codimension κ * initialGreen κ z =
+      (uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ * initialGreen κ z) *
+        coverRadius T n ^ codimension κ := by
+    calc
+      _ = uniformInverseAngleConstant κ 1 *
+          (Real.exp (-codimension κ * (w : ℝ)) * ((2 * z.im) * Real.exp (-u)) ^ codimension κ) *
+          initialGreen κ z := by ring
+      _ = uniformInverseAngleConstant κ 1 *
+          ((2 * z.im) * Real.exp (-V)) ^ codimension κ * initialGreen κ z := by
+        rw [radius_gap_power (2 * z.im) u (w : ℝ) (codimension κ) (mul_pos (by norm_num) hzim), huv]
+      _ = _ := by
+        rw [coverVisitLevel_radius z hz T n,
+          Real.mul_rpow (by positivity : 0 ≤ 512 * Real.pi) (coverRadius_pos T n).le]
+        ring
+  rw [hK] at happroach
+  have hKnonneg : 0 ≤ (uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ *
+      initialGreen κ z) * coverRadius T n ^ codimension κ := by
+    have hC := uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
+    have hI := initialGreen_pos κ z hz
+    exact mul_nonneg (mul_nonneg
+      (mul_nonneg hC.le (Real.rpow_nonneg (by positivity) _)) hI.le)
+      (Real.rpow_nonneg (coverRadius_pos T n).le _)
+  calc
+    _ ≤ P (A ∩ {ω | radiusLevelTime (γ ω) (G ω) z V < ∞}) := measure_mono hsubset
+    _ ≤ _ := happroach
+    _ ≤ ENNReal.ofReal ((uniformInverseAngleConstant κ 1 * (512 * Real.pi) ^ codimension κ *
+          initialGreen κ z) * coverRadius T n ^ codimension κ) *
+        ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) :=
+      mul_le_mul' le_rfl hfailure
+    _ = _ := (ENNReal.ofReal_mul hKnonneg).symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualPastThreshold.lean b/lean/OAI/Probability/SLE/Gauge/ActualPastThreshold.lean
index dfa5997d4854aba235e19ca9e4d2d04fbaee53ee..a3c93f34a1e62e3139e05e7d6bdcdd3be48176b4 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualPastThreshold.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualPastThreshold.lean
@@ -1,69 +1,69 @@
-import OAI.Probability.SLE.Mass.PastThresholdMass
-
-namespace OAI
-
-/-! Specialization of the half-mass threshold comparison to canonical mass. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-attribute [local instance] Classical.propDecidable
-
-namespace SLEExactGauge.Multipoint
-
-lemma compactWeightedMass_one (K : Set upperHalfPlane) (μ : Measure upperHalfPlane) :
-    compactWeightedMass (ContinuousMap.const _ (1:ℝ≥0)) K μ=(μ K).toReal := by
-  simp [compactWeightedMass,measureReal_def]
-
-namespace CanonicalMass
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-
-lemma past_threshold_detects_half_terminal
-    (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (q : ℕ → ℝ≥0) (hq : DenseRange q) (K : Set upperHalfPlane) (hK : IsCompact K)
-    (H : ℝ) (hH : 0<H) :
-    ∀ᵐ ω ∂P, ENNReal.ofReal (2*H)<C.rational.terminal ω K →
-      C.rational.terminal ω K≤2*C.rational.terminal ω (K ∩ ⋃ n,
-        if H<(C.process (q n) ω K).toReal then
-          {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)}ᶜ else ∅) := by
-  have hp : ∀ᵐ ω ∂P, ∀ n, C.process (q n) ω=(C.rational.terminal ω).restrict
-      {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)} := by
-    apply ae_all_iff.mpr
-    intro n
-    simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe]
-      using C.atTime_eq_restrict_past hB hBm hc h0 hκ hκ8 F hw (q n)
-  have hs := C.compactWeightedMass_paths (ContinuousMap.const _ (1:ℝ≥0)) K hK
-  filter_upwards [hp,hs] with ω hpω hsω
-  intro hlarge
-  let μ := C.rational.terminal ω
-  let := (C.rational.terminal_radon ω).2
-  let f : ℝ≥0 → ℝ := fun t => compactWeightedMass (ContinuousMap.const _ (1:ℝ≥0)) K (C.process t ω)
-  have hf : Continuous f := hsω.2.2.1
-  have hf0 : f 0=0 := hsω.2.1
-  have hflim : Tendsto f atTop (𝓝 (μ K).toReal) := by
-    simpa only [f,μ,compactWeightedMass_one] using hsω.2.2.2.1
-  have hfpast (n : ℕ) : μ (K ∩ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)})=
-      ENNReal.ofReal (f (q n)) := by
-    have hfin : C.process (q n) ω K≠∞ := by
-      let := (C.radon (q n) ω).2
-      exact hK.measure_lt_top.ne
-    rw [show f (q n)=(C.process (q n) ω K).toReal from compactWeightedMass_one K _,
-      ENNReal.ofReal_toReal hfin,hpω n,Measure.restrict_apply hK.measurableSet]
-  have h := past_threshold_detects_half μ K hK.measure_lt_top.ne
-    (fun t => {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})
-    (fun t => ((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet)
-    q hq f hf hf0 hflim hfpast H hH hlarge
-  simpa only [f,compactWeightedMass_one] using h
-
-end CanonicalMass
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.PastThresholdMass
+
+namespace OAI
+
+/-! Specialization of the half-mass threshold comparison to canonical mass. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+attribute [local instance] Classical.propDecidable
+
+namespace SLEExactGauge.Multipoint
+
+lemma compactWeightedMass_one (K : Set upperHalfPlane) (μ : Measure upperHalfPlane) :
+    compactWeightedMass (ContinuousMap.const _ (1:ℝ≥0)) K μ=(μ K).toReal := by
+  simp [compactWeightedMass,measureReal_def]
+
+namespace CanonicalMass
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+
+lemma past_threshold_detects_half_terminal
+    (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (q : ℕ → ℝ≥0) (hq : DenseRange q) (K : Set upperHalfPlane) (hK : IsCompact K)
+    (H : ℝ) (hH : 0<H) :
+    ∀ᵐ ω ∂P, ENNReal.ofReal (2*H)<C.rational.terminal ω K →
+      C.rational.terminal ω K≤2*C.rational.terminal ω (K ∩ ⋃ n,
+        if H<(C.process (q n) ω K).toReal then
+          {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)}ᶜ else ∅) := by
+  have hp : ∀ᵐ ω ∂P, ∀ n, C.process (q n) ω=(C.rational.terminal ω).restrict
+      {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)} := by
+    apply ae_all_iff.mpr
+    intro n
+    simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe]
+      using C.atTime_eq_restrict_past hB hBm hc h0 hκ hκ8 F hw (q n)
+  have hs := C.compactWeightedMass_paths (ContinuousMap.const _ (1:ℝ≥0)) K hK
+  filter_upwards [hp,hs] with ω hpω hsω
+  intro hlarge
+  let μ := C.rational.terminal ω
+  let := (C.rational.terminal_radon ω).2
+  let f : ℝ≥0 → ℝ := fun t => compactWeightedMass (ContinuousMap.const _ (1:ℝ≥0)) K (C.process t ω)
+  have hf : Continuous f := hsω.2.2.1
+  have hf0 : f 0=0 := hsω.2.1
+  have hflim : Tendsto f atTop (𝓝 (μ K).toReal) := by
+    simpa only [f,μ,compactWeightedMass_one] using hsω.2.2.2.1
+  have hfpast (n : ℕ) : μ (K ∩ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 (q n)})=
+      ENNReal.ofReal (f (q n)) := by
+    have hfin : C.process (q n) ω K≠∞ := by
+      let := (C.radon (q n) ω).2
+      exact hK.measure_lt_top.ne
+    rw [show f (q n)=(C.process (q n) ω K).toReal from compactWeightedMass_one K _,
+      ENNReal.ofReal_toReal hfin,hpω n,Measure.restrict_apply hK.measurableSet]
+  have h := past_threshold_detects_half μ K hK.measure_lt_top.ne
+    (fun t => {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t})
+    (fun t => ((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet)
+    q hq f hf hf0 hflim hfpast H hH hlarge
+  simpa only [f,compactWeightedMass_one] using h
+
+end CanonicalMass
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualRibbonHistory.lean b/lean/OAI/Probability/SLE/Gauge/ActualRibbonHistory.lean
index f87361ed6aedcb2a44d1bd3bc5de559d2e21b2b3..b81accc57889e667ce111be87bb15f879374ad06 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualRibbonHistory.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualRibbonHistory.lean
@@ -1,108 +1,108 @@
-import OAI.Probability.SLE.Gauge.RibbonFrontStops
-
-namespace OAI
-
-/-! Construct the history required by the uniform passage theorem from
-an actual global first-front event and physical prefix containment. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates FirstFrontPassage
-
-theorem actual_history_at_global_front
-    {C L : ℝ≥0} {ε : ℝ}
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (P : ℂ → ℂ)
-    (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
-    (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
-    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
-    (hescape : Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop)
-    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re)
-    (hg0 : g (γ 0) = 0) (k : ℝ) (hk : 1 ≤ k) (T : ℝ≥0)
-    (hfirst : firstFrontTime (fun t => g (γ t)) k = (T : WithTop ℝ≥0))
-    (hpast : MapsTo γ (Icc 0 T) (P '' {z ∈ closedRibbon | |z.im| ≤ ε}))
-    (hoffset : |(coordinateInverse P closedRibbon (γ T)).im| ≤ ε / 4)
-    (hupper : LipschitzOnWith C P (ribbonStepPatch k))
-    (hlower : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k,
-      dist z w ≤ L * dist (P z) (P w)) :
-    ∃ d : RibbonHistory C L ε, d.U = U ∧ d.trace = γ ∧ d.forward = G ∧
-      d.inverse = F ∧ d.chart = P ∧ d.front = k ∧ d.time = T := by
-  have hhit := FrontSuccessMeasurability.firstFrontTime_attained
-    (fun t => g (γ t)) (g.continuous.comp hw.1) k T hfirst
-  let xp := fun t => coordinateInverse P closedRibbon (γ t)
-  have hmap : MapsTo γ (Icc 0 T) (P '' closedRibbon) :=
-    fun _ ht => image_mono inter_subset_left (hpast ht)
-  have hcoords : MapsTo xp (Icc 0 T) closedRibbon :=
-    fun _ ht => coordinateInverse_mem (hmap ht)
-  have hxpc : ContinuousOn xp (Icc 0 T) :=
-    (coordinateInverse_continuousOn_of_embedding P hemb.isEmbedding).comp
-      hw.1.continuousOn hmap
-  have hγeq : EqOn γ (P ∘ xp) (Icc 0 T) :=
-    fun _ ht => (coordinateInverse_right (hmap ht)).symm
-  have hgtrace (t : ℝ≥0) (ht : t ∈ Icc 0 T) : g (γ t) = (xp t).re := by
-    rw [hγeq ht]
-    exact hg _ (hcoords ht)
-  have hT : 0 < T := by
-    apply lt_of_le_of_ne bot_le
-    intro he
-    have hh := hhit
-    rw [← he] at hh
-    change g (γ 0) = k at hh
-    rw [hg0] at hh
-    linarith
-  have hxpre : (xp T).re = k := (hgtrace T ⟨bot_le, le_rfl⟩).symm.trans hhit
-  have hattain : xp T = (k : ℂ) + ((xp T).im : ℂ) * Complex.I := by
-    apply Complex.ext <;> simp [hxpre]
-  have hstrict : ∀ t : ℝ≥0, t < T → (xp t).re < k := by
-    have hh := strict_before_attained_front (fun t => g (γ t)) k T
-      (g.continuous.comp hw.1).continuousOn (by rw [hg0]; linarith) hfirst
-    intro t ht
-    rw [← hgtrace t ⟨bot_le, ht.le⟩]
-    exact hh t ht
-  have hcore : ∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε := by
-    intro t ht
-    refine ⟨?_, ?_⟩
-    · rcases ht.2.eq_or_lt with rfl | hlt
-      · exact hxpre.le
-      · exact (hstrict t hlt).le
-    · obtain ⟨z, ⟨hz, hiε⟩, he⟩ := hpast ht
-      simpa only [xp, ← he, coordinateInverse_left hi hz] using hiε
-  let d : RibbonHistory C L ε := {
-    U := U
-    trace := γ
-    forward := G
-    inverse := F
-    driver_continuous := hU
-    witness := hw
-    chart := P
-    chart_continuous := hP
-    chart_injective := hi
-    chart_upper := hH
-    chart_escape := hescape
-    front := k
-    offset := (xp T).im
-    front_ge_one := hk
-    offset_bound := hoffset
-    patch_upper := hupper
-    patch_lower := hlower
-    time := T
-    time_pos := hT
-    coordinates := xp
-    coordinates_continuous := hxpc
-    trace_coordinates := hγeq
-    initial_front := by rw [← hgtrace 0 ⟨bot_le, bot_le⟩, hg0]
-    attained := hattain
-    coordinates_ribbon := hcoords
-    past_core := hcore
-    strict_past := hstrict }
-  exact ⟨d, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RibbonFrontStops
+
+namespace OAI
+
+/-! Construct the history required by the uniform passage theorem from
+an actual global first-front event and physical prefix containment. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates FirstFrontPassage
+
+theorem actual_history_at_global_front
+    {C L : ℝ≥0} {ε : ℝ}
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (P : ℂ → ℂ)
+    (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
+    (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
+    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
+    (hescape : Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop)
+    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re)
+    (hg0 : g (γ 0) = 0) (k : ℝ) (hk : 1 ≤ k) (T : ℝ≥0)
+    (hfirst : firstFrontTime (fun t => g (γ t)) k = (T : WithTop ℝ≥0))
+    (hpast : MapsTo γ (Icc 0 T) (P '' {z ∈ closedRibbon | |z.im| ≤ ε}))
+    (hoffset : |(coordinateInverse P closedRibbon (γ T)).im| ≤ ε / 4)
+    (hupper : LipschitzOnWith C P (ribbonStepPatch k))
+    (hlower : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k,
+      dist z w ≤ L * dist (P z) (P w)) :
+    ∃ d : RibbonHistory C L ε, d.U = U ∧ d.trace = γ ∧ d.forward = G ∧
+      d.inverse = F ∧ d.chart = P ∧ d.front = k ∧ d.time = T := by
+  have hhit := FrontSuccessMeasurability.firstFrontTime_attained
+    (fun t => g (γ t)) (g.continuous.comp hw.1) k T hfirst
+  let xp := fun t => coordinateInverse P closedRibbon (γ t)
+  have hmap : MapsTo γ (Icc 0 T) (P '' closedRibbon) :=
+    fun _ ht => image_mono inter_subset_left (hpast ht)
+  have hcoords : MapsTo xp (Icc 0 T) closedRibbon :=
+    fun _ ht => coordinateInverse_mem (hmap ht)
+  have hxpc : ContinuousOn xp (Icc 0 T) :=
+    (coordinateInverse_continuousOn_of_embedding P hemb.isEmbedding).comp
+      hw.1.continuousOn hmap
+  have hγeq : EqOn γ (P ∘ xp) (Icc 0 T) :=
+    fun _ ht => (coordinateInverse_right (hmap ht)).symm
+  have hgtrace (t : ℝ≥0) (ht : t ∈ Icc 0 T) : g (γ t) = (xp t).re := by
+    rw [hγeq ht]
+    exact hg _ (hcoords ht)
+  have hT : 0 < T := by
+    apply lt_of_le_of_ne bot_le
+    intro he
+    have hh := hhit
+    rw [← he] at hh
+    change g (γ 0) = k at hh
+    rw [hg0] at hh
+    linarith
+  have hxpre : (xp T).re = k := (hgtrace T ⟨bot_le, le_rfl⟩).symm.trans hhit
+  have hattain : xp T = (k : ℂ) + ((xp T).im : ℂ) * Complex.I := by
+    apply Complex.ext <;> simp [hxpre]
+  have hstrict : ∀ t : ℝ≥0, t < T → (xp t).re < k := by
+    have hh := strict_before_attained_front (fun t => g (γ t)) k T
+      (g.continuous.comp hw.1).continuousOn (by rw [hg0]; linarith) hfirst
+    intro t ht
+    rw [← hgtrace t ⟨bot_le, ht.le⟩]
+    exact hh t ht
+  have hcore : ∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε := by
+    intro t ht
+    refine ⟨?_, ?_⟩
+    · rcases ht.2.eq_or_lt with rfl | hlt
+      · exact hxpre.le
+      · exact (hstrict t hlt).le
+    · obtain ⟨z, ⟨hz, hiε⟩, he⟩ := hpast ht
+      simpa only [xp, ← he, coordinateInverse_left hi hz] using hiε
+  let d : RibbonHistory C L ε := {
+    U := U
+    trace := γ
+    forward := G
+    inverse := F
+    driver_continuous := hU
+    witness := hw
+    chart := P
+    chart_continuous := hP
+    chart_injective := hi
+    chart_upper := hH
+    chart_escape := hescape
+    front := k
+    offset := (xp T).im
+    front_ge_one := hk
+    offset_bound := hoffset
+    patch_upper := hupper
+    patch_lower := hlower
+    time := T
+    time_pos := hT
+    coordinates := xp
+    coordinates_continuous := hxpc
+    trace_coordinates := hγeq
+    initial_front := by rw [← hgtrace 0 ⟨bot_le, bot_le⟩, hg0]
+    attained := hattain
+    coordinates_ribbon := hcoords
+    past_core := hcore
+    strict_past := hstrict }
+  exact ⟨d, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualRibbonIteration.lean b/lean/OAI/Probability/SLE/Gauge/ActualRibbonIteration.lean
index 7c5eaf877ada293e10dc4ea2a666a5b2456e4222..2a81329b338f451e1e8cc111c7c215179f28ed72 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualRibbonIteration.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualRibbonIteration.lean
@@ -1,78 +1,78 @@
-import OAI.Probability.SLE.Gauge.ActualRibbonPrefixBound
-import OAI.Probability.SLE.Gauge.RibbonExperimentAssembly
-
-namespace OAI
-
-/-! Iteration along the actual global integer fronts. Previous successes
-supply admissibility; no independence or deterministic completion bound is
-required on failed samples. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates FirstFrontPassage Target
-
-local instance ribbonIterationPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance ribbonIterationPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-def ribbonIntegerFront {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ)
-    (n : ℕ) : Ω → WithTop ℝ≥0 := ribbonFrontStop g γ ((n : ℝ) + 1)
-
-lemma ribbonIntegerFront_monotone {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ)
-    (hc : ∀ ω, Continuous (γ ω)) (h0 : ∀ ω, g (γ ω 0) = 0) :
-    Monotone (ribbonIntegerFront g γ) := by
-  intro n k hnk ω
-  exact firstFrontTime_mono_of_initial _ (g.continuous.comp (hc ω))
-    (by change g (γ ω 0) ≤ (n : ℝ) + 1; rw [h0 ω]; positivity) (by exact_mod_cast Nat.add_le_add_right hnk 1)
-
-def actualRibbonSuccess {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
-    (g : C(ℂ, ℝ)) (P : ℂ → ℂ) (ε κ v c : ℝ) (n : ℕ) : Set Ω :=
-  RibbonStepSuccess γ (ribbonIntegerFront g γ n) (ribbonIntegerFront g γ (n + 1))
-    P ε κ v c (physicalMiddleRegion P closedRibbon ((n : ℝ) + 1) ε)
-
-/-- Actual fronts turn the quantitative step estimate into a product
-lower bound for the exact physical passage-and-mass events. -/
-theorem actual_ribbon_successes_lower_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ)
-    (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t))
-    (hc : ∀ ω, Continuous (γ ω)) (g : C(ℂ, ℝ)) (hg0 : ∀ ω, g (γ ω 0) = 0)
-    (P : ℂ → ℂ) (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
-    (ε κ v c : ℝ) (q : ℝ≥0∞)
-    (hstep : ∀ (n : ℕ) (A : Set Ω),
-      MeasurableSet[(ribbonFrontStop_stopping ℱ g γ ha hc ((n : ℝ) + 1)).measurableSpace] A →
-      A ⊆ GoodRibbonPrefix γ (ribbonIntegerFront g γ n) P ε →
-        q * μ A ≤ μ (A ∩ actualRibbonSuccess γ g P ε κ v c n)) :
-    ∀ N : ℕ, q ^ N * μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε) ≤
-      μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
-        ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) := by
-  let τ := ribbonIntegerFront g γ
-  have hτ (n : ℕ) : IsStoppingTime ℱ (τ n) :=
-    ribbonFrontStop_stopping ℱ g γ ha hc ((n : ℝ) + 1)
-  have hmono : Monotone τ := ribbonIntegerFront_monotone g γ hc hg0
-  let 𝒢 : Filtration ℕ mΩ := ⟨fun n => (hτ n).measurableSpace,
-    fun _ _ h => IsStoppingTime.measurableSpace_mono _ _ (hmono h),
-    fun n => (hτ n).measurableSpace_le⟩
-  let init := GoodRibbonPrefix γ (τ 0) P ε
-  have hinit : MeasurableSet[𝒢 0] init :=
-    goodRibbonPrefix_measurable ℱ γ ha hc (τ 0) (hτ 0) P hemb ε
-  have hsuccess (n : ℕ) : MeasurableSet[𝒢 (n + 1)] (actualRibbonSuccess γ g P ε κ v c n) :=
-    ribbonStepSuccess_measurable ℱ γ ha hc (τ n) (τ (n + 1)) (hτ n) (hτ (n + 1))
-      (hmono (Nat.le_succ n)) P hemb ε κ v c _
-  apply adapted_successes_lower_bound_on_previous μ 𝒢 init hinit _ hsuccess q
-  intro n A hA hprev
-  apply hstep n A hA
-  intro ω hω
-  have hωprev := hprev hω
-  cases n with
-  | zero => exact hωprev.1
-  | succ n =>
-    have hh := mem_iInter.mp (mem_iInter.mp hωprev.2 n) (Finset.mem_range.mpr (Nat.lt_succ_self n))
-    exact hh.1
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualRibbonPrefixBound
+import OAI.Probability.SLE.Gauge.RibbonExperimentAssembly
+
+namespace OAI
+
+/-! Iteration along the actual global integer fronts. Previous successes
+supply admissibility; no independence or deterministic completion bound is
+required on failed samples. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates FirstFrontPassage Target
+
+local instance ribbonIterationPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance ribbonIterationPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+def ribbonIntegerFront {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ)
+    (n : ℕ) : Ω → WithTop ℝ≥0 := ribbonFrontStop g γ ((n : ℝ) + 1)
+
+lemma ribbonIntegerFront_monotone {Ω : Type*} (g : C(ℂ, ℝ)) (γ : Ω → ℝ≥0 → ℂ)
+    (hc : ∀ ω, Continuous (γ ω)) (h0 : ∀ ω, g (γ ω 0) = 0) :
+    Monotone (ribbonIntegerFront g γ) := by
+  intro n k hnk ω
+  exact firstFrontTime_mono_of_initial _ (g.continuous.comp (hc ω))
+    (by change g (γ ω 0) ≤ (n : ℝ) + 1; rw [h0 ω]; positivity) (by exact_mod_cast Nat.add_le_add_right hnk 1)
+
+def actualRibbonSuccess {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
+    (g : C(ℂ, ℝ)) (P : ℂ → ℂ) (ε κ v c : ℝ) (n : ℕ) : Set Ω :=
+  RibbonStepSuccess γ (ribbonIntegerFront g γ n) (ribbonIntegerFront g γ (n + 1))
+    P ε κ v c (physicalMiddleRegion P closedRibbon ((n : ℝ) + 1) ε)
+
+/-- Actual fronts turn the quantitative step estimate into a product
+lower bound for the exact physical passage-and-mass events. -/
+theorem actual_ribbon_successes_lower_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ)
+    (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t))
+    (hc : ∀ ω, Continuous (γ ω)) (g : C(ℂ, ℝ)) (hg0 : ∀ ω, g (γ ω 0) = 0)
+    (P : ℂ → ℂ) (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
+    (ε κ v c : ℝ) (q : ℝ≥0∞)
+    (hstep : ∀ (n : ℕ) (A : Set Ω),
+      MeasurableSet[(ribbonFrontStop_stopping ℱ g γ ha hc ((n : ℝ) + 1)).measurableSpace] A →
+      A ⊆ GoodRibbonPrefix γ (ribbonIntegerFront g γ n) P ε →
+        q * μ A ≤ μ (A ∩ actualRibbonSuccess γ g P ε κ v c n)) :
+    ∀ N : ℕ, q ^ N * μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε) ≤
+      μ (GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
+        ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) := by
+  let τ := ribbonIntegerFront g γ
+  have hτ (n : ℕ) : IsStoppingTime ℱ (τ n) :=
+    ribbonFrontStop_stopping ℱ g γ ha hc ((n : ℝ) + 1)
+  have hmono : Monotone τ := ribbonIntegerFront_monotone g γ hc hg0
+  let 𝒢 : Filtration ℕ mΩ := ⟨fun n => (hτ n).measurableSpace,
+    fun _ _ h => IsStoppingTime.measurableSpace_mono _ _ (hmono h),
+    fun n => (hτ n).measurableSpace_le⟩
+  let init := GoodRibbonPrefix γ (τ 0) P ε
+  have hinit : MeasurableSet[𝒢 0] init :=
+    goodRibbonPrefix_measurable ℱ γ ha hc (τ 0) (hτ 0) P hemb ε
+  have hsuccess (n : ℕ) : MeasurableSet[𝒢 (n + 1)] (actualRibbonSuccess γ g P ε κ v c n) :=
+    ribbonStepSuccess_measurable ℱ γ ha hc (τ n) (τ (n + 1)) (hτ n) (hτ (n + 1))
+      (hmono (Nat.le_succ n)) P hemb ε κ v c _
+  apply adapted_successes_lower_bound_on_previous μ 𝒢 init hinit _ hsuccess q
+  intro n A hA hprev
+  apply hstep n A hA
+  intro ω hω
+  have hωprev := hprev hω
+  cases n with
+  | zero => exact hωprev.1
+  | succ n =>
+    have hh := mem_iInter.mp (mem_iInter.mp hωprev.2 n) (Finset.mem_range.mpr (Nat.lt_succ_self n))
+    exact hh.1
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualRibbonMass.lean b/lean/OAI/Probability/SLE/Gauge/ActualRibbonMass.lean
index 84591e3187140263cf8e5e11973312d7c8dc10ae..c8fcfbd1d0a7c5352bffceed7d14c5fde4e27083 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualRibbonMass.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualRibbonMass.lean
@@ -1,55 +1,55 @@
-import OAI.Probability.SLE.Gauge.PublishedRibbonIteration
-
-namespace OAI
-
-/-! Charge selected actual successive-front windows to a single sausage,
-retaining the successful terminal prefix for clearance and density transfer. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates
-
-lemma actual_successes_terminal_prefix
-    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (g : C(ℂ, ℝ)) (P : ℂ → ℂ)
-    (ε κ v c : ℝ) (N : ℕ) (ω : Ω)
-    (hω : ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
-      ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) :
-    ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ N) P ε := by
-  cases N with
-  | zero => exact hω.1
-  | succ N =>
-    exact (mem_iInter.mp (mem_iInter.mp hω.2 N) (Finset.mem_range.mpr (Nat.lt_succ_self N))).1
-
-/-- Only selected middle strips are charged. The trace windows are the
-original actual fronts, including their exact starting and ending times. -/
-theorem actual_successes_force_whole_mass
-    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (g : C(ℂ, ℝ)) (P : ℂ → ℂ)
-    (hplanar : PublishedPlanarDiskOpenness) (hP : ContinuousOn P closedRibbon)
-    (hi : InjOn P closedRibbon) (ε κ v c : ℝ) (hε : ε ≤ 1)
-    (N : ℕ) (I : Finset ℕ) (hI : ∀ i ∈ I, 0 < i ∧ i ≤ N) (D : Set ℂ)
-    (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ D)
-    (ω : Ω) (hω : ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
-      ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) :
-    ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ N) P ε ∧
-      (I.card : ℝ≥0∞) * ENNReal.ofReal c ≤ wholeSausageCharge κ (γ ω) v D := by
-  refine ⟨actual_successes_terminal_prefix γ g P ε κ v c N ω hω, ?_⟩
-  apply whole_sausage_charge_from_ribbon_steps hplanar P hP hi hε I κ (γ ω)
-    (fun i => (ribbonIntegerFront g γ (i - 1) ω).untopD 0)
-    (fun i => (ribbonIntegerFront g γ i ω).untopD 0) v c D hregions
-  intro i hiI
-  obtain ⟨hi0, hiN⟩ := hI i hiI
-  have hsub : i - 1 + 1 = i := Nat.sub_add_cancel hi0
-  have hreal : ((i - 1 : ℕ) : ℝ) + 1 = (i : ℝ) := by exact_mod_cast hsub
-  have hn : i - 1 < N := by omega
-  have hs := (mem_iInter.mp (mem_iInter.mp hω.2 (i - 1)) (Finset.mem_range.mpr hn)).2
-  simpa only [actualRibbonSuccess, RibbonStepSuccess, mem_inter_iff, mem_ofPred_eq,
-    hsub, hreal] using hs
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PublishedRibbonIteration
+
+namespace OAI
+
+/-! Charge selected actual successive-front windows to a single sausage,
+retaining the successful terminal prefix for clearance and density transfer. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates
+
+lemma actual_successes_terminal_prefix
+    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (g : C(ℂ, ℝ)) (P : ℂ → ℂ)
+    (ε κ v c : ℝ) (N : ℕ) (ω : Ω)
+    (hω : ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
+      ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) :
+    ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ N) P ε := by
+  cases N with
+  | zero => exact hω.1
+  | succ N =>
+    exact (mem_iInter.mp (mem_iInter.mp hω.2 N) (Finset.mem_range.mpr (Nat.lt_succ_self N))).1
+
+/-- Only selected middle strips are charged. The trace windows are the
+original actual fronts, including their exact starting and ending times. -/
+theorem actual_successes_force_whole_mass
+    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (g : C(ℂ, ℝ)) (P : ℂ → ℂ)
+    (hplanar : PublishedPlanarDiskOpenness) (hP : ContinuousOn P closedRibbon)
+    (hi : InjOn P closedRibbon) (ε κ v c : ℝ) (hε : ε ≤ 1)
+    (N : ℕ) (I : Finset ℕ) (hI : ∀ i ∈ I, 0 < i ∧ i ≤ N) (D : Set ℂ)
+    (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ D)
+    (ω : Ω) (hω : ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
+      ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) :
+    ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ N) P ε ∧
+      (I.card : ℝ≥0∞) * ENNReal.ofReal c ≤ wholeSausageCharge κ (γ ω) v D := by
+  refine ⟨actual_successes_terminal_prefix γ g P ε κ v c N ω hω, ?_⟩
+  apply whole_sausage_charge_from_ribbon_steps hplanar P hP hi hε I κ (γ ω)
+    (fun i => (ribbonIntegerFront g γ (i - 1) ω).untopD 0)
+    (fun i => (ribbonIntegerFront g γ i ω).untopD 0) v c D hregions
+  intro i hiI
+  obtain ⟨hi0, hiN⟩ := hI i hiI
+  have hsub : i - 1 + 1 = i := Nat.sub_add_cancel hi0
+  have hreal : ((i - 1 : ℕ) : ℝ) + 1 = (i : ℝ) := by exact_mod_cast hsub
+  have hn : i - 1 < N := by omega
+  have hs := (mem_iInter.mp (mem_iInter.mp hω.2 (i - 1)) (Finset.mem_range.mpr hn)).2
+  simpa only [actualRibbonSuccess, RibbonStepSuccess, mem_inter_iff, mem_ofPred_eq,
+    hsub, hreal] using hs
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualRibbonPrefixBound.lean b/lean/OAI/Probability/SLE/Gauge/ActualRibbonPrefixBound.lean
index d3ee23895acd5f3628eaf4d0e68416e9c278d57a..7494851a09fd2c80919412296a0f47b8ad42e654 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualRibbonPrefixBound.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualRibbonPrefixBound.lean
@@ -1,83 +1,83 @@
-import OAI.Probability.SLE.Gauge.ActualRibbonStep
-
-namespace OAI
-
-/-! Apply the proved actual conditional ribbon estimate on the measurable
-physical prefix, then charge its mass to the two original global fronts. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates FirstFrontPassage
-
-local instance prefixBoundPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance prefixBoundPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem actual_ribbon_step_on_good_prefix
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ)
-    (U : Ω → ℝ≥0 → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hU : ∀ ω, Continuous (U ω)) (hw : ∀ ω, IsCapacityTwoWitness (U ω) (γ ω) (G ω) (F ω))
-    (C L : ℝ≥0) (hL : 0 < L) (ε κ v c : ℝ) (hε : ε ≤ 1 / 8) (q : ℝ≥0∞)
-    (P : ℂ → ℂ) (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
-    (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
-    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
-    (hescape : Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop)
-    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re)
-    (hg0 : ∀ ω, g (γ ω 0) = 0) (k : ℝ) (hk : 1 ≤ k)
-    (hu : ∀ j ∈ ({k, k + 1} : Set ℝ), LipschitzOnWith C P (ribbonStepPatch j))
-    (hl : ∀ j ∈ ({k, k + 1} : Set ℝ), ∀ z ∈ ribbonStepPatch j,
-      ∀ w ∈ ribbonStepPatch j, dist z w ≤ L * dist (P z) (P w))
-    (hτ : IsStoppingTime ℱ (ribbonFrontStop g γ k))
-    (hbound : ∀ E : Set Ω,
-      (∀ᵐ ω ∂μ, ω ∈ E → ribbonFrontStop g γ k ω ≠ ⊤) →
-      (∀ᵐ ω ∂μ, ω ∈ E → ActualRibbonAdmissible U γ G F C L ε κ v c
-        (ribbonPathStep P C L ε k κ v c) ω ((ribbonFrontStop g γ k ω).untopD 0)) →
-      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → A ⊆ E →
-        q * μ A ≤ μ (A ∩ {ω | shiftedActualTrace γ (fun ω => (hw ω).1) ω
-          ((ribbonFrontStop g γ k ω).untopD 0) ∈ ribbonPathStep P C L ε k κ v c}))
-    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A)
-    (hAE : A ⊆ GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε) :
-    q * μ A ≤ μ (A ∩ RibbonStepSuccess γ (ribbonFrontStop g γ k)
-      (ribbonFrontStop g γ (k + 1)) P ε κ v c (physicalMiddleRegion P closedRibbon k ε)) := by
-  have hcert (ω : Ω) (hω : ω ∈ GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε) :
-      ∃ d : RibbonHistory C L ε, d.U = U ω ∧ d.trace = γ ω ∧ d.forward = G ω ∧
-        d.inverse = F ω ∧ d.chart = P ∧ d.front = k ∧
-        d.time = (ribbonFrontStop g γ k ω).untopD 0 := by
-    obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ :=
-      actual_history_of_goodRibbonPrefix (U ω) (γ ω) (G ω) (F ω) (hU ω) (hw ω)
-        P hemb hP hi hH hescape g hg (hg0 ω) k hk hω (hu k (by simp)) (hl k (by simp))
-    refine ⟨d, hU', hγ', hG', hF', hP', hk', ?_⟩
-    have hh := congrArg (fun t : WithTop ℝ≥0 => t.untopD 0) ht
-    simpa only [WithTop.untopD_coe, ribbonFrontStop] using hh
-  have hstates : ∀ᵐ ω ∂μ, ω ∈ GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε →
-      ActualRibbonAdmissible U γ G F C L ε κ v c (ribbonPathStep P C L ε k κ v c)
-        ω ((ribbonFrontStop g γ k ω).untopD 0) := by
-    apply Filter.Eventually.of_forall
-    intro ω hω
-    obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ := hcert ω hω
-    exact ⟨d, hU', hγ', hG', hF', ht, by rw [d.stepEvent_eq_ribbonPathStep, hP', hk']⟩
-  have hb := hbound (GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε)
-    (Filter.Eventually.of_forall (fun _ hω => hω.1)) hstates A hA hAE
-  apply hb.trans (measure_mono ?_)
-  intro ω hω
-  refine ⟨hω.1, ?_⟩
-  obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ := hcert ω (hAE hω.1)
-  have he : d.futureTrace = shiftedActualTrace γ (fun ω => (hw ω).1) ω
-      ((ribbonFrontStop g γ k ω).untopD 0) := by
-    ext t
-    simp only [RibbonHistory.futureTrace, shiftedActualTrace, ContinuousMap.coe_mk, hγ', ht]
-  have hstep : d.futureTrace ∈ d.stepEvent κ v c := by
-    rw [he, d.stepEvent_eq_ribbonPathStep, hP', hk']
-    exact hω.2
-  have hh := d.actual_step_success hε hL κ v c g (by rw [hP']; exact hg) hstep
-    (by rw [hP', hk']; exact hu _ (by simp)) (by rw [hP', hk']; exact hl _ (by simp))
-  simpa only [RibbonStepSuccess, GoodRibbonPrefix, mem_inter_iff, mem_ofPred_eq,
-    hγ', hP', hk', ribbonFrontStop] using hh
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualRibbonStep
+
+namespace OAI
+
+/-! Apply the proved actual conditional ribbon estimate on the measurable
+physical prefix, then charge its mass to the two original global fronts. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates FirstFrontPassage
+
+local instance prefixBoundPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance prefixBoundPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem actual_ribbon_step_on_good_prefix
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ)
+    (U : Ω → ℝ≥0 → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hU : ∀ ω, Continuous (U ω)) (hw : ∀ ω, IsCapacityTwoWitness (U ω) (γ ω) (G ω) (F ω))
+    (C L : ℝ≥0) (hL : 0 < L) (ε κ v c : ℝ) (hε : ε ≤ 1 / 8) (q : ℝ≥0∞)
+    (P : ℂ → ℂ) (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
+    (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
+    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
+    (hescape : Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop)
+    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re)
+    (hg0 : ∀ ω, g (γ ω 0) = 0) (k : ℝ) (hk : 1 ≤ k)
+    (hu : ∀ j ∈ ({k, k + 1} : Set ℝ), LipschitzOnWith C P (ribbonStepPatch j))
+    (hl : ∀ j ∈ ({k, k + 1} : Set ℝ), ∀ z ∈ ribbonStepPatch j,
+      ∀ w ∈ ribbonStepPatch j, dist z w ≤ L * dist (P z) (P w))
+    (hτ : IsStoppingTime ℱ (ribbonFrontStop g γ k))
+    (hbound : ∀ E : Set Ω,
+      (∀ᵐ ω ∂μ, ω ∈ E → ribbonFrontStop g γ k ω ≠ ⊤) →
+      (∀ᵐ ω ∂μ, ω ∈ E → ActualRibbonAdmissible U γ G F C L ε κ v c
+        (ribbonPathStep P C L ε k κ v c) ω ((ribbonFrontStop g γ k ω).untopD 0)) →
+      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → A ⊆ E →
+        q * μ A ≤ μ (A ∩ {ω | shiftedActualTrace γ (fun ω => (hw ω).1) ω
+          ((ribbonFrontStop g γ k ω).untopD 0) ∈ ribbonPathStep P C L ε k κ v c}))
+    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A)
+    (hAE : A ⊆ GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε) :
+    q * μ A ≤ μ (A ∩ RibbonStepSuccess γ (ribbonFrontStop g γ k)
+      (ribbonFrontStop g γ (k + 1)) P ε κ v c (physicalMiddleRegion P closedRibbon k ε)) := by
+  have hcert (ω : Ω) (hω : ω ∈ GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε) :
+      ∃ d : RibbonHistory C L ε, d.U = U ω ∧ d.trace = γ ω ∧ d.forward = G ω ∧
+        d.inverse = F ω ∧ d.chart = P ∧ d.front = k ∧
+        d.time = (ribbonFrontStop g γ k ω).untopD 0 := by
+    obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ :=
+      actual_history_of_goodRibbonPrefix (U ω) (γ ω) (G ω) (F ω) (hU ω) (hw ω)
+        P hemb hP hi hH hescape g hg (hg0 ω) k hk hω (hu k (by simp)) (hl k (by simp))
+    refine ⟨d, hU', hγ', hG', hF', hP', hk', ?_⟩
+    have hh := congrArg (fun t : WithTop ℝ≥0 => t.untopD 0) ht
+    simpa only [WithTop.untopD_coe, ribbonFrontStop] using hh
+  have hstates : ∀ᵐ ω ∂μ, ω ∈ GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε →
+      ActualRibbonAdmissible U γ G F C L ε κ v c (ribbonPathStep P C L ε k κ v c)
+        ω ((ribbonFrontStop g γ k ω).untopD 0) := by
+    apply Filter.Eventually.of_forall
+    intro ω hω
+    obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ := hcert ω hω
+    exact ⟨d, hU', hγ', hG', hF', ht, by rw [d.stepEvent_eq_ribbonPathStep, hP', hk']⟩
+  have hb := hbound (GoodRibbonPrefix γ (ribbonFrontStop g γ k) P ε)
+    (Filter.Eventually.of_forall (fun _ hω => hω.1)) hstates A hA hAE
+  apply hb.trans (measure_mono ?_)
+  intro ω hω
+  refine ⟨hω.1, ?_⟩
+  obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ := hcert ω (hAE hω.1)
+  have he : d.futureTrace = shiftedActualTrace γ (fun ω => (hw ω).1) ω
+      ((ribbonFrontStop g γ k ω).untopD 0) := by
+    ext t
+    simp only [RibbonHistory.futureTrace, shiftedActualTrace, ContinuousMap.coe_mk, hγ', ht]
+  have hstep : d.futureTrace ∈ d.stepEvent κ v c := by
+    rw [he, d.stepEvent_eq_ribbonPathStep, hP', hk']
+    exact hω.2
+  have hh := d.actual_step_success hε hL κ v c g (by rw [hP']; exact hg) hstep
+    (by rw [hP', hk']; exact hu _ (by simp)) (by rw [hP', hk']; exact hl _ (by simp))
+  simpa only [RibbonStepSuccess, GoodRibbonPrefix, mem_inter_iff, mem_ofPred_eq,
+    hγ', hP', hk', ribbonFrontStop] using hh
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualRibbonStep.lean b/lean/OAI/Probability/SLE/Gauge/ActualRibbonStep.lean
index cf15755f28e43ca2c4b1d5111aac7fb3b89a0909..efb53b10c8d5dd27df134182f4d3d3929ab4da44 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualRibbonStep.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualRibbonStep.lean
@@ -1,138 +1,138 @@
-import OAI.Probability.SLE.Gauge.RibbonStepSuccess
-import OAI.Probability.SLE.Gauge.ExtendedRibbonStep
-
-namespace OAI
-
-/-! A measurable narrow physical prefix yields the exact actual history.
-A successful uniform step yields the next such prefix and an actual mass
-window between successive global first-front times. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates FirstFrontPassage BoundaryScaleLaw
-
-/-- The path-space step event is deterministic once the chart and integer
-front are fixed, although the actual Loewner history is random. -/
-def ribbonPathStep (P : ℂ → ℂ) (C L : ℝ≥0) (ε k κ v c : ℝ) : Set C(ℝ≥0, ℂ) :=
-  frontMassEvent (longitudinalCoordinate P closedRibbon k)
-    (physicalPassageRegion (fun z => P ((k : ℂ) + z)) ribbonStepCompact C ε (ε / L))
-    (P ((k : ℂ) + 1)) (ε / L) κ v c (physicalMiddleRegion P closedRibbon k ε)
-
-lemma RibbonHistory.stepEvent_eq_ribbonPathStep
-    {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (κ v c : ℝ) :
-    d.stepEvent κ v c = ribbonPathStep d.chart C L ε d.front κ v c := rfl
-
-lemma ribbonPathStep_measurable
-    (P : ℂ → ℂ) (C L : ℝ≥0) (ε k κ v c : ℝ)
-    (hplanar : PublishedPlanarDiskOpenness) (hε : ε ≤ 1 / 8) (hk : 1 ≤ k)
-    (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
-    (hP : ContinuousOn P closedRibbon)
-    (hlower : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k,
-      dist z w ≤ L * dist (P z) (P w)) :
-    MeasurableSet (ribbonPathStep P C L ε k κ v c) := by
-  let Q := fun z : ℂ => P ((k : ℂ) + z)
-  have hmap : MapsTo (fun z : ℂ => (k : ℂ) + z) ribbonStepCompact closedRibbon := by
-    intro z hz
-    change 0 ≤ k + z.re ∧ |0 + z.im| ≤ 1
-    exact ⟨by linarith [hz.1], by simpa using hz.2.2.trans (by norm_num)⟩
-  have hQ : ContinuousOn Q ribbonStepCompact :=
-    hP.comp (continuous_const.add continuous_id).continuousOn hmap
-  have hQl : ∀ u ∈ ribbonStepCompact, ∀ w ∈ ribbonStepCompact,
-      dist u w ≤ L * dist (Q u) (Q w) := by
-    intro u hu w hw
-    simpa only [Q, dist_add_left] using
-      hlower _ (translate_ribbonStepCompact hu) _ (translate_ribbonStepCompact hw)
-  have hK := ribbonStepCompact_isCompact.image_of_continuousOn hQ
-  have hp : ContinuousOn (longitudinalCoordinate P closedRibbon k) (Q '' ribbonStepCompact) :=
-    ((Complex.continuous_re.comp_continuousOn
-      (coordinateInverse_continuousOn_of_embedding P hemb.isEmbedding)).sub continuousOn_const).mono
-      (by rintro _ ⟨z, hz, rfl⟩; exact ⟨(k : ℂ) + z, hmap hz, rfl⟩)
-  have hV : IsOpen (physicalPassageRegion Q ribbonStepCompact C ε (ε / L)) := by
-    have he : ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) =
-        interior ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) := by
-      apply Subset.antisymm
-      · exact fun z hz => ⟨ribbonStepCompact_passage hε C (ε / L) hz, hz.2⟩
-      · exact inter_subset_inter_left _ interior_subset
-    unfold physicalPassageRegion
-    rw [he]
-    exact isOpen_image_coordinate_subset hplanar Q ribbonStepCompact _ hQ
-      (lower_lipschitz_injOn _ _ L hQl)
-      (isOpen_interior.inter (passageCoordinateRegion_isOpen C ε (ε / L)))
-      (inter_subset_left.trans interior_subset)
-  exact measurableSet_frontMassEvent _ _ _ _ κ v c _ _ hK.isClosed hp hV
-    (image_mono inter_subset_left)
-
-theorem actual_history_of_goodRibbonPrefix
-    {C L : ℝ≥0} {ε : ℝ}
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (P : ℂ → ℂ)
-    (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
-    (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
-    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
-    (hescape : Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop)
-    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re)
-    (hg0 : g (γ 0) = 0) (k : ℝ) (hk : 1 ≤ k)
-    (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ)
-      (fun _ => firstFrontTime (fun t => g (γ t)) k) P ε)
-    (hupper : LipschitzOnWith C P (ribbonStepPatch k))
-    (hlower : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k,
-      dist z w ≤ L * dist (P z) (P w)) :
-    ∃ d : RibbonHistory C L ε, d.U = U ∧ d.trace = γ ∧ d.forward = G ∧
-      d.inverse = F ∧ d.chart = P ∧ d.front = k ∧
-      (d.time : WithTop ℝ≥0) = firstFrontTime (fun t => g (γ t)) k := by
-  let τ := firstFrontTime (fun t => g (γ t)) k
-  have hfinite : τ ≠ ⊤ := hpref.1
-  lift τ to ℝ≥0 using hfinite with T hT
-  have hfirst : firstFrontTime (fun t => g (γ t)) k = (T : WithTop ℝ≥0) := hT.symm
-  have hprefix : MapsTo γ (Icc 0 T) (ribbonCore P ε) := by
-    simpa only [hfirst, WithTop.untopD_coe] using hpref.2.1
-  have hend : γ T ∈ ribbonCore P (ε / 4) := by
-    simpa only [hfirst, WithTop.untopD_coe] using hpref.2.2
-  have hoff : |(coordinateInverse P closedRibbon (γ T)).im| ≤ ε / 4 := by
-    obtain ⟨z, ⟨hz, hiε⟩, he⟩ := hend
-    simpa only [← he, coordinateInverse_left hi hz] using hiε
-  obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ := actual_history_at_global_front
-    U γ G F hU hw P hemb hP hi hH hescape g hg hg0 k hk T hfirst hprefix hoff hupper hlower
-  exact ⟨d, hU', hγ', hG', hF', hP', hk', by rw [ht]; exact hfirst.symm⟩
-
-/-- The uniform step's mass is charged to the actual successive global
-fronts, and its continuation is again a good measurable physical prefix. -/
-theorem RibbonHistory.actual_step_success
-    {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε)
-    (hε : ε ≤ 1 / 8) (hL : 0 < L) (κ v c : ℝ)
-    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (d.chart z) = z.re)
-    (hstep : d.futureTrace ∈ d.stepEvent κ v c)
-    (hupper : LipschitzOnWith C d.chart (ribbonStepPatch (d.front + 1)))
-    (hlower : ∀ u ∈ ribbonStepPatch (d.front + 1),
-      ∀ v ∈ ribbonStepPatch (d.front + 1), dist u v ≤ L * dist (d.chart u) (d.chart v)) :
-    () ∈ RibbonStepSuccess (fun _ : Unit => d.trace)
-      (fun _ => firstFrontTime (fun t => g (d.trace t)) d.front)
-      (fun _ => firstFrontTime (fun t => g (d.trace t)) (d.front + 1))
-      d.chart ε κ v c (physicalMiddleRegion d.chart closedRibbon d.front ε) := by
-  obtain ⟨d', hf, ht, hchart, _, htrace, _, _, _⟩ :=
-    d.next_history_of_stepEvent hε hL κ v c hstep hupper hlower
-  have hnext := d'.global_frontStop_eq g (by rw [hchart]; exact hg)
-  rw [htrace, hf] at hnext
-  have hprev := d.global_frontStop_eq g hg
-  change _ ∧ _
-  constructor
-  · have hh := d'.goodRibbonPrefix
-    simpa only [GoodRibbonPrefix, mem_ofPred_eq, htrace, hchart, hnext,
-      WithTop.untopD_coe] using hh
-  · have hm : ENNReal.ofReal c ≤ segmentSausageCharge κ d.futureTrace 0
-        (frontTime (fun t => longitudinalCoordinate d.chart closedRibbon d.front
-          (d.futureTrace t)) 1) v (physicalMiddleRegion d.chart closedRibbon d.front ε) := hstep.2
-    change ENNReal.ofReal c ≤ segmentSausageCharge κ
-      (fun t => d.trace (d.time + t)) 0 _ v _ at hm
-    rw [segmentSausageCharge_shift, ← ht] at hm
-    simpa only [mem_ofPred_eq, hprev, hnext, WithTop.untopD_coe] using hm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RibbonStepSuccess
+import OAI.Probability.SLE.Gauge.ExtendedRibbonStep
+
+namespace OAI
+
+/-! A measurable narrow physical prefix yields the exact actual history.
+A successful uniform step yields the next such prefix and an actual mass
+window between successive global first-front times. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates FirstFrontPassage BoundaryScaleLaw
+
+/-- The path-space step event is deterministic once the chart and integer
+front are fixed, although the actual Loewner history is random. -/
+def ribbonPathStep (P : ℂ → ℂ) (C L : ℝ≥0) (ε k κ v c : ℝ) : Set C(ℝ≥0, ℂ) :=
+  frontMassEvent (longitudinalCoordinate P closedRibbon k)
+    (physicalPassageRegion (fun z => P ((k : ℂ) + z)) ribbonStepCompact C ε (ε / L))
+    (P ((k : ℂ) + 1)) (ε / L) κ v c (physicalMiddleRegion P closedRibbon k ε)
+
+lemma RibbonHistory.stepEvent_eq_ribbonPathStep
+    {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε) (κ v c : ℝ) :
+    d.stepEvent κ v c = ribbonPathStep d.chart C L ε d.front κ v c := rfl
+
+lemma ribbonPathStep_measurable
+    (P : ℂ → ℂ) (C L : ℝ≥0) (ε k κ v c : ℝ)
+    (hplanar : PublishedPlanarDiskOpenness) (hε : ε ≤ 1 / 8) (hk : 1 ≤ k)
+    (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
+    (hP : ContinuousOn P closedRibbon)
+    (hlower : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k,
+      dist z w ≤ L * dist (P z) (P w)) :
+    MeasurableSet (ribbonPathStep P C L ε k κ v c) := by
+  let Q := fun z : ℂ => P ((k : ℂ) + z)
+  have hmap : MapsTo (fun z : ℂ => (k : ℂ) + z) ribbonStepCompact closedRibbon := by
+    intro z hz
+    change 0 ≤ k + z.re ∧ |0 + z.im| ≤ 1
+    exact ⟨by linarith [hz.1], by simpa using hz.2.2.trans (by norm_num)⟩
+  have hQ : ContinuousOn Q ribbonStepCompact :=
+    hP.comp (continuous_const.add continuous_id).continuousOn hmap
+  have hQl : ∀ u ∈ ribbonStepCompact, ∀ w ∈ ribbonStepCompact,
+      dist u w ≤ L * dist (Q u) (Q w) := by
+    intro u hu w hw
+    simpa only [Q, dist_add_left] using
+      hlower _ (translate_ribbonStepCompact hu) _ (translate_ribbonStepCompact hw)
+  have hK := ribbonStepCompact_isCompact.image_of_continuousOn hQ
+  have hp : ContinuousOn (longitudinalCoordinate P closedRibbon k) (Q '' ribbonStepCompact) :=
+    ((Complex.continuous_re.comp_continuousOn
+      (coordinateInverse_continuousOn_of_embedding P hemb.isEmbedding)).sub continuousOn_const).mono
+      (by rintro _ ⟨z, hz, rfl⟩; exact ⟨(k : ℂ) + z, hmap hz, rfl⟩)
+  have hV : IsOpen (physicalPassageRegion Q ribbonStepCompact C ε (ε / L)) := by
+    have he : ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) =
+        interior ribbonStepCompact ∩ passageCoordinateRegion C ε (ε / L) := by
+      apply Subset.antisymm
+      · exact fun z hz => ⟨ribbonStepCompact_passage hε C (ε / L) hz, hz.2⟩
+      · exact inter_subset_inter_left _ interior_subset
+    unfold physicalPassageRegion
+    rw [he]
+    exact isOpen_image_coordinate_subset hplanar Q ribbonStepCompact _ hQ
+      (lower_lipschitz_injOn _ _ L hQl)
+      (isOpen_interior.inter (passageCoordinateRegion_isOpen C ε (ε / L)))
+      (inter_subset_left.trans interior_subset)
+  exact measurableSet_frontMassEvent _ _ _ _ κ v c _ _ hK.isClosed hp hV
+    (image_mono inter_subset_left)
+
+theorem actual_history_of_goodRibbonPrefix
+    {C L : ℝ≥0} {ε : ℝ}
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (P : ℂ → ℂ)
+    (hemb : Topology.IsClosedEmbedding (fun z : closedRibbon => P z))
+    (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
+    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
+    (hescape : Tendsto (fun t : ℝ => ‖P (t : ℂ)‖) atTop atTop)
+    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (P z) = z.re)
+    (hg0 : g (γ 0) = 0) (k : ℝ) (hk : 1 ≤ k)
+    (hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ)
+      (fun _ => firstFrontTime (fun t => g (γ t)) k) P ε)
+    (hupper : LipschitzOnWith C P (ribbonStepPatch k))
+    (hlower : ∀ z ∈ ribbonStepPatch k, ∀ w ∈ ribbonStepPatch k,
+      dist z w ≤ L * dist (P z) (P w)) :
+    ∃ d : RibbonHistory C L ε, d.U = U ∧ d.trace = γ ∧ d.forward = G ∧
+      d.inverse = F ∧ d.chart = P ∧ d.front = k ∧
+      (d.time : WithTop ℝ≥0) = firstFrontTime (fun t => g (γ t)) k := by
+  let τ := firstFrontTime (fun t => g (γ t)) k
+  have hfinite : τ ≠ ⊤ := hpref.1
+  lift τ to ℝ≥0 using hfinite with T hT
+  have hfirst : firstFrontTime (fun t => g (γ t)) k = (T : WithTop ℝ≥0) := hT.symm
+  have hprefix : MapsTo γ (Icc 0 T) (ribbonCore P ε) := by
+    simpa only [hfirst, WithTop.untopD_coe] using hpref.2.1
+  have hend : γ T ∈ ribbonCore P (ε / 4) := by
+    simpa only [hfirst, WithTop.untopD_coe] using hpref.2.2
+  have hoff : |(coordinateInverse P closedRibbon (γ T)).im| ≤ ε / 4 := by
+    obtain ⟨z, ⟨hz, hiε⟩, he⟩ := hend
+    simpa only [← he, coordinateInverse_left hi hz] using hiε
+  obtain ⟨d, hU', hγ', hG', hF', hP', hk', ht⟩ := actual_history_at_global_front
+    U γ G F hU hw P hemb hP hi hH hescape g hg hg0 k hk T hfirst hprefix hoff hupper hlower
+  exact ⟨d, hU', hγ', hG', hF', hP', hk', by rw [ht]; exact hfirst.symm⟩
+
+/-- The uniform step's mass is charged to the actual successive global
+fronts, and its continuation is again a good measurable physical prefix. -/
+theorem RibbonHistory.actual_step_success
+    {C L : ℝ≥0} {ε : ℝ} (d : RibbonHistory C L ε)
+    (hε : ε ≤ 1 / 8) (hL : 0 < L) (κ v c : ℝ)
+    (g : C(ℂ, ℝ)) (hg : ∀ z ∈ closedRibbon, g (d.chart z) = z.re)
+    (hstep : d.futureTrace ∈ d.stepEvent κ v c)
+    (hupper : LipschitzOnWith C d.chart (ribbonStepPatch (d.front + 1)))
+    (hlower : ∀ u ∈ ribbonStepPatch (d.front + 1),
+      ∀ v ∈ ribbonStepPatch (d.front + 1), dist u v ≤ L * dist (d.chart u) (d.chart v)) :
+    () ∈ RibbonStepSuccess (fun _ : Unit => d.trace)
+      (fun _ => firstFrontTime (fun t => g (d.trace t)) d.front)
+      (fun _ => firstFrontTime (fun t => g (d.trace t)) (d.front + 1))
+      d.chart ε κ v c (physicalMiddleRegion d.chart closedRibbon d.front ε) := by
+  obtain ⟨d', hf, ht, hchart, _, htrace, _, _, _⟩ :=
+    d.next_history_of_stepEvent hε hL κ v c hstep hupper hlower
+  have hnext := d'.global_frontStop_eq g (by rw [hchart]; exact hg)
+  rw [htrace, hf] at hnext
+  have hprev := d.global_frontStop_eq g hg
+  change _ ∧ _
+  constructor
+  · have hh := d'.goodRibbonPrefix
+    simpa only [GoodRibbonPrefix, mem_ofPred_eq, htrace, hchart, hnext,
+      WithTop.untopD_coe] using hh
+  · have hm : ENNReal.ofReal c ≤ segmentSausageCharge κ d.futureTrace 0
+        (frontTime (fun t => longitudinalCoordinate d.chart closedRibbon d.front
+          (d.futureTrace t)) 1) v (physicalMiddleRegion d.chart closedRibbon d.front ε) := hstep.2
+    change ENNReal.ofReal c ≤ segmentSausageCharge κ
+      (fun t => d.trace (d.time + t)) 0 _ v _ at hm
+    rw [segmentSausageCharge_shift, ← ht] at hm
+    simpa only [mem_ofPred_eq, hprev, hnext, WithTop.untopD_coe] using hm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualRibbonStoppedMass.lean b/lean/OAI/Probability/SLE/Gauge/ActualRibbonStoppedMass.lean
index 98f4752be57b582063f5e3738d7cbe2610e00e88..6988e1a0bf7f2fe66e3991682071c00a09d0b9ee 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualRibbonStoppedMass.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualRibbonStoppedMass.lean
@@ -1,80 +1,80 @@
-import OAI.Probability.SLE.Gauge.ActualRibbonMass
-
-namespace OAI
-
-/-! The retained mass is present at successful completion and is observable
-at that stopping time. It supplies the adapted cover tests and the
-successful-completion change of measure. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates FrontSuccessMeasurability
-
-lemma untopD_le_of_le_finite {a b : WithTop ℝ≥0} (hab : a ≤ b) (hb : b ≠ ⊤) :
-    a.untopD 0 ≤ b.untopD 0 := by
-  cases a with
-  | top => exact False.elim (hb (top_le_iff.mp hab))
-  | coe s =>
-    cases b with
-    | top => exact False.elim (hb rfl)
-    | coe t => simpa only [WithTop.untopD_coe, WithTop.coe_le_coe] using hab
-
-/-- All selected mass is measurable by the successful completion time;
-no later portion of the trace is used to create a dense candidate. -/
-theorem actual_successes_force_stopped_mass
-    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (hc : ∀ ω, Continuous (γ ω))
-    (g : C(ℂ, ℝ)) (hg0 : ∀ ω, g (γ ω 0) = 0) (P : ℂ → ℂ)
-    (hplanar : PublishedPlanarDiskOpenness) (hP : ContinuousOn P closedRibbon)
-    (hi : InjOn P closedRibbon) (ε κ v c : ℝ) (hε : ε ≤ 1)
-    (N : ℕ) (I : Finset ℕ) (hI : ∀ i ∈ I, 0 < i ∧ i ≤ N) (D : Set ℂ)
-    (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ D)
-    (ω : Ω) (hω : ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
-      ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) :
-    ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ N) P ε ∧
-      (I.card : ℝ≥0∞) * ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω) 0
-        ((ribbonIntegerFront g γ N ω).untopD 0) v D := by
-  have hpref := actual_successes_terminal_prefix γ g P ε κ v c N ω hω
-  have hmono := ribbonIntegerFront_monotone g γ hc hg0
-  let T := (ribbonIntegerFront g γ N ω).untopD 0
-  let f := sourceTracePath γ hc ω
-  let p := clampedPath f 0 T
-  have htbound (i : ℕ) (hiN : i ≤ N) : (ribbonIntegerFront g γ i ω).untopD 0 ≤ T :=
-    untopD_le_of_le_finite (hmono hiN ω) hpref.1
-  have hmass := whole_sausage_charge_from_ribbon_steps hplanar P hP hi hε I κ p
-    (fun i => (ribbonIntegerFront g γ (i - 1) ω).untopD 0)
-    (fun i => (ribbonIntegerFront g γ i ω).untopD 0) v c D hregions
-  have hselected (i : ℕ) (hiI : i ∈ I) : ENNReal.ofReal c ≤
-      segmentSausageCharge κ p ((ribbonIntegerFront g γ (i - 1) ω).untopD 0)
-        ((ribbonIntegerFront g γ i ω).untopD 0) v (physicalMiddleRegion P closedRibbon i ε) := by
-    obtain ⟨hi0, hiN⟩ := hI i hiI
-    have hsub : i - 1 + 1 = i := Nat.sub_add_cancel hi0
-    have hreal : ((i - 1 : ℕ) : ℝ) + 1 = (i : ℝ) := by exact_mod_cast hsub
-    have hn : i - 1 < N := by omega
-    have hs := (mem_iInter.mp (mem_iInter.mp hω.2 (i - 1)) (Finset.mem_range.mpr hn)).2
-    have hcharge : ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω)
-        ((ribbonIntegerFront g γ (i - 1) ω).untopD 0) ((ribbonIntegerFront g γ i ω).untopD 0) v
-        (physicalMiddleRegion P closedRibbon i ε) := by
-      simpa only [mem_ofPred_eq, hsub, hreal] using hs
-    have he : EqOn p (γ ω) (Icc ((ribbonIntegerFront g γ (i - 1) ω).untopD 0)
-        ((ribbonIntegerFront g γ i ω).untopD 0)) := by
-      intro u hu
-      have huT := hu.2.trans (htbound i hiN)
-      simp only [p, clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
-        min_eq_left huT, zero_le, max_eq_right, f, sourceTracePath]
-    simpa only [segmentSausageCharge, sourceSausageSet_congr_interval he v] using hcharge
-  have hh := hmass hselected
-  have he := sourceSausageSet_eq_clamped f 0 T (bot_le : (0 : ℝ≥0) ≤ T) v
-  refine ⟨hpref, ?_⟩
-  change (I.card : ℝ≥0∞) * ENNReal.ofReal c ≤ wholeSausageCharge κ (clampedPath f 0 T) v D at hh
-  unfold wholeSausageCharge at hh
-  rw [← he] at hh
-  simpa only [segmentSausageCharge, f, sourceTracePath, ContinuousMap.coe_mk] using hh
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualRibbonMass
+
+namespace OAI
+
+/-! The retained mass is present at successful completion and is observable
+at that stopping time. It supplies the adapted cover tests and the
+successful-completion change of measure. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates FrontSuccessMeasurability
+
+lemma untopD_le_of_le_finite {a b : WithTop ℝ≥0} (hab : a ≤ b) (hb : b ≠ ⊤) :
+    a.untopD 0 ≤ b.untopD 0 := by
+  cases a with
+  | top => exact False.elim (hb (top_le_iff.mp hab))
+  | coe s =>
+    cases b with
+    | top => exact False.elim (hb rfl)
+    | coe t => simpa only [WithTop.untopD_coe, WithTop.coe_le_coe] using hab
+
+/-- All selected mass is measurable by the successful completion time;
+no later portion of the trace is used to create a dense candidate. -/
+theorem actual_successes_force_stopped_mass
+    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (hc : ∀ ω, Continuous (γ ω))
+    (g : C(ℂ, ℝ)) (hg0 : ∀ ω, g (γ ω 0) = 0) (P : ℂ → ℂ)
+    (hplanar : PublishedPlanarDiskOpenness) (hP : ContinuousOn P closedRibbon)
+    (hi : InjOn P closedRibbon) (ε κ v c : ℝ) (hε : ε ≤ 1)
+    (N : ℕ) (I : Finset ℕ) (hI : ∀ i ∈ I, 0 < i ∧ i ≤ N) (D : Set ℂ)
+    (hregions : ∀ i ∈ I, physicalMiddleRegion P closedRibbon i ε ⊆ D)
+    (ω : Ω) (hω : ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ 0) P ε ∩
+      ⋂ n ∈ Finset.range N, actualRibbonSuccess γ g P ε κ v c n) :
+    ω ∈ GoodRibbonPrefix γ (ribbonIntegerFront g γ N) P ε ∧
+      (I.card : ℝ≥0∞) * ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω) 0
+        ((ribbonIntegerFront g γ N ω).untopD 0) v D := by
+  have hpref := actual_successes_terminal_prefix γ g P ε κ v c N ω hω
+  have hmono := ribbonIntegerFront_monotone g γ hc hg0
+  let T := (ribbonIntegerFront g γ N ω).untopD 0
+  let f := sourceTracePath γ hc ω
+  let p := clampedPath f 0 T
+  have htbound (i : ℕ) (hiN : i ≤ N) : (ribbonIntegerFront g γ i ω).untopD 0 ≤ T :=
+    untopD_le_of_le_finite (hmono hiN ω) hpref.1
+  have hmass := whole_sausage_charge_from_ribbon_steps hplanar P hP hi hε I κ p
+    (fun i => (ribbonIntegerFront g γ (i - 1) ω).untopD 0)
+    (fun i => (ribbonIntegerFront g γ i ω).untopD 0) v c D hregions
+  have hselected (i : ℕ) (hiI : i ∈ I) : ENNReal.ofReal c ≤
+      segmentSausageCharge κ p ((ribbonIntegerFront g γ (i - 1) ω).untopD 0)
+        ((ribbonIntegerFront g γ i ω).untopD 0) v (physicalMiddleRegion P closedRibbon i ε) := by
+    obtain ⟨hi0, hiN⟩ := hI i hiI
+    have hsub : i - 1 + 1 = i := Nat.sub_add_cancel hi0
+    have hreal : ((i - 1 : ℕ) : ℝ) + 1 = (i : ℝ) := by exact_mod_cast hsub
+    have hn : i - 1 < N := by omega
+    have hs := (mem_iInter.mp (mem_iInter.mp hω.2 (i - 1)) (Finset.mem_range.mpr hn)).2
+    have hcharge : ENNReal.ofReal c ≤ segmentSausageCharge κ (γ ω)
+        ((ribbonIntegerFront g γ (i - 1) ω).untopD 0) ((ribbonIntegerFront g γ i ω).untopD 0) v
+        (physicalMiddleRegion P closedRibbon i ε) := by
+      simpa only [mem_ofPred_eq, hsub, hreal] using hs
+    have he : EqOn p (γ ω) (Icc ((ribbonIntegerFront g γ (i - 1) ω).untopD 0)
+        ((ribbonIntegerFront g γ i ω).untopD 0)) := by
+      intro u hu
+      have huT := hu.2.trans (htbound i hiN)
+      simp only [p, clampedPath, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
+        min_eq_left huT, zero_le, max_eq_right, f, sourceTracePath]
+    simpa only [segmentSausageCharge, sourceSausageSet_congr_interval he v] using hcharge
+  have hh := hmass hselected
+  have he := sourceSausageSet_eq_clamped f 0 T (bot_le : (0 : ℝ≥0) ≤ T) v
+  refine ⟨hpref, ?_⟩
+  change (I.card : ℝ≥0∞) * ENNReal.ofReal c ≤ wholeSausageCharge κ (clampedPath f 0 T) v D at hh
+  unfold wholeSausageCharge at hh
+  rw [← he] at hh
+  simpa only [segmentSausageCharge, f, sourceTracePath, ContinuousMap.coe_mk] using hh
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualSequencePassage.lean b/lean/OAI/Probability/SLE/Gauge/ActualSequencePassage.lean
index 6e881f5089294487c2bc78f6bf076adcf4f2ff24..7a8a567b04056aec094003de80219c9f8e56a9bb 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualSequencePassage.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualSequencePassage.lean
@@ -1,117 +1,117 @@
-import OAI.Probability.SLE.Geometry.ActualGuideCompactness
-import OAI.Probability.SLE.Gauge.SequencePassageSuccess
-
-namespace OAI
-
-/-! The actual joint guide compactness theorem retains the whole normalized
-parameter-guide family. Uniform convergence of that family supplies the
-common access point and the small physical radius needed by the proved
-sequence passage theorem. No separate access-radius hypothesis is needed. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.SequencePassage
-open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess
-
-/-- Uniform guide convergence and simplicity of the limiting guide give a
-nonzero common access point at an arbitrarily small positive guide time. -/
-theorem small_initial_access_segment
-    (C : ℝ≥0) (r : ℝ) (hr : 0 < r)
-    (βₙ : ℕ → ℝ → ℂ) (β : ℝ → ℂ)
-    (hβ0 : β 0 = 0) (hβi : InjOn β (Icc 0 2))
-    (hconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2)) :
-    ∃ δ : ℝ, 0 < δ ∧ δ ≤ 2 ∧ (C : ℝ) * δ ≤ r ∧ β δ ≠ 0 ∧
-      Tendsto (fun n => βₙ n δ) atTop (𝓝 (β δ)) := by
-  let δ : ℝ := min 1 (r / ((C : ℝ) + 1))
-  have hden : 0 < (C : ℝ) + 1 := by positivity
-  have hδ : 0 < δ := lt_min (by norm_num) (div_pos hr hden)
-  have hδ2 : δ ≤ 2 := (min_le_left _ _).trans (by norm_num)
-  have hprod : δ * ((C : ℝ) + 1) ≤ r := (le_div_iff₀ hden).mp (min_le_right _ _)
-  have hsmall : (C : ℝ) * δ ≤ r := by nlinarith
-  have hne : β δ ≠ 0 := by
-    intro he
-    have hδ0 : δ = 0 := hβi ⟨hδ.le, hδ2⟩ ⟨le_rfl, by norm_num⟩ (he.trans hβ0.symm)
-    exact hδ.ne' hδ0
-  exact ⟨δ, hδ, hδ2, hsmall, hne, hconv.tendsto_at ⟨hδ.le, hδ2⟩⟩
-
-/-- A convergent normalized guide family gives the full sequence passage
-estimate, with hypotheses on deterministic continuation tubes and ribbon
-coordinate regions. -/
-theorem localized_sequence_success_of_uniform_guide_limit
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P)
-    (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω)
-      (γ ω) (G ω) (F₀ ω))
-    (hboundary : PublishedLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane)
-    (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ))
-    (βₙ : ℕ → ℝ → ℂ)
-    (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 2)) (hβ0 : ∀ n, βₙ n 0 = 0)
-    (hβHn : ∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im)
-    (β : ℝ → ℂ) (hβlim0 : β 0 = 0) (hβlimi : InjOn β (Icc 0 2))
-    (hβconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2))
-    (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0)
-    (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) (hα0 : ∀ n, αₙ n 0 = aₙ n)
-    (hmap : ∀ n t, t ∈ Icc (0 : ℝ) 2 → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ))
-    (r₀ : ℝ) (hr₀ : 0 < r₀) :
-    ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ,
-      0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧
-      ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 →
-      ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b →
-      ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane →
-      ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W →
-      ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T →
-      ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) →
-      (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B →
-      (∀ t ∈ Ioc tB T, B < ‖β t‖) →
-      (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) →
-      (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) →
-      F '' (β '' Icc tB T) ⊆ O → β T ∈ W →
-      (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) →
-      TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane →
-      ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U →
-      ContinuousOn p U → (∀ n, ContinuousOn (pₙ n) U) →
-      TendstoLocallyUniformlyOn pₙ p atTop U →
-      ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) →
-      (∀ w ∈ ball a (r / 4), p w < 1 / 3) →
-      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) →
-      ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V →
-      ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) →
-      ∀ (bₙ : ℕ → ℂ) (ε : ℝ),
-      (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) →
-      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) →
-      ∀ (R : Set ℂ), IsOpen R →
-      (∀ w ∈ ball a (r / 2) ∪ O, 1 / 3 < p w → p w < 2 / 3 → w ∈ R) →
-      ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) →
-      ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) →
-      (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) →
-      ∀ (Rₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, R ⊆ Rₙ n) →
-      ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω |
-        ω ∈ FrontSuccess η pₙ
-          (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧
-        ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω)
-          (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2))
-          (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
-  obtain ⟨δ, hδ, hδ2, hsmall, hz, hpoint⟩ :=
-    small_initial_access_segment Cg r₀ hr₀ βₙ β hβlim0 hβlimi hβconv
-  have hsub : Icc (0 : ℝ) δ ⊆ Icc 0 2 := Icc_subset_Icc_right hδ2
-  exact localized_sequence_passage_success κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw
-    hboundary hsupport Fₙ Fbar hFn hinj hbar hext δ hδ βₙ
-    (fun n => (hβc n).mono hsub) hβ0 (fun n t ht => hβHn n t (hsub ht))
-    (β δ) hz hpoint αₙ aₙ Cg (fun n => (hα n).mono hsub) hα0
-    (fun n t ht => hmap n t (hsub ht)) r₀ hr₀ hsmall
-
-end SLEExactGauge.SequencePassage
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.ActualGuideCompactness
+import OAI.Probability.SLE.Gauge.SequencePassageSuccess
+
+namespace OAI
+
+/-! The actual joint guide compactness theorem retains the whole normalized
+parameter-guide family. Uniform convergence of that family supplies the
+common access point and the small physical radius needed by the proved
+sequence passage theorem. No separate access-radius hypothesis is needed. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.SequencePassage
+open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess
+
+/-- Uniform guide convergence and simplicity of the limiting guide give a
+nonzero common access point at an arbitrarily small positive guide time. -/
+theorem small_initial_access_segment
+    (C : ℝ≥0) (r : ℝ) (hr : 0 < r)
+    (βₙ : ℕ → ℝ → ℂ) (β : ℝ → ℂ)
+    (hβ0 : β 0 = 0) (hβi : InjOn β (Icc 0 2))
+    (hconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2)) :
+    ∃ δ : ℝ, 0 < δ ∧ δ ≤ 2 ∧ (C : ℝ) * δ ≤ r ∧ β δ ≠ 0 ∧
+      Tendsto (fun n => βₙ n δ) atTop (𝓝 (β δ)) := by
+  let δ : ℝ := min 1 (r / ((C : ℝ) + 1))
+  have hden : 0 < (C : ℝ) + 1 := by positivity
+  have hδ : 0 < δ := lt_min (by norm_num) (div_pos hr hden)
+  have hδ2 : δ ≤ 2 := (min_le_left _ _).trans (by norm_num)
+  have hprod : δ * ((C : ℝ) + 1) ≤ r := (le_div_iff₀ hden).mp (min_le_right _ _)
+  have hsmall : (C : ℝ) * δ ≤ r := by nlinarith
+  have hne : β δ ≠ 0 := by
+    intro he
+    have hδ0 : δ = 0 := hβi ⟨hδ.le, hδ2⟩ ⟨le_rfl, by norm_num⟩ (he.trans hβ0.symm)
+    exact hδ.ne' hδ0
+  exact ⟨δ, hδ, hδ2, hsmall, hne, hconv.tendsto_at ⟨hδ.le, hδ2⟩⟩
+
+/-- A convergent normalized guide family gives the full sequence passage
+estimate, with hypotheses on deterministic continuation tubes and ribbon
+coordinate regions. -/
+theorem localized_sequence_success_of_uniform_guide_limit
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P)
+    (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω)
+      (γ ω) (G ω) (F₀ ω))
+    (hboundary : PublishedLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane)
+    (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ))
+    (βₙ : ℕ → ℝ → ℂ)
+    (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 2)) (hβ0 : ∀ n, βₙ n 0 = 0)
+    (hβHn : ∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im)
+    (β : ℝ → ℂ) (hβlim0 : β 0 = 0) (hβlimi : InjOn β (Icc 0 2))
+    (hβconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2))
+    (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0)
+    (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) (hα0 : ∀ n, αₙ n 0 = aₙ n)
+    (hmap : ∀ n t, t ∈ Icc (0 : ℝ) 2 → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ))
+    (r₀ : ℝ) (hr₀ : 0 < r₀) :
+    ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ,
+      0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧
+      ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 →
+      ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b →
+      ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane →
+      ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W →
+      ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T →
+      ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) →
+      (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B →
+      (∀ t ∈ Ioc tB T, B < ‖β t‖) →
+      (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) →
+      (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) →
+      F '' (β '' Icc tB T) ⊆ O → β T ∈ W →
+      (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) →
+      TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane →
+      ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U →
+      ContinuousOn p U → (∀ n, ContinuousOn (pₙ n) U) →
+      TendstoLocallyUniformlyOn pₙ p atTop U →
+      ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) →
+      (∀ w ∈ ball a (r / 4), p w < 1 / 3) →
+      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) →
+      ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V →
+      ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) →
+      ∀ (bₙ : ℕ → ℂ) (ε : ℝ),
+      (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) →
+      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) →
+      ∀ (R : Set ℂ), IsOpen R →
+      (∀ w ∈ ball a (r / 2) ∪ O, 1 / 3 < p w → p w < 2 / 3 → w ∈ R) →
+      ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) →
+      ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) →
+      (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) →
+      ∀ (Rₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, R ⊆ Rₙ n) →
+      ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω |
+        ω ∈ FrontSuccess η pₙ
+          (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧
+        ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω)
+          (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2))
+          (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
+  obtain ⟨δ, hδ, hδ2, hsmall, hz, hpoint⟩ :=
+    small_initial_access_segment Cg r₀ hr₀ βₙ β hβlim0 hβlimi hβconv
+  have hsub : Icc (0 : ℝ) δ ⊆ Icc 0 2 := Icc_subset_Icc_right hδ2
+  exact localized_sequence_passage_success κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw
+    hboundary hsupport Fₙ Fbar hFn hinj hbar hext δ hδ βₙ
+    (fun n => (hβc n).mono hsub) hβ0 (fun n t ht => hβHn n t (hsub ht))
+    (β δ) hz hpoint αₙ aₙ Cg (fun n => (hα n).mono hsub) hα0
+    (fun n t ht => hmap n t (hsub ht)) r₀ hr₀ hsmall
+
+end SLEExactGauge.SequencePassage
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ActualSequencePassageRegions.lean b/lean/OAI/Probability/SLE/Gauge/ActualSequencePassageRegions.lean
index e7e230b692a1df0e4a4c5d3df804fb158b54eb4e..3f5126fd6bc8e4115603b981bfb33c8f19adae79 100644
--- a/lean/OAI/Probability/SLE/Gauge/ActualSequencePassageRegions.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ActualSequencePassageRegions.lean
@@ -1,93 +1,93 @@
-import OAI.Probability.SLE.Gauge.ActualSequencePassage
-import OAI.Probability.SLE.Gauge.SequencePassageRegions
-
-namespace OAI
-
-/-! Actual convergent guide families imply sequence passage using only
-compact-dependent middle-strip margins. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.SequencePassage
-open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess
-
-/-- A convergent normalized guide family gives the full sequence passage
-estimate, with hypotheses on deterministic continuation tubes and ribbon
-coordinate regions. -/
-theorem localized_sequence_success_of_uniform_guide_limit_middle_regions
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P)
-    (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω)
-      (γ ω) (G ω) (F₀ ω))
-    (hboundary : PublishedLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane)
-    (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ))
-    (βₙ : ℕ → ℝ → ℂ)
-    (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 2)) (hβ0 : ∀ n, βₙ n 0 = 0)
-    (hβHn : ∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im)
-    (β : ℝ → ℂ) (hβlim0 : β 0 = 0) (hβlimi : InjOn β (Icc 0 2))
-    (hβconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2))
-    (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0)
-    (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) (hα0 : ∀ n, αₙ n 0 = aₙ n)
-    (hmap : ∀ n t, t ∈ Icc (0 : ℝ) 2 → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ))
-    (r₀ : ℝ) (hr₀ : 0 < r₀) :
-    ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ,
-      0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧
-      ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 →
-      ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b →
-      ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane →
-      ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W →
-      ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T →
-      ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) →
-      (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B →
-      (∀ t ∈ Ioc tB T, B < ‖β t‖) →
-      (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) →
-      (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) →
-      F '' (β '' Icc tB T) ⊆ O → β T ∈ W →
-      (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) →
-      TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane →
-      ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U →
-      ContinuousOn p U → (∀ᶠ n in atTop, ContinuousOn (pₙ n) U) →
-      TendstoLocallyUniformlyOn pₙ p atTop U →
-      ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) →
-      (∀ w ∈ ball a (r / 4), p w < 1 / 3) →
-      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) →
-      ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V →
-      ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) →
-      ∀ (bₙ : ℕ → ℂ) (ε : ℝ),
-      (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) →
-      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) →
-      ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) →
-      ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) →
-      (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) →
-      ∀ (Rₙ : ℕ → Set ℂ),
-      (∀ᶠ n in atTop, U ∩ {w | (1 / 3 : ℝ) < pₙ n w ∧ pₙ n w < 2 / 3} ⊆ Rₙ n) →
-      ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω |
-        ω ∈ FrontSuccess η pₙ
-          (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧
-        ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω)
-          (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2))
-          (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
-  obtain ⟨δ, hδ, hδ2, hsmall, hz, hpoint⟩ :=
-    small_initial_access_segment Cg r₀ hr₀ βₙ β hβlim0 hβlimi hβconv
-  have hsub : Icc (0 : ℝ) δ ⊆ Icc 0 2 := Icc_subset_Icc_right hδ2
-  exact localized_sequence_passage_success_middle_regions κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw
-    hboundary hsupport Fₙ Fbar hFn hinj hbar hext δ hδ βₙ
-    (fun n => (hβc n).mono hsub) hβ0 (fun n t ht => hβHn n t (hsub ht))
-    (β δ) hz hpoint αₙ aₙ Cg (fun n => (hα n).mono hsub) hα0
-    (fun n t ht => hmap n t (hsub ht)) r₀ hr₀ hsmall
-
-end SLEExactGauge.SequencePassage
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualSequencePassage
+import OAI.Probability.SLE.Gauge.SequencePassageRegions
+
+namespace OAI
+
+/-! Actual convergent guide families imply sequence passage using only
+compact-dependent middle-strip margins. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.SequencePassage
+open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess
+
+/-- A convergent normalized guide family gives the full sequence passage
+estimate, with hypotheses on deterministic continuation tubes and ribbon
+coordinate regions. -/
+theorem localized_sequence_success_of_uniform_guide_limit_middle_regions
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P)
+    (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω)
+      (γ ω) (G ω) (F₀ ω))
+    (hboundary : PublishedLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane)
+    (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ))
+    (βₙ : ℕ → ℝ → ℂ)
+    (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 2)) (hβ0 : ∀ n, βₙ n 0 = 0)
+    (hβHn : ∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im)
+    (β : ℝ → ℂ) (hβlim0 : β 0 = 0) (hβlimi : InjOn β (Icc 0 2))
+    (hβconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2))
+    (αₙ : ℕ → ℝ → ℂ) (aₙ : ℕ → ℂ) (Cg : ℝ≥0)
+    (hα : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2)) (hα0 : ∀ n, αₙ n 0 = aₙ n)
+    (hmap : ∀ n t, t ∈ Icc (0 : ℝ) 2 → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ))
+    (r₀ : ℝ) (hr₀ : 0 < r₀) :
+    ∃ T₀ : ℝ≥0, 0 < T₀ ∧ ∃ q₀ b₀ b A d L₀ : ℝ,
+      0 < q₀ ∧ 0 < b₀ ∧ 0 < b ∧ 0 < A ∧ 0 < d ∧ 1 < L₀ ∧
+      ∀ (L : ℝ), L₀ ≤ L → A * L ^ (-d) ≤ q₀ / 2 →
+      ∀ (B r : ℝ), 0 < B → 0 < r → B / 2 ≤ b₀ * L ^ b →
+      ∀ (F : ℂ → ℂ), DifferentiableOn ℂ F upperHalfPlane →
+      ∀ (a : ℂ) (O W : Set ℂ), IsOpen O → IsOpen W →
+      ∀ (β : ℝ → ℂ) (tB T : ℝ), 0 < tB → tB < T →
+      ContinuousOn β (Icc 0 T) → InjOn β (Icc 0 T) →
+      (∀ t ∈ Ioc 0 T, 0 < (β t).im) → ‖β tB‖ = B →
+      (∀ t ∈ Ioc tB T, B < ‖β t‖) →
+      (∀ θ ∈ Ioo 0 Real.pi, ‖F (GuideSplice.circlePoint B θ) - a‖ < r / 4) →
+      (∀ w : ℂ, B / 8 ≤ ‖w‖ → ‖w‖ ≤ B → ‖w‖ / 2 ≤ w.im → ‖F w - a‖ < r / 4) →
+      F '' (β '' Icc tB T) ⊆ O → β T ∈ W →
+      (∀ w ∈ upperHalfPlane, deriv F w ≠ 0) →
+      TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane →
+      ∀ (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ), IsOpen U →
+      ContinuousOn p U → (∀ᶠ n in atTop, ContinuousOn (pₙ n) U) →
+      TendstoLocallyUniformlyOn pₙ p atTop U →
+      ball a (r / 2) ∪ O ⊆ U → (∀ w ∈ W, 1 < p (F w)) →
+      (∀ w ∈ ball a (r / 4), p w < 1 / 3) →
+      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ U ∩ {w | pₙ n w < 1}) →
+      ∀ (V : Set ℂ), IsOpen V → ball a (r / 2) ∪ O ⊆ V →
+      ∀ (Vₙ : ℕ → Set ℂ), (∀ᶠ n in atTop, V ⊆ Vₙ n) →
+      ∀ (bₙ : ℕ → ℂ) (ε : ℝ),
+      (∀ n w, w ∈ Vₙ n → pₙ n w = 1 → ‖w - bₙ n‖ < ε / 4) →
+      (∀ᶠ n in atTop, ball (aₙ n) (L * r₀) ⊆ Vₙ n) →
+      ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) → Tendsto v atTop (𝓝 0) →
+      ∀ (η : ℕ → Ω → ℝ≥0 → ℂ), (∀ n, ∀ᵐ ω ∂P, Continuous (η n ω)) →
+      (∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) →
+      ∀ (Rₙ : ℕ → Set ℂ),
+      (∀ᶠ n in atTop, U ∩ {w | (1 / 3 : ℝ) < pₙ n w ∧ pₙ n w < 2 / 3} ⊆ Rₙ n) →
+      ∃ c q : ℝ, 0 < c ∧ 0 < q ∧ ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω |
+        ω ∈ FrontSuccess η pₙ
+          (fun ω => boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2)) Vₙ bₙ ε n ∧
+        ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω)
+          (boundedFirstHeightTime (γ ω) B (T₀ * (Real.toNNReal B) ^ 2))
+          (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
+  obtain ⟨δ, hδ, hδ2, hsmall, hz, hpoint⟩ :=
+    small_initial_access_segment Cg r₀ hr₀ βₙ β hβlim0 hβlimi hβconv
+  have hsub : Icc (0 : ℝ) δ ⊆ Icc 0 2 := Icc_subset_Icc_right hδ2
+  exact localized_sequence_passage_success_middle_regions κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw
+    hboundary hsupport Fₙ Fbar hFn hinj hbar hext δ hδ βₙ
+    (fun n => (hβc n).mono hsub) hβ0 (fun n t ht => hβHn n t (hsub ht))
+    (β δ) hz hpoint αₙ aₙ Cg (fun n => (hα n).mono hsub) hα0
+    (fun n t ht => hmap n t (hsub ht)) r₀ hr₀ hsmall
+
+end SLEExactGauge.SequencePassage
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/BatchAdaptedFailure.lean b/lean/OAI/Probability/SLE/Gauge/BatchAdaptedFailure.lean
index f02f3840aaa089bf4c9cf01117371ee8184cc2a5..1d3e4edcf4a2e100c832c9cd4d748430c0881fba 100644
--- a/lean/OAI/Probability/SLE/Gauge/BatchAdaptedFailure.lean
+++ b/lean/OAI/Probability/SLE/Gauge/BatchAdaptedFailure.lean
@@ -1,75 +1,75 @@
-import OAI.Probability.SLE.Gauge.DenseCoverAdaptedTests
-
-namespace OAI
-
-/-! The prescribed batch uses unequal success probabilities. Their sum,
-not independence, controls the probability that every adapted test fails. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem adapted_failures_product_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω)
-    (F : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
-    (hmeas : ∀ n, MeasurableSet[F (n+1)] (failure n)) (q : ℕ → ℝ≥0∞)
-    (hstep : ∀ n (A : Set Ω), MeasurableSet[F n] A → P (A ∩ failure n) ≤ q n*P A)
-    (N : ℕ) :
-    P (⋂ n ∈ Finset.range N, failure n) ≤ (∏ n∈Finset.range N, q n)*P univ := by
-  induction N with
-  | zero => simp
-  | succ N ih =>
-    let A : Set Ω := ⋂ n∈Finset.range N, failure n
-    have hA : MeasurableSet[F N] A := by
-      apply MeasurableSet.iInter
-      intro n
-      apply MeasurableSet.iInter
-      intro hn
-      exact (F.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n)
-    have he : (⋂ n∈Finset.range (N+1), failure n)=A∩failure N := by
-      ext ω
-      simp only [mem_iInter,Finset.mem_range,mem_inter_iff,A]
-      constructor
-      · intro h
-        exact ⟨fun n hn => h n (Nat.lt_succ_of_lt hn),h N (Nat.lt_succ_self N)⟩
-      · rintro ⟨h,hN⟩ n hn
-        rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl
-        · exact h n hn
-        · exact hN
-    rw [he,Finset.prod_range_succ]
-    exact (hstep N A hA).trans ((mul_le_mul' le_rfl ih).trans_eq (by ac_rfl))
-
-lemma finite_failure_product_le_exp (p : ℕ → ℝ) (hp1 : ∀ n, p n ≤ 1) (N : ℕ) :
-    (∏ n∈Finset.range N, ENNReal.ofReal (1-p n)) ≤
-      ENNReal.ofReal (Real.exp (-(∑ n∈Finset.range N, p n))) := by
-  rw [← ENNReal.ofReal_prod_of_nonneg (fun n _ => sub_nonneg.mpr (hp1 n))]
-  apply ENNReal.ofReal_le_ofReal
-  calc
-    _ ≤ ∏ n∈Finset.range N, Real.exp (-p n) :=
-      Finset.prod_le_prod₀ (fun n _ => sub_nonneg.mpr (hp1 n))
-        (fun n _ => Real.one_sub_le_exp_neg (p n))
-    _ = _ := by rw [← Real.exp_sum,Finset.sum_neg_distrib]
-
-theorem adapted_successes_failure_exp_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (F : Filtration ℕ mΩ) (success : ℕ → Set Ω)
-    (hmeas : ∀ n, MeasurableSet[F (n+1)] (success n))
-    (p : ℕ → ℝ) (hp0 : ∀ n, 0 ≤ p n) (hp1 : ∀ n, p n ≤ 1)
-    (hstep : ∀ n (A : Set Ω), MeasurableSet[F n] A →
-      ENNReal.ofReal (p n)*P A ≤ P (A∩success n))
-    (N : ℕ) :
-    P (⋂ n∈Finset.range N, (success n)ᶜ) ≤
-      ENNReal.ofReal (Real.exp (-(∑ n∈Finset.range N, p n))) := by
-  have h := adapted_failures_product_bound P F (fun n => (success n)ᶜ)
-    (fun n => (hmeas n).compl) (fun n => ENNReal.ofReal (1-p n))
-    (fun n A hA => complement_inter_bound_of_success_lower P
-      (F.le (n+1) _ (hmeas n)) (hp0 n) (hp1 n) (hstep n A hA)) N
-  rw [measure_univ,mul_one] at h
-  exact h.trans (finite_failure_product_le_exp p hp1 N)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.DenseCoverAdaptedTests
+
+namespace OAI
+
+/-! The prescribed batch uses unequal success probabilities. Their sum,
+not independence, controls the probability that every adapted test fails. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem adapted_failures_product_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω)
+    (F : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
+    (hmeas : ∀ n, MeasurableSet[F (n+1)] (failure n)) (q : ℕ → ℝ≥0∞)
+    (hstep : ∀ n (A : Set Ω), MeasurableSet[F n] A → P (A ∩ failure n) ≤ q n*P A)
+    (N : ℕ) :
+    P (⋂ n ∈ Finset.range N, failure n) ≤ (∏ n∈Finset.range N, q n)*P univ := by
+  induction N with
+  | zero => simp
+  | succ N ih =>
+    let A : Set Ω := ⋂ n∈Finset.range N, failure n
+    have hA : MeasurableSet[F N] A := by
+      apply MeasurableSet.iInter
+      intro n
+      apply MeasurableSet.iInter
+      intro hn
+      exact (F.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n)
+    have he : (⋂ n∈Finset.range (N+1), failure n)=A∩failure N := by
+      ext ω
+      simp only [mem_iInter,Finset.mem_range,mem_inter_iff,A]
+      constructor
+      · intro h
+        exact ⟨fun n hn => h n (Nat.lt_succ_of_lt hn),h N (Nat.lt_succ_self N)⟩
+      · rintro ⟨h,hN⟩ n hn
+        rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl
+        · exact h n hn
+        · exact hN
+    rw [he,Finset.prod_range_succ]
+    exact (hstep N A hA).trans ((mul_le_mul' le_rfl ih).trans_eq (by ac_rfl))
+
+lemma finite_failure_product_le_exp (p : ℕ → ℝ) (hp1 : ∀ n, p n ≤ 1) (N : ℕ) :
+    (∏ n∈Finset.range N, ENNReal.ofReal (1-p n)) ≤
+      ENNReal.ofReal (Real.exp (-(∑ n∈Finset.range N, p n))) := by
+  rw [← ENNReal.ofReal_prod_of_nonneg (fun n _ => sub_nonneg.mpr (hp1 n))]
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    _ ≤ ∏ n∈Finset.range N, Real.exp (-p n) :=
+      Finset.prod_le_prod₀ (fun n _ => sub_nonneg.mpr (hp1 n))
+        (fun n _ => Real.one_sub_le_exp_neg (p n))
+    _ = _ := by rw [← Real.exp_sum,Finset.sum_neg_distrib]
+
+theorem adapted_successes_failure_exp_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (F : Filtration ℕ mΩ) (success : ℕ → Set Ω)
+    (hmeas : ∀ n, MeasurableSet[F (n+1)] (success n))
+    (p : ℕ → ℝ) (hp0 : ∀ n, 0 ≤ p n) (hp1 : ∀ n, p n ≤ 1)
+    (hstep : ∀ n (A : Set Ω), MeasurableSet[F n] A →
+      ENNReal.ofReal (p n)*P A ≤ P (A∩success n))
+    (N : ℕ) :
+    P (⋂ n∈Finset.range N, (success n)ᶜ) ≤
+      ENNReal.ofReal (Real.exp (-(∑ n∈Finset.range N, p n))) := by
+  have h := adapted_failures_product_bound P F (fun n => (success n)ᶜ)
+    (fun n => (hmeas n).compl) (fun n => ENNReal.ofReal (1-p n))
+    (fun n A hA => complement_inter_bound_of_success_lower P
+      (F.le (n+1) _ (hmeas n)) (hp0 n) (hp1 n) (hstep n A hA)) N
+  rw [measure_univ,mul_one] at h
+  exact h.trans (finite_failure_product_le_exp p hp1 N)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/BatchEntryOrder.lean b/lean/OAI/Probability/SLE/Gauge/BatchEntryOrder.lean
index c39305231720c6d92e009b5102ee3110538c319b..ad8757dd83256ef8710f8a51e1cc61a9547cabea 100644
--- a/lean/OAI/Probability/SLE/Gauge/BatchEntryOrder.lean
+++ b/lean/OAI/Probability/SLE/Gauge/BatchEntryOrder.lean
@@ -1,130 +1,130 @@
-import OAI.Probability.SLE.MomentGauge
-import Mathlib.Data.Finset.Sort
-import Mathlib.Data.Prod.Lex
-
-namespace OAI
-
-/-! The prescribed repeated entries occupy distinct successive positions.
-Their position order agrees with decreasing prescribed radii. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge.Target
-
-private lemma prefix_eq_sum (c : ℤ → ℕ) (hc : Set.Finite {k | 0 < c k}) (b : ℤ) :
-    (∑' k : ℤ, if k < b then c k else 0) =
-      ∑ k∈hc.toFinset, if k < b then c k else 0 := by
-  apply tsum_eq_sum
-  intro k hk
-  have hz : c k=0 := by
-    have hn : ¬0<c k := by
-      intro h
-      exact hk (by simpa only [Set.Finite.mem_toFinset,mem_ofPred_eq] using h)
-    omega
-  simp only [hz,ite_self]
-
-private lemma prefix_add_le (c : ℤ → ℕ) (hc : Set.Finite {k | 0 < c k})
-    {k l : ℤ} (hkl : k < l) (hk : 0 < c k) :
-    (∑' i : ℤ, if i < k then c i else 0)+c k ≤
-      ∑' i : ℤ, if i < l then c i else 0 := by
-  have hkS : k∈hc.toFinset := by simpa only [Set.Finite.mem_toFinset,mem_ofPred_eq] using hk
-  rw [prefix_eq_sum c hc k,prefix_eq_sum c hc l]
-  have hK := Finset.sum_erase_add hc.toFinset (fun i => if i < k then c i else 0) hkS
-  have hL := Finset.sum_erase_add hc.toFinset (fun i => if i < l then c i else 0) hkS
-  simp only [lt_self_iff_false,ite_false,add_zero] at hK
-  rw [ite_eq_left hkl] at hL
-  rw [← hK,← hL]
-  apply Nat.add_le_add_right
-  apply Finset.sum_le_sum
-  intro i _
-  by_cases hik : i < k
-  · rw [ite_eq_left hik,ite_eq_left (hik.trans hkl)]
-  · rw [ite_eq_right hik]
-    exact Nat.zero_le _
-
-lemma entryPosition_strict_of_key_lt (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n)
-    {k l : ℤ} {j j' : ℕ} (hkl : k < l) (hj : j < multiplicity κ W n k) :
-    entryPosition κ W (n,some (k,j)) < entryPosition κ W (n,some (l,j')) := by
-  have h := prefix_add_le (multiplicity κ W n) (multiplicity_support_finite κ W hn hm1 hJ)
-    hkl (Nat.zero_le j |>.trans_lt hj)
-  simp only [entryPosition]
-  omega
-
-lemma entryPosition_injective_on_batch (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n)
-    {k l : ℤ} {j j' : ℕ} (hj : j < multiplicity κ W n k)
-    (hj' : j' < multiplicity κ W n l)
-    (he : entryPosition κ W (n,some (k,j))=entryPosition κ W (n,some (l,j'))) :
-    (k,j)=(l,j') := by
-  rcases lt_trichotomy k l with hkl | rfl | hlk
-  · exact False.elim ((ne_of_lt (entryPosition_strict_of_key_lt κ W hn hm1 hJ hkl hj)) he)
-  · simp only [entryPosition] at he
-    exact Prod.ext rfl (Nat.add_left_cancel he)
-  · exact False.elim ((ne_of_lt (entryPosition_strict_of_key_lt κ W hn hm1 hJ hlk hj')) he.symm)
-
-lemma entryRadius_lt_of_position_lt {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw)
-    {n : ℕ} (hn : 1 ≤ n) {o o' : Option (ℤ × ℕ)}
-    (hpos : entryPosition κ W (n,o) < entryPosition κ W (n,o')) :
-    entryRadius κ W (n,o') < entryRadius κ W (n,o) := by
-  have hnR : (0:ℝ)<n := by exact_mod_cast (Nat.zero_lt_of_lt hn)
-  have hp : (entryPosition κ W (n,o):ℝ)<entryPosition κ W (n,o') := by exact_mod_cast hpos
-  have hcoef : 0 < 2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ)) := by
-    exact mul_pos (mul_pos (by norm_num) (internal_a_pos hκ)) (by positivity)
-  unfold entryRadius
-  apply Real.exp_lt_exp.mpr
-  nlinarith [mul_pos hcoef (sub_pos.mpr hp)]
-
-/-- The repeated tests have a finite chronological enumeration, without
-discarding entries whose dyadic thresholds coincide. -/
-theorem ordered_repeated_batch (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n) :
-    ∃ (J : ℕ) (e : Fin J → ℤ × ℕ),
-      Function.Injective e ∧
-      (∀ kj, kj ∈ Set.range e ↔ kj.2 < multiplicity κ W n kj.1) ∧
-      StrictMono (fun i => entryPosition κ W (n,some (e i))) := by
-  classical
-  let hs := (repeated_entries_finite κ W hn hm1 hJ).image toLex
-  let s : Finset (ℤ ×ₗ ℕ) := hs.toFinset
-  let e : Fin s.card → ℤ × ℕ := fun i => ofLex (s.orderEmbOfFin rfl i)
-  have he (i : Fin s.card) : (e i).2 < multiplicity κ W n (e i).1 := by
-    have h := s.orderEmbOfFin_mem rfl i
-    change s.orderEmbOfFin rfl i ∈ hs.toFinset at h
-    rw [Set.Finite.mem_toFinset] at h
-    obtain ⟨kj,hkj,heq⟩ := h
-    simpa only [e,← heq,ofLex_toLex,mem_ofPred_eq] using hkj
-  refine ⟨s.card,e,?_,?_,?_⟩
-  · exact ofLex.injective.comp (s.orderEmbOfFin rfl).injective
-  · intro kj
-    constructor
-    · rintro ⟨i,rfl⟩
-      exact he i
-    · intro hkj
-      have hm : toLex kj ∈ s := by
-        change toLex kj ∈ hs.toFinset
-        rw [Set.Finite.mem_toFinset]
-        exact ⟨kj,hkj,rfl⟩
-      have hr : toLex kj ∈ Set.range (s.orderEmbOfFin rfl) := by
-        rw [Finset.range_orderEmbOfFin]
-        exact hm
-      obtain ⟨i,hi⟩ := hr
-      exact ⟨i,by simp only [e,hi,ofLex_toLex]⟩
-  · intro i j hij
-    have hij' := (s.orderEmbOfFin rfl).strictMono hij
-    change toLex (e i) < toLex (e j) at hij'
-    rw [Prod.Lex.toLex_lt_toLex] at hij'
-    rcases hij' with hlt | ⟨heq,hlt⟩
-    · exact entryPosition_strict_of_key_lt κ W hn hm1 hJ hlt (he i)
-    · simp only [entryPosition]
-      rw [heq]
-      exact Nat.add_lt_add_left hlt _
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.MomentGauge
+import Mathlib.Data.Finset.Sort
+import Mathlib.Data.Prod.Lex
+
+namespace OAI
+
+/-! The prescribed repeated entries occupy distinct successive positions.
+Their position order agrees with decreasing prescribed radii. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge.Target
+
+private lemma prefix_eq_sum (c : ℤ → ℕ) (hc : Set.Finite {k | 0 < c k}) (b : ℤ) :
+    (∑' k : ℤ, if k < b then c k else 0) =
+      ∑ k∈hc.toFinset, if k < b then c k else 0 := by
+  apply tsum_eq_sum
+  intro k hk
+  have hz : c k=0 := by
+    have hn : ¬0<c k := by
+      intro h
+      exact hk (by simpa only [Set.Finite.mem_toFinset,mem_ofPred_eq] using h)
+    omega
+  simp only [hz,ite_self]
+
+private lemma prefix_add_le (c : ℤ → ℕ) (hc : Set.Finite {k | 0 < c k})
+    {k l : ℤ} (hkl : k < l) (hk : 0 < c k) :
+    (∑' i : ℤ, if i < k then c i else 0)+c k ≤
+      ∑' i : ℤ, if i < l then c i else 0 := by
+  have hkS : k∈hc.toFinset := by simpa only [Set.Finite.mem_toFinset,mem_ofPred_eq] using hk
+  rw [prefix_eq_sum c hc k,prefix_eq_sum c hc l]
+  have hK := Finset.sum_erase_add hc.toFinset (fun i => if i < k then c i else 0) hkS
+  have hL := Finset.sum_erase_add hc.toFinset (fun i => if i < l then c i else 0) hkS
+  simp only [lt_self_iff_false,ite_false,add_zero] at hK
+  rw [ite_eq_left hkl] at hL
+  rw [← hK,← hL]
+  apply Nat.add_le_add_right
+  apply Finset.sum_le_sum
+  intro i _
+  by_cases hik : i < k
+  · rw [ite_eq_left hik,ite_eq_left (hik.trans hkl)]
+  · rw [ite_eq_right hik]
+    exact Nat.zero_le _
+
+lemma entryPosition_strict_of_key_lt (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n)
+    {k l : ℤ} {j j' : ℕ} (hkl : k < l) (hj : j < multiplicity κ W n k) :
+    entryPosition κ W (n,some (k,j)) < entryPosition κ W (n,some (l,j')) := by
+  have h := prefix_add_le (multiplicity κ W n) (multiplicity_support_finite κ W hn hm1 hJ)
+    hkl (Nat.zero_le j |>.trans_lt hj)
+  simp only [entryPosition]
+  omega
+
+lemma entryPosition_injective_on_batch (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n)
+    {k l : ℤ} {j j' : ℕ} (hj : j < multiplicity κ W n k)
+    (hj' : j' < multiplicity κ W n l)
+    (he : entryPosition κ W (n,some (k,j))=entryPosition κ W (n,some (l,j'))) :
+    (k,j)=(l,j') := by
+  rcases lt_trichotomy k l with hkl | rfl | hlk
+  · exact False.elim ((ne_of_lt (entryPosition_strict_of_key_lt κ W hn hm1 hJ hkl hj)) he)
+  · simp only [entryPosition] at he
+    exact Prod.ext rfl (Nat.add_left_cancel he)
+  · exact False.elim ((ne_of_lt (entryPosition_strict_of_key_lt κ W hn hm1 hJ hlk hj')) he.symm)
+
+lemma entryRadius_lt_of_position_lt {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw)
+    {n : ℕ} (hn : 1 ≤ n) {o o' : Option (ℤ × ℕ)}
+    (hpos : entryPosition κ W (n,o) < entryPosition κ W (n,o')) :
+    entryRadius κ W (n,o') < entryRadius κ W (n,o) := by
+  have hnR : (0:ℝ)<n := by exact_mod_cast (Nat.zero_lt_of_lt hn)
+  have hp : (entryPosition κ W (n,o):ℝ)<entryPosition κ W (n,o') := by exact_mod_cast hpos
+  have hcoef : 0 < 2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ)) := by
+    exact mul_pos (mul_pos (by norm_num) (internal_a_pos hκ)) (by positivity)
+  unfold entryRadius
+  apply Real.exp_lt_exp.mpr
+  nlinarith [mul_pos hcoef (sub_pos.mpr hp)]
+
+/-- The repeated tests have a finite chronological enumeration, without
+discarding entries whose dyadic thresholds coincide. -/
+theorem ordered_repeated_batch (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n) :
+    ∃ (J : ℕ) (e : Fin J → ℤ × ℕ),
+      Function.Injective e ∧
+      (∀ kj, kj ∈ Set.range e ↔ kj.2 < multiplicity κ W n kj.1) ∧
+      StrictMono (fun i => entryPosition κ W (n,some (e i))) := by
+  classical
+  let hs := (repeated_entries_finite κ W hn hm1 hJ).image toLex
+  let s : Finset (ℤ ×ₗ ℕ) := hs.toFinset
+  let e : Fin s.card → ℤ × ℕ := fun i => ofLex (s.orderEmbOfFin rfl i)
+  have he (i : Fin s.card) : (e i).2 < multiplicity κ W n (e i).1 := by
+    have h := s.orderEmbOfFin_mem rfl i
+    change s.orderEmbOfFin rfl i ∈ hs.toFinset at h
+    rw [Set.Finite.mem_toFinset] at h
+    obtain ⟨kj,hkj,heq⟩ := h
+    simpa only [e,← heq,ofLex_toLex,mem_ofPred_eq] using hkj
+  refine ⟨s.card,e,?_,?_,?_⟩
+  · exact ofLex.injective.comp (s.orderEmbOfFin rfl).injective
+  · intro kj
+    constructor
+    · rintro ⟨i,rfl⟩
+      exact he i
+    · intro hkj
+      have hm : toLex kj ∈ s := by
+        change toLex kj ∈ hs.toFinset
+        rw [Set.Finite.mem_toFinset]
+        exact ⟨kj,hkj,rfl⟩
+      have hr : toLex kj ∈ Set.range (s.orderEmbOfFin rfl) := by
+        rw [Finset.range_orderEmbOfFin]
+        exact hm
+      obtain ⟨i,hi⟩ := hr
+      exact ⟨i,by simp only [e,hi,ofLex_toLex]⟩
+  · intro i j hij
+    have hij' := (s.orderEmbOfFin rfl).strictMono hij
+    change toLex (e i) < toLex (e j) at hij'
+    rw [Prod.Lex.toLex_lt_toLex] at hij'
+    rcases hij' with hlt | ⟨heq,hlt⟩
+    · exact entryPosition_strict_of_key_lt κ W hn hm1 hJ hlt (he i)
+    · simp only [entryPosition]
+      rw [heq]
+      exact Nat.add_lt_add_left hlt _
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/BatchProbability.lean b/lean/OAI/Probability/SLE/Gauge/BatchProbability.lean
index 167629621320047a386c7ee5e4cd602c990a1a27..9a425e78b5c85a17af3c93d9c037d16167927149 100644
--- a/lean/OAI/Probability/SLE/Gauge/BatchProbability.lean
+++ b/lean/OAI/Probability/SLE/Gauge/BatchProbability.lean
@@ -1,126 +1,126 @@
-import OAI.Probability.SLE.Gauge.BatchEntryOrder
-import OAI.Probability.SLE.Gauge.BatchAdaptedFailure
-
-namespace OAI
-
-/-! Reindexing the prescribed repeated tests preserves their probability
-budget. The finite adapted failure estimate therefore applies to the actual
-batch, with its repetitions, rather than just to distinct thresholds. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge
-
-lemma adapted_fin_successes_failure_exp_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (F : Filtration ℕ mΩ) {N : ℕ} (success : Fin N → Set Ω)
-    (hmeas : ∀ i, MeasurableSet[F (i.val+1)] (success i))
-    (p : Fin N → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hp1 : ∀ i, p i ≤ 1)
-    (hstep : ∀ i (A : Set Ω), MeasurableSet[F i.val] A →
-      ENNReal.ofReal (p i)*P A ≤ P (A∩success i)) :
-    P (⋂ i, (success i)ᶜ) ≤ ENNReal.ofReal (Real.exp (-(∑ i, p i))) := by
-  classical
-  let s : ℕ → Set Ω := fun i => if h : i<N then success ⟨i,h⟩ else ∅
-  let q : ℕ → ℝ := fun i => if h : i<N then p ⟨i,h⟩ else 0
-  have hm (i) : MeasurableSet[F (i+1)] (s i) := by
-    dsimp only [s]
-    split_ifs with h
-    · exact hmeas ⟨i,h⟩
-    · exact @MeasurableSet.empty Ω (F (i+1))
-  have h0 (i) : 0 ≤ q i := by
-    dsimp only [q]
-    split_ifs with h
-    · exact hp0 ⟨i,h⟩
-    · exact le_rfl
-  have h1 (i) : q i ≤ 1 := by
-    dsimp only [q]
-    split_ifs with h
-    · exact hp1 ⟨i,h⟩
-    · norm_num
-  have hs (i) (A : Set Ω) (hA : MeasurableSet[F i] A) :
-      ENNReal.ofReal (q i)*P A ≤ P (A∩s i) := by
-    by_cases hi : i<N
-    · simpa only [q,s,dite_eq_left hi] using hstep ⟨i,hi⟩ A hA
-    · simp [q,s,hi]
-  have h := adapted_successes_failure_exp_bound P F s hm q h0 h1 hs N
-  have he : (⋂ i∈Finset.range N, (s i)ᶜ) = ⋂ i, (success i)ᶜ := by
-    ext ω
-    simp only [mem_iInter,Finset.mem_range,mem_compl_iff]
-    constructor
-    · intro h i
-      simpa only [s,dite_eq_left i.isLt] using h i.val i.isLt
-    · intro h i hi
-      simpa only [s,dite_eq_left hi] using h ⟨i,hi⟩
-  have hsum : (∑ i∈Finset.range N, q i) = ∑ i : Fin N, p i := by
-    rw [← Fin.sum_univ_eq_sum_range]
-    apply Finset.sum_congr rfl
-    intro i _
-    simp only [q,dite_eq_left i.isLt]
-  rwa [he,hsum] at h
-
-namespace Target
-
-lemma repeated_batch_sum_eq (κ : ℝ) (W : WienerLaw) (n J : ℕ)
-    (e : Fin J → ℤ × ℕ) (hi : Function.Injective e)
-    (he : ∀ kj, kj ∈ Set.range e ↔ kj.2 < multiplicity κ W n kj.1)
-    (q : ℤ → ℝ) :
-    (∑ i, q (e i).1) = ∑' k : ℤ, (multiplicity κ W n k : ℝ)*q k := by
-  classical
-  let f : Fin J → Σ k : ℤ, Fin (multiplicity κ W n k) := fun i =>
-    ⟨(e i).1,(e i).2,(he (e i)).mp ⟨i,rfl⟩⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro i j hij
-      apply hi
-      have h := congrArg (fun p : Σ k : ℤ, Fin (multiplicity κ W n k) => (p.1,p.2.val)) hij
-      simpa only [f] using h
-    · rintro ⟨k,j⟩
-      obtain ⟨i,hi⟩ := (he (k,j.val)).mpr j.isLt
-      refine ⟨i,?_⟩
-      dsimp only [f]
-      refine Sigma.ext (congrArg Prod.fst hi) ?_
-      exact (Fin.heq_ext_iff (congrArg (multiplicity κ W n) (congrArg Prod.fst hi))).2
-        (congrArg Prod.snd hi)
-  let E := Equiv.ofBijective f hf
-  let : Fintype (Σ k : ℤ, Fin (multiplicity κ W n k)) := Fintype.ofEquiv (Fin J) E
-  calc
-    (∑ i, q (e i).1) = ∑' i, q (e i).1 := (tsum_fintype _).symm
-    _ = ∑' p : Σ k : ℤ, Fin (multiplicity κ W n k), q p.1 :=
-      E.tsum_eq (fun p : Σ k : ℤ, Fin (multiplicity κ W n k) => q p.1)
-    _ = ∑' k : ℤ, ∑' _j : Fin (multiplicity κ W n k), q k :=
-      (Summable.of_finite).tsum_sigma' (fun _ => Summable.of_finite)
-    _ = _ := by simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
-
-end Target
-
-/-- Half of each centered success probability is enough for the paper's
-exponent, even when the tests are dependent and the thresholds repeat. -/
-theorem repeated_batch_failure_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (F : Filtration ℕ mΩ) (κ : ℝ) (W : Target.WienerLaw) (n J : ℕ)
-    (e : Fin J → ℤ × ℕ) (hi : Function.Injective e)
-    (he : ∀ kj, kj ∈ Set.range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
-    (q : ℤ → ℝ) (hq0 : ∀ k, 0 ≤ q k) (hq1 : ∀ k, q k ≤ 1)
-    (hbudget : (2:ℝ)^n/8 ≤ ∑' k, (Target.multiplicity κ W n k:ℝ)*q k)
-    (success : Fin J → Set Ω)
-    (hmeas : ∀ i, MeasurableSet[F (i.val+1)] (success i))
-    (hstep : ∀ i (A : Set Ω), MeasurableSet[F i.val] A →
-      ENNReal.ofReal (q (e i).1/2)*P A ≤ P (A∩success i)) :
-    P (⋂ i, (success i)ᶜ) ≤ ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
-  have hb : (2:ℝ)^n/8 ≤ ∑ i, q (e i).1 := by
-    rwa [Target.repeated_batch_sum_eq κ W n J e hi he q]
-  have hp0 (i) : 0 ≤ q (e i).1/2 := div_nonneg (hq0 _) (by norm_num)
-  have hp1 (i) : q (e i).1/2 ≤ 1 := by linarith [hq1 (e i).1]
-  refine (adapted_fin_successes_failure_exp_bound P F success hmeas
-    (fun i => q (e i).1/2) hp0 hp1 hstep).trans ?_
-  apply ENNReal.ofReal_le_ofReal
-  apply Real.exp_le_exp.mpr
-  rw [← Finset.sum_div]
-  linarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.BatchEntryOrder
+import OAI.Probability.SLE.Gauge.BatchAdaptedFailure
+
+namespace OAI
+
+/-! Reindexing the prescribed repeated tests preserves their probability
+budget. The finite adapted failure estimate therefore applies to the actual
+batch, with its repetitions, rather than just to distinct thresholds. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge
+
+lemma adapted_fin_successes_failure_exp_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (F : Filtration ℕ mΩ) {N : ℕ} (success : Fin N → Set Ω)
+    (hmeas : ∀ i, MeasurableSet[F (i.val+1)] (success i))
+    (p : Fin N → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hp1 : ∀ i, p i ≤ 1)
+    (hstep : ∀ i (A : Set Ω), MeasurableSet[F i.val] A →
+      ENNReal.ofReal (p i)*P A ≤ P (A∩success i)) :
+    P (⋂ i, (success i)ᶜ) ≤ ENNReal.ofReal (Real.exp (-(∑ i, p i))) := by
+  classical
+  let s : ℕ → Set Ω := fun i => if h : i<N then success ⟨i,h⟩ else ∅
+  let q : ℕ → ℝ := fun i => if h : i<N then p ⟨i,h⟩ else 0
+  have hm (i) : MeasurableSet[F (i+1)] (s i) := by
+    dsimp only [s]
+    split_ifs with h
+    · exact hmeas ⟨i,h⟩
+    · exact @MeasurableSet.empty Ω (F (i+1))
+  have h0 (i) : 0 ≤ q i := by
+    dsimp only [q]
+    split_ifs with h
+    · exact hp0 ⟨i,h⟩
+    · exact le_rfl
+  have h1 (i) : q i ≤ 1 := by
+    dsimp only [q]
+    split_ifs with h
+    · exact hp1 ⟨i,h⟩
+    · norm_num
+  have hs (i) (A : Set Ω) (hA : MeasurableSet[F i] A) :
+      ENNReal.ofReal (q i)*P A ≤ P (A∩s i) := by
+    by_cases hi : i<N
+    · simpa only [q,s,dite_eq_left hi] using hstep ⟨i,hi⟩ A hA
+    · simp [q,s,hi]
+  have h := adapted_successes_failure_exp_bound P F s hm q h0 h1 hs N
+  have he : (⋂ i∈Finset.range N, (s i)ᶜ) = ⋂ i, (success i)ᶜ := by
+    ext ω
+    simp only [mem_iInter,Finset.mem_range,mem_compl_iff]
+    constructor
+    · intro h i
+      simpa only [s,dite_eq_left i.isLt] using h i.val i.isLt
+    · intro h i hi
+      simpa only [s,dite_eq_left hi] using h ⟨i,hi⟩
+  have hsum : (∑ i∈Finset.range N, q i) = ∑ i : Fin N, p i := by
+    rw [← Fin.sum_univ_eq_sum_range]
+    apply Finset.sum_congr rfl
+    intro i _
+    simp only [q,dite_eq_left i.isLt]
+  rwa [he,hsum] at h
+
+namespace Target
+
+lemma repeated_batch_sum_eq (κ : ℝ) (W : WienerLaw) (n J : ℕ)
+    (e : Fin J → ℤ × ℕ) (hi : Function.Injective e)
+    (he : ∀ kj, kj ∈ Set.range e ↔ kj.2 < multiplicity κ W n kj.1)
+    (q : ℤ → ℝ) :
+    (∑ i, q (e i).1) = ∑' k : ℤ, (multiplicity κ W n k : ℝ)*q k := by
+  classical
+  let f : Fin J → Σ k : ℤ, Fin (multiplicity κ W n k) := fun i =>
+    ⟨(e i).1,(e i).2,(he (e i)).mp ⟨i,rfl⟩⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro i j hij
+      apply hi
+      have h := congrArg (fun p : Σ k : ℤ, Fin (multiplicity κ W n k) => (p.1,p.2.val)) hij
+      simpa only [f] using h
+    · rintro ⟨k,j⟩
+      obtain ⟨i,hi⟩ := (he (k,j.val)).mpr j.isLt
+      refine ⟨i,?_⟩
+      dsimp only [f]
+      refine Sigma.ext (congrArg Prod.fst hi) ?_
+      exact (Fin.heq_ext_iff (congrArg (multiplicity κ W n) (congrArg Prod.fst hi))).2
+        (congrArg Prod.snd hi)
+  let E := Equiv.ofBijective f hf
+  let : Fintype (Σ k : ℤ, Fin (multiplicity κ W n k)) := Fintype.ofEquiv (Fin J) E
+  calc
+    (∑ i, q (e i).1) = ∑' i, q (e i).1 := (tsum_fintype _).symm
+    _ = ∑' p : Σ k : ℤ, Fin (multiplicity κ W n k), q p.1 :=
+      E.tsum_eq (fun p : Σ k : ℤ, Fin (multiplicity κ W n k) => q p.1)
+    _ = ∑' k : ℤ, ∑' _j : Fin (multiplicity κ W n k), q k :=
+      (Summable.of_finite).tsum_sigma' (fun _ => Summable.of_finite)
+    _ = _ := by simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+
+end Target
+
+/-- Half of each centered success probability is enough for the paper's
+exponent, even when the tests are dependent and the thresholds repeat. -/
+theorem repeated_batch_failure_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (F : Filtration ℕ mΩ) (κ : ℝ) (W : Target.WienerLaw) (n J : ℕ)
+    (e : Fin J → ℤ × ℕ) (hi : Function.Injective e)
+    (he : ∀ kj, kj ∈ Set.range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
+    (q : ℤ → ℝ) (hq0 : ∀ k, 0 ≤ q k) (hq1 : ∀ k, q k ≤ 1)
+    (hbudget : (2:ℝ)^n/8 ≤ ∑' k, (Target.multiplicity κ W n k:ℝ)*q k)
+    (success : Fin J → Set Ω)
+    (hmeas : ∀ i, MeasurableSet[F (i.val+1)] (success i))
+    (hstep : ∀ i (A : Set Ω), MeasurableSet[F i.val] A →
+      ENNReal.ofReal (q (e i).1/2)*P A ≤ P (A∩success i)) :
+    P (⋂ i, (success i)ᶜ) ≤ ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
+  have hb : (2:ℝ)^n/8 ≤ ∑ i, q (e i).1 := by
+    rwa [Target.repeated_batch_sum_eq κ W n J e hi he q]
+  have hp0 (i) : 0 ≤ q (e i).1/2 := div_nonneg (hq0 _) (by norm_num)
+  have hp1 (i) : q (e i).1/2 ≤ 1 := by linarith [hq1 (e i).1]
+  refine (adapted_fin_successes_failure_exp_bound P F success hmeas
+    (fun i => q (e i).1/2) hp0 hp1 hstep).trans ?_
+  apply ENNReal.ofReal_le_ofReal
+  apply Real.exp_le_exp.mpr
+  rw [← Finset.sum_div]
+  linarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/BatchRadiusGaps.lean b/lean/OAI/Probability/SLE/Gauge/BatchRadiusGaps.lean
index a8ada06c69c00173b459393cbb92a88a9bc70b3b..7081fd9dbb85d858ab6c5b1401548acf8d4e4d33 100644
--- a/lean/OAI/Probability/SLE/Gauge/BatchRadiusGaps.lean
+++ b/lean/OAI/Probability/SLE/Gauge/BatchRadiusGaps.lean
@@ -1,75 +1,75 @@
-import OAI.Probability.SLE.Gauge.BatchEntryOrder
-
-namespace OAI
-
-/-! The exact radius prescription leaves a full test interval and the
-subsequent mixing gap between consecutive tests and before cleanup. -/
-noncomputable section
-open Set
-
-namespace SLEExactGauge.Target
-
-lemma entryPosition_lt_cleanup (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n)
-    {k : ℤ} {j : ℕ} (hj : j < multiplicity κ W n k) :
-    entryPosition κ W (n,some (k,j)) < entryPosition κ W (n,none) := by
-  classical
-  let c := multiplicity κ W n
-  let hs := multiplicity_support_finite κ W hn hm1 hJ
-  have hk : k∈hs.toFinset := by
-    rw [Set.Finite.mem_toFinset]
-    exact (Nat.zero_le j).trans_lt hj
-  have hz (i : ℤ) (hi : i∉hs.toFinset) : c i=0 := by
-    have : ¬0<c i := by
-      intro h
-      exact hi (hs.mem_toFinset.mpr h)
-    omega
-  have hp : (∑' i, if i<k then c i else 0) = ∑ i∈hs.toFinset, if i<k then c i else 0 := by
-    apply tsum_eq_sum
-    intro i hi
-    simp only [hz i hi,ite_self]
-  have ht : (∑' i, c i) = ∑ i∈hs.toFinset, c i := by
-    exact tsum_eq_sum (fun i hi => hz i hi)
-  have hb : (∑' i, if i<k then c i else 0)+c k ≤ ∑' i, c i := by
-    rw [hp,ht]
-    have hP := Finset.sum_erase_add hs.toFinset (fun i => if i<k then c i else 0) hk
-    have hT := Finset.sum_erase_add hs.toFinset c hk
-    simp only [lt_self_iff_false,ite_false,add_zero] at hP
-    rw [← hP,← hT]
-    apply Nat.add_le_add_right
-    apply Finset.sum_le_sum
-    intro i _
-    split_ifs
-    · exact le_rfl
-    · exact Nat.zero_le _
-  exact (Nat.add_lt_add_left hj _).trans_le hb
-
-lemma entryRadius_gap_of_position_lt {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
-    {n : ℕ} (hn : 1≤n) {o o' : Option (ℤ × ℕ)}
-    (hpos : entryPosition κ W (n,o) < entryPosition κ W (n,o')) :
-    entryRadius κ W (n,o') ≤ entryRadius κ W (n,o) *
-      Real.exp (-2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ))) := by
-  have hpN : entryPosition κ W (n,o)+1 ≤ entryPosition κ W (n,o') := hpos
-  have hp : (entryPosition κ W (n,o):ℝ)+1 ≤ entryPosition κ W (n,o') := by exact_mod_cast hpN
-  have hnR : (0:ℝ)<n := by exact_mod_cast (Nat.zero_lt_of_lt hn)
-  have hcoef : 0 < 2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ)) := by
-    exact mul_pos (mul_pos (by norm_num) (internal_a_pos hκ)) (by positivity)
-  unfold entryRadius
-  rw [← Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  nlinarith [mul_nonneg hcoef.le (sub_nonneg.mpr hp)]
-
-lemma entryRadius_cleanup_gap {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
-    {n : ℕ} (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n)
-    {k : ℤ} {j : ℕ} (hj : j < multiplicity κ W n k) :
-    entryRadius κ W (n,none) ≤ entryRadius κ W (n,some (k,j)) *
-      Real.exp (-2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ))) :=
-  entryRadius_gap_of_position_lt hκ W hn (entryPosition_lt_cleanup κ W hn hm1 hJ hj)
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.BatchEntryOrder
+
+namespace OAI
+
+/-! The exact radius prescription leaves a full test interval and the
+subsequent mixing gap between consecutive tests and before cleanup. -/
+noncomputable section
+open Set
+
+namespace SLEExactGauge.Target
+
+lemma entryPosition_lt_cleanup (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n)
+    {k : ℤ} {j : ℕ} (hj : j < multiplicity κ W n k) :
+    entryPosition κ W (n,some (k,j)) < entryPosition κ W (n,none) := by
+  classical
+  let c := multiplicity κ W n
+  let hs := multiplicity_support_finite κ W hn hm1 hJ
+  have hk : k∈hs.toFinset := by
+    rw [Set.Finite.mem_toFinset]
+    exact (Nat.zero_le j).trans_lt hj
+  have hz (i : ℤ) (hi : i∉hs.toFinset) : c i=0 := by
+    have : ¬0<c i := by
+      intro h
+      exact hi (hs.mem_toFinset.mpr h)
+    omega
+  have hp : (∑' i, if i<k then c i else 0) = ∑ i∈hs.toFinset, if i<k then c i else 0 := by
+    apply tsum_eq_sum
+    intro i hi
+    simp only [hz i hi,ite_self]
+  have ht : (∑' i, c i) = ∑ i∈hs.toFinset, c i := by
+    exact tsum_eq_sum (fun i hi => hz i hi)
+  have hb : (∑' i, if i<k then c i else 0)+c k ≤ ∑' i, c i := by
+    rw [hp,ht]
+    have hP := Finset.sum_erase_add hs.toFinset (fun i => if i<k then c i else 0) hk
+    have hT := Finset.sum_erase_add hs.toFinset c hk
+    simp only [lt_self_iff_false,ite_false,add_zero] at hP
+    rw [← hP,← hT]
+    apply Nat.add_le_add_right
+    apply Finset.sum_le_sum
+    intro i _
+    split_ifs
+    · exact le_rfl
+    · exact Nat.zero_le _
+  exact (Nat.add_lt_add_left hj _).trans_le hb
+
+lemma entryRadius_gap_of_position_lt {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
+    {n : ℕ} (hn : 1≤n) {o o' : Option (ℤ × ℕ)}
+    (hpos : entryPosition κ W (n,o) < entryPosition κ W (n,o')) :
+    entryRadius κ W (n,o') ≤ entryRadius κ W (n,o) *
+      Real.exp (-2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ))) := by
+  have hpN : entryPosition κ W (n,o)+1 ≤ entryPosition κ W (n,o') := hpos
+  have hp : (entryPosition κ W (n,o):ℝ)+1 ≤ entryPosition κ W (n,o') := by exact_mod_cast hpN
+  have hnR : (0:ℝ)<n := by exact_mod_cast (Nat.zero_lt_of_lt hn)
+  have hcoef : 0 < 2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ)) := by
+    exact mul_pos (mul_pos (by norm_num) (internal_a_pos hκ)) (by positivity)
+  unfold entryRadius
+  rw [← Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  nlinarith [mul_nonneg hcoef.le (sub_nonneg.mpr hp)]
+
+lemma entryRadius_cleanup_gap {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
+    {n : ℕ} (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n)
+    {k : ℤ} {j : ℕ} (hj : j < multiplicity κ W n k) :
+    entryRadius κ W (n,none) ≤ entryRadius κ W (n,some (k,j)) *
+      Real.exp (-2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ))) :=
+  entryRadius_gap_of_position_lt hκ W hn (entryPosition_lt_cleanup κ W hn hm1 hJ hj)
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalBatchFailure.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalBatchFailure.lean
index d23defd077f805ec3fd44b435f81aec1823e1095..a1ff431977a135f9e4b8b8920ba0535da58582a6 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalBatchFailure.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalBatchFailure.lean
@@ -1,80 +1,80 @@
-import OAI.Probability.SLE.Gauge.CanonicalBatchSuccess
-import OAI.Probability.SLE.Gauge.BatchProbability
-
-namespace OAI
-
-/-! The exact prescribed batch has a nonempty chronological enumeration
-and enough total centered success probability. The finite failure estimate
-requires only the corresponding adapted half-probability inequalities. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-  (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-  (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-  (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-
-include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW in
-theorem ordered_batch_success_budget {n : ℕ} (hn : 3≤n) :
-    ∃ (J : ℕ) (e : Fin J → ℤ × ℕ), 0<J ∧ Function.Injective e ∧
-      (∀ kj, kj ∈ Set.range e ↔ kj.2 < Target.multiplicity κ W n kj.1) ∧
-      StrictMono (fun i => Target.entryPosition κ W (n,some (e i))) ∧
-      (2:ℝ)^n/8 ≤ ∑ i,
-        (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
-          {p | MomentBatches.dyadic (e i).1 ≤
-            C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2} := by
-  obtain ⟨J,e,hi,he,horder⟩ := Target.ordered_repeated_batch κ W (by omega)
-    (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1)
-    (C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
-  let q : ℤ → ℝ := fun k =>
-    (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
-      {p | MomentBatches.dyadic k ≤ C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}
-  have hb : (2:ℝ)^n/8 ≤ ∑ i, q (e i).1 := by
-    rw [Target.repeated_batch_sum_eq κ W n J e hi he q]
-    exact C.centeredDiskLaw_batch_success hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW hn
-  have hJ : 0<J := by
-    by_contra h
-    have hz : J=0 := by omega
-    subst J
-    simp only [Finset.univ_eq_empty,Finset.sum_empty] at hb
-    exact (div_pos (pow_pos (by norm_num : (0:ℝ)<2) n) (by norm_num)).not_ge hb
-  exact ⟨J,e,hJ,hi,he,horder,hb⟩
-
-include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW in
-theorem ordered_batch_failure_bound
-    {Ξ : Type*} [mΞ : MeasurableSpace Ξ] (Q : Measure Ξ) [IsProbabilityMeasure Q]
-    (ℱ : Filtration ℕ mΞ) {n J : ℕ} (hn : 3≤n)
-    (e : Fin J → ℤ × ℕ) (hi : Function.Injective e)
-    (he : ∀ kj, kj ∈ Set.range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
-    (success : Fin J → Set Ξ)
-    (hmeas : ∀ i, MeasurableSet[ℱ (i.val+1)] (success i))
-    (hstep : ∀ i (A : Set Ξ), MeasurableSet[ℱ i.val] A →
-      ENNReal.ofReal
-        ((centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
-          {p | MomentBatches.dyadic (e i).1 ≤
-            C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}/2)*Q A ≤
-        Q (A∩success i)) :
-    Q (⋂ i, (success i)ᶜ) ≤ ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
-  let ℓ := 4*(Target.batchLevel κ W n:ℝ)/κ
-  have hℓ : 0≤ℓ := div_nonneg (by positivity) hκ.le
-  let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw ℓ hℓ
-  exact repeated_batch_failure_bound Q ℱ κ W n J e hi he
-    (fun k => (centeredDiskLaw P κ B hc ℓ).real
-      {p | MomentBatches.dyadic k ≤ C.diskAtRadius p.1 ℓ p.2})
-    (fun _ => measureReal_nonneg) (fun _ => measureReal_le_one)
-    (C.centeredDiskLaw_batch_success hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW hn)
-    success hmeas hstep
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CanonicalBatchSuccess
+import OAI.Probability.SLE.Gauge.BatchProbability
+
+namespace OAI
+
+/-! The exact prescribed batch has a nonempty chronological enumeration
+and enough total centered success probability. The finite failure estimate
+requires only the corresponding adapted half-probability inequalities. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+  (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+  (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+  (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+
+include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW in
+theorem ordered_batch_success_budget {n : ℕ} (hn : 3≤n) :
+    ∃ (J : ℕ) (e : Fin J → ℤ × ℕ), 0<J ∧ Function.Injective e ∧
+      (∀ kj, kj ∈ Set.range e ↔ kj.2 < Target.multiplicity κ W n kj.1) ∧
+      StrictMono (fun i => Target.entryPosition κ W (n,some (e i))) ∧
+      (2:ℝ)^n/8 ≤ ∑ i,
+        (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
+          {p | MomentBatches.dyadic (e i).1 ≤
+            C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2} := by
+  obtain ⟨J,e,hi,he,horder⟩ := Target.ordered_repeated_batch κ W (by omega)
+    (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1)
+    (C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
+  let q : ℤ → ℝ := fun k =>
+    (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
+      {p | MomentBatches.dyadic k ≤ C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}
+  have hb : (2:ℝ)^n/8 ≤ ∑ i, q (e i).1 := by
+    rw [Target.repeated_batch_sum_eq κ W n J e hi he q]
+    exact C.centeredDiskLaw_batch_success hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW hn
+  have hJ : 0<J := by
+    by_contra h
+    have hz : J=0 := by omega
+    subst J
+    simp only [Finset.univ_eq_empty,Finset.sum_empty] at hb
+    exact (div_pos (pow_pos (by norm_num : (0:ℝ)<2) n) (by norm_num)).not_ge hb
+  exact ⟨J,e,hJ,hi,he,horder,hb⟩
+
+include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW in
+theorem ordered_batch_failure_bound
+    {Ξ : Type*} [mΞ : MeasurableSpace Ξ] (Q : Measure Ξ) [IsProbabilityMeasure Q]
+    (ℱ : Filtration ℕ mΞ) {n J : ℕ} (hn : 3≤n)
+    (e : Fin J → ℤ × ℕ) (hi : Function.Injective e)
+    (he : ∀ kj, kj ∈ Set.range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
+    (success : Fin J → Set Ξ)
+    (hmeas : ∀ i, MeasurableSet[ℱ (i.val+1)] (success i))
+    (hstep : ∀ i (A : Set Ξ), MeasurableSet[ℱ i.val] A →
+      ENNReal.ofReal
+        ((centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
+          {p | MomentBatches.dyadic (e i).1 ≤
+            C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}/2)*Q A ≤
+        Q (A∩success i)) :
+    Q (⋂ i, (success i)ᶜ) ≤ ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
+  let ℓ := 4*(Target.batchLevel κ W n:ℝ)/κ
+  have hℓ : 0≤ℓ := div_nonneg (by positivity) hκ.le
+  let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw ℓ hℓ
+  exact repeated_batch_failure_bound Q ℱ κ W n J e hi he
+    (fun k => (centeredDiskLaw P κ B hc ℓ).real
+      {p | MomentBatches.dyadic k ≤ C.diskAtRadius p.1 ℓ p.2})
+    (fun _ => measureReal_nonneg) (fun _ => measureReal_le_one)
+    (C.centeredDiskLaw_batch_success hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW hn)
+    success hmeas hstep
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalBatchSuccess.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalBatchSuccess.lean
index d5cdf7938a4fc99f605b9ea1a14a3ffb3bc60657..610ae526db1041fff413d79b70ed6f149e5d1fd8 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalBatchSuccess.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalBatchSuccess.lean
@@ -1,50 +1,50 @@
-import OAI.Probability.SLE.Gauge.CanonicalDiskMomentTails
-
-namespace OAI
-
-/-! The prescribed finite batch level supplies the successful-test budget.
-This uses its exact retained moment and the next limiting moment bound. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-theorem centeredDiskLaw_batch_success
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    {n : ℕ} (hn : 3≤n) :
-    (2:ℝ)^n/8 ≤ ∑' k : ℤ, (Target.multiplicity κ W n k : ℝ) *
-      (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
-        {p | MomentBatches.dyadic k ≤ C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2} := by
-  let ℓ := Target.batchLevel κ W n
-  have hu : 0≤4*(ℓ:ℝ)/κ := div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg ℓ)) hκ.le
-  let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw _ hu
-  apply MomentBatches.target_lower_batch_probability
-    (centeredDiskLaw P κ B hc (4*(ℓ:ℝ)/κ))
-    (fun p => C.diskAtRadius p.1 (4*(ℓ:ℝ)/κ) p.2)
-    (C.diskAtRadius_joint_measurable hc F hw _ hu)
-    (fun p => C.diskAtRadius_nonneg _ _ _) κ W hn
-    (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW n)
-    (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW (n+1))
-    (C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) n)
-    (C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) (n+1))
-  · rw [C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) n]
-    exact (C.prescribed_batch_retains_moment hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW n).2.le
-  · rw [C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) (n+1)]
-    exact C.finiteMoment_nat_le_moment hB hc h0 hκ hκ8 F hw hgrowth W hW (n+1) ℓ
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CanonicalDiskMomentTails
+
+namespace OAI
+
+/-! The prescribed finite batch level supplies the successful-test budget.
+This uses its exact retained moment and the next limiting moment bound. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+theorem centeredDiskLaw_batch_success
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    {n : ℕ} (hn : 3≤n) :
+    (2:ℝ)^n/8 ≤ ∑' k : ℤ, (Target.multiplicity κ W n k : ℝ) *
+      (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
+        {p | MomentBatches.dyadic k ≤ C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2} := by
+  let ℓ := Target.batchLevel κ W n
+  have hu : 0≤4*(ℓ:ℝ)/κ := div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg ℓ)) hκ.le
+  let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw _ hu
+  apply MomentBatches.target_lower_batch_probability
+    (centeredDiskLaw P κ B hc (4*(ℓ:ℝ)/κ))
+    (fun p => C.diskAtRadius p.1 (4*(ℓ:ℝ)/κ) p.2)
+    (C.diskAtRadius_joint_measurable hc F hw _ hu)
+    (fun p => C.diskAtRadius_nonneg _ _ _) κ W hn
+    (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW n)
+    (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW (n+1))
+    (C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) n)
+    (C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) (n+1))
+  · rw [C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) n]
+    exact (C.prescribed_batch_retains_moment hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW n).2.le
+  · rw [C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ (Nat.cast_nonneg ℓ) (n+1)]
+    exact C.finiteMoment_nat_le_moment hB hc h0 hκ hκ8 F hw hgrowth W hW (n+1) ℓ
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalCoverExpectation.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalCoverExpectation.lean
index 07f8de61e272d3dd40f0d4b5424870b311d917e9..f095ba9029a0e3c88f2203b52720a7f1f1b8fb6f 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalCoverExpectation.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalCoverExpectation.lean
@@ -1,43 +1,43 @@
-import OAI.Probability.SLE.Gauge.PrescribedCoverExpectation
-import OAI.Probability.SLE.Mass.CanonicalPhysicalMass
-
-namespace OAI
-
-/-! Applying the actual finite-cover reduction to canonical terminal mass
-leaves precisely the expected missed-cell error to be estimated. -/
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance canonicalCoverPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance canonicalCoverPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-theorem RationalCanonicalMass.prescribed_cover_finite_expectation
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (M : RationalCanonicalMass P κ B hBm γ G) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Target.WienerLaw) (hg : Target.IsHausdorffGauge (Target.gauge κ W))
-    (hm1 : 0<Target.moment κ W 1)
-    (hJ : ∀ n, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
-    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    (C c R : ℝ) (hC : 3 ≤ C) (hc : 0<c) (hR : 0 ≤ R)
-    (K : Set ℂ) (hK : IsClosed K) (hKR : K ⊆ closedBall 0 R)
-    (hH : ∀ z∈K, 0 ≤ z.im)
-    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
-    (he : ∀ n j, (e n j).2<Target.multiplicity κ W (n+1) (e n j).1)
-    (herr : Tendsto (fun n => ∫⁻ ω,
-      prescribedMissedCost f M.physicalTerminal κ W (n+1) (e n) C c R K ω ∂P)
-        atTop (𝓝 0)) :
-    (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (f ω)∩K) ∂P) < ∞ := by
-  refine lt_of_le_of_lt (expected_prescribed_hausdorff_le_of_missed_cost P hκ W hg hm1 hJ
-    f hf M.physicalTerminal M.physicalTerminal_measurable C c R hC hc hR K hK hKR hH J e he herr) ?_
-  exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top
-    (M.physicalTerminal_bounded_mean_lt_top hκ hκ8 (R+3+C))
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedCoverExpectation
+import OAI.Probability.SLE.Mass.CanonicalPhysicalMass
+
+namespace OAI
+
+/-! Applying the actual finite-cover reduction to canonical terminal mass
+leaves precisely the expected missed-cell error to be estimated. -/
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance canonicalCoverPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance canonicalCoverPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+theorem RationalCanonicalMass.prescribed_cover_finite_expectation
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (M : RationalCanonicalMass P κ B hBm γ G) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Target.WienerLaw) (hg : Target.IsHausdorffGauge (Target.gauge κ W))
+    (hm1 : 0<Target.moment κ W 1)
+    (hJ : ∀ n, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
+    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    (C c R : ℝ) (hC : 3 ≤ C) (hc : 0<c) (hR : 0 ≤ R)
+    (K : Set ℂ) (hK : IsClosed K) (hKR : K ⊆ closedBall 0 R)
+    (hH : ∀ z∈K, 0 ≤ z.im)
+    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
+    (he : ∀ n j, (e n j).2<Target.multiplicity κ W (n+1) (e n j).1)
+    (herr : Tendsto (fun n => ∫⁻ ω,
+      prescribedMissedCost f M.physicalTerminal κ W (n+1) (e n) C c R K ω ∂P)
+        atTop (𝓝 0)) :
+    (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (f ω)∩K) ∂P) < ∞ := by
+  refine lt_of_le_of_lt (expected_prescribed_hausdorff_le_of_missed_cost P hκ W hg hm1 hJ
+    f hf M.physicalTerminal M.physicalTerminal_measurable C c R hC hc hR K hK hKR hH J e he herr) ?_
+  exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top
+    (M.physicalTerminal_bounded_mean_lt_top hκ hκ8 (R+3+C))
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalDenseIntegration.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalDenseIntegration.lean
index 96f5103d023b2a2c23539f7faccf00d0942917a4..f670b174e6621cac8c6c423b853a9667b3d12d38 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalDenseIntegration.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalDenseIntegration.lean
@@ -1,94 +1,94 @@
-import OAI.Probability.SLE.Gauge.CanonicalDenseSections
-import OAI.Probability.SLE.Geometry.RestartWeightedKernel
-
-namespace OAI
-
-/-! Integration of the proved canonical dense sections over a genuinely
-random stopped past. The actual Loewner comparison supplies the final
-density domination; there is no independence assumption between tests. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance denseIntegrationRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance denseIntegrationRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance denseIntegrationComplexMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance denseIntegrationComplexBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem canonical_dense_sections_integrate
-    {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (P : Measure Ω) [IsFiniteMeasure P]
-    (ν : Measure C(ℝ≥0, ℝ)) [IsFiniteMeasure ν]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (Y : Ω → C(ℝ≥0, ℝ)) (hY : Measurable[mΩ] Y)
-    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
-      @Measure.map Ω C(ℝ≥0, ℝ) mΩ _ Y (P.restrict A) = P A • ν)
-    (κ b r v u : ℝ) (N : ℕ) (z : ℂ)
-    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
-    (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable[m] Fb)
-    (w : Ω → ℂ) (hw : Measurable[m] w)
-    (E : Set Ω) (hE : MeasurableSet[m] E)
-    (M : Ω → ℝ≥0∞) (hM : Measurable[m] M) (hMfinite : ∀ ω, M ω ≠ ∞)
-    (q : ℝ≥0∞) (hq : q ≠ ∞)
-    (hsections : ∀ ω ∈ E, q ≤
-      (ν.withDensity (fun p => canonicalFiniteDensity κ (u, w ω, p)))
-        (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z))
-    (Z : Ω → ℝ≥0∞) (S : Set Ω) (hS : MeasurableSet[mΩ] S)
-    (hactual : ∀ᵐ ω ∂P, ω ∈ E →
-      (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z).indicator
-        (fun p => M ω * canonicalFiniteDensity κ (u, w ω, p)) (Y ω) ≤ S.indicator Z ω)
-    (A : Set Ω) (hA : MeasurableSet[m] A) :
-    q * (∫⁻ ω in A ∩ E, M ω ∂P) ≤ P.withDensity Z (A ∩ S) := by
-  let : MeasurableSpace Ω := mΩ
-  let C : Set (Ω × C(ℝ≥0, ℝ)) :=
-    {p | p.1 ∈ E ∧ p.2 ∈ canonicalDenseEvent κ Φ (Fb p.1) u (w p.1) b r N v z}
-  let K : Ω × C(ℝ≥0, ℝ) → ℝ≥0∞ :=
-    fun p => M p.1 * canonicalFiniteDensity κ (u, w p.1, p.2)
-  have hC : @MeasurableSet (Ω × C(ℝ≥0, ℝ)) (m.prod _) C :=
-    (hE.preimage measurable_fst).inter
-      (@measurableSet_canonicalDenseEvent_product Ω m κ b v N Φ hΦ Fb hFb
-        (fun _ => u) measurable_const w hw (fun _ => r) measurable_const
-        (fun _ => z) measurable_const)
-  have hK : @Measurable (Ω × C(ℝ≥0, ℝ)) ℝ≥0∞ (m.prod _) _ K :=
-    (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp
-      (measurable_const.prodMk ((hw.comp measurable_fst).prodMk measurable_snd)))
-  have hsec : ∀ᵐ ω ∂P, q * E.indicator M ω ≤ ∫⁻ p, C.indicator K (ω, p) ∂ν := by
-    apply Eventually.of_forall
-    intro ω
-    by_cases hω : ω ∈ E
-    · have hf : (fun p => C.indicator K (ω, p)) =
-          fun p => M ω * (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z).indicator
-            (fun p => canonicalFiniteDensity κ (u, w ω, p)) p := by
-        funext p
-        by_cases hp : p ∈ canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z <;>
-          simp [C, K, hω, hp]
-      rw [hf, lintegral_const_mul' _ _ (hMfinite ω), indicator_of_mem hω]
-      rw [lintegral_indicator (measurableSet_canonicalDenseEvent κ b r v N Φ hΦ (Fb ω) u (w ω) z),
-        ← withDensity_apply]
-      · simpa only [mul_comm q] using mul_le_mul' le_rfl (hsections ω hω)
-      · exact measurableSet_canonicalDenseEvent κ b r v N Φ hΦ (Fb ω) u (w ω) z
-    · simp [C, hω]
-  have hdom : ∀ᵐ ω ∂P, C.indicator K (ω, Y ω) ≤ S.indicator Z ω := by
-    filter_upwards [hactual] with ω hω
-    by_cases hωE : ω ∈ E
-    · have he : C.indicator K (ω, Y ω) =
-          (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z).indicator
-            (fun p => M ω * canonicalFiniteDensity κ (u, w ω, p)) (Y ω) := by
-        by_cases hp : Y ω ∈ canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z <;>
-          simp [C, K, hωE, hp]
-      rw [he]
-      exact hω hωE
-    · simp [C, hωE]
-  have h := restart_weighted_event_lower_bound P ν m hm Y hY hlaw K hK C hC
-    (E.indicator M) Z q hq S hS hsec hdom A hA
-  rw [setLIntegral_indicator (hm E hE), inter_comm E A] at h
-  exact h
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CanonicalDenseSections
+import OAI.Probability.SLE.Geometry.RestartWeightedKernel
+
+namespace OAI
+
+/-! Integration of the proved canonical dense sections over a genuinely
+random stopped past. The actual Loewner comparison supplies the final
+density domination; there is no independence assumption between tests. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance denseIntegrationRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance denseIntegrationRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance denseIntegrationComplexMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance denseIntegrationComplexBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem canonical_dense_sections_integrate
+    {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (P : Measure Ω) [IsFiniteMeasure P]
+    (ν : Measure C(ℝ≥0, ℝ)) [IsFiniteMeasure ν]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (Y : Ω → C(ℝ≥0, ℝ)) (hY : Measurable[mΩ] Y)
+    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
+      @Measure.map Ω C(ℝ≥0, ℝ) mΩ _ Y (P.restrict A) = P A • ν)
+    (κ b r v u : ℝ) (N : ℕ) (z : ℂ)
+    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
+    (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable[m] Fb)
+    (w : Ω → ℂ) (hw : Measurable[m] w)
+    (E : Set Ω) (hE : MeasurableSet[m] E)
+    (M : Ω → ℝ≥0∞) (hM : Measurable[m] M) (hMfinite : ∀ ω, M ω ≠ ∞)
+    (q : ℝ≥0∞) (hq : q ≠ ∞)
+    (hsections : ∀ ω ∈ E, q ≤
+      (ν.withDensity (fun p => canonicalFiniteDensity κ (u, w ω, p)))
+        (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z))
+    (Z : Ω → ℝ≥0∞) (S : Set Ω) (hS : MeasurableSet[mΩ] S)
+    (hactual : ∀ᵐ ω ∂P, ω ∈ E →
+      (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z).indicator
+        (fun p => M ω * canonicalFiniteDensity κ (u, w ω, p)) (Y ω) ≤ S.indicator Z ω)
+    (A : Set Ω) (hA : MeasurableSet[m] A) :
+    q * (∫⁻ ω in A ∩ E, M ω ∂P) ≤ P.withDensity Z (A ∩ S) := by
+  let : MeasurableSpace Ω := mΩ
+  let C : Set (Ω × C(ℝ≥0, ℝ)) :=
+    {p | p.1 ∈ E ∧ p.2 ∈ canonicalDenseEvent κ Φ (Fb p.1) u (w p.1) b r N v z}
+  let K : Ω × C(ℝ≥0, ℝ) → ℝ≥0∞ :=
+    fun p => M p.1 * canonicalFiniteDensity κ (u, w p.1, p.2)
+  have hC : @MeasurableSet (Ω × C(ℝ≥0, ℝ)) (m.prod _) C :=
+    (hE.preimage measurable_fst).inter
+      (@measurableSet_canonicalDenseEvent_product Ω m κ b v N Φ hΦ Fb hFb
+        (fun _ => u) measurable_const w hw (fun _ => r) measurable_const
+        (fun _ => z) measurable_const)
+  have hK : @Measurable (Ω × C(ℝ≥0, ℝ)) ℝ≥0∞ (m.prod _) _ K :=
+    (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp
+      (measurable_const.prodMk ((hw.comp measurable_fst).prodMk measurable_snd)))
+  have hsec : ∀ᵐ ω ∂P, q * E.indicator M ω ≤ ∫⁻ p, C.indicator K (ω, p) ∂ν := by
+    apply Eventually.of_forall
+    intro ω
+    by_cases hω : ω ∈ E
+    · have hf : (fun p => C.indicator K (ω, p)) =
+          fun p => M ω * (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z).indicator
+            (fun p => canonicalFiniteDensity κ (u, w ω, p)) p := by
+        funext p
+        by_cases hp : p ∈ canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z <;>
+          simp [C, K, hω, hp]
+      rw [hf, lintegral_const_mul' _ _ (hMfinite ω), indicator_of_mem hω]
+      rw [lintegral_indicator (measurableSet_canonicalDenseEvent κ b r v N Φ hΦ (Fb ω) u (w ω) z),
+        ← withDensity_apply]
+      · simpa only [mul_comm q] using mul_le_mul' le_rfl (hsections ω hω)
+      · exact measurableSet_canonicalDenseEvent κ b r v N Φ hΦ (Fb ω) u (w ω) z
+    · simp [C, hω]
+  have hdom : ∀ᵐ ω ∂P, C.indicator K (ω, Y ω) ≤ S.indicator Z ω := by
+    filter_upwards [hactual] with ω hω
+    by_cases hωE : ω ∈ E
+    · have he : C.indicator K (ω, Y ω) =
+          (canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z).indicator
+            (fun p => M ω * canonicalFiniteDensity κ (u, w ω, p)) (Y ω) := by
+        by_cases hp : Y ω ∈ canonicalDenseEvent κ Φ (Fb ω) u (w ω) b r N v z <;>
+          simp [C, K, hωE, hp]
+      rw [he]
+      exact hω hωE
+    · simp [C, hωE]
+  have h := restart_weighted_event_lower_bound P ν m hm Y hY hlaw K hK C hC
+    (E.indicator M) Z q hq S hS hsec hdom A hA
+  rw [setLIntegral_indicator (hm E hE), inter_comm E A] at h
+  exact h
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalDenseSections.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalDenseSections.lean
index 39508dd8a74fe56dba48909d7b82522b1dd98ceb..9ac8126217ddde122000fda891bec5a8134f7419 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalDenseSections.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalDenseSections.lean
@@ -1,242 +1,242 @@
-import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel
-import OAI.Probability.SLE.Gauge.ConformalDenseExperiment
-import OAI.Probability.SLE.Gauge.RandomPassageBounds
-
-namespace OAI
-
-/-! Measurable canonical sections for the proved conformal dense experiment.
-The probability estimate is transported from the actual SLE theorem; the
-canonical path and stopping kernels introduce no further stochastic input. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open FrontSuccessMeasurability
-
-local instance denseSectionDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance denseSectionDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance denseSectionPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance denseSectionPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-lemma measurable_stoppedAtPath
-    {X : Type*} [MeasurableSpace X]
-    (f : X → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (τ : X → WithTop ℝ≥0) (hτ : Measurable τ) :
-    Measurable (fun x => stoppedAtPath (f x) (τ x)) := by
-  apply source_path_measurable
-  intro t
-  have hm : Measurable (fun x => f x
-      ((min (τ x) (t : WithTop ℝ≥0)).untopD (Classical.arbitrary ℝ≥0))) :=
-    continuous_eval.measurable.comp
-      (hf.prodMk ((hτ.min measurable_const).untopD (Classical.arbitrary ℝ≥0)))
-  simpa only [stoppedAtPath_apply, WithTop.untopA] using hm
-
-lemma measurable_wholeSausageCharge_closedBall
-    {X : Type*} [MeasurableSpace X]
-    (κ v : ℝ) (f : X → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (a : X → ℂ) (ha : Measurable a) (r : X → ℝ) (hr : Measurable r) :
-    Measurable (fun x => wholeSausageCharge κ (f x) v (closedBall (a x) (r x))) := by
-  let I : X × ℂ → ℝ≥0∞ := fun p =>
-    if p.2 ∈ closedBall (a p.1) (r p.1) ∩ wholeSausageSet (f p.1) v then 1 else 0
-  have hball : MeasurableSet {p : X × ℂ | p.2 ∈ closedBall (a p.1) (r p.1)} :=
-    measurableSet_le (measurable_snd.dist (ha.comp measurable_fst)) (hr.comp measurable_fst)
-  have hsausage : MeasurableSet {p : X × ℂ | p.2 ∈ wholeSausageSet (f p.1) v} :=
-    (wholeSausage_joint_open v).measurableSet.preimage
-      ((hf.comp measurable_fst).prodMk measurable_snd)
-  have hI : Measurable I := Measurable.ite (hball.inter hsausage)
-    measurable_const measurable_const
-  have he (x : X) : wholeSausageCharge κ (f x) v (closedBall (a x) (r x)) =
-      ENNReal.ofReal (v ^ (-codimension κ)) * ∫⁻ z, I (x, z) ∂volume := by
-    unfold wholeSausageCharge
-    congr 1
-    have hIeq : (fun z => I (x, z)) =
-        (closedBall (a x) (r x) ∩ wholeSausageSet (f x) v).indicator (fun _ => 1) := by
-      funext z
-      by_cases hz : z ∈ closedBall (a x) (r x) ∩ wholeSausageSet (f x) v
-      · simp only [I, ite_eq_left hz, indicator_of_mem hz]
-      · simp only [I, ite_eq_right hz, indicator_of_notMem hz]
-    rw [hIeq]
-    exact (lintegral_indicator_one (μ := volume) (isClosed_closedBall.measurableSet.inter
-      (wholeSausageSet_isOpen (f x) v).measurableSet)).symm
-  simp_rw [he]
-  exact measurable_const.mul (hI.lintegral_prod_right' (ν := volume))
-
-def canonicalDenseEvent (κ : ℝ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ))
-    (Fb : C(closedHalfPlane, ℂ)) (u : ℝ) (w : ℂ) (b r : ℝ)
-    (m : ℕ) (v : ℝ) (a : ℂ) : Set C(ℝ≥0, ℝ) :=
-  {p | ENNReal.ofReal (b * r ^ dimension κ * (m : ℝ) ^ codimension κ) ≤
-    stoppedWholeSausageCharge κ (finiteBoundaryPath Fb (Φ p))
-      (canonicalRadiusLevel κ (u, w, p)) v (closedBall a r)}
-
-/-- All geometric marked parameters may be random. In particular this
-applies to the product of the stopped past with a fresh driver path. -/
-lemma measurableSet_canonicalDenseEvent_random
-    {X : Type*} [MeasurableSpace X]
-    (κ b v : ℝ) (m : ℕ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
-    (Fb : X → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb)
-    (u : X → ℝ) (hu : Measurable u) (w : X → ℂ) (hw : Measurable w)
-    (r : X → ℝ) (hr : Measurable r) (a : X → ℂ) (ha : Measurable a)
-    (Y : X → C(ℝ≥0, ℝ)) (hY : Measurable Y) :
-    MeasurableSet {x | Y x ∈ canonicalDenseEvent κ Φ (Fb x) (u x) (w x)
-      b (r x) m v (a x)} := by
-  have hpath : Measurable (fun x => finiteBoundaryPath (Fb x) (Φ (Y x))) :=
-    (measurable_random_finiteBoundaryPath Fb hFb).comp
-      (measurable_id.prodMk (hΦ.comp hY))
-  have hstop := measurable_stoppedAtPath _ hpath _
-    (measurable_canonicalRadiusLevel_random_state κ hY hu hw)
-  have hmass := measurable_wholeSausageCharge_closedBall κ v _ hstop a ha r hr
-  have hthresh : Measurable (fun x => ENNReal.ofReal
-      (b * r x ^ dimension κ * (m : ℝ) ^ codimension κ)) := by fun_prop
-  exact measurableSet_le hthresh hmass
-
-lemma measurableSet_canonicalDenseEvent
-    (κ b r v : ℝ) (m : ℕ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
-    (Fb : C(closedHalfPlane, ℂ)) (u : ℝ) (w a : ℂ) :
-    MeasurableSet (canonicalDenseEvent κ Φ Fb u w b r m v a) := by
-  exact measurableSet_canonicalDenseEvent_random κ b v m Φ hΦ
-    (fun _ => Fb) measurable_const (fun _ => u) measurable_const
-    (fun _ => w) measurable_const (fun _ => r) measurable_const
-    (fun _ => a) measurable_const id measurable_id
-
-lemma measurableSet_canonicalDenseEvent_product
-    {X : Type*} [MeasurableSpace X]
-    (κ b v : ℝ) (m : ℕ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
-    (Fb : X → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb)
-    (u : X → ℝ) (hu : Measurable u) (w : X → ℂ) (hw : Measurable w)
-    (r : X → ℝ) (hr : Measurable r) (a : X → ℂ) (ha : Measurable a) :
-    MeasurableSet {x : X × C(ℝ≥0, ℝ) |
-      x.2 ∈ canonicalDenseEvent κ Φ (Fb x.1) (u x.1) (w x.1)
-        b (r x.1) m v (a x.1)} :=
-  measurableSet_canonicalDenseEvent_random κ b v m Φ hΦ
-    _ (hFb.comp measurable_fst) _ (hu.comp measurable_fst)
-    _ (hw.comp measurable_fst) _ (hr.comp measurable_fst)
-    _ (ha.comp measurable_fst) _ measurable_snd
-
-lemma map_withDensity_comp
-    {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
-    (μ : Measure X) (f : X → Y) (hf : Measurable f)
-    (d : Y → ℝ≥0∞) (hd : Measurable d) :
-    (μ.withDensity (fun x => d (f x))).map f = (μ.map f).withDensity d := by
-  apply Measure.ext
-  intro S hS
-  rw [Measure.map_apply hf hS, withDensity_apply _ (hf hS), withDensity_apply _ hS,
-    setLIntegral_map hS hd hf]
-
-/-- The canonical density is exactly the pushforward of the original
-finite centered law, simultaneously for any chosen nonnegative level. -/
-theorem canonical_finiteCenteredLaw_map
-    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂μ, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (u : ℝ) (hu : 0 ≤ u) (w : ℂ) :
-    (finiteCenteredLaw μ κ B γ G w u).map (sourceDriverPath B hc) =
-      (μ.map (sourceDriverPath B hc)).withDensity
-        (fun p => canonicalFiniteDensity κ (u, w, p)) := by
-  have hbp := source_path_measurable (sourceDriverPath B hc) hm
-  have hd : Measurable (fun p => canonicalFiniteDensity κ (u, w, p)) :=
-    (canonicalFiniteDensity_measurable κ).comp
-      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
-  have he : finiteCenteredLaw μ κ B γ G w u =
-      μ.withDensity (fun ω => canonicalFiniteDensity κ (u, w, sourceDriverPath B hc ω)) := by
-    unfold finiteCenteredLaw
-    apply withDensity_congr_ae
-    filter_upwards [hw] with ω hω
-    exact (canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) hω u hu w).symm
-  rw [he, map_withDensity_comp μ _ hbp _ hd]
-
-namespace Serpentine
-open Target
-
-/-- The conformal dense-experiment theorem on canonical driver space.
-The constants are those of the proved actual SLE theorem, with half the
-height margin reserved to use the finite boundary chart on the interior. -/
-theorem uniform_canonical_conformal_dense_sections
-    {Ω : Type} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
-    (hΦγ : ∀ᵐ ω ∂μ, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω)
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport μ γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W)
-    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8)
-    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
-    ∃ h b C : ℝ, 0 < h ∧ h ≤ s / 2 ∧ 0 < b ∧ 0 < C ∧
-      ∃ δ : ℕ → ℝ, (∀ m, 0 < m → 0 < δ m) ∧
-      ∀ (f : ℂ → ℂ) (Fb : C(closedHalfPlane, ℂ)) (w : ℂ),
-        DifferentiableOn ℂ f upperHalfPlane → InjOn f upperHalfPlane →
-        (∀ z : closedHalfPlane, (z : ℂ) ∈ upperHalfPlane → Fb z = f z) →
-        ‖w‖ = 1 → s ≤ w.im → deriv f w ≠ 0 →
-        ∀ m : ℕ, 0 < m → ∀ v : ℝ, 0 < v →
-          v < δ m * (2 * w.im * ‖deriv f w‖) →
-          ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤
-            ((μ.map (sourceDriverPath B hc)).withDensity
-              (fun p => canonicalFiniteDensity κ (experimentRadiusLevel h, w, p)))
-              (canonicalDenseEvent κ Φ Fb (experimentRadiusLevel h) w b
-                (2 * w.im * ‖deriv f w‖) m v (f w)) := by
-  have hs2 : 0 < s / 2 := by positivity
-  have hs21 : s / 2 ≤ 1 := by linarith
-  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hmass⟩ :=
-    uniform_conformal_dense_stopped_mass μ κ hκ hκ8 B hB hBm hc hB0 γ G F hw
-      hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall hs2 hs21
-  refine ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, ?_⟩
-  intro f Fb w hf hfi hFb hwn hwy hne m hm v hv hvcut
-  let bp := sourceDriverPath B hc
-  let gp := sourceTracePath γ (fun ω => (hw ω).1)
-  let η : Ω → ℝ≥0 → ℂ := fun ω => finiteBoundaryPath Fb (gp ω)
-  have hη : ∀ ω, Continuous (η ω) := fun ω => (finiteBoundaryPath Fb (gp ω)).continuous
-  have hηeq (ω : Ω) (t : ℝ≥0) (ht : γ ω t ∈ closedBall w h) : η ω t = f (γ ω t) := by
-    have hib : w.im - (γ ω t).im ≤ h := by
-      calc
-        _ = (w - γ ω t).im := rfl
-        _ ≤ ‖w - γ ω t‖ := Complex.im_le_norm _
-        _ = dist (γ ω t) w := by rw [dist_eq_norm, norm_sub_rev]
-        _ ≤ h := ht
-    have hpos : γ ω t ∈ upperHalfPlane := by change 0 < (γ ω t).im; linarith
-    have hclosed : ClosedPathProperty (gp ω) := by
-      intro t
-      exact (hw ω).2.2.1 t
-    change finiteBoundaryPath Fb (gp ω) t = _
-    rw [finiteBoundaryPath_apply Fb (gp ω) hclosed t]
-    exact hFb _ hpos
-  let ρ : Ω → WithTop ℝ≥0 := fun ω => canonicalRadiusLevel κ (experimentRadiusLevel h, w, bp ω)
-  have hu : 0 ≤ experimentRadiusLevel h := experimentRadiusLevel_nonneg hh (by linarith)
-  have hρ : ∀ᵐ ω ∂μ, ρ ω = radiusLevelTime (γ ω) (G ω) w (experimentRadiusLevel h) :=
-    Eventually.of_forall fun ω => canonicalRadiusLevel_eq κ (bp ω) (hw ω) _ hu w
-  have hp := hmass f w hf hfi hwn (by linarith) hne η hη hηeq ρ hρ m hm v hv hvcut
-  let E := canonicalDenseEvent κ Φ Fb (experimentRadiusLevel h) w b
-    (2 * w.im * ‖deriv f w‖) m v (f w)
-  have hE : MeasurableSet E := measurableSet_canonicalDenseEvent _ _ _ _ _ Φ hΦ Fb _ _ _
-  have hmB : ∀ t, Measurable (B t) := fun t => hBm.comp (measurable_const.prodMk measurable_id)
-  rw [← canonical_finiteCenteredLaw_map μ κ B hmB hc γ G F
-    (Eventually.of_forall hw) _ hu w, Measure.map_apply (source_path_measurable bp hmB) hE]
-  apply hp.trans_eq
-  apply measure_congr
-  apply (withDensity_absolutelyContinuous μ _).ae_le
-  filter_upwards [hΦγ] with ω hω
-  apply propext
-  change (ENNReal.ofReal _ ≤ stoppedWholeSausageCharge κ (sourceTracePath η hη ω) (ρ ω) v _) ↔
-    (ENNReal.ofReal _ ≤ stoppedWholeSausageCharge κ (finiteBoundaryPath Fb (Φ (bp ω))) (ρ ω) v _)
-  rw [hω]
-  rfl
-
-end Serpentine
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel
+import OAI.Probability.SLE.Gauge.ConformalDenseExperiment
+import OAI.Probability.SLE.Gauge.RandomPassageBounds
+
+namespace OAI
+
+/-! Measurable canonical sections for the proved conformal dense experiment.
+The probability estimate is transported from the actual SLE theorem; the
+canonical path and stopping kernels introduce no further stochastic input. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open FrontSuccessMeasurability
+
+local instance denseSectionDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance denseSectionDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance denseSectionPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance denseSectionPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+lemma measurable_stoppedAtPath
+    {X : Type*} [MeasurableSpace X]
+    (f : X → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (τ : X → WithTop ℝ≥0) (hτ : Measurable τ) :
+    Measurable (fun x => stoppedAtPath (f x) (τ x)) := by
+  apply source_path_measurable
+  intro t
+  have hm : Measurable (fun x => f x
+      ((min (τ x) (t : WithTop ℝ≥0)).untopD (Classical.arbitrary ℝ≥0))) :=
+    continuous_eval.measurable.comp
+      (hf.prodMk ((hτ.min measurable_const).untopD (Classical.arbitrary ℝ≥0)))
+  simpa only [stoppedAtPath_apply, WithTop.untopA] using hm
+
+lemma measurable_wholeSausageCharge_closedBall
+    {X : Type*} [MeasurableSpace X]
+    (κ v : ℝ) (f : X → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (a : X → ℂ) (ha : Measurable a) (r : X → ℝ) (hr : Measurable r) :
+    Measurable (fun x => wholeSausageCharge κ (f x) v (closedBall (a x) (r x))) := by
+  let I : X × ℂ → ℝ≥0∞ := fun p =>
+    if p.2 ∈ closedBall (a p.1) (r p.1) ∩ wholeSausageSet (f p.1) v then 1 else 0
+  have hball : MeasurableSet {p : X × ℂ | p.2 ∈ closedBall (a p.1) (r p.1)} :=
+    measurableSet_le (measurable_snd.dist (ha.comp measurable_fst)) (hr.comp measurable_fst)
+  have hsausage : MeasurableSet {p : X × ℂ | p.2 ∈ wholeSausageSet (f p.1) v} :=
+    (wholeSausage_joint_open v).measurableSet.preimage
+      ((hf.comp measurable_fst).prodMk measurable_snd)
+  have hI : Measurable I := Measurable.ite (hball.inter hsausage)
+    measurable_const measurable_const
+  have he (x : X) : wholeSausageCharge κ (f x) v (closedBall (a x) (r x)) =
+      ENNReal.ofReal (v ^ (-codimension κ)) * ∫⁻ z, I (x, z) ∂volume := by
+    unfold wholeSausageCharge
+    congr 1
+    have hIeq : (fun z => I (x, z)) =
+        (closedBall (a x) (r x) ∩ wholeSausageSet (f x) v).indicator (fun _ => 1) := by
+      funext z
+      by_cases hz : z ∈ closedBall (a x) (r x) ∩ wholeSausageSet (f x) v
+      · simp only [I, ite_eq_left hz, indicator_of_mem hz]
+      · simp only [I, ite_eq_right hz, indicator_of_notMem hz]
+    rw [hIeq]
+    exact (lintegral_indicator_one (μ := volume) (isClosed_closedBall.measurableSet.inter
+      (wholeSausageSet_isOpen (f x) v).measurableSet)).symm
+  simp_rw [he]
+  exact measurable_const.mul (hI.lintegral_prod_right' (ν := volume))
+
+def canonicalDenseEvent (κ : ℝ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ))
+    (Fb : C(closedHalfPlane, ℂ)) (u : ℝ) (w : ℂ) (b r : ℝ)
+    (m : ℕ) (v : ℝ) (a : ℂ) : Set C(ℝ≥0, ℝ) :=
+  {p | ENNReal.ofReal (b * r ^ dimension κ * (m : ℝ) ^ codimension κ) ≤
+    stoppedWholeSausageCharge κ (finiteBoundaryPath Fb (Φ p))
+      (canonicalRadiusLevel κ (u, w, p)) v (closedBall a r)}
+
+/-- All geometric marked parameters may be random. In particular this
+applies to the product of the stopped past with a fresh driver path. -/
+lemma measurableSet_canonicalDenseEvent_random
+    {X : Type*} [MeasurableSpace X]
+    (κ b v : ℝ) (m : ℕ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
+    (Fb : X → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb)
+    (u : X → ℝ) (hu : Measurable u) (w : X → ℂ) (hw : Measurable w)
+    (r : X → ℝ) (hr : Measurable r) (a : X → ℂ) (ha : Measurable a)
+    (Y : X → C(ℝ≥0, ℝ)) (hY : Measurable Y) :
+    MeasurableSet {x | Y x ∈ canonicalDenseEvent κ Φ (Fb x) (u x) (w x)
+      b (r x) m v (a x)} := by
+  have hpath : Measurable (fun x => finiteBoundaryPath (Fb x) (Φ (Y x))) :=
+    (measurable_random_finiteBoundaryPath Fb hFb).comp
+      (measurable_id.prodMk (hΦ.comp hY))
+  have hstop := measurable_stoppedAtPath _ hpath _
+    (measurable_canonicalRadiusLevel_random_state κ hY hu hw)
+  have hmass := measurable_wholeSausageCharge_closedBall κ v _ hstop a ha r hr
+  have hthresh : Measurable (fun x => ENNReal.ofReal
+      (b * r x ^ dimension κ * (m : ℝ) ^ codimension κ)) := by fun_prop
+  exact measurableSet_le hthresh hmass
+
+lemma measurableSet_canonicalDenseEvent
+    (κ b r v : ℝ) (m : ℕ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
+    (Fb : C(closedHalfPlane, ℂ)) (u : ℝ) (w a : ℂ) :
+    MeasurableSet (canonicalDenseEvent κ Φ Fb u w b r m v a) := by
+  exact measurableSet_canonicalDenseEvent_random κ b v m Φ hΦ
+    (fun _ => Fb) measurable_const (fun _ => u) measurable_const
+    (fun _ => w) measurable_const (fun _ => r) measurable_const
+    (fun _ => a) measurable_const id measurable_id
+
+lemma measurableSet_canonicalDenseEvent_product
+    {X : Type*} [MeasurableSpace X]
+    (κ b v : ℝ) (m : ℕ) (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
+    (Fb : X → C(closedHalfPlane, ℂ)) (hFb : Measurable Fb)
+    (u : X → ℝ) (hu : Measurable u) (w : X → ℂ) (hw : Measurable w)
+    (r : X → ℝ) (hr : Measurable r) (a : X → ℂ) (ha : Measurable a) :
+    MeasurableSet {x : X × C(ℝ≥0, ℝ) |
+      x.2 ∈ canonicalDenseEvent κ Φ (Fb x.1) (u x.1) (w x.1)
+        b (r x.1) m v (a x.1)} :=
+  measurableSet_canonicalDenseEvent_random κ b v m Φ hΦ
+    _ (hFb.comp measurable_fst) _ (hu.comp measurable_fst)
+    _ (hw.comp measurable_fst) _ (hr.comp measurable_fst)
+    _ (ha.comp measurable_fst) _ measurable_snd
+
+lemma map_withDensity_comp
+    {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
+    (μ : Measure X) (f : X → Y) (hf : Measurable f)
+    (d : Y → ℝ≥0∞) (hd : Measurable d) :
+    (μ.withDensity (fun x => d (f x))).map f = (μ.map f).withDensity d := by
+  apply Measure.ext
+  intro S hS
+  rw [Measure.map_apply hf hS, withDensity_apply _ (hf hS), withDensity_apply _ hS,
+    setLIntegral_map hS hd hf]
+
+/-- The canonical density is exactly the pushforward of the original
+finite centered law, simultaneously for any chosen nonnegative level. -/
+theorem canonical_finiteCenteredLaw_map
+    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂μ, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (u : ℝ) (hu : 0 ≤ u) (w : ℂ) :
+    (finiteCenteredLaw μ κ B γ G w u).map (sourceDriverPath B hc) =
+      (μ.map (sourceDriverPath B hc)).withDensity
+        (fun p => canonicalFiniteDensity κ (u, w, p)) := by
+  have hbp := source_path_measurable (sourceDriverPath B hc) hm
+  have hd : Measurable (fun p => canonicalFiniteDensity κ (u, w, p)) :=
+    (canonicalFiniteDensity_measurable κ).comp
+      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
+  have he : finiteCenteredLaw μ κ B γ G w u =
+      μ.withDensity (fun ω => canonicalFiniteDensity κ (u, w, sourceDriverPath B hc ω)) := by
+    unfold finiteCenteredLaw
+    apply withDensity_congr_ae
+    filter_upwards [hw] with ω hω
+    exact (canonicalFiniteDensity_eq κ (sourceDriverPath B hc ω) hω u hu w).symm
+  rw [he, map_withDensity_comp μ _ hbp _ hd]
+
+namespace Serpentine
+open Target
+
+/-- The conformal dense-experiment theorem on canonical driver space.
+The constants are those of the proved actual SLE theorem, with half the
+height margin reserved to use the finite boundary chart on the interior. -/
+theorem uniform_canonical_conformal_dense_sections
+    {Ω : Type} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ)) (hΦ : Measurable Φ)
+    (hΦγ : ∀ᵐ ω ∂μ, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω)
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport μ γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W)
+    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8)
+    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
+    ∃ h b C : ℝ, 0 < h ∧ h ≤ s / 2 ∧ 0 < b ∧ 0 < C ∧
+      ∃ δ : ℕ → ℝ, (∀ m, 0 < m → 0 < δ m) ∧
+      ∀ (f : ℂ → ℂ) (Fb : C(closedHalfPlane, ℂ)) (w : ℂ),
+        DifferentiableOn ℂ f upperHalfPlane → InjOn f upperHalfPlane →
+        (∀ z : closedHalfPlane, (z : ℂ) ∈ upperHalfPlane → Fb z = f z) →
+        ‖w‖ = 1 → s ≤ w.im → deriv f w ≠ 0 →
+        ∀ m : ℕ, 0 < m → ∀ v : ℝ, 0 < v →
+          v < δ m * (2 * w.im * ‖deriv f w‖) →
+          ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤
+            ((μ.map (sourceDriverPath B hc)).withDensity
+              (fun p => canonicalFiniteDensity κ (experimentRadiusLevel h, w, p)))
+              (canonicalDenseEvent κ Φ Fb (experimentRadiusLevel h) w b
+                (2 * w.im * ‖deriv f w‖) m v (f w)) := by
+  have hs2 : 0 < s / 2 := by positivity
+  have hs21 : s / 2 ≤ 1 := by linarith
+  obtain ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, hmass⟩ :=
+    uniform_conformal_dense_stopped_mass μ κ hκ hκ8 B hB hBm hc hB0 γ G F hw
+      hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall hs2 hs21
+  refine ⟨h, b, C, hh, hhs, hb, hC, δ, hδ, ?_⟩
+  intro f Fb w hf hfi hFb hwn hwy hne m hm v hv hvcut
+  let bp := sourceDriverPath B hc
+  let gp := sourceTracePath γ (fun ω => (hw ω).1)
+  let η : Ω → ℝ≥0 → ℂ := fun ω => finiteBoundaryPath Fb (gp ω)
+  have hη : ∀ ω, Continuous (η ω) := fun ω => (finiteBoundaryPath Fb (gp ω)).continuous
+  have hηeq (ω : Ω) (t : ℝ≥0) (ht : γ ω t ∈ closedBall w h) : η ω t = f (γ ω t) := by
+    have hib : w.im - (γ ω t).im ≤ h := by
+      calc
+        _ = (w - γ ω t).im := rfl
+        _ ≤ ‖w - γ ω t‖ := Complex.im_le_norm _
+        _ = dist (γ ω t) w := by rw [dist_eq_norm, norm_sub_rev]
+        _ ≤ h := ht
+    have hpos : γ ω t ∈ upperHalfPlane := by change 0 < (γ ω t).im; linarith
+    have hclosed : ClosedPathProperty (gp ω) := by
+      intro t
+      exact (hw ω).2.2.1 t
+    change finiteBoundaryPath Fb (gp ω) t = _
+    rw [finiteBoundaryPath_apply Fb (gp ω) hclosed t]
+    exact hFb _ hpos
+  let ρ : Ω → WithTop ℝ≥0 := fun ω => canonicalRadiusLevel κ (experimentRadiusLevel h, w, bp ω)
+  have hu : 0 ≤ experimentRadiusLevel h := experimentRadiusLevel_nonneg hh (by linarith)
+  have hρ : ∀ᵐ ω ∂μ, ρ ω = radiusLevelTime (γ ω) (G ω) w (experimentRadiusLevel h) :=
+    Eventually.of_forall fun ω => canonicalRadiusLevel_eq κ (bp ω) (hw ω) _ hu w
+  have hp := hmass f w hf hfi hwn (by linarith) hne η hη hηeq ρ hρ m hm v hv hvcut
+  let E := canonicalDenseEvent κ Φ Fb (experimentRadiusLevel h) w b
+    (2 * w.im * ‖deriv f w‖) m v (f w)
+  have hE : MeasurableSet E := measurableSet_canonicalDenseEvent _ _ _ _ _ Φ hΦ Fb _ _ _
+  have hmB : ∀ t, Measurable (B t) := fun t => hBm.comp (measurable_const.prodMk measurable_id)
+  rw [← canonical_finiteCenteredLaw_map μ κ B hmB hc γ G F
+    (Eventually.of_forall hw) _ hu w, Measure.map_apply (source_path_measurable bp hmB) hE]
+  apply hp.trans_eq
+  apply measure_congr
+  apply (withDensity_absolutelyContinuous μ _).ae_le
+  filter_upwards [hΦγ] with ω hω
+  apply propext
+  change (ENNReal.ofReal _ ≤ stoppedWholeSausageCharge κ (sourceTracePath η hη ω) (ρ ω) v _) ↔
+    (ENNReal.ofReal _ ≤ stoppedWholeSausageCharge κ (finiteBoundaryPath Fb (Φ (bp ω))) (ρ ω) v _)
+  rw [hω]
+  rfl
+
+end Serpentine
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalDiskMomentTails.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalDiskMomentTails.lean
index 844828a739de44ae667d8a09a93754bc9cfb8432..34a8ed129c3ca8f93d6744f9cd2cdaecdf7519cb 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalDiskMomentTails.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalDiskMomentTails.lean
@@ -1,126 +1,126 @@
-import OAI.Probability.SLE.Geometry.CanonicalDiskTails
-import OAI.Probability.SLE.Geometry.FiniteLawTailENN
-
-namespace OAI
-
-/-! Summable tails for the exact prescribed moments, obtained from the actual
-finite centered laws. The supremum is taken over finite radius levels. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-  (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-  (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-
-include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
-lemma finiteMoment_nat_le_moment (n j : ℕ) :
-    Target.finiteMoment κ W n j ≤ Target.moment κ W n := by
-  apply ge_of_tendsto (C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW n)
-  filter_upwards [eventually_ge_atTop j] with k hk
-  exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW n
-    (by change (0:ℝ)≤(j:ℝ); exact Nat.cast_nonneg j)
-    (by change (0:ℝ)≤(k:ℝ); exact Nat.cast_nonneg k) (Nat.cast_le.mpr hk)
-
-include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
-lemma centeredDiskLaw_integrable_pow (ℓ : ℝ) (hℓ : 0≤ℓ) (n : ℕ) :
-    Integrable (fun p : ℝ × Ω => C.diskAtRadius p.1 (4*ℓ/κ) p.2 ^ n)
-      (centeredDiskLaw P κ B hc (4*ℓ/κ)) := by
-  have hu : 0≤4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
-  have hm := (C.diskAtRadius_joint_measurable hc F hw _ hu).pow_const n
-  have hnonneg : ∀ p : ℝ × Ω, 0≤C.diskAtRadius p.1 (4*ℓ/κ) p.2^n :=
-    fun p => pow_nonneg (C.diskAtRadius_nonneg _ _ _) n
-  apply (lintegral_ofReal_ne_top_iff_integrable hm.aestronglyMeasurable
-    (Eventually.of_forall hnonneg)).mp
-  simp_rw [ENNReal.ofReal_pow (C.diskAtRadius_nonneg _ _ _)]
-  rw [C.centeredDiskLaw_moment hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ n]
-  exact ENNReal.ofReal_ne_top
-
-include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
-lemma centeredDiskLaw_integral_pow (ℓ : ℝ) (hℓ : 0≤ℓ) (n : ℕ) :
-    (∫ p : ℝ × Ω, C.diskAtRadius p.1 (4*ℓ/κ) p.2 ^ n
-      ∂centeredDiskLaw P κ B hc (4*ℓ/κ)) = Target.finiteMoment κ W n ℓ := by
-  obtain ⟨M,_,hM⟩ := C.uniform_finiteMoment_bound hB hc h0 hκ hκ8 F hw hgrowth W hW n
-  apply (ENNReal.ofReal_eq_ofReal_iff
-    (integral_nonneg (fun p => pow_nonneg (C.diskAtRadius_nonneg _ _ _) n))
-    (hM ℓ hℓ).1).mp
-  rw [ofReal_integral_eq_lintegral_ofReal
-    (C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ n)
-    (Eventually.of_forall (fun p => pow_nonneg (C.diskAtRadius_nonneg _ _ _) n))]
-  simp_rw [ENNReal.ofReal_pow (C.diskAtRadius_nonneg _ _ _)]
-  exact C.centeredDiskLaw_moment hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ n
-
-def limitingDiskTail (hc : ∀ ω, Continuous (fun t => B t ω)) (b : ℝ) : ℝ≥0∞ :=
-  ⨆ j : ℕ, C.centeredDiskTail hc b (4*(j:ℝ)/κ)
-
-include C hB hc h0 hκ hκ8 F hw in
-lemma centeredDiskTail_nat_mono (b : ℝ) :
-    Monotone (fun j : ℕ => C.centeredDiskTail hc b (4*(j:ℝ)/κ)) := by
-  intro i j hij
-  apply C.centeredDiskTail_mono hB hc h0 hκ hκ8 F hw b
-  · exact div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg i)) hκ.le
-  · exact div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le
-  · exact div_le_div_of_nonneg_right
-      (mul_le_mul_of_nonneg_left (Nat.cast_le.mpr hij) (by norm_num)) hκ.le
-
-include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
-lemma limitingDiskTail_batch_bound {n : ℕ} (hn : 1≤n)
-    (hm : 0<Target.moment κ W n) :
-    (∑' k : ℤ, (Target.multiplicity κ W n k : ℝ≥0∞) *
-      C.limitingDiskTail hc (16*MomentBatches.dyadic k)) ≤
-        ENNReal.ofReal ((4:ℝ)^n/16^n/(1-(1/2:ℝ)^n)) := by
-  let Q := fun j : ℕ => centeredDiskLaw P κ B hc (4*(j:ℝ)/κ)
-  let X := fun j : ℕ => fun p : ℝ × Ω => C.diskAtRadius p.1 (4*(j:ℝ)/κ) p.2
-  have hQ (j) : IsFiniteMeasure (Q j) := by
-    let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw (4*(j:ℝ)/κ)
-      (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
-    infer_instance
-  have hX (j) : Measurable (X j) := C.diskAtRadius_joint_measurable hc F hw _
-    (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
-  simp_rw [← MomentBatches.copies_eq_target]
-  exact MomentBatches.limiting_upper_batch_bound Q hQ X hX
-    (fun _ p => C.diskAtRadius_nonneg _ _ _)
-    (fun b => C.centeredDiskTail_nat_mono hB hc h0 hκ hκ8 F hw b) hn hm
-    (fun j => C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW
-      j (Nat.cast_nonneg j) n)
-    (fun j => (C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW
-      j (Nat.cast_nonneg j) n).trans_le
-      (C.finiteMoment_nat_le_moment hB hc h0 hκ hκ8 F hw hgrowth W hW n j))
-
-include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
-lemma limitingDiskTail_cleanup_bound :
-    (∑' n : ℕ, C.limitingDiskTail hc (16*((n:ℝ)+1))) ≤
-      ENNReal.ofReal (Target.moment κ W 1/16) := by
-  let Q := fun j : ℕ => centeredDiskLaw P κ B hc (4*(j:ℝ)/κ)
-  let X := fun j : ℕ => fun p : ℝ × Ω => C.diskAtRadius p.1 (4*(j:ℝ)/κ) p.2
-  have hQ (j) : IsFiniteMeasure (Q j) := by
-    let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw (4*(j:ℝ)/κ)
-      (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
-    infer_instance
-  have hX (j) : Measurable (X j) := C.diskAtRadius_joint_measurable hc F hw _
-    (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
-  apply MomentBatches.limiting_cleanup_tail_bound Q hQ X hX
-    (fun _ p => C.diskAtRadius_nonneg _ _ _)
-    (fun b => C.centeredDiskTail_nat_mono hB hc h0 hκ hκ8 F hw b)
-  · intro j
-    simpa only [pow_one] using C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw
-      hgrowth W hW j (Nat.cast_nonneg j) 1
-  · intro j
-    simpa only [pow_one] using
-      (C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW
-        j (Nat.cast_nonneg j) 1).trans_le
-        (C.finiteMoment_nat_le_moment hB hc h0 hκ hκ8 F hw hgrowth W hW 1 j)
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CanonicalDiskTails
+import OAI.Probability.SLE.Geometry.FiniteLawTailENN
+
+namespace OAI
+
+/-! Summable tails for the exact prescribed moments, obtained from the actual
+finite centered laws. The supremum is taken over finite radius levels. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+  (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+  (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+
+include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
+lemma finiteMoment_nat_le_moment (n j : ℕ) :
+    Target.finiteMoment κ W n j ≤ Target.moment κ W n := by
+  apply ge_of_tendsto (C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW n)
+  filter_upwards [eventually_ge_atTop j] with k hk
+  exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW n
+    (by change (0:ℝ)≤(j:ℝ); exact Nat.cast_nonneg j)
+    (by change (0:ℝ)≤(k:ℝ); exact Nat.cast_nonneg k) (Nat.cast_le.mpr hk)
+
+include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
+lemma centeredDiskLaw_integrable_pow (ℓ : ℝ) (hℓ : 0≤ℓ) (n : ℕ) :
+    Integrable (fun p : ℝ × Ω => C.diskAtRadius p.1 (4*ℓ/κ) p.2 ^ n)
+      (centeredDiskLaw P κ B hc (4*ℓ/κ)) := by
+  have hu : 0≤4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
+  have hm := (C.diskAtRadius_joint_measurable hc F hw _ hu).pow_const n
+  have hnonneg : ∀ p : ℝ × Ω, 0≤C.diskAtRadius p.1 (4*ℓ/κ) p.2^n :=
+    fun p => pow_nonneg (C.diskAtRadius_nonneg _ _ _) n
+  apply (lintegral_ofReal_ne_top_iff_integrable hm.aestronglyMeasurable
+    (Eventually.of_forall hnonneg)).mp
+  simp_rw [ENNReal.ofReal_pow (C.diskAtRadius_nonneg _ _ _)]
+  rw [C.centeredDiskLaw_moment hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ n]
+  exact ENNReal.ofReal_ne_top
+
+include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
+lemma centeredDiskLaw_integral_pow (ℓ : ℝ) (hℓ : 0≤ℓ) (n : ℕ) :
+    (∫ p : ℝ × Ω, C.diskAtRadius p.1 (4*ℓ/κ) p.2 ^ n
+      ∂centeredDiskLaw P κ B hc (4*ℓ/κ)) = Target.finiteMoment κ W n ℓ := by
+  obtain ⟨M,_,hM⟩ := C.uniform_finiteMoment_bound hB hc h0 hκ hκ8 F hw hgrowth W hW n
+  apply (ENNReal.ofReal_eq_ofReal_iff
+    (integral_nonneg (fun p => pow_nonneg (C.diskAtRadius_nonneg _ _ _) n))
+    (hM ℓ hℓ).1).mp
+  rw [ofReal_integral_eq_lintegral_ofReal
+    (C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ n)
+    (Eventually.of_forall (fun p => pow_nonneg (C.diskAtRadius_nonneg _ _ _) n))]
+  simp_rw [ENNReal.ofReal_pow (C.diskAtRadius_nonneg _ _ _)]
+  exact C.centeredDiskLaw_moment hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ n
+
+def limitingDiskTail (hc : ∀ ω, Continuous (fun t => B t ω)) (b : ℝ) : ℝ≥0∞ :=
+  ⨆ j : ℕ, C.centeredDiskTail hc b (4*(j:ℝ)/κ)
+
+include C hB hc h0 hκ hκ8 F hw in
+lemma centeredDiskTail_nat_mono (b : ℝ) :
+    Monotone (fun j : ℕ => C.centeredDiskTail hc b (4*(j:ℝ)/κ)) := by
+  intro i j hij
+  apply C.centeredDiskTail_mono hB hc h0 hκ hκ8 F hw b
+  · exact div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg i)) hκ.le
+  · exact div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le
+  · exact div_le_div_of_nonneg_right
+      (mul_le_mul_of_nonneg_left (Nat.cast_le.mpr hij) (by norm_num)) hκ.le
+
+include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
+lemma limitingDiskTail_batch_bound {n : ℕ} (hn : 1≤n)
+    (hm : 0<Target.moment κ W n) :
+    (∑' k : ℤ, (Target.multiplicity κ W n k : ℝ≥0∞) *
+      C.limitingDiskTail hc (16*MomentBatches.dyadic k)) ≤
+        ENNReal.ofReal ((4:ℝ)^n/16^n/(1-(1/2:ℝ)^n)) := by
+  let Q := fun j : ℕ => centeredDiskLaw P κ B hc (4*(j:ℝ)/κ)
+  let X := fun j : ℕ => fun p : ℝ × Ω => C.diskAtRadius p.1 (4*(j:ℝ)/κ) p.2
+  have hQ (j) : IsFiniteMeasure (Q j) := by
+    let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw (4*(j:ℝ)/κ)
+      (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
+    infer_instance
+  have hX (j) : Measurable (X j) := C.diskAtRadius_joint_measurable hc F hw _
+    (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
+  simp_rw [← MomentBatches.copies_eq_target]
+  exact MomentBatches.limiting_upper_batch_bound Q hQ X hX
+    (fun _ p => C.diskAtRadius_nonneg _ _ _)
+    (fun b => C.centeredDiskTail_nat_mono hB hc h0 hκ hκ8 F hw b) hn hm
+    (fun j => C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw hgrowth W hW
+      j (Nat.cast_nonneg j) n)
+    (fun j => (C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW
+      j (Nat.cast_nonneg j) n).trans_le
+      (C.finiteMoment_nat_le_moment hB hc h0 hκ hκ8 F hw hgrowth W hW n j))
+
+include C hB hc h0 hκ hκ8 F hw hgrowth W hW in
+lemma limitingDiskTail_cleanup_bound :
+    (∑' n : ℕ, C.limitingDiskTail hc (16*((n:ℝ)+1))) ≤
+      ENNReal.ofReal (Target.moment κ W 1/16) := by
+  let Q := fun j : ℕ => centeredDiskLaw P κ B hc (4*(j:ℝ)/κ)
+  let X := fun j : ℕ => fun p : ℝ × Ω => C.diskAtRadius p.1 (4*(j:ℝ)/κ) p.2
+  have hQ (j) : IsFiniteMeasure (Q j) := by
+    let := centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw (4*(j:ℝ)/κ)
+      (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
+    infer_instance
+  have hX (j) : Measurable (X j) := C.diskAtRadius_joint_measurable hc F hw _
+    (div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg j)) hκ.le)
+  apply MomentBatches.limiting_cleanup_tail_bound Q hQ X hX
+    (fun _ p => C.diskAtRadius_nonneg _ _ _)
+    (fun b => C.centeredDiskTail_nat_mono hB hc h0 hκ hκ8 F hw b)
+  · intro j
+    simpa only [pow_one] using C.centeredDiskLaw_integrable_pow hB hc h0 hκ hκ8 F hw
+      hgrowth W hW j (Nat.cast_nonneg j) 1
+  · intro j
+    simpa only [pow_one] using
+      (C.centeredDiskLaw_integral_pow hB hc h0 hκ hκ8 F hw hgrowth W hW
+        j (Nat.cast_nonneg j) 1).trans_le
+        (C.finiteMoment_nat_le_moment hB hc h0 hκ hκ8 F hw hgrowth W hW 1 j)
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalGaugeTails.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalGaugeTails.lean
index 55aa758bbb560ddfb8de131a481d7da7bbea18fa..cbaf00b6e618c55f9e19409e5695a580e897055b 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalGaugeTails.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalGaugeTails.lean
@@ -1,59 +1,59 @@
-import OAI.Probability.SLE.Gauge.CanonicalDiskMomentTails
-import OAI.Probability.SLE.Gauge.GaugeTailSum
-
-namespace OAI
-
-/-! All prescribed gauge entries have a summable limiting centered-disk tail,
-including repeated dyadic entries and the cleanup sequence. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-  (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-  (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-  (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-
-include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
-
-lemma limitingDiskTail_all_batches_bound :
-    (∑' e : Target.GaugeIndex κ W, C.limitingDiskTail hc (16*Target.weights κ W e)) ≤
-      ENNReal.ofReal (Target.moment κ W 1/16) + ENNReal.ofReal (2/3:ℝ) := by
-  rw [Target.gauge_tail_sum_eq]
-  apply add_le_add (C.limitingDiskTail_cleanup_bound hB hc h0 hκ hκ8 F hw hgrowth W hW)
-  calc
-    _ ≤ ∑' n : ℕ, ENNReal.ofReal (2*(1/4:ℝ)^(n+1)) := by
-      apply ENNReal.tsum_le_tsum
-      intro n
-      exact (C.limitingDiskTail_batch_bound hB hc h0 hκ hκ8 F hw hgrowth W hW
-        (by omega) (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW (n+1))).trans
-        (ENNReal.ofReal_le_ofReal (MomentBatches.upper_constant_le_geometric (by omega)))
-    _ = ENNReal.ofReal (2/3:ℝ) := by
-      simp_rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2),
-        ENNReal.ofReal_pow (by norm_num : (0:ℝ)≤1/4)]
-      rw [ENNReal.tsum_mul_left,ENNReal.tsum_geometric_add_one]
-      rw [← ENNReal.ofReal_one,
-        ← ENNReal.ofReal_sub 1 (by norm_num : (0:ℝ)≤1/4),
-        ← ENNReal.ofReal_inv_of_pos (by norm_num : (0:ℝ)<1-1/4),
-        ← ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤1/4),
-        ← ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2)]
-      norm_num
-
-theorem limitingDiskTail_summable :
-    (∑' e : Target.GaugeIndex κ W, C.limitingDiskTail hc (16*Target.weights κ W e)) ≠ ∞ := by
-  exact ne_top_of_le_ne_top (by finiteness)
-    (C.limitingDiskTail_all_batches_bound hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CanonicalDiskMomentTails
+import OAI.Probability.SLE.Gauge.GaugeTailSum
+
+namespace OAI
+
+/-! All prescribed gauge entries have a summable limiting centered-disk tail,
+including repeated dyadic entries and the cleanup sequence. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+  (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+  (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+  (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+
+include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
+
+lemma limitingDiskTail_all_batches_bound :
+    (∑' e : Target.GaugeIndex κ W, C.limitingDiskTail hc (16*Target.weights κ W e)) ≤
+      ENNReal.ofReal (Target.moment κ W 1/16) + ENNReal.ofReal (2/3:ℝ) := by
+  rw [Target.gauge_tail_sum_eq]
+  apply add_le_add (C.limitingDiskTail_cleanup_bound hB hc h0 hκ hκ8 F hw hgrowth W hW)
+  calc
+    _ ≤ ∑' n : ℕ, ENNReal.ofReal (2*(1/4:ℝ)^(n+1)) := by
+      apply ENNReal.tsum_le_tsum
+      intro n
+      exact (C.limitingDiskTail_batch_bound hB hc h0 hκ hκ8 F hw hgrowth W hW
+        (by omega) (C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW (n+1))).trans
+        (ENNReal.ofReal_le_ofReal (MomentBatches.upper_constant_le_geometric (by omega)))
+    _ = ENNReal.ofReal (2/3:ℝ) := by
+      simp_rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2),
+        ENNReal.ofReal_pow (by norm_num : (0:ℝ)≤1/4)]
+      rw [ENNReal.tsum_mul_left,ENNReal.tsum_geometric_add_one]
+      rw [← ENNReal.ofReal_one,
+        ← ENNReal.ofReal_sub 1 (by norm_num : (0:ℝ)≤1/4),
+        ← ENNReal.ofReal_inv_of_pos (by norm_num : (0:ℝ)<1-1/4),
+        ← ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤1/4),
+        ← ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2)]
+      norm_num
+
+theorem limitingDiskTail_summable :
+    (∑' e : Target.GaugeIndex κ W, C.limitingDiskTail hc (16*Target.weights κ W e)) ≠ ∞ := by
+  exact ne_top_of_le_ne_top (by finiteness)
+    (C.limitingDiskTail_all_batches_bound hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalGridLower.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalGridLower.lean
index 92659e380031329ab96599f477c6b30ddcab1a26..6b4aec48953d4d5ce9855cfb607e0aae26624477 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalGridLower.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalGridLower.lean
@@ -1,113 +1,113 @@
-import OAI.Probability.SLE.Mass.CompactSpatialMass
-import OAI.Probability.SLE.Geometry.CountablePastBound
-import OAI.Probability.SLE.Gauge.ExpectedBadGrid
-import OAI.Probability.SLE.Gauge.ExactMomentGauge
-
-namespace OAI
-
-/-! The finite centered-law tests which suffice for the prescribed lower bound.
-The half-threshold and the factor two come from actual past-mass detection. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-attribute [local instance] Classical.propDecidable
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
-  (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0)
-  (hκ : 0 < κ) (hκ8 : κ < 8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-
-include C hB hc h0 hκ hκ8 F hw
-
-lemma compact_spatial_bad_charge_of_finite_tests
-    (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u)
-    (hlim : Tendsto u atTop atTop) (q : ℕ → ℝ≥0) (hq : DenseRange q)
-    (K : Set upperHalfPlane) (hK : IsCompact K) (z : ℂ) (r H : ℝ) (hH : 0 < H)
-    (A : ℝ≥0∞)
-    (hbound : ∀ j, ∀ᶠ n in atTop,
-      (∫⁻ x in (Subtype.val : upperHalfPlane → ℂ) ''
-          (K ∩ Subtype.val ⁻¹' closedBall z r),
-        ENNReal.ofReal (initialGreen κ x) * finiteCenteredLaw P κ B γ G x (u n)
-          (finiteStrictPastEvent ((Finset.range j).image q)
-            (fun t => {ω | H < (C.process t ω (K ∩ Subtype.val ⁻¹' closedBall z r)).toReal})
-            (fun ω => radiusLevelTime (γ ω) (G ω) x (u n))) ∂volume) ≤ A) :
-    (∫⁻ ω, if ENNReal.ofReal (2*H) <
-        compactSpatialMass (C.rational.terminal ω) K (closedBall z r) then
-      compactSpatialMass (C.rational.terminal ω) K (closedBall z r) else 0 ∂P) ≤ 2*A := by
-  have hb := C.bad_cell_mass_le_of_finite_bound hB hc h0 hκ hκ8 F hw
-    u hu hmono hlim q hq (K ∩ Subtype.val ⁻¹' closedBall z r)
-    (compactSpatialMass_compact_cell K hK z r) H hH A hbound
-  simpa only [compactSpatialMass_apply _ _ _ isClosed_closedBall.measurableSet,
-    Set.indicator,Set.mem_ofPred_eq] using hb
-
-/-- Finite centered-law bounds imply the prescribed lower-bound criterion.
-The gauge and its positive moments are the prescribed nested Wiener coefficients. -/
-theorem ae_positive_prescribed_of_finite_tests [IsProbabilityMeasure P]
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u)
-    (hlim : Tendsto u atTop atTop) (q : ℕ → ℝ≥0) (hq : DenseRange q)
-    (K : Set upperHalfPlane) (hK : IsCompact K) (D : ℝ) (hD : 0 < D)
-    (p : Target.GaugeIndex κ W → ℝ≥0∞) (hp : (∑' e, p e) ≠ ∞)
-    (hbound : ∀ e : Target.GaugeIndex κ W, ∀ i : ℤ×ℤ, ∀ j, ∀ᶠ n in atTop,
-      (∫⁻ x in (Subtype.val : upperHalfPlane → ℂ) ''
-          (K ∩ Subtype.val ⁻¹' closedBall (coverGridCenter (Target.radii κ W e) i)
-            (2*Target.radii κ W e)),
-        ENNReal.ofReal (initialGreen κ x) * finiteCenteredLaw P κ B γ G x (u n)
-          (finiteStrictPastEvent ((Finset.range j).image q)
-            (fun t => {ω | D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ <
-              (C.process t ω (K ∩ Subtype.val ⁻¹' closedBall
-                (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e))).toReal})
-            (fun ω => radiusLevelTime (γ ω) (G ω) x (u n))) ∂volume) ≤
-        p e * compactSpatialMass (initialGreenMeasure κ) K
-          (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e))) :
-    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E →
-      0 < C.rational.terminal ω (K ∩ Subtype.val ⁻¹' E) →
-      0 < Target.hausdorff (Target.gauge κ W) E := by
-  let μ (ω : Ω) := compactSpatialMass (C.rational.terminal ω) K
-  let η := compactSpatialMass (initialGreenMeasure κ) K
-  let : IsFiniteMeasure η := compact_spatial_initial_finite (κ := κ) K hK
-  have hsum : (∑' e : Target.GaugeIndex κ W, 2*p e) ≠ ∞ := by
-    rw [ENNReal.tsum_mul_left]
-    exact (ENNReal.mul_lt_top (by norm_num) (lt_top_iff_ne_top.mpr hp)).ne
-  have hcell (e : Target.GaugeIndex κ W) (i : ℤ×ℤ) :
-      (∫⁻ ω, if ENNReal.ofReal
-          ((2*D)*Target.weights κ W e*(Target.radii κ W e)^Target.d κ) <
-            μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) then
-          μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) else 0 ∂P) ≤
-        (2*p e)*η (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) := by
-    have hH : 0 < D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ :=
-      mul_pos (mul_pos hD (Target.weights_pos κ W e))
-        (Real.rpow_pos_of_pos (Target.radii_pos κ W e) _)
-    have hb := C.compact_spatial_bad_charge_of_finite_tests hB hc h0 hκ hκ8 F hw
-      u hu hmono hlim q hq K hK (coverGridCenter (Target.radii κ W e) i)
-      (2*Target.radii κ W e) _ hH _ (hbound e i)
-    have he : (2*D)*Target.weights κ W e*(Target.radii κ W e)^Target.d κ =
-        2*(D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ) := by ring
-    simpa only [he,mul_assoc] using hb
-  have ha := ae_positive_prescribed_hausdorff_of_cell_charges P μ
-    (C.compact_spatial_terminal_measurable K hK)
-    (C.compact_spatial_terminal_finite K hK) η κ W
-    (C.prescribed_gauge_admissible hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
-    (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
-    (fun n _ => C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
-    (2*D) (by positivity) (fun e => 2*p e) hsum (by
-      intro e i
-      convert hcell e i using 1
-      apply lintegral_congr
-      intro ω
-      split_ifs <;> rfl)
-  filter_upwards [ha] with ω hω E hE hpE
-  exact hω E hE (by simpa only [μ,compactSpatialMass_apply _ _ _ hE] using hpE)
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.CompactSpatialMass
+import OAI.Probability.SLE.Geometry.CountablePastBound
+import OAI.Probability.SLE.Gauge.ExpectedBadGrid
+import OAI.Probability.SLE.Gauge.ExactMomentGauge
+
+namespace OAI
+
+/-! The finite centered-law tests which suffice for the prescribed lower bound.
+The half-threshold and the factor two come from actual past-mass detection. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+attribute [local instance] Classical.propDecidable
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
+  (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω = 0)
+  (hκ : 0 < κ) (hκ8 : κ < 8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+
+include C hB hc h0 hκ hκ8 F hw
+
+lemma compact_spatial_bad_charge_of_finite_tests
+    (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u)
+    (hlim : Tendsto u atTop atTop) (q : ℕ → ℝ≥0) (hq : DenseRange q)
+    (K : Set upperHalfPlane) (hK : IsCompact K) (z : ℂ) (r H : ℝ) (hH : 0 < H)
+    (A : ℝ≥0∞)
+    (hbound : ∀ j, ∀ᶠ n in atTop,
+      (∫⁻ x in (Subtype.val : upperHalfPlane → ℂ) ''
+          (K ∩ Subtype.val ⁻¹' closedBall z r),
+        ENNReal.ofReal (initialGreen κ x) * finiteCenteredLaw P κ B γ G x (u n)
+          (finiteStrictPastEvent ((Finset.range j).image q)
+            (fun t => {ω | H < (C.process t ω (K ∩ Subtype.val ⁻¹' closedBall z r)).toReal})
+            (fun ω => radiusLevelTime (γ ω) (G ω) x (u n))) ∂volume) ≤ A) :
+    (∫⁻ ω, if ENNReal.ofReal (2*H) <
+        compactSpatialMass (C.rational.terminal ω) K (closedBall z r) then
+      compactSpatialMass (C.rational.terminal ω) K (closedBall z r) else 0 ∂P) ≤ 2*A := by
+  have hb := C.bad_cell_mass_le_of_finite_bound hB hc h0 hκ hκ8 F hw
+    u hu hmono hlim q hq (K ∩ Subtype.val ⁻¹' closedBall z r)
+    (compactSpatialMass_compact_cell K hK z r) H hH A hbound
+  simpa only [compactSpatialMass_apply _ _ _ isClosed_closedBall.measurableSet,
+    Set.indicator,Set.mem_ofPred_eq] using hb
+
+/-- Finite centered-law bounds imply the prescribed lower-bound criterion.
+The gauge and its positive moments are the prescribed nested Wiener coefficients. -/
+theorem ae_positive_prescribed_of_finite_tests [IsProbabilityMeasure P]
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u)
+    (hlim : Tendsto u atTop atTop) (q : ℕ → ℝ≥0) (hq : DenseRange q)
+    (K : Set upperHalfPlane) (hK : IsCompact K) (D : ℝ) (hD : 0 < D)
+    (p : Target.GaugeIndex κ W → ℝ≥0∞) (hp : (∑' e, p e) ≠ ∞)
+    (hbound : ∀ e : Target.GaugeIndex κ W, ∀ i : ℤ×ℤ, ∀ j, ∀ᶠ n in atTop,
+      (∫⁻ x in (Subtype.val : upperHalfPlane → ℂ) ''
+          (K ∩ Subtype.val ⁻¹' closedBall (coverGridCenter (Target.radii κ W e) i)
+            (2*Target.radii κ W e)),
+        ENNReal.ofReal (initialGreen κ x) * finiteCenteredLaw P κ B γ G x (u n)
+          (finiteStrictPastEvent ((Finset.range j).image q)
+            (fun t => {ω | D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ <
+              (C.process t ω (K ∩ Subtype.val ⁻¹' closedBall
+                (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e))).toReal})
+            (fun ω => radiusLevelTime (γ ω) (G ω) x (u n))) ∂volume) ≤
+        p e * compactSpatialMass (initialGreenMeasure κ) K
+          (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e))) :
+    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E →
+      0 < C.rational.terminal ω (K ∩ Subtype.val ⁻¹' E) →
+      0 < Target.hausdorff (Target.gauge κ W) E := by
+  let μ (ω : Ω) := compactSpatialMass (C.rational.terminal ω) K
+  let η := compactSpatialMass (initialGreenMeasure κ) K
+  let : IsFiniteMeasure η := compact_spatial_initial_finite (κ := κ) K hK
+  have hsum : (∑' e : Target.GaugeIndex κ W, 2*p e) ≠ ∞ := by
+    rw [ENNReal.tsum_mul_left]
+    exact (ENNReal.mul_lt_top (by norm_num) (lt_top_iff_ne_top.mpr hp)).ne
+  have hcell (e : Target.GaugeIndex κ W) (i : ℤ×ℤ) :
+      (∫⁻ ω, if ENNReal.ofReal
+          ((2*D)*Target.weights κ W e*(Target.radii κ W e)^Target.d κ) <
+            μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) then
+          μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) else 0 ∂P) ≤
+        (2*p e)*η (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) := by
+    have hH : 0 < D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ :=
+      mul_pos (mul_pos hD (Target.weights_pos κ W e))
+        (Real.rpow_pos_of_pos (Target.radii_pos κ W e) _)
+    have hb := C.compact_spatial_bad_charge_of_finite_tests hB hc h0 hκ hκ8 F hw
+      u hu hmono hlim q hq K hK (coverGridCenter (Target.radii κ W e) i)
+      (2*Target.radii κ W e) _ hH _ (hbound e i)
+    have he : (2*D)*Target.weights κ W e*(Target.radii κ W e)^Target.d κ =
+        2*(D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ) := by ring
+    simpa only [he,mul_assoc] using hb
+  have ha := ae_positive_prescribed_hausdorff_of_cell_charges P μ
+    (C.compact_spatial_terminal_measurable K hK)
+    (C.compact_spatial_terminal_finite K hK) η κ W
+    (C.prescribed_gauge_admissible hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
+    (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
+    (fun n _ => C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
+    (2*D) (by positivity) (fun e => 2*p e) hsum (by
+      intro e i
+      convert hcell e i using 1
+      apply lintegral_congr
+      intro ω
+      split_ifs <;> rfl)
+  filter_upwards [ha] with ω hω E hE hpE
+  exact hω E hE (by simpa only [μ,compactSpatialMass_apply _ _ _ hE] using hpE)
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalLateGridLower.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalLateGridLower.lean
index c500547ea66735803017499828a6d07cc3b2b33e..82a2429fd0b3f7cf9e0cd064241320139e63a7be 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalLateGridLower.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalLateGridLower.lean
@@ -1,99 +1,99 @@
-import OAI.Probability.SLE.Gauge.CanonicalGaugeTails
-import OAI.Probability.SLE.Geometry.FinitePastTrivialBound
-
-namespace OAI
-
-/-! The lower bound only needs the centered tail comparison in sufficiently
-late batches. Every earlier entry is controlled by its finite probability law. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-attribute [local instance] Classical.propDecidable
-
-namespace SLEExactGauge
-
-def prescribedCompactGridCell (κ : ℝ) (W : Target.WienerLaw) (K : Set upperHalfPlane)
-    (e : Target.GaugeIndex κ W) (i : ℤ×ℤ) : Set upperHalfPlane :=
-  K ∩ Subtype.val ⁻¹' closedBall (coverGridCenter (Target.radii κ W e) i)
-    (2*Target.radii κ W e)
-
-namespace Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-theorem ae_positive_prescribed_of_late_disk_tails
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (u : ℕ → ℝ) (hu : ∀ n, 0≤u n) (hmono : Monotone u) (hlim : Tendsto u atTop atTop)
-    (q : ℕ → ℝ≥0) (hq : DenseRange q) (K : Set upperHalfPlane) (hK : IsCompact K)
-    (D : ℝ) (hD : 0<D) (N : ℕ)
-    (hlate : ∀ e : Target.GaugeIndex κ W, N≤e.val.1 → ∀ (i : ℤ×ℤ) (j : ℕ),
-      ∀ᶠ n in atTop,
-      C.compactFinitePastTest (prescribedCompactGridCell κ W K e i)
-        (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ)
-        ((Finset.range j).image q) (u n) ≤
-      (2*C.limitingDiskTail hc (16*Target.weights κ W e)) *
-        initialGreenMeasure κ (prescribedCompactGridCell κ W K e i)) :
-    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E →
-      0 < C.rational.terminal ω (K ∩ Subtype.val ⁻¹' E) →
-      0 < Target.hausdorff (Target.gauge κ W) E := by
-  let p : Target.GaugeIndex κ W → ℝ≥0∞ := fun e =>
-    if e.val.1<N then 1 else 2*C.limitingDiskTail hc (16*Target.weights κ W e)
-  have hE : Set.Finite {e : Target.GaugeIndex κ W | e.val.1<N} := by
-    apply ((Target.early_entries_finite κ W
-      (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
-      (fun n _ => C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n) N).preimage
-      (f := fun e : Target.GaugeIndex κ W => e.val) Subtype.val_injective.injOn).subset
-    intro e he
-    exact ⟨e.property,he⟩
-  have hsE : (∑' e : Target.GaugeIndex κ W, if e.val.1<N then (1:ℝ≥0∞) else 0) ≠ ∞ := by
-    rw [tsum_eq_sum (s := hE.toFinset)]
-    · apply ne_of_lt
-      apply ENNReal.sum_lt_top.mpr
-      intro e he
-      split_ifs <;> norm_num
-    · intro e he
-      exact ite_eq_right (by simpa only [Set.Finite.mem_toFinset,Set.mem_ofPred_eq] using he)
-  have hsTail : (∑' e : Target.GaugeIndex κ W,
-      2*C.limitingDiskTail hc (16*Target.weights κ W e)) ≠ ∞ := by
-    rw [ENNReal.tsum_mul_left]
-    exact ENNReal.mul_ne_top (by norm_num)
-      (C.limitingDiskTail_summable hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
-  have hp : (∑' e, p e) ≠ ∞ := by
-    apply ne_top_of_le_ne_top _ (ENNReal.tsum_le_tsum (fun e =>
-      show p e ≤ (if e.val.1<N then (1:ℝ≥0∞) else 0) +
-        2*C.limitingDiskTail hc (16*Target.weights κ W e) by
-          dsimp only [p]; split_ifs <;> simp))
-    rw [ENNReal.tsum_add]
-    exact ENNReal.add_ne_top.mpr ⟨hsE,hsTail⟩
-  apply C.ae_positive_prescribed_of_finite_tests hB hc h0 hκ hκ8 F hw hpublished hgrowth
-    W hW u hu hmono hlim q hq K hK D hD p hp
-  intro e i j
-  rw [compactSpatialMass_apply _ _ _ isClosed_closedBall.measurableSet]
-  change ∀ᶠ n in atTop, C.compactFinitePastTest (prescribedCompactGridCell κ W K e i)
-    (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ)
-    ((Finset.range j).image q) (u n) ≤ p e *
-      initialGreenMeasure κ (prescribedCompactGridCell κ W K e i)
-  by_cases he : e.val.1<N
-  · apply Eventually.of_forall
-    intro n
-    have hcell : IsCompact (prescribedCompactGridCell κ W K e i) :=
-      compactSpatialMass_compact_cell K hK _ _
-    simpa only [p,he,ite_true,one_mul] using
-      C.compactFinitePastTest_le_initial hB hκ hκ8 F hw
-        (prescribedCompactGridCell κ W K e i) hcell _ ((Finset.range j).image q) (u n) (hu n)
-  · simpa only [p,he,ite_false] using hlate e (le_of_not_gt he) i j
-
-end Multipoint.CanonicalMass
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CanonicalGaugeTails
+import OAI.Probability.SLE.Geometry.FinitePastTrivialBound
+
+namespace OAI
+
+/-! The lower bound only needs the centered tail comparison in sufficiently
+late batches. Every earlier entry is controlled by its finite probability law. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+attribute [local instance] Classical.propDecidable
+
+namespace SLEExactGauge
+
+def prescribedCompactGridCell (κ : ℝ) (W : Target.WienerLaw) (K : Set upperHalfPlane)
+    (e : Target.GaugeIndex κ W) (i : ℤ×ℤ) : Set upperHalfPlane :=
+  K ∩ Subtype.val ⁻¹' closedBall (coverGridCenter (Target.radii κ W e) i)
+    (2*Target.radii κ W e)
+
+namespace Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+theorem ae_positive_prescribed_of_late_disk_tails
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (u : ℕ → ℝ) (hu : ∀ n, 0≤u n) (hmono : Monotone u) (hlim : Tendsto u atTop atTop)
+    (q : ℕ → ℝ≥0) (hq : DenseRange q) (K : Set upperHalfPlane) (hK : IsCompact K)
+    (D : ℝ) (hD : 0<D) (N : ℕ)
+    (hlate : ∀ e : Target.GaugeIndex κ W, N≤e.val.1 → ∀ (i : ℤ×ℤ) (j : ℕ),
+      ∀ᶠ n in atTop,
+      C.compactFinitePastTest (prescribedCompactGridCell κ W K e i)
+        (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ)
+        ((Finset.range j).image q) (u n) ≤
+      (2*C.limitingDiskTail hc (16*Target.weights κ W e)) *
+        initialGreenMeasure κ (prescribedCompactGridCell κ W K e i)) :
+    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E →
+      0 < C.rational.terminal ω (K ∩ Subtype.val ⁻¹' E) →
+      0 < Target.hausdorff (Target.gauge κ W) E := by
+  let p : Target.GaugeIndex κ W → ℝ≥0∞ := fun e =>
+    if e.val.1<N then 1 else 2*C.limitingDiskTail hc (16*Target.weights κ W e)
+  have hE : Set.Finite {e : Target.GaugeIndex κ W | e.val.1<N} := by
+    apply ((Target.early_entries_finite κ W
+      (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
+      (fun n _ => C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n) N).preimage
+      (f := fun e : Target.GaugeIndex κ W => e.val) Subtype.val_injective.injOn).subset
+    intro e he
+    exact ⟨e.property,he⟩
+  have hsE : (∑' e : Target.GaugeIndex κ W, if e.val.1<N then (1:ℝ≥0∞) else 0) ≠ ∞ := by
+    rw [tsum_eq_sum (s := hE.toFinset)]
+    · apply ne_of_lt
+      apply ENNReal.sum_lt_top.mpr
+      intro e he
+      split_ifs <;> norm_num
+    · intro e he
+      exact ite_eq_right (by simpa only [Set.Finite.mem_toFinset,Set.mem_ofPred_eq] using he)
+  have hsTail : (∑' e : Target.GaugeIndex κ W,
+      2*C.limitingDiskTail hc (16*Target.weights κ W e)) ≠ ∞ := by
+    rw [ENNReal.tsum_mul_left]
+    exact ENNReal.mul_ne_top (by norm_num)
+      (C.limitingDiskTail_summable hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
+  have hp : (∑' e, p e) ≠ ∞ := by
+    apply ne_top_of_le_ne_top _ (ENNReal.tsum_le_tsum (fun e =>
+      show p e ≤ (if e.val.1<N then (1:ℝ≥0∞) else 0) +
+        2*C.limitingDiskTail hc (16*Target.weights κ W e) by
+          dsimp only [p]; split_ifs <;> simp))
+    rw [ENNReal.tsum_add]
+    exact ENNReal.add_ne_top.mpr ⟨hsE,hsTail⟩
+  apply C.ae_positive_prescribed_of_finite_tests hB hc h0 hκ hκ8 F hw hpublished hgrowth
+    W hW u hu hmono hlim q hq K hK D hD p hp
+  intro e i j
+  rw [compactSpatialMass_apply _ _ _ isClosed_closedBall.measurableSet]
+  change ∀ᶠ n in atTop, C.compactFinitePastTest (prescribedCompactGridCell κ W K e i)
+    (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ)
+    ((Finset.range j).image q) (u n) ≤ p e *
+      initialGreenMeasure κ (prescribedCompactGridCell κ W K e i)
+  by_cases he : e.val.1<N
+  · apply Eventually.of_forall
+    intro n
+    have hcell : IsCompact (prescribedCompactGridCell κ W K e i) :=
+      compactSpatialMass_compact_cell K hK _ _
+    simpa only [p,he,ite_true,one_mul] using
+      C.compactFinitePastTest_le_initial hB hκ hκ8 F hw
+        (prescribedCompactGridCell κ W K e i) hcell _ ((Finset.range j).image q) (u n) (hu n)
+  · simpa only [p,he,ite_false] using hlate e (le_of_not_gt he) i j
+
+end Multipoint.CanonicalMass
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CanonicalPrescribedUpper.lean b/lean/OAI/Probability/SLE/Gauge/CanonicalPrescribedUpper.lean
index 39c6081c34856a7cc5eda71114a2959e747c8639..eea5cd98ebbb2fc3697a01fd3b417f929c65e578 100644
--- a/lean/OAI/Probability/SLE/Gauge/CanonicalPrescribedUpper.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CanonicalPrescribedUpper.lean
@@ -1,59 +1,59 @@
-import OAI.Probability.SLE.Geometry.CanonicalInteriorBound
-import OAI.Probability.SLE.Gauge.PrescribedInteriorCost
-import OAI.Probability.SLE.Gauge.CanonicalCoverExpectation
-import OAI.Probability.SLE.Gauge.ExactMomentGauge
-
-namespace OAI
-
-/-! Concrete finite-law canonical test failures imply the prescribed-gauge
-upper bound through boundary estimates, covers, and expectations. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-local instance prescribedUpperPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance prescribedUpperPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (M : CanonicalMass P κ B hBm γ G)
-
-theorem prescribed_expectation_of_finite_failures
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hγ : ∀ t, Measurable (fun ω => γ ω t))
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
-    (he : ∀ n i, (e n i).2 < Target.multiplicity κ W n (e n i).1)
-    (C c R : ℝ) (hC : 3 ≤ C) (hcpos : 0<c) (hR : 0 ≤ R)
-    (K : Set ℂ) (hK : IsClosed K) (hKR : K ⊆ closedBall 0 R) (hH : ∀ z∈K, 0 ≤ z.im)
-    (hfailure : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
-      Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im →
-      M.prescribedCenteredFailure W n (e n) C c
-        (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij) ≤
-        ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16)))) :
-    (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (γ ω)∩K) ∂P) < ∞ := by
-  let f := sourceTracePath γ (fun ω => (hw ω).1)
-  have hf : Measurable f := source_path_measurable f hγ
-  have hm1 := M.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
-  have hJm (n : ℕ) (_ : 1≤n) : Target.moment κ W 1^n ≤ Target.moment κ W n :=
-    M.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
-  have hI := M.eventually_prescribed_interior_bound hB hc h0 hκ hκ8 F hw hpublished W
-    hm1 hJm J e he C c R K hfailure
-  have hcost := prescribed_missed_cost_tendsto_zero_of_interior P f hf hB hκ hκ8 G F
-    (Eventually.of_forall hw) M.rational.physicalTerminal M.rational.physicalTerminal_measurable W
-    J e C c R hR K hK _ hI
-  exact M.rational.prescribed_cover_finite_expectation hκ hκ8 W
-    (M.prescribed_gauge_admissible hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW) hm1 hJm
-    f hf C c R hC hcpos hR K hK hKR hH (fun n => J (n+1)) (fun n => e (n+1))
-    (fun n i => he (n+1) i) (hcost.comp (tendsto_add_atTop_nat 1))
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CanonicalInteriorBound
+import OAI.Probability.SLE.Gauge.PrescribedInteriorCost
+import OAI.Probability.SLE.Gauge.CanonicalCoverExpectation
+import OAI.Probability.SLE.Gauge.ExactMomentGauge
+
+namespace OAI
+
+/-! Concrete finite-law canonical test failures imply the prescribed-gauge
+upper bound through boundary estimates, covers, and expectations. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+local instance prescribedUpperPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance prescribedUpperPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (M : CanonicalMass P κ B hBm γ G)
+
+theorem prescribed_expectation_of_finite_failures
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hγ : ∀ t, Measurable (fun ω => γ ω t))
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
+    (he : ∀ n i, (e n i).2 < Target.multiplicity κ W n (e n i).1)
+    (C c R : ℝ) (hC : 3 ≤ C) (hcpos : 0<c) (hR : 0 ≤ R)
+    (K : Set ℂ) (hK : IsClosed K) (hKR : K ⊆ closedBall 0 R) (hH : ∀ z∈K, 0 ≤ z.im)
+    (hfailure : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
+      Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im →
+      M.prescribedCenteredFailure W n (e n) C c
+        (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij) ≤
+        ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16)))) :
+    (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (γ ω)∩K) ∂P) < ∞ := by
+  let f := sourceTracePath γ (fun ω => (hw ω).1)
+  have hf : Measurable f := source_path_measurable f hγ
+  have hm1 := M.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
+  have hJm (n : ℕ) (_ : 1≤n) : Target.moment κ W 1^n ≤ Target.moment κ W n :=
+    M.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
+  have hI := M.eventually_prescribed_interior_bound hB hc h0 hκ hκ8 F hw hpublished W
+    hm1 hJm J e he C c R K hfailure
+  have hcost := prescribed_missed_cost_tendsto_zero_of_interior P f hf hB hκ hκ8 G F
+    (Eventually.of_forall hw) M.rational.physicalTerminal M.rational.physicalTerminal_measurable W
+    J e C c R hR K hK _ hI
+  exact M.rational.prescribed_cover_finite_expectation hκ hκ8 W
+    (M.prescribed_gauge_admissible hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW) hm1 hJm
+    f hf C c R hC hcpos hR K hK hKR hH (fun n => J (n+1)) (fun n => e (n+1))
+    (fun n i => he (n+1) i) (hcost.comp (tendsto_add_atTop_nat 1))
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CenteredMomentJensen.lean b/lean/OAI/Probability/SLE/Gauge/CenteredMomentJensen.lean
index 909db79d0f44a85b8ef852e68b6230fbb1480791..c17193eb4338fcc503a1bf6a12a0486be0cf1fe4 100644
--- a/lean/OAI/Probability/SLE/Gauge/CenteredMomentJensen.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CenteredMomentJensen.lean
@@ -1,141 +1,141 @@
-import OAI.Probability.SLE.Gauge.FiniteMomentMonotonicity
-import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular
-
-namespace OAI
-
-/-! Jensen inequalities for the exact stopped coefficients and their
-prescribed limit. No law after the centered terminal time is used. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-lemma centered_diskScalar_integrable_eq
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ)
-    (σ : Ω → WithTop ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B
-      (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ)
-    (heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ)) :
-    let Q := finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ)
-    let X := finiteStoppedScalar (C.diskScalar θ hθ) σ
-    Integrable (fun ω => X ω^m) Q ∧
-      Target.levelCoefficient κ W ℓ m θ=(∫ ω, X ω^m ∂Q) := by
-  dsimp only
-  let Q := finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ)
-  let X := finiteStoppedScalar (C.diskScalar θ hθ) σ
-  have hX : Measurable X := (finiteStoppedScalar_measurable _ _ (C.diskScalar_progressive θ hθ) σ hσ).mono
-    hσ.measurableSpace_le le_rfl
-  have hX0 (ω) : 0≤X ω := by
-    dsimp only [X,finiteStoppedScalar]
-    split_ifs
-    · exact le_rfl
-    · exact compactWeightedMass_nonneg _ _ _
-  have he := C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m σ hσ heσ
-  have hp (ω) : ENNReal.ofReal (X ω^m)=ENNReal.ofReal (X ω)^m := ENNReal.ofReal_pow (hX0 ω) m
-  have hi : Integrable (fun ω => X ω^m) Q := by
-    apply (lintegral_ofReal_ne_top_iff_integrable (hX.pow_const m).aestronglyMeasurable
-      (Eventually.of_forall (fun ω => pow_nonneg (hX0 ω) m))).mp
-    simp_rw [hp]
-    rw [← he]
-    exact ENNReal.ofReal_ne_top
-  refine ⟨hi,?_⟩
-  have he' : ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ)=
-      ENNReal.ofReal (∫ ω, X ω^m ∂Q) := by
-    rw [ofReal_integral_eq_lintegral_ofReal hi
-      (Eventually.of_forall (fun ω => pow_nonneg (hX0 ω) m))]
-    simpa only [hp] using he
-  have hc0 := (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).1 θ hθ
-  have hi0 : 0≤∫ ω, X ω^m ∂Q := integral_nonneg (fun ω => pow_nonneg (hX0 ω) m)
-  simpa only [ENNReal.toReal_ofReal hc0,ENNReal.toReal_ofReal hi0] using congrArg ENNReal.toReal he'
-
-include C in
-lemma levelCoefficient_jensen
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
-    (Target.levelCoefficient κ W ℓ 1 θ)^m≤Target.levelCoefficient κ W ℓ m θ := by
-  let σ := fun ω => (radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ) : WithTop ℝ≥0)
-  have hz := Target.diskMapInv_maps hθ (show (0:ℂ)∈Metric.ball 0 1 by simp)
-  have hu : 0≤4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
-  have hs := actual_radiusLevelTime_stopping hB
-    (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)))
-    hBm hc h0 κ hκ hκ8 γ G F hw (Target.diskMapInv θ 0) hz (4*ℓ/κ) hu
-  have heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ) := Eventually.of_forall (fun _ => rfl)
-  let Q := finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ)
-  let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F
-    (Eventually.of_forall hw) _ hz _ hu
-  obtain ⟨hi1,he1⟩ := C.centered_diskScalar_integrable_eq hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ 1 σ hs heσ
-  obtain ⟨him,hem⟩ := C.centered_diskScalar_integrable_eq hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m σ hs heσ
-  rw [he1,hem]
-  simp only [pow_one] at hi1 ⊢
-  refine (convexOn_pow (𝕜:=ℝ) m).map_integral_le (μ:=Q)
-    (continuous_pow m).continuousOn isClosed_Ici ?_ hi1 him
-  filter_upwards [] with ω
-  change 0 ≤ finiteStoppedScalar (C.diskScalar θ hθ) σ ω
-  dsimp only [finiteStoppedScalar]
-  split_ifs
-  · exact le_rfl
-  · exact compactWeightedMass_nonneg _ _ _
-
-include C in
-theorem finiteMoment_jensen
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
-    (Target.finiteMoment κ W 1 ℓ)^m≤Target.finiteMoment κ W m ℓ := by
-  let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
-  obtain ⟨hn1,hi1⟩ := C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1
-  obtain ⟨hnm,him⟩ := C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m
-  have hj : ∀ᵐ θ ∂Target.angleLaw κ,
-      (Target.levelCoefficient κ W ℓ 1 θ)^m≤Target.levelCoefficient κ W ℓ m θ := by
-    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-    exact C.levelCoefficient_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
-  have hp : Integrable (fun θ => (Target.levelCoefficient κ W ℓ 1 θ)^m) (Target.angleLaw κ) := by
-    apply him.mono' (hi1.aestronglyMeasurable.pow m)
-    filter_upwards [hj,Target.angleLaw_ae_mem κ] with θ hθ hθI
-    change ‖(Target.levelCoefficient κ W ℓ 1 θ)^m‖ ≤ _
-    rw [Real.norm_of_nonneg (pow_nonneg (hn1 θ hθI) m)]
-    exact hθ
-  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1,
-    C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m]
-  apply le_trans ((convexOn_pow (𝕜:=ℝ) m).map_integral_le (continuous_pow m).continuousOn isClosed_Ici
-    (by filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ; exact hn1 θ hθ) hi1 hp)
-  exact integral_mono_ae hp him hj
-
-include C in
-theorem moment_jensen
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (m : ℕ) : (Target.moment κ W 1)^m≤Target.moment κ W m := by
-  exact le_of_tendsto_of_tendsto
-    ((C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW 1).pow m)
-    (C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW m)
-    (Eventually.of_forall (fun ℓ => C.finiteMoment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW
-      ℓ (Nat.cast_nonneg ℓ) m))
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.FiniteMomentMonotonicity
+import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular
+
+namespace OAI
+
+/-! Jensen inequalities for the exact stopped coefficients and their
+prescribed limit. No law after the centered terminal time is used. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+lemma centered_diskScalar_integrable_eq
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ)
+    (σ : Ω → WithTop ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B
+      (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) σ)
+    (heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ)) :
+    let Q := finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ)
+    let X := finiteStoppedScalar (C.diskScalar θ hθ) σ
+    Integrable (fun ω => X ω^m) Q ∧
+      Target.levelCoefficient κ W ℓ m θ=(∫ ω, X ω^m ∂Q) := by
+  dsimp only
+  let Q := finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ)
+  let X := finiteStoppedScalar (C.diskScalar θ hθ) σ
+  have hX : Measurable X := (finiteStoppedScalar_measurable _ _ (C.diskScalar_progressive θ hθ) σ hσ).mono
+    hσ.measurableSpace_le le_rfl
+  have hX0 (ω) : 0≤X ω := by
+    dsimp only [X,finiteStoppedScalar]
+    split_ifs
+    · exact le_rfl
+    · exact compactWeightedMass_nonneg _ _ _
+  have he := C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m σ hσ heσ
+  have hp (ω) : ENNReal.ofReal (X ω^m)=ENNReal.ofReal (X ω)^m := ENNReal.ofReal_pow (hX0 ω) m
+  have hi : Integrable (fun ω => X ω^m) Q := by
+    apply (lintegral_ofReal_ne_top_iff_integrable (hX.pow_const m).aestronglyMeasurable
+      (Eventually.of_forall (fun ω => pow_nonneg (hX0 ω) m))).mp
+    simp_rw [hp]
+    rw [← he]
+    exact ENNReal.ofReal_ne_top
+  refine ⟨hi,?_⟩
+  have he' : ENNReal.ofReal (Target.levelCoefficient κ W ℓ m θ)=
+      ENNReal.ofReal (∫ ω, X ω^m ∂Q) := by
+    rw [ofReal_integral_eq_lintegral_ofReal hi
+      (Eventually.of_forall (fun ω => pow_nonneg (hX0 ω) m))]
+    simpa only [hp] using he
+  have hc0 := (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).1 θ hθ
+  have hi0 : 0≤∫ ω, X ω^m ∂Q := integral_nonneg (fun ω => pow_nonneg (hX0 ω) m)
+  simpa only [ENNReal.toReal_ofReal hc0,ENNReal.toReal_ofReal hi0] using congrArg ENNReal.toReal he'
+
+include C in
+lemma levelCoefficient_jensen
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
+    (Target.levelCoefficient κ W ℓ 1 θ)^m≤Target.levelCoefficient κ W ℓ m θ := by
+  let σ := fun ω => (radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ) : WithTop ℝ≥0)
+  have hz := Target.diskMapInv_maps hθ (show (0:ℂ)∈Metric.ball 0 1 by simp)
+  have hu : 0≤4*ℓ/κ := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
+  have hs := actual_radiusLevelTime_stopping hB
+    (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)))
+    hBm hc h0 κ hκ hκ8 γ G F hw (Target.diskMapInv θ 0) hz (4*ℓ/κ) hu
+  have heσ : ∀ᵐ ω ∂P, σ ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (4*ℓ/κ) := Eventually.of_forall (fun _ => rfl)
+  let Q := finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) (4*ℓ/κ)
+  let : IsProbabilityMeasure Q := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F
+    (Eventually.of_forall hw) _ hz _ hu
+  obtain ⟨hi1,he1⟩ := C.centered_diskScalar_integrable_eq hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ 1 σ hs heσ
+  obtain ⟨him,hem⟩ := C.centered_diskScalar_integrable_eq hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m σ hs heσ
+  rw [he1,hem]
+  simp only [pow_one] at hi1 ⊢
+  refine (convexOn_pow (𝕜:=ℝ) m).map_integral_le (μ:=Q)
+    (continuous_pow m).continuousOn isClosed_Ici ?_ hi1 him
+  filter_upwards [] with ω
+  change 0 ≤ finiteStoppedScalar (C.diskScalar θ hθ) σ ω
+  dsimp only [finiteStoppedScalar]
+  split_ifs
+  · exact le_rfl
+  · exact compactWeightedMass_nonneg _ _ _
+
+include C in
+theorem finiteMoment_jensen
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
+    (Target.finiteMoment κ W 1 ℓ)^m≤Target.finiteMoment κ W m ℓ := by
+  let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
+  obtain ⟨hn1,hi1⟩ := C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1
+  obtain ⟨hnm,him⟩ := C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m
+  have hj : ∀ᵐ θ ∂Target.angleLaw κ,
+      (Target.levelCoefficient κ W ℓ 1 θ)^m≤Target.levelCoefficient κ W ℓ m θ := by
+    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+    exact C.levelCoefficient_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
+  have hp : Integrable (fun θ => (Target.levelCoefficient κ W ℓ 1 θ)^m) (Target.angleLaw κ) := by
+    apply him.mono' (hi1.aestronglyMeasurable.pow m)
+    filter_upwards [hj,Target.angleLaw_ae_mem κ] with θ hθ hθI
+    change ‖(Target.levelCoefficient κ W ℓ 1 θ)^m‖ ≤ _
+    rw [Real.norm_of_nonneg (pow_nonneg (hn1 θ hθI) m)]
+    exact hθ
+  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1,
+    C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m]
+  apply le_trans ((convexOn_pow (𝕜:=ℝ) m).map_integral_le (continuous_pow m).continuousOn isClosed_Ici
+    (by filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ; exact hn1 θ hθ) hi1 hp)
+  exact integral_mono_ae hp him hj
+
+include C in
+theorem moment_jensen
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (m : ℕ) : (Target.moment κ W 1)^m≤Target.moment κ W m := by
+  exact le_of_tendsto_of_tendsto
+    ((C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW 1).pow m)
+    (C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW m)
+    (Eventually.of_forall (fun ℓ => C.finiteMoment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW
+      ℓ (Nat.cast_nonneg ℓ) m))
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ConditionalRibbonStep.lean b/lean/OAI/Probability/SLE/Gauge/ConditionalRibbonStep.lean
index c60afa7e516cf9bed4aa5ede498369af65390d3a..fd06b3bee9428a2d55cc33cae0ae1f091ef5f46d 100644
--- a/lean/OAI/Probability/SLE/Gauge/ConditionalRibbonStep.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ConditionalRibbonStep.lean
@@ -1,143 +1,143 @@
-import OAI.Probability.SLE.Gauge.UniformRibbonStep
-import OAI.Probability.SLE.Gauge.RandomPassageBounds
-
-namespace OAI
-
-/-! The actual uniform ribbon step integrates against the measurable
-stopped restart law. The estimate holds on every event in the past where
-an actual admissible ribbon history has been certified. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates BoundaryScaleLaw Target
-
-local instance conditionalRibbonPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance conditionalRibbonPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-local instance conditionalRibbonDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance conditionalRibbonDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-
-lemma ribbon_section_restart_lower_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (p : Ω → C(ℝ≥0, ℂ)) (hp : Measurable p)
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (Y : Ω → C(ℝ≥0, ℂ)) (hY : Measurable[mΩ] Y)
-    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
-      @Measure.map Ω C(ℝ≥0, ℂ) mΩ (borel _) Y (P.restrict A) =
-        P A • @Measure.map Ω C(ℝ≥0, ℂ) mΩ (borel _) p P)
-    (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable[m] Fb)
-    {C L : ℝ≥0} {ε : ℝ} (κ v c q : ℝ)
-    (S : Set C(ℝ≥0, ℂ)) (hS : MeasurableSet S)
-    (E : Set Ω)
-    (hstates : ∀ᵐ ω ∂P, ω ∈ E → ∃ d : RibbonHistory C L ε,
-      (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane →
-        Fb ω z = d.inverse d.time ((d.U d.time : ℂ) + z)) ∧ d.stepEvent κ v c = S)
-    (hstep : ∀ (d : RibbonHistory C L ε) (f : C(closedHalfPlane, ℂ)),
-      (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane →
-        f z = d.inverse d.time ((d.U d.time : ℂ) + z)) →
-      ENNReal.ofReal q ≤ P {ω | finiteBoundaryPath f (p ω) ∈ d.stepEvent κ v c})
-    (A : Set Ω) (hA : MeasurableSet[m] A) (hAE : A ⊆ E) :
-    ENNReal.ofReal q * P A ≤ P (A ∩ {ω | finiteBoundaryPath (Fb ω) (Y ω) ∈ S}) := by
-  let : MeasurableSpace Ω := mΩ
-  have : IsProbabilityMeasure (P.map p) :=
-    (Measure.isProbabilityMeasure_map_iff hp.aemeasurable).mpr inferInstance
-  let T : Set (Ω × C(ℝ≥0, ℂ)) := {x | finiteBoundaryPath (Fb x.1) x.2 ∈ S}
-  have hT : @MeasurableSet (Ω × C(ℝ≥0, ℂ)) (m.prod (borel _)) T := by
-    let : MeasurableSpace Ω := m
-    exact hS.preimage (measurable_random_finiteBoundaryPath Fb hFb)
-  have hsections : ∀ᵐ ω ∂P, ω ∈ E →
-      ENNReal.ofReal q ≤ P.map p {x | (ω, x) ∈ T} := by
-    filter_upwards [hstates] with ω hω hωE
-    obtain ⟨d, hf, he⟩ := hω hωE
-    have hsec : MeasurableSet {x | finiteBoundaryPath (Fb ω) x ∈ S} :=
-      hS.preimage (measurable_finiteBoundaryPath (Fb ω))
-    rw [show {x | (ω, x) ∈ T} = {x | finiteBoundaryPath (Fb ω) x ∈ S} from rfl,
-      Measure.map_apply hp hsec]
-    change ENNReal.ofReal q ≤ P {ω' | finiteBoundaryPath (Fb ω) (p ω') ∈ S}
-    simpa only [he] using hstep d (Fb ω) hf
-  exact restart_event_lower_bound P (P.map p) m hm Y hY hlaw T hT E
-    (ENNReal.ofReal q) hsections A hA hAE
-
-/-- The genuine Brownian restart and the actual uniform passage estimate
-give the adapted lower bound needed by concatenation. Histories are
-required only on the event where the preceding steps succeeded. -/
-theorem uniform_conditional_actual_ribbon_step
-    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω))
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W)
-    (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L)
-    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) :
-    ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧
-      ∀ (v : ℝ), 0 < v → v < v₀ → ∀ (τ : Ω → ℝ≥0)
-      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (τ ω : WithTop ℝ≥0))),
-      ∀ (S : Set C(ℝ≥0, ℂ)), MeasurableSet S → ∀ E : Set Ω,
-      (∀ᵐ ω ∂P, ω ∈ E → ∃ d : RibbonHistory C L ε,
-        d.U = (fun t => Real.sqrt κ * B t ω) ∧ d.trace = γ ω ∧
-        d.forward = G ω ∧ d.inverse = F ω ∧ d.time = τ ω ∧ d.stepEvent κ v c = S) →
-      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → A ⊆ E →
-        ENNReal.ofReal q * P A ≤ P (A ∩ {ω |
-          (⟨fun t => γ ω (τ ω + t), (hw ω).1.comp (continuous_const.add continuous_id)⟩ :
-            C(ℝ≥0, ℂ)) ∈ S}) := by
-  obtain ⟨q, c, v₀, hq, hcm, hv₀, hstep⟩ :=
-    uniform_actual_ribbon_step P κ hκ hκ8 B hB hm hc γ G F hw hcompact hboundary
-      hsupport hplanar hstraight hexit hwiener W hW C L hC hL ε hε hεsmall
-  refine ⟨q, c, v₀, hq, hcm, hv₀, ?_⟩
-  intro v hv hvcut τ hτ S hS E hstates A hA hAE
-  obtain ⟨Φ, Fb, hΦm, hΦ, hFbm, hFb, _, hlaw, hactual⟩ :=
-    source_measurable_finite_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw hboundary τ hτ
-  let p := sourceTracePath γ (fun ω => (hw ω).1)
-  have hp : Measurable p := by
-    have he : p = fun ω => Φ (sourceDriverPath B hc ω) := funext fun ω => (hΦ ω).symm
-    rw [he]
-    exact hΦm.comp (source_path_measurable (sourceDriverPath B hc) hm)
-  let Y := fun ω => Φ (sourceRestartDriverPath B hc τ ω)
-  have hY : Measurable Y := hΦm.comp
-    (source_path_measurable _ (fun t => (measurable_pi_apply t).comp
-      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))))
-  have hcert : ∀ᵐ ω ∂P, ω ∈ E → ∃ d : RibbonHistory C L ε,
-      (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane →
-        Fb ω z = d.inverse d.time ((d.U d.time : ℂ) + z)) ∧ d.stepEvent κ v c = S := by
-    filter_upwards [hstates] with ω hω hωE
-    obtain ⟨d, hU, _, _, hF, ht, he⟩ := hω hωE
-    refine ⟨d, ?_, he⟩
-    intro z hz
-    rw [hU, hF, ht, hFb ω z hz]
-    congr 1
-    simp only [Complex.ofReal_mul, add_comm]
-  have hh := ribbon_section_restart_lower_bound P p hp hτ.measurableSpace
-    hτ.measurableSpace_le Y hY hlaw Fb hFbm κ v c q S hS E hcert
-    (fun d f hf => hstep d f hf v hv hvcut) A hA hAE
-  apply hh.trans
-  apply measure_mono_ae
-  filter_upwards [hactual] with ω hω hωA
-  refine ⟨hωA.1, ?_⟩
-  have he : (⟨fun t => γ ω (τ ω + t),
-      (hw ω).1.comp (continuous_const.add continuous_id)⟩ : C(ℝ≥0, ℂ)) =
-      finiteBoundaryPath (Fb ω) (Y ω) := by
-    ext t
-    exact hω t
-  change (⟨fun t => γ ω (τ ω + t),
-    (hw ω).1.comp (continuous_const.add continuous_id)⟩ : C(ℝ≥0, ℂ)) ∈ S
-  rw [he]
-  exact hωA.2
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.UniformRibbonStep
+import OAI.Probability.SLE.Gauge.RandomPassageBounds
+
+namespace OAI
+
+/-! The actual uniform ribbon step integrates against the measurable
+stopped restart law. The estimate holds on every event in the past where
+an actual admissible ribbon history has been certified. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates BoundaryScaleLaw Target
+
+local instance conditionalRibbonPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance conditionalRibbonPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+local instance conditionalRibbonDriverMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance conditionalRibbonDriverBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+
+lemma ribbon_section_restart_lower_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (p : Ω → C(ℝ≥0, ℂ)) (hp : Measurable p)
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (Y : Ω → C(ℝ≥0, ℂ)) (hY : Measurable[mΩ] Y)
+    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
+      @Measure.map Ω C(ℝ≥0, ℂ) mΩ (borel _) Y (P.restrict A) =
+        P A • @Measure.map Ω C(ℝ≥0, ℂ) mΩ (borel _) p P)
+    (Fb : Ω → C(closedHalfPlane, ℂ)) (hFb : Measurable[m] Fb)
+    {C L : ℝ≥0} {ε : ℝ} (κ v c q : ℝ)
+    (S : Set C(ℝ≥0, ℂ)) (hS : MeasurableSet S)
+    (E : Set Ω)
+    (hstates : ∀ᵐ ω ∂P, ω ∈ E → ∃ d : RibbonHistory C L ε,
+      (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane →
+        Fb ω z = d.inverse d.time ((d.U d.time : ℂ) + z)) ∧ d.stepEvent κ v c = S)
+    (hstep : ∀ (d : RibbonHistory C L ε) (f : C(closedHalfPlane, ℂ)),
+      (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane →
+        f z = d.inverse d.time ((d.U d.time : ℂ) + z)) →
+      ENNReal.ofReal q ≤ P {ω | finiteBoundaryPath f (p ω) ∈ d.stepEvent κ v c})
+    (A : Set Ω) (hA : MeasurableSet[m] A) (hAE : A ⊆ E) :
+    ENNReal.ofReal q * P A ≤ P (A ∩ {ω | finiteBoundaryPath (Fb ω) (Y ω) ∈ S}) := by
+  let : MeasurableSpace Ω := mΩ
+  have : IsProbabilityMeasure (P.map p) :=
+    (Measure.isProbabilityMeasure_map_iff hp.aemeasurable).mpr inferInstance
+  let T : Set (Ω × C(ℝ≥0, ℂ)) := {x | finiteBoundaryPath (Fb x.1) x.2 ∈ S}
+  have hT : @MeasurableSet (Ω × C(ℝ≥0, ℂ)) (m.prod (borel _)) T := by
+    let : MeasurableSpace Ω := m
+    exact hS.preimage (measurable_random_finiteBoundaryPath Fb hFb)
+  have hsections : ∀ᵐ ω ∂P, ω ∈ E →
+      ENNReal.ofReal q ≤ P.map p {x | (ω, x) ∈ T} := by
+    filter_upwards [hstates] with ω hω hωE
+    obtain ⟨d, hf, he⟩ := hω hωE
+    have hsec : MeasurableSet {x | finiteBoundaryPath (Fb ω) x ∈ S} :=
+      hS.preimage (measurable_finiteBoundaryPath (Fb ω))
+    rw [show {x | (ω, x) ∈ T} = {x | finiteBoundaryPath (Fb ω) x ∈ S} from rfl,
+      Measure.map_apply hp hsec]
+    change ENNReal.ofReal q ≤ P {ω' | finiteBoundaryPath (Fb ω) (p ω') ∈ S}
+    simpa only [he] using hstep d (Fb ω) hf
+  exact restart_event_lower_bound P (P.map p) m hm Y hY hlaw T hT E
+    (ENNReal.ofReal q) hsections A hA hAE
+
+/-- The genuine Brownian restart and the actual uniform passage estimate
+give the adapted lower bound needed by concatenation. Histories are
+required only on the event where the preceding steps succeeded. -/
+theorem uniform_conditional_actual_ribbon_step
+    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω))
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W)
+    (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L)
+    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) :
+    ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧
+      ∀ (v : ℝ), 0 < v → v < v₀ → ∀ (τ : Ω → ℝ≥0)
+      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (τ ω : WithTop ℝ≥0))),
+      ∀ (S : Set C(ℝ≥0, ℂ)), MeasurableSet S → ∀ E : Set Ω,
+      (∀ᵐ ω ∂P, ω ∈ E → ∃ d : RibbonHistory C L ε,
+        d.U = (fun t => Real.sqrt κ * B t ω) ∧ d.trace = γ ω ∧
+        d.forward = G ω ∧ d.inverse = F ω ∧ d.time = τ ω ∧ d.stepEvent κ v c = S) →
+      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → A ⊆ E →
+        ENNReal.ofReal q * P A ≤ P (A ∩ {ω |
+          (⟨fun t => γ ω (τ ω + t), (hw ω).1.comp (continuous_const.add continuous_id)⟩ :
+            C(ℝ≥0, ℂ)) ∈ S}) := by
+  obtain ⟨q, c, v₀, hq, hcm, hv₀, hstep⟩ :=
+    uniform_actual_ribbon_step P κ hκ hκ8 B hB hm hc γ G F hw hcompact hboundary
+      hsupport hplanar hstraight hexit hwiener W hW C L hC hL ε hε hεsmall
+  refine ⟨q, c, v₀, hq, hcm, hv₀, ?_⟩
+  intro v hv hvcut τ hτ S hS E hstates A hA hAE
+  obtain ⟨Φ, Fb, hΦm, hΦ, hFbm, hFb, _, hlaw, hactual⟩ :=
+    source_measurable_finite_boundary_restart κ hκ hκ8 B hB hm hc γ G F hw hboundary τ hτ
+  let p := sourceTracePath γ (fun ω => (hw ω).1)
+  have hp : Measurable p := by
+    have he : p = fun ω => Φ (sourceDriverPath B hc ω) := funext fun ω => (hΦ ω).symm
+    rw [he]
+    exact hΦm.comp (source_path_measurable (sourceDriverPath B hc) hm)
+  let Y := fun ω => Φ (sourceRestartDriverPath B hc τ ω)
+  have hY : Measurable Y := hΦm.comp
+    (source_path_measurable _ (fun t => (measurable_pi_apply t).comp
+      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))))
+  have hcert : ∀ᵐ ω ∂P, ω ∈ E → ∃ d : RibbonHistory C L ε,
+      (∀ z : closedHalfPlane, z.1 ∈ upperHalfPlane →
+        Fb ω z = d.inverse d.time ((d.U d.time : ℂ) + z)) ∧ d.stepEvent κ v c = S := by
+    filter_upwards [hstates] with ω hω hωE
+    obtain ⟨d, hU, _, _, hF, ht, he⟩ := hω hωE
+    refine ⟨d, ?_, he⟩
+    intro z hz
+    rw [hU, hF, ht, hFb ω z hz]
+    congr 1
+    simp only [Complex.ofReal_mul, add_comm]
+  have hh := ribbon_section_restart_lower_bound P p hp hτ.measurableSpace
+    hτ.measurableSpace_le Y hY hlaw Fb hFbm κ v c q S hS E hcert
+    (fun d f hf => hstep d f hf v hv hvcut) A hA hAE
+  apply hh.trans
+  apply measure_mono_ae
+  filter_upwards [hactual] with ω hω hωA
+  refine ⟨hωA.1, ?_⟩
+  have he : (⟨fun t => γ ω (τ ω + t),
+      (hw ω).1.comp (continuous_const.add continuous_id)⟩ : C(ℝ≥0, ℂ)) =
+      finiteBoundaryPath (Fb ω) (Y ω) := by
+    ext t
+    exact hω t
+  change (⟨fun t => γ ω (τ ω + t),
+    (hw ω).1.comp (continuous_const.add continuous_id)⟩ : C(ℝ≥0, ℂ)) ∈ S
+  rw [he]
+  exact hωA.2
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ConformalCompactSausage.lean b/lean/OAI/Probability/SLE/Gauge/ConformalCompactSausage.lean
index de0f52cb87fc8d02c631cd60bdb9170217f9cfeb..7ed1cb17e7a1a6863b0d8a9aa2945a1361535d63 100644
--- a/lean/OAI/Probability/SLE/Gauge/ConformalCompactSausage.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ConformalCompactSausage.lean
@@ -1,141 +1,141 @@
-import OAI.Probability.SLE.Gauge.ConformalSausageTransfer
-import Mathlib.Analysis.Complex.LocallyUniformLimit
-
-namespace OAI
-
-/-! Uniform sausage transport on retained compact parameter sets. Local
-uniform convergence supplies derivative bounds and a common positive mesh;
-the actual Jacobian and convex local patches supply the area comparison. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-theorem eventually_derivative_norm_bounds_on_compact
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
-    (hne : ∀ z ∈ K, deriv F z ≠ 0) :
-    ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∀ᶠ n in atTop,
-      ∀ z ∈ K, c ≤ ‖deriv (Fₙ n) z‖ ∧ ‖deriv (Fₙ n) z‖ ≤ C := by
-  have hc : ContinuousOn (fun z => ‖deriv F z‖) K :=
-    ((hF.deriv isOpen_upperHalfPlane).continuousOn.mono hKH).norm
-  obtain ⟨z₀, hz₀, hmin⟩ := hK.exists_isMinOn hKn hc
-  let c₀ := ‖deriv F z₀‖
-  have hc₀ : 0 < c₀ := norm_pos_iff.mpr (hne z₀ hz₀)
-  obtain ⟨C₀, hC₀, hmax⟩ :=
-    (hK.image_of_continuousOn ((hF.deriv isOpen_upperHalfPlane).continuousOn.mono hKH)).isBounded.exists_pos_norm_lt
-  have hderconv := hconv.deriv (Eventually.of_forall hFn) isOpen_upperHalfPlane
-  have hu : TendstoUniformlyOn (fun n => deriv (Fₙ n)) (deriv F) atTop K :=
-    (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK).mp (hderconv.mono hKH)
-  have he : ∀ᶠ n in atTop, ∀ z ∈ K,
-      dist (deriv F z) (deriv (Fₙ n) z) < min (c₀ / 2) 1 :=
-    hu _ (Metric.dist_mem_uniformity (lt_min (by positivity) (by norm_num)))
-  refine ⟨c₀ / 2, C₀ + 1, by positivity, by positivity, ?_⟩
-  filter_upwards [he] with n hn
-  intro z hz
-  have hd := hn z hz
-  have hlow := hmin hz
-  change c₀ ≤ ‖deriv F z‖ at hlow
-  have hup := hmax _ (mem_image_of_mem (deriv F) hz)
-  have hdiff := norm_sub_norm_le (deriv F z) (deriv (Fₙ n) z)
-  have hdiff' := norm_sub_norm_le (deriv (Fₙ n) z) (deriv F z)
-  rw [← dist_eq_norm] at hdiff hdiff'
-  rw [dist_comm] at hdiff'
-  constructor <;> dsimp only [c₀] at * <;>
-    linarith [min_le_left (c₀ / 2) 1, min_le_right (c₀ / 2) 1]
-
-theorem eventually_image_compact_subset_open
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (hF : ContinuousOn F upperHalfPlane)
-    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
-    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R) :
-    ∀ᶠ n in atTop, Fₙ n '' K ⊆ R := by
-  have hFK : IsCompact (F '' K) := hK.image_of_continuousOn (hF.mono hKH)
-  obtain ⟨ε, hε, hεR⟩ := hFK.exists_thickening_subset_open hR himage
-  have hu : TendstoUniformlyOn Fₙ F atTop K :=
-    (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK).mp (hconv.mono hKH)
-  have he : ∀ᶠ n in atTop, ∀ z ∈ K, dist (F z) (Fₙ n z) < ε :=
-    hu _ (Metric.dist_mem_uniformity hε)
-  filter_upwards [he] with n hn
-  rintro w ⟨z, hz, rfl⟩
-  apply hεR
-  apply mem_thickening_iff.mpr
-  exact ⟨F z, mem_image_of_mem F hz, by simpa only [dist_comm] using hn z hz⟩
-
-/-- All sufficiently late conformal maps transport every interval retained
-in `K`, at every sufficiently small deterministic mesh, with common constants.
-The compact set itself need not be convex. -/
-theorem eventually_compact_sausage_transfer
-    (κ : ℝ) (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
-    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R) :
-    ∃ c ell v₀ : ℝ, 0 < c ∧ 0 < ell ∧ 0 < v₀ ∧
-      ∀ᶠ n in atTop, ∀ (γ : ℝ≥0 → ℂ) (s t : ℝ≥0), γ '' Icc s t ⊆ K →
-        ∀ v : ℝ, 0 < v → v < v₀ →
-        ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) *
-          sourceSausageMass κ γ s t (ell * v) ≤
-        ENNReal.ofReal (v ^ (-codimension κ)) *
-          volume (R ∩ sourceSausageSet (fun u => Fₙ n (γ u)) s t v) := by
-  let U := upperHalfPlane ∩ F ⁻¹' R
-  have hU : IsOpen U := hF.continuousOn.isOpen_inter_preimage isOpen_upperHalfPlane hR
-  have hKU : K ⊆ U := fun z hz => ⟨hKH hz, himage (mem_image_of_mem F hz)⟩
-  obtain ⟨δ, hδ, hδU⟩ := hK.exists_cthickening_subset_open hU hKU
-  let L := cthickening δ K
-  have hL : IsCompact L := hK.cthickening
-  have hKL : K ⊆ L := self_subset_cthickening K
-  have hLn : L.Nonempty := hKn.mono hKL
-  have hLH : L ⊆ upperHalfPlane := fun z hz => (hδU hz).1
-  have hLR : F '' L ⊆ R := by
-    rintro w ⟨z, hz, rfl⟩
-    exact (hδU hz).2
-  obtain ⟨c, C, hc, hC, hbounds⟩ := eventually_derivative_norm_bounds_on_compact
-    Fₙ F hFn hF hconv L hL hLn hLH (fun z hz => hne z (hLH hz))
-  have hmaps := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
-    L hL hLH R hR hLR
-  let ell := min 1 (1 / (C + 1))
-  have hell : 0 < ell := lt_min zero_lt_one (by positivity)
-  have hell1 : ell ≤ 1 := min_le_left _ _
-  have hscale : C * ell ≤ 1 := by
-    have he := (le_div_iff₀ (by positivity : 0 < C + 1)).mp (min_le_right 1 (1 / (C + 1)))
-    dsimp only [ell] at *
-    nlinarith
-  refine ⟨c, ell, δ, hc, hell, hδ, ?_⟩
-  filter_upwards [hbounds, hmaps] with n hn hmap
-  intro γ s t hγK v hv hvδ
-  have hsmall : ell * v < δ := (mul_le_of_le_one_left hv.le hell1).trans_lt hvδ
-  have hSL : sourceSausageSet γ s t (ell * v) ⊆ L := by
-    rintro z ⟨u, hu, hdist⟩
-    exact mem_cthickening_of_dist_le z (γ u) δ K
-      (hγK (mem_image_of_mem γ hu)) (hdist.le.trans hsmall.le)
-  apply sourceSausageMass_conformal_transfer κ (Fₙ n) (deriv (Fₙ n)) γ s t R
-    hc.le hv hell ((hinj n).mono (hSL.trans hLH))
-    (fun z hz => ((hFn n).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hLH (hSL hz)))).hasDerivAt)
-    (fun z hz => (hn z (hSL hz)).1) ⟨C, hC.le⟩ hscale
-    (image_mono hSL |>.trans hmap)
-  rintro z ⟨u, hu, hdist⟩
-  have huK := hγK (mem_image_of_mem γ hu)
-  have hballL : closedBall (γ u) δ ⊆ L := closedBall_subset_cthickening huK δ
-  refine ⟨u, hu, hdist, closedBall (γ u) δ, convex_closedBall _ _,
-    mem_closedBall.mpr (hdist.le.trans hsmall.le), mem_closedBall_self hδ.le, ?_, ?_⟩
-  · intro w hw
-    exact ((hFn n).differentiableAt
-      (isOpen_upperHalfPlane.mem_nhds (hLH (hballL hw)))).hasDerivAt.hasDerivWithinAt
-  · intro w hw
-    exact_mod_cast (hn w (hballL hw)).2
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ConformalSausageTransfer
+import Mathlib.Analysis.Complex.LocallyUniformLimit
+
+namespace OAI
+
+/-! Uniform sausage transport on retained compact parameter sets. Local
+uniform convergence supplies derivative bounds and a common positive mesh;
+the actual Jacobian and convex local patches supply the area comparison. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+theorem eventually_derivative_norm_bounds_on_compact
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
+    (hne : ∀ z ∈ K, deriv F z ≠ 0) :
+    ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∀ᶠ n in atTop,
+      ∀ z ∈ K, c ≤ ‖deriv (Fₙ n) z‖ ∧ ‖deriv (Fₙ n) z‖ ≤ C := by
+  have hc : ContinuousOn (fun z => ‖deriv F z‖) K :=
+    ((hF.deriv isOpen_upperHalfPlane).continuousOn.mono hKH).norm
+  obtain ⟨z₀, hz₀, hmin⟩ := hK.exists_isMinOn hKn hc
+  let c₀ := ‖deriv F z₀‖
+  have hc₀ : 0 < c₀ := norm_pos_iff.mpr (hne z₀ hz₀)
+  obtain ⟨C₀, hC₀, hmax⟩ :=
+    (hK.image_of_continuousOn ((hF.deriv isOpen_upperHalfPlane).continuousOn.mono hKH)).isBounded.exists_pos_norm_lt
+  have hderconv := hconv.deriv (Eventually.of_forall hFn) isOpen_upperHalfPlane
+  have hu : TendstoUniformlyOn (fun n => deriv (Fₙ n)) (deriv F) atTop K :=
+    (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK).mp (hderconv.mono hKH)
+  have he : ∀ᶠ n in atTop, ∀ z ∈ K,
+      dist (deriv F z) (deriv (Fₙ n) z) < min (c₀ / 2) 1 :=
+    hu _ (Metric.dist_mem_uniformity (lt_min (by positivity) (by norm_num)))
+  refine ⟨c₀ / 2, C₀ + 1, by positivity, by positivity, ?_⟩
+  filter_upwards [he] with n hn
+  intro z hz
+  have hd := hn z hz
+  have hlow := hmin hz
+  change c₀ ≤ ‖deriv F z‖ at hlow
+  have hup := hmax _ (mem_image_of_mem (deriv F) hz)
+  have hdiff := norm_sub_norm_le (deriv F z) (deriv (Fₙ n) z)
+  have hdiff' := norm_sub_norm_le (deriv (Fₙ n) z) (deriv F z)
+  rw [← dist_eq_norm] at hdiff hdiff'
+  rw [dist_comm] at hdiff'
+  constructor <;> dsimp only [c₀] at * <;>
+    linarith [min_le_left (c₀ / 2) 1, min_le_right (c₀ / 2) 1]
+
+theorem eventually_image_compact_subset_open
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (hF : ContinuousOn F upperHalfPlane)
+    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
+    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R) :
+    ∀ᶠ n in atTop, Fₙ n '' K ⊆ R := by
+  have hFK : IsCompact (F '' K) := hK.image_of_continuousOn (hF.mono hKH)
+  obtain ⟨ε, hε, hεR⟩ := hFK.exists_thickening_subset_open hR himage
+  have hu : TendstoUniformlyOn Fₙ F atTop K :=
+    (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK).mp (hconv.mono hKH)
+  have he : ∀ᶠ n in atTop, ∀ z ∈ K, dist (F z) (Fₙ n z) < ε :=
+    hu _ (Metric.dist_mem_uniformity hε)
+  filter_upwards [he] with n hn
+  rintro w ⟨z, hz, rfl⟩
+  apply hεR
+  apply mem_thickening_iff.mpr
+  exact ⟨F z, mem_image_of_mem F hz, by simpa only [dist_comm] using hn z hz⟩
+
+/-- All sufficiently late conformal maps transport every interval retained
+in `K`, at every sufficiently small deterministic mesh, with common constants.
+The compact set itself need not be convex. -/
+theorem eventually_compact_sausage_transfer
+    (κ : ℝ) (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
+    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R) :
+    ∃ c ell v₀ : ℝ, 0 < c ∧ 0 < ell ∧ 0 < v₀ ∧
+      ∀ᶠ n in atTop, ∀ (γ : ℝ≥0 → ℂ) (s t : ℝ≥0), γ '' Icc s t ⊆ K →
+        ∀ v : ℝ, 0 < v → v < v₀ →
+        ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) *
+          sourceSausageMass κ γ s t (ell * v) ≤
+        ENNReal.ofReal (v ^ (-codimension κ)) *
+          volume (R ∩ sourceSausageSet (fun u => Fₙ n (γ u)) s t v) := by
+  let U := upperHalfPlane ∩ F ⁻¹' R
+  have hU : IsOpen U := hF.continuousOn.isOpen_inter_preimage isOpen_upperHalfPlane hR
+  have hKU : K ⊆ U := fun z hz => ⟨hKH hz, himage (mem_image_of_mem F hz)⟩
+  obtain ⟨δ, hδ, hδU⟩ := hK.exists_cthickening_subset_open hU hKU
+  let L := cthickening δ K
+  have hL : IsCompact L := hK.cthickening
+  have hKL : K ⊆ L := self_subset_cthickening K
+  have hLn : L.Nonempty := hKn.mono hKL
+  have hLH : L ⊆ upperHalfPlane := fun z hz => (hδU hz).1
+  have hLR : F '' L ⊆ R := by
+    rintro w ⟨z, hz, rfl⟩
+    exact (hδU hz).2
+  obtain ⟨c, C, hc, hC, hbounds⟩ := eventually_derivative_norm_bounds_on_compact
+    Fₙ F hFn hF hconv L hL hLn hLH (fun z hz => hne z (hLH hz))
+  have hmaps := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
+    L hL hLH R hR hLR
+  let ell := min 1 (1 / (C + 1))
+  have hell : 0 < ell := lt_min zero_lt_one (by positivity)
+  have hell1 : ell ≤ 1 := min_le_left _ _
+  have hscale : C * ell ≤ 1 := by
+    have he := (le_div_iff₀ (by positivity : 0 < C + 1)).mp (min_le_right 1 (1 / (C + 1)))
+    dsimp only [ell] at *
+    nlinarith
+  refine ⟨c, ell, δ, hc, hell, hδ, ?_⟩
+  filter_upwards [hbounds, hmaps] with n hn hmap
+  intro γ s t hγK v hv hvδ
+  have hsmall : ell * v < δ := (mul_le_of_le_one_left hv.le hell1).trans_lt hvδ
+  have hSL : sourceSausageSet γ s t (ell * v) ⊆ L := by
+    rintro z ⟨u, hu, hdist⟩
+    exact mem_cthickening_of_dist_le z (γ u) δ K
+      (hγK (mem_image_of_mem γ hu)) (hdist.le.trans hsmall.le)
+  apply sourceSausageMass_conformal_transfer κ (Fₙ n) (deriv (Fₙ n)) γ s t R
+    hc.le hv hell ((hinj n).mono (hSL.trans hLH))
+    (fun z hz => ((hFn n).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hLH (hSL hz)))).hasDerivAt)
+    (fun z hz => (hn z (hSL hz)).1) ⟨C, hC.le⟩ hscale
+    (image_mono hSL |>.trans hmap)
+  rintro z ⟨u, hu, hdist⟩
+  have huK := hγK (mem_image_of_mem γ hu)
+  have hballL : closedBall (γ u) δ ⊆ L := closedBall_subset_cthickening huK δ
+  refine ⟨u, hu, hdist, closedBall (γ u) δ, convex_closedBall _ _,
+    mem_closedBall.mpr (hdist.le.trans hsmall.le), mem_closedBall_self hδ.le, ?_, ?_⟩
+  · intro w hw
+    exact ((hFn n).differentiableAt
+      (isOpen_upperHalfPlane.mem_nhds (hLH (hballL hw)))).hasDerivAt.hasDerivWithinAt
+  · intro w hw
+    exact_mod_cast (hn w (hballL hw)).2
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ConformalDenseExperiment.lean b/lean/OAI/Probability/SLE/Gauge/ConformalDenseExperiment.lean
index 8d298f0959c52bc6f85b8f28977119a9db34ffa9..97db081e655b56b1ce4e8c3360fac203c36f8e06 100644
--- a/lean/OAI/Probability/SLE/Gauge/ConformalDenseExperiment.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ConformalDenseExperiment.lean
@@ -1,156 +1,156 @@
-import OAI.Probability.SLE.Gauge.SerpentineCenteredMass
-import OAI.Probability.SLE.Gauge.SerpentineConformalGeometry
-
-namespace OAI
-
-/-! The actual dense experiment transported through an arbitrary inverse
-chart. Its lower bound is uniform over normalized marked states and its
-cutoff is proportional to the physical conformal radius. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates FirstFrontPassage FrontSuccessMeasurability
-
-lemma segmentSausageCharge_le_stoppedWhole
-    (κ : ℝ) (f : C(ℝ≥0, ℂ)) (T : ℝ≥0) (ρ : WithTop ℝ≥0)
-    (hT : (T : WithTop ℝ≥0) ≤ ρ) (v : ℝ) (D : Set ℂ) :
-    segmentSausageCharge κ f 0 T v D ≤ stoppedWholeSausageCharge κ f ρ v D := by
-  cases hρ : ρ with
-  | top =>
-    simp only [stoppedWholeSausageCharge, stoppedAtPath, ite_true]
-    apply mul_le_mul' le_rfl
-    apply measure_mono
-    rintro z ⟨hz, u, _hu, hdu⟩
-    exact ⟨hz, u, by simpa only [dist_comm] using hdu⟩
-  | coe t =>
-    rw [stoppedWholeSausageCharge_finite]
-    exact segmentSausageCharge_mono κ f le_rfl (WithTop.coe_le_coe.mp (hρ ▸ hT)) v Subset.rfl
-
-namespace Serpentine
-open Target
-
-lemma dense_conformal_mass_threshold
-    (κ A c h a : ℝ) (hκ : 0 < κ) (hA : 0 < A) (hc : 0 < c)
-    (hh : 0 < h) (ha : 0 < a) (w : ℂ) (hwy : 0 < w.im) (hwn : ‖w‖ ≤ 1)
-    (m : ℕ) :
-    (A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ) *
-        (2 * w.im * a) ^ dimension κ * (m : ℝ) ^ codimension κ ≤
-      (A * a ^ dimension κ) * (c * (h / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := by
-  have hwi := (Complex.im_le_norm w).trans hwn
-  have hr : (2 * w.im * a) ^ dimension κ ≤ 2 ^ dimension κ * a ^ dimension κ := by
-    calc
-      _ ≤ (2 * a) ^ dimension κ := Real.rpow_le_rpow (by positivity)
-        (by nlinarith) (by dsimp [dimension]; positivity)
-      _ = _ := Real.mul_rpow (by norm_num) ha.le
-  have hbase : 0 ≤ A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ := by positivity
-  have hm : 0 ≤ (m : ℝ) ^ codimension κ := Real.rpow_nonneg (Nat.cast_nonneg _) _
-  calc
-    _ ≤ (A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ) *
-        (2 ^ dimension κ * a ^ dimension κ) * (m : ℝ) ^ codimension κ :=
-      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hr hbase) hm
-    _ = _ := by field_simp
-
-theorem uniform_conformal_dense_stopped_mass
-    {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport μ γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W)
-    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8)
-    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
-    ∃ h b C : ℝ, 0 < h ∧ h ≤ s ∧ 0 < b ∧ 0 < C ∧
-      ∃ δ : ℕ → ℝ, (∀ m, 0 < m → 0 < δ m) ∧
-      ∀ (f : ℂ → ℂ) (w : ℂ), DifferentiableOn ℂ f upperHalfPlane →
-        InjOn f upperHalfPlane → ‖w‖ = 1 → s ≤ w.im → deriv f w ≠ 0 →
-        ∀ (η : Ω → ℝ≥0 → ℂ) (hη : ∀ ω, Continuous (η ω)),
-        (∀ ω t, γ ω t ∈ closedBall w h → η ω t = f (γ ω t)) →
-        ∀ ρ : Ω → WithTop ℝ≥0,
-        (∀ᵐ ω ∂μ, ρ ω = radiusLevelTime (γ ω) (G ω) w (experimentRadiusLevel h)) →
-        ∀ m : ℕ, 0 < m → ∀ v : ℝ, 0 < v →
-        v < δ m * (2 * w.im * ‖deriv f w‖) →
-        ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤
-          finiteCenteredLaw μ κ B γ G w (experimentRadiusLevel h)
-            {ω | ENNReal.ofReal (b * (2 * w.im * ‖deriv f w‖) ^ dimension κ *
-                (m : ℝ) ^ codimension κ) ≤
-              stoppedWholeSausageCharge κ (sourceTracePath η hη ω) (ρ ω) v
-                (closedBall (f w) (2 * w.im * ‖deriv f w‖))} := by
-  obtain ⟨c, v₀, hcm, hv₀, hmass⟩ := uniform_serpentine_centered_stopped_mass μ κ hκ hκ8
-    B hB hBm hc hB0 γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall
-  obtain ⟨h, A, L, δ₀, hh, hhs, hA, hL, hδ₀, htransfer⟩ :=
-    uniform_conformal_experiment_restricted_transfer κ hcompact hs hs1
-  have hh1 := hhs.trans hs1
-  obtain ⟨C, hC, hmass⟩ := hmass h hh hh1
-  let b := A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ
-  let δ := fun m => min δ₀ (L * (ribbonWidth (halfHeight h) m * v₀) / 2)
-  refine ⟨h, b, C, hh, hhs, by dsimp [b]; positivity, hC, δ, ?_, ?_⟩
-  · intro m hm
-    have hwidth := fit_width_pos hh hm
-    exact lt_min hδ₀ (by positivity)
-  intro f w hf hfi hwn hwy hne η hη heq ρ hρ m hm v hv hvcut
-  have hwh : h ≤ w.im := hhs.trans hwy
-  have hwpos : 0 < w.im := hs.trans_le hwy
-  have hnorm : 0 < ‖deriv f w‖ := norm_pos_iff.mpr hne
-  have hwi : w.im ≤ 1 := (Complex.im_le_norm w).trans hwn.le
-  have hvsource : 0 < v / (L * ‖deriv f w‖) := div_pos hv (mul_pos hL hnorm)
-  have hvcut' : v / (L * ‖deriv f w‖) < ribbonWidth (halfHeight h) m * v₀ := by
-    apply conformal_experiment_mesh_cutoff hL hnorm (mul_pos (fit_width_pos hh hm) hv₀) w hwi
-    exact hvcut.trans_le (mul_le_mul_of_nonneg_right (min_le_right _ _) (by positivity))
-  obtain ⟨g, hg⟩ := exists_global_ribbon_progress _ (physicalExperimentChart_closedEmbedding hh w hwh hm)
-  let τ := ribbonFrontStop g γ (fitCompletion h w m)
-  let E := RibbonStepSuccess γ (fun _ => 0) τ (physicalExperimentChart h w m) ε κ
-    (v / (L * ‖deriv f w‖)) (c * (h / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ)
-    (closedBall (center h w) (h / 20))
-  have hp : ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤
-      finiteCenteredLaw μ κ B γ G w (experimentRadiusLevel h) E :=
-    hmass w hwn.le hwh m hm (v / (L * ‖deriv f w‖)) hvsource hvcut' g hg
-  apply hp.trans
-  apply measure_mono_ae
-  apply (withDensity_absolutelyContinuous μ _).ae_le
-  filter_upwards [hρ] with ω hρω hω
-  have hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ ω) (fun _ => τ ω)
-      (physicalExperimentChart h w m) ε := hω.1
-  have hbefore := (good_physicalExperiment_before_radius ((hc ω).const_mul _) (hw ω)
-    hh w hwn.le hwh hm (τ ω) ε hpref).1
-  have hTρ : (((τ ω).untopD 0 : ℝ≥0) : WithTop ℝ≥0) ≤ ρ ω := by
-    cases ht : τ ω with
-    | top => exact (hpref.1 ht).elim
-    | coe T =>
-      simp only [ht, WithTop.untopD_coe] at hbefore ⊢
-      rw [hρω]
-      exact_mod_cast hbefore.le
-  have htrans := htransfer f w hf hfi hwn hwy hne (γ ω) (η ω) 0 ((τ ω).untopD 0)
-    (fun t _ ht => heq ω t ht) v hv
-    (hvcut.trans_le (mul_le_mul_of_nonneg_right (min_le_left _ _) (by positivity)))
-  have hthresh := dense_conformal_mass_threshold κ A c h ‖deriv f w‖ hκ hA hcm hh hnorm w
-    (lt_of_lt_of_le hs hwy) hwn.le m
-  calc
-    _ ≤ ENNReal.ofReal (A * ‖deriv f w‖ ^ dimension κ) *
-        ENNReal.ofReal (c * (h / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := by
-      rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ A * ‖deriv f w‖ ^ dimension κ)]
-      exact ENNReal.ofReal_le_ofReal hthresh
-    _ ≤ ENNReal.ofReal (A * ‖deriv f w‖ ^ dimension κ) *
-        segmentSausageCharge κ (γ ω) 0 ((τ ω).untopD 0) (v / (L * ‖deriv f w‖))
-          (closedBall (center h w) (h / 20)) := mul_le_mul' le_rfl hω.2
-    _ ≤ segmentSausageCharge κ (η ω) 0 ((τ ω).untopD 0) v
-        (closedBall (f w) (2 * w.im * ‖deriv f w‖)) := htrans
-    _ ≤ _ := segmentSausageCharge_le_stoppedWhole κ (sourceTracePath η hη ω) _ _ hTρ _ _
-
-end Serpentine
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.SerpentineCenteredMass
+import OAI.Probability.SLE.Gauge.SerpentineConformalGeometry
+
+namespace OAI
+
+/-! The actual dense experiment transported through an arbitrary inverse
+chart. Its lower bound is uniform over normalized marked states and its
+cutoff is proportional to the physical conformal radius. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates FirstFrontPassage FrontSuccessMeasurability
+
+lemma segmentSausageCharge_le_stoppedWhole
+    (κ : ℝ) (f : C(ℝ≥0, ℂ)) (T : ℝ≥0) (ρ : WithTop ℝ≥0)
+    (hT : (T : WithTop ℝ≥0) ≤ ρ) (v : ℝ) (D : Set ℂ) :
+    segmentSausageCharge κ f 0 T v D ≤ stoppedWholeSausageCharge κ f ρ v D := by
+  cases hρ : ρ with
+  | top =>
+    simp only [stoppedWholeSausageCharge, stoppedAtPath, ite_true]
+    apply mul_le_mul' le_rfl
+    apply measure_mono
+    rintro z ⟨hz, u, _hu, hdu⟩
+    exact ⟨hz, u, by simpa only [dist_comm] using hdu⟩
+  | coe t =>
+    rw [stoppedWholeSausageCharge_finite]
+    exact segmentSausageCharge_mono κ f le_rfl (WithTop.coe_le_coe.mp (hρ ▸ hT)) v Subset.rfl
+
+namespace Serpentine
+open Target
+
+lemma dense_conformal_mass_threshold
+    (κ A c h a : ℝ) (hκ : 0 < κ) (hA : 0 < A) (hc : 0 < c)
+    (hh : 0 < h) (ha : 0 < a) (w : ℂ) (hwy : 0 < w.im) (hwn : ‖w‖ ≤ 1)
+    (m : ℕ) :
+    (A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ) *
+        (2 * w.im * a) ^ dimension κ * (m : ℝ) ^ codimension κ ≤
+      (A * a ^ dimension κ) * (c * (h / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := by
+  have hwi := (Complex.im_le_norm w).trans hwn
+  have hr : (2 * w.im * a) ^ dimension κ ≤ 2 ^ dimension κ * a ^ dimension κ := by
+    calc
+      _ ≤ (2 * a) ^ dimension κ := Real.rpow_le_rpow (by positivity)
+        (by nlinarith) (by dsimp [dimension]; positivity)
+      _ = _ := Real.mul_rpow (by norm_num) ha.le
+  have hbase : 0 ≤ A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ := by positivity
+  have hm : 0 ≤ (m : ℝ) ^ codimension κ := Real.rpow_nonneg (Nat.cast_nonneg _) _
+  calc
+    _ ≤ (A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ) *
+        (2 ^ dimension κ * a ^ dimension κ) * (m : ℝ) ^ codimension κ :=
+      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hr hbase) hm
+    _ = _ := by field_simp
+
+theorem uniform_conformal_dense_stopped_mass
+    {Ω : Type} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B μ)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport μ γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W)
+    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8)
+    {s : ℝ} (hs : 0 < s) (hs1 : s ≤ 1) :
+    ∃ h b C : ℝ, 0 < h ∧ h ≤ s ∧ 0 < b ∧ 0 < C ∧
+      ∃ δ : ℕ → ℝ, (∀ m, 0 < m → 0 < δ m) ∧
+      ∀ (f : ℂ → ℂ) (w : ℂ), DifferentiableOn ℂ f upperHalfPlane →
+        InjOn f upperHalfPlane → ‖w‖ = 1 → s ≤ w.im → deriv f w ≠ 0 →
+        ∀ (η : Ω → ℝ≥0 → ℂ) (hη : ∀ ω, Continuous (η ω)),
+        (∀ ω t, γ ω t ∈ closedBall w h → η ω t = f (γ ω t)) →
+        ∀ ρ : Ω → WithTop ℝ≥0,
+        (∀ᵐ ω ∂μ, ρ ω = radiusLevelTime (γ ω) (G ω) w (experimentRadiusLevel h)) →
+        ∀ m : ℕ, 0 < m → ∀ v : ℝ, 0 < v →
+        v < δ m * (2 * w.im * ‖deriv f w‖) →
+        ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤
+          finiteCenteredLaw μ κ B γ G w (experimentRadiusLevel h)
+            {ω | ENNReal.ofReal (b * (2 * w.im * ‖deriv f w‖) ^ dimension κ *
+                (m : ℝ) ^ codimension κ) ≤
+              stoppedWholeSausageCharge κ (sourceTracePath η hη ω) (ρ ω) v
+                (closedBall (f w) (2 * w.im * ‖deriv f w‖))} := by
+  obtain ⟨c, v₀, hcm, hv₀, hmass⟩ := uniform_serpentine_centered_stopped_mass μ κ hκ hκ8
+    B hB hBm hc hB0 γ G F hw hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW ε hε hεsmall
+  obtain ⟨h, A, L, δ₀, hh, hhs, hA, hL, hδ₀, htransfer⟩ :=
+    uniform_conformal_experiment_restricted_transfer κ hcompact hs hs1
+  have hh1 := hhs.trans hs1
+  obtain ⟨C, hC, hmass⟩ := hmass h hh hh1
+  let b := A * c * (h / 20000) ^ dimension κ / 2 ^ dimension κ
+  let δ := fun m => min δ₀ (L * (ribbonWidth (halfHeight h) m * v₀) / 2)
+  refine ⟨h, b, C, hh, hhs, by dsimp [b]; positivity, hC, δ, ?_, ?_⟩
+  · intro m hm
+    have hwidth := fit_width_pos hh hm
+    exact lt_min hδ₀ (by positivity)
+  intro f w hf hfi hwn hwy hne η hη heq ρ hρ m hm v hv hvcut
+  have hwh : h ≤ w.im := hhs.trans hwy
+  have hwpos : 0 < w.im := hs.trans_le hwy
+  have hnorm : 0 < ‖deriv f w‖ := norm_pos_iff.mpr hne
+  have hwi : w.im ≤ 1 := (Complex.im_le_norm w).trans hwn.le
+  have hvsource : 0 < v / (L * ‖deriv f w‖) := div_pos hv (mul_pos hL hnorm)
+  have hvcut' : v / (L * ‖deriv f w‖) < ribbonWidth (halfHeight h) m * v₀ := by
+    apply conformal_experiment_mesh_cutoff hL hnorm (mul_pos (fit_width_pos hh hm) hv₀) w hwi
+    exact hvcut.trans_le (mul_le_mul_of_nonneg_right (min_le_right _ _) (by positivity))
+  obtain ⟨g, hg⟩ := exists_global_ribbon_progress _ (physicalExperimentChart_closedEmbedding hh w hwh hm)
+  let τ := ribbonFrontStop g γ (fitCompletion h w m)
+  let E := RibbonStepSuccess γ (fun _ => 0) τ (physicalExperimentChart h w m) ε κ
+    (v / (L * ‖deriv f w‖)) (c * (h / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ)
+    (closedBall (center h w) (h / 20))
+  have hp : ENNReal.ofReal (Real.exp (-C * (m : ℝ) ^ 2)) ≤
+      finiteCenteredLaw μ κ B γ G w (experimentRadiusLevel h) E :=
+    hmass w hwn.le hwh m hm (v / (L * ‖deriv f w‖)) hvsource hvcut' g hg
+  apply hp.trans
+  apply measure_mono_ae
+  apply (withDensity_absolutelyContinuous μ _).ae_le
+  filter_upwards [hρ] with ω hρω hω
+  have hpref : () ∈ GoodRibbonPrefix (fun _ : Unit => γ ω) (fun _ => τ ω)
+      (physicalExperimentChart h w m) ε := hω.1
+  have hbefore := (good_physicalExperiment_before_radius ((hc ω).const_mul _) (hw ω)
+    hh w hwn.le hwh hm (τ ω) ε hpref).1
+  have hTρ : (((τ ω).untopD 0 : ℝ≥0) : WithTop ℝ≥0) ≤ ρ ω := by
+    cases ht : τ ω with
+    | top => exact (hpref.1 ht).elim
+    | coe T =>
+      simp only [ht, WithTop.untopD_coe] at hbefore ⊢
+      rw [hρω]
+      exact_mod_cast hbefore.le
+  have htrans := htransfer f w hf hfi hwn hwy hne (γ ω) (η ω) 0 ((τ ω).untopD 0)
+    (fun t _ ht => heq ω t ht) v hv
+    (hvcut.trans_le (mul_le_mul_of_nonneg_right (min_le_left _ _) (by positivity)))
+  have hthresh := dense_conformal_mass_threshold κ A c h ‖deriv f w‖ hκ hA hcm hh hnorm w
+    (lt_of_lt_of_le hs hwy) hwn.le m
+  calc
+    _ ≤ ENNReal.ofReal (A * ‖deriv f w‖ ^ dimension κ) *
+        ENNReal.ofReal (c * (h / 20000) ^ dimension κ * (m : ℝ) ^ codimension κ) := by
+      rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ A * ‖deriv f w‖ ^ dimension κ)]
+      exact ENNReal.ofReal_le_ofReal hthresh
+    _ ≤ ENNReal.ofReal (A * ‖deriv f w‖ ^ dimension κ) *
+        segmentSausageCharge κ (γ ω) 0 ((τ ω).untopD 0) (v / (L * ‖deriv f w‖))
+          (closedBall (center h w) (h / 20)) := mul_le_mul' le_rfl hω.2
+    _ ≤ segmentSausageCharge κ (η ω) 0 ((τ ω).untopD 0) v
+        (closedBall (f w) (2 * w.im * ‖deriv f w‖)) := htrans
+    _ ≤ _ := segmentSausageCharge_le_stoppedWhole κ (sourceTracePath η hη ω) _ _ hTρ _ _
+
+end Serpentine
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ConformalSausageTransfer.lean b/lean/OAI/Probability/SLE/Gauge/ConformalSausageTransfer.lean
index 7015a040f3e55f6dfd376dcbb1e00e6852b49686..ab6fd8ed33320fc481939c9e0330b59f6712c826 100644
--- a/lean/OAI/Probability/SLE/Gauge/ConformalSausageTransfer.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ConformalSausageTransfer.lean
@@ -1,128 +1,128 @@
-import OAI.Probability.SLE.SausageScaling
-
-namespace OAI
-
-/-!
-Deterministic conformal area and sausage transfer in `rc:compact-sausage-transfer`
-of *An explicit exact Hausdorff gauge for SLE*. The local patches below may
-depend on the sausage point; their union need not be convex.
--/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped ENNReal NNReal
-
-namespace SLEExactGauge
-
-lemma complex_derivative_real_det (a : ℂ) :
-    ((ContinuousLinearMap.toSpanSingleton ℂ a).restrictScalars ℝ).det = ‖a‖ ^ 2 := by
-  simp [ContinuousLinearMap.det, LinearMap.det_restrictScalars,
-    Algebra.norm_complex_eq, Complex.normSq_eq_norm_sq]
-
-/-- An injective complex differentiable map with derivative bounded below
-expands area by at least the square of that lower bound. -/
-theorem conformal_area_image_lower
-    {F F' : ℂ → ℂ} {S : Set ℂ} (hS : MeasurableSet S)
-    (hinj : InjOn F S) (hF : ∀ z ∈ S, HasDerivAt F (F' z) z)
-    {c : ℝ} (hc : 0 ≤ c) (hbound : ∀ z ∈ S, c ≤ ‖F' z‖) :
-    ENNReal.ofReal (c ^ 2) * volume S ≤ volume (F '' S) := by
-  have harea := lintegral_abs_det_fderiv_eq_addHaar_image volume hS
-    (fun z hz => ((hF z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hinj
-  rw [← harea, ← setLIntegral_const]
-  apply setLIntegral_mono' hS
-  intro z hz
-  rw [complex_derivative_real_det, abs_of_nonneg (sq_nonneg _)]
-  exact ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ hc (hbound z hz) 2)
-
-/-- A point in a small sausage and its witnessing trace point only need to
-lie in one common Lipschitz patch. No global convexity is assumed. -/
-theorem sourceSausage_image_subset_of_lipschitz_patches
-    (F : ℂ → ℂ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0)
-    {v ell : ℝ} (hv : 0 < v) (C : ℝ≥0) (hscale : (C : ℝ) * ell ≤ 1)
-    (hpatch : ∀ z ∈ sourceSausageSet γ s t (ell * v),
-      ∃ u ∈ Icc s t, dist z (γ u) < ell * v ∧
-        ∃ K : Set ℂ, z ∈ K ∧ γ u ∈ K ∧ LipschitzOnWith C F K) :
-    F '' sourceSausageSet γ s t (ell * v) ⊆
-      sourceSausageSet (fun u => F (γ u)) s t v := by
-  rintro _ ⟨z, hz, rfl⟩
-  obtain ⟨u, hu, hdist, K, hzK, huK, hLip⟩ := hpatch z hz
-  refine ⟨u, hu, (hLip.dist_le_mul z hzK (γ u) huK).trans_lt ?_⟩
-  by_cases hC : C = 0
-  · simpa only [hC, NNReal.coe_zero, zero_mul] using hv
-  · have hCpos : 0 < (C : ℝ) := by exact_mod_cast (pos_iff_ne_zero.mpr hC)
-    calc
-      (C : ℝ) * dist z (γ u) < (C : ℝ) * (ell * v) :=
-        mul_lt_mul_of_pos_left hdist hCpos
-      _ ≤ v := by nlinarith [mul_le_mul_of_nonneg_right hscale hv.le]
-
-/-- Complex derivative bounds on convex patches give the local Lipschitz
-comparison required to send the small sausage into the physical sausage. -/
-theorem sourceSausage_image_subset_of_derivative_patches
-    (F F' : ℂ → ℂ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0)
-    {v ell : ℝ} (hv : 0 < v) (C : ℝ≥0) (hscale : (C : ℝ) * ell ≤ 1)
-    (hpatch : ∀ z ∈ sourceSausageSet γ s t (ell * v),
-      ∃ u ∈ Icc s t, dist z (γ u) < ell * v ∧
-        ∃ K : Set ℂ, Convex ℝ K ∧ z ∈ K ∧ γ u ∈ K ∧
-          (∀ w ∈ K, HasDerivWithinAt F (F' w) K w) ∧
-          (∀ w ∈ K, ‖F' w‖₊ ≤ C)) :
-    F '' sourceSausageSet γ s t (ell * v) ⊆
-      sourceSausageSet (fun u => F (γ u)) s t v := by
-  apply sourceSausage_image_subset_of_lipschitz_patches F γ s t hv C hscale
-  intro z hz
-  obtain ⟨u, hu, hdist, K, hconv, hzK, huK, hF, hbound⟩ := hpatch z hz
-  exact ⟨u, hu, hdist, K, hzK, huK,
-    hconv.lipschitzOnWith_of_nnnorm_hasDerivWithin_le hF hbound⟩
-
-/-- Normalize the area inequality at physical mesh `v`, expressing the
-small-mesh mass at `ell * v`. This is the exact factor in the manuscript. -/
-theorem normalized_conformal_area_lower
-    {F F' : ℂ → ℂ} {S A : Set ℂ} (hS : MeasurableSet S)
-    (hinj : InjOn F S) (hF : ∀ z ∈ S, HasDerivAt F (F' z) z)
-    {c v ell p : ℝ} (hc : 0 ≤ c) (hv : 0 < v) (hell : 0 < ell)
-    (hbound : ∀ z ∈ S, c ≤ ‖F' z‖) (himage : F '' S ⊆ A) :
-    ENNReal.ofReal (c ^ 2 * ell ^ p) *
-        (ENNReal.ofReal ((ell * v) ^ (-p)) * volume S) ≤
-      ENNReal.ofReal (v ^ (-p)) * volume A := by
-  have harea := (conformal_area_image_lower hS hinj hF hc hbound).trans
-    (measure_mono himage)
-  have hcoeff : c ^ 2 * ell ^ p * (ell * v) ^ (-p) = v ^ (-p) * c ^ 2 := by
-    rw [Real.mul_rpow hell.le hv.le]
-    calc
-      _ = c ^ 2 * (ell ^ p * ell ^ (-p)) * v ^ (-p) := by ring
-      _ = v ^ (-p) * c ^ 2 := by rw [← Real.rpow_add hell, add_neg_cancel, Real.rpow_zero]; ring
-  have he : ENNReal.ofReal (c ^ 2 * ell ^ p) * ENNReal.ofReal ((ell * v) ^ (-p)) =
-      ENNReal.ofReal (v ^ (-p)) * ENNReal.ofReal (c ^ 2) := by
-    rw [← ENNReal.ofReal_mul (mul_nonneg (sq_nonneg c) (Real.rpow_nonneg hell.le _)),
-      hcoeff, ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _)]
-  rw [← mul_assoc, he, mul_assoc]
-  exact mul_le_mul' le_rfl harea
-
-/-- The physical sausage inside the retained middle region receives at
-least `c² ell^p` times the small-mesh normalized mass. -/
-theorem sourceSausageMass_conformal_transfer
-    (κ : ℝ) (F F' : ℂ → ℂ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0)
-    (R : Set ℂ) {c v ell : ℝ} (hc : 0 ≤ c) (hv : 0 < v) (hell : 0 < ell)
-    (hinj : InjOn F (sourceSausageSet γ s t (ell * v)))
-    (hF : ∀ z ∈ sourceSausageSet γ s t (ell * v), HasDerivAt F (F' z) z)
-    (hbound : ∀ z ∈ sourceSausageSet γ s t (ell * v), c ≤ ‖F' z‖)
-    (C : ℝ≥0) (hscale : (C : ℝ) * ell ≤ 1)
-    (hregion : F '' sourceSausageSet γ s t (ell * v) ⊆ R)
-    (hpatch : ∀ z ∈ sourceSausageSet γ s t (ell * v),
-      ∃ u ∈ Icc s t, dist z (γ u) < ell * v ∧
-        ∃ K : Set ℂ, Convex ℝ K ∧ z ∈ K ∧ γ u ∈ K ∧
-          (∀ w ∈ K, HasDerivWithinAt F (F' w) K w) ∧
-          (∀ w ∈ K, ‖F' w‖₊ ≤ C)) :
-    ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) *
-        sourceSausageMass κ γ s t (ell * v) ≤
-      ENNReal.ofReal (v ^ (-codimension κ)) *
-        volume (R ∩ sourceSausageSet (fun u => F (γ u)) s t v) := by
-  apply normalized_conformal_area_lower (source_sausage_isOpen γ s t (ell * v)).measurableSet
-    hinj hF hc hv hell hbound
-  exact subset_inter hregion
-    (sourceSausage_image_subset_of_derivative_patches F F' γ s t hv C hscale hpatch)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.SausageScaling
+
+namespace OAI
+
+/-!
+Deterministic conformal area and sausage transfer in `rc:compact-sausage-transfer`
+of *An explicit exact Hausdorff gauge for SLE*. The local patches below may
+depend on the sausage point; their union need not be convex.
+-/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped ENNReal NNReal
+
+namespace SLEExactGauge
+
+lemma complex_derivative_real_det (a : ℂ) :
+    ((ContinuousLinearMap.toSpanSingleton ℂ a).restrictScalars ℝ).det = ‖a‖ ^ 2 := by
+  simp [ContinuousLinearMap.det, LinearMap.det_restrictScalars,
+    Algebra.norm_complex_eq, Complex.normSq_eq_norm_sq]
+
+/-- An injective complex differentiable map with derivative bounded below
+expands area by at least the square of that lower bound. -/
+theorem conformal_area_image_lower
+    {F F' : ℂ → ℂ} {S : Set ℂ} (hS : MeasurableSet S)
+    (hinj : InjOn F S) (hF : ∀ z ∈ S, HasDerivAt F (F' z) z)
+    {c : ℝ} (hc : 0 ≤ c) (hbound : ∀ z ∈ S, c ≤ ‖F' z‖) :
+    ENNReal.ofReal (c ^ 2) * volume S ≤ volume (F '' S) := by
+  have harea := lintegral_abs_det_fderiv_eq_addHaar_image volume hS
+    (fun z hz => ((hF z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hinj
+  rw [← harea, ← setLIntegral_const]
+  apply setLIntegral_mono' hS
+  intro z hz
+  rw [complex_derivative_real_det, abs_of_nonneg (sq_nonneg _)]
+  exact ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ hc (hbound z hz) 2)
+
+/-- A point in a small sausage and its witnessing trace point only need to
+lie in one common Lipschitz patch. No global convexity is assumed. -/
+theorem sourceSausage_image_subset_of_lipschitz_patches
+    (F : ℂ → ℂ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0)
+    {v ell : ℝ} (hv : 0 < v) (C : ℝ≥0) (hscale : (C : ℝ) * ell ≤ 1)
+    (hpatch : ∀ z ∈ sourceSausageSet γ s t (ell * v),
+      ∃ u ∈ Icc s t, dist z (γ u) < ell * v ∧
+        ∃ K : Set ℂ, z ∈ K ∧ γ u ∈ K ∧ LipschitzOnWith C F K) :
+    F '' sourceSausageSet γ s t (ell * v) ⊆
+      sourceSausageSet (fun u => F (γ u)) s t v := by
+  rintro _ ⟨z, hz, rfl⟩
+  obtain ⟨u, hu, hdist, K, hzK, huK, hLip⟩ := hpatch z hz
+  refine ⟨u, hu, (hLip.dist_le_mul z hzK (γ u) huK).trans_lt ?_⟩
+  by_cases hC : C = 0
+  · simpa only [hC, NNReal.coe_zero, zero_mul] using hv
+  · have hCpos : 0 < (C : ℝ) := by exact_mod_cast (pos_iff_ne_zero.mpr hC)
+    calc
+      (C : ℝ) * dist z (γ u) < (C : ℝ) * (ell * v) :=
+        mul_lt_mul_of_pos_left hdist hCpos
+      _ ≤ v := by nlinarith [mul_le_mul_of_nonneg_right hscale hv.le]
+
+/-- Complex derivative bounds on convex patches give the local Lipschitz
+comparison required to send the small sausage into the physical sausage. -/
+theorem sourceSausage_image_subset_of_derivative_patches
+    (F F' : ℂ → ℂ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0)
+    {v ell : ℝ} (hv : 0 < v) (C : ℝ≥0) (hscale : (C : ℝ) * ell ≤ 1)
+    (hpatch : ∀ z ∈ sourceSausageSet γ s t (ell * v),
+      ∃ u ∈ Icc s t, dist z (γ u) < ell * v ∧
+        ∃ K : Set ℂ, Convex ℝ K ∧ z ∈ K ∧ γ u ∈ K ∧
+          (∀ w ∈ K, HasDerivWithinAt F (F' w) K w) ∧
+          (∀ w ∈ K, ‖F' w‖₊ ≤ C)) :
+    F '' sourceSausageSet γ s t (ell * v) ⊆
+      sourceSausageSet (fun u => F (γ u)) s t v := by
+  apply sourceSausage_image_subset_of_lipschitz_patches F γ s t hv C hscale
+  intro z hz
+  obtain ⟨u, hu, hdist, K, hconv, hzK, huK, hF, hbound⟩ := hpatch z hz
+  exact ⟨u, hu, hdist, K, hzK, huK,
+    hconv.lipschitzOnWith_of_nnnorm_hasDerivWithin_le hF hbound⟩
+
+/-- Normalize the area inequality at physical mesh `v`, expressing the
+small-mesh mass at `ell * v`. This is the exact factor in the manuscript. -/
+theorem normalized_conformal_area_lower
+    {F F' : ℂ → ℂ} {S A : Set ℂ} (hS : MeasurableSet S)
+    (hinj : InjOn F S) (hF : ∀ z ∈ S, HasDerivAt F (F' z) z)
+    {c v ell p : ℝ} (hc : 0 ≤ c) (hv : 0 < v) (hell : 0 < ell)
+    (hbound : ∀ z ∈ S, c ≤ ‖F' z‖) (himage : F '' S ⊆ A) :
+    ENNReal.ofReal (c ^ 2 * ell ^ p) *
+        (ENNReal.ofReal ((ell * v) ^ (-p)) * volume S) ≤
+      ENNReal.ofReal (v ^ (-p)) * volume A := by
+  have harea := (conformal_area_image_lower hS hinj hF hc hbound).trans
+    (measure_mono himage)
+  have hcoeff : c ^ 2 * ell ^ p * (ell * v) ^ (-p) = v ^ (-p) * c ^ 2 := by
+    rw [Real.mul_rpow hell.le hv.le]
+    calc
+      _ = c ^ 2 * (ell ^ p * ell ^ (-p)) * v ^ (-p) := by ring
+      _ = v ^ (-p) * c ^ 2 := by rw [← Real.rpow_add hell, add_neg_cancel, Real.rpow_zero]; ring
+  have he : ENNReal.ofReal (c ^ 2 * ell ^ p) * ENNReal.ofReal ((ell * v) ^ (-p)) =
+      ENNReal.ofReal (v ^ (-p)) * ENNReal.ofReal (c ^ 2) := by
+    rw [← ENNReal.ofReal_mul (mul_nonneg (sq_nonneg c) (Real.rpow_nonneg hell.le _)),
+      hcoeff, ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _)]
+  rw [← mul_assoc, he, mul_assoc]
+  exact mul_le_mul' le_rfl harea
+
+/-- The physical sausage inside the retained middle region receives at
+least `c² ell^p` times the small-mesh normalized mass. -/
+theorem sourceSausageMass_conformal_transfer
+    (κ : ℝ) (F F' : ℂ → ℂ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0)
+    (R : Set ℂ) {c v ell : ℝ} (hc : 0 ≤ c) (hv : 0 < v) (hell : 0 < ell)
+    (hinj : InjOn F (sourceSausageSet γ s t (ell * v)))
+    (hF : ∀ z ∈ sourceSausageSet γ s t (ell * v), HasDerivAt F (F' z) z)
+    (hbound : ∀ z ∈ sourceSausageSet γ s t (ell * v), c ≤ ‖F' z‖)
+    (C : ℝ≥0) (hscale : (C : ℝ) * ell ≤ 1)
+    (hregion : F '' sourceSausageSet γ s t (ell * v) ⊆ R)
+    (hpatch : ∀ z ∈ sourceSausageSet γ s t (ell * v),
+      ∃ u ∈ Icc s t, dist z (γ u) < ell * v ∧
+        ∃ K : Set ℂ, Convex ℝ K ∧ z ∈ K ∧ γ u ∈ K ∧
+          (∀ w ∈ K, HasDerivWithinAt F (F' w) K w) ∧
+          (∀ w ∈ K, ‖F' w‖₊ ≤ C)) :
+    ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) *
+        sourceSausageMass κ γ s t (ell * v) ≤
+      ENNReal.ofReal (v ^ (-codimension κ)) *
+        volume (R ∩ sourceSausageSet (fun u => F (γ u)) s t v) := by
+  apply normalized_conformal_area_lower (source_sausage_isOpen γ s t (ell * v)).measurableSet
+    hinj hF hc hv hell hbound
+  exact subset_inter hregion
+    (sourceSausage_image_subset_of_derivative_patches F F' γ s t hv C hscale hpatch)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverChangeOfLaw.lean b/lean/OAI/Probability/SLE/Gauge/CoverChangeOfLaw.lean
index 7a5c0fbaf2b5453e0075acffa7f8a716ca0c7909..662b77ad91bf6cb47fa3aa160d0ed48a1864d6a2 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverChangeOfLaw.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverChangeOfLaw.lean
@@ -1,142 +1,142 @@
-import OAI.Probability.SLE.FiniteStoppedLaw
-import OAI.Probability.SLE.Gauge.DenseCoverCore
-
-namespace OAI
-
-/-!
-The finite-stop change of measure in the missed-cell estimate of
-`prop:dense-cover-upper`, *An explicit exact Hausdorff gauge for SLE*.
-
-A conditional approach estimate is the hypothesis for the conditioning
-and change-of-law steps.
--/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- Restricting a normalized density to an event exactly recovers its
-unnormalized integral. No measurability of the density is needed here. -/
-theorem setLIntegral_eq_mul_normalizedDensity
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (w : Ω → ℝ) {a : ℝ} (ha : 0 < a) {A : Set Ω} (hA : MeasurableSet A) :
-    (∫⁻ ω in A, ENNReal.ofReal (w ω) ∂P) =
-      ENNReal.ofReal a *
-        P.withDensity (fun ω => ENNReal.ofReal (w ω / a)) A := by
-  rw [withDensity_apply _ hA,
-    ← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
-  apply lintegral_congr
-  intro ω
-  rw [ENNReal.ofReal_div_of_pos ha,
-    ENNReal.mul_div_cancel (ENNReal.ofReal_ne_zero_iff.mpr ha) ENNReal.ofReal_ne_top]
-
-/-- The precise event identity for the finite centered law, at a fixed
-logarithmic conformal-radius stop. -/
-theorem finiteCenteredLaw_event_identity
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ)
-    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (u : ℝ) {A : Set Ω} (hA : MeasurableSet A) :
-    (∫⁻ ω in A, ENNReal.ofReal
-      (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u) ∂P) =
-      ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z u A :=
-  setLIntegral_eq_mul_normalizedDensity P _ (initialGreen_pos κ z hz) hA
-
-/-- A conditional approach bound can be integrated on any event determined
-by the conditioning sigma algebra. The smaller event need not be measurable. -/
-theorem measure_le_setLIntegral_of_conditional_approach
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    {E V A : Set Ω} (hV : MeasurableSet[mΩ] V) (hA : MeasurableSet[m] A)
-    (hE : E ⊆ V ∩ A) (W : Ω → ℝ≥0∞) (K : ℝ≥0∞) (hK : K ≠ ∞)
-    (happroach : ∀ᵐ ω ∂P, ω ∈ A →
-      P⁻[V.indicator (fun _ => 1) | m] ω ≤ K * W ω) :
-    P E ≤ K * ∫⁻ ω in A, W ω ∂P := by
-  let : MeasurableSpace Ω := mΩ
-  calc
-    P E ≤ P (V ∩ A) := measure_mono hE
-    _ = ∫⁻ ω in A, V.indicator (fun _ => 1) ω ∂P := by
-      rw [setLIntegral_indicator hV, setLIntegral_const, one_mul]
-    _ = ∫⁻ ω in A, P⁻[V.indicator (fun _ => 1) | m] ω ∂P :=
-      (setLIntegral_condLExp hm P _ hA).symm
-    _ ≤ ∫⁻ ω in A, K * W ω ∂P :=
-      lintegral_mono_ae ((ae_restrict_iff' (hm A hA)).mpr happroach)
-    _ = K * ∫⁻ ω in A, W ω ∂P := lintegral_const_mul' K W hK
-
-/-- Transfer a visited cell with no candidate from the original law to a
-finite centered law. `A` is the adapted failure event and `V` the visit event.
-The displayed conditional approach bound is the only probabilistic estimate
-assumed by this reduction. -/
-theorem finiteCenteredLaw_missed_cell_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ)
-    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ)
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    {E V A : Set Ω} (hV : MeasurableSet[mΩ] V) (hA : MeasurableSet[m] A)
-    (hE : E ⊆ V ∩ A) (C r p : ℝ)
-    (happroach : ∀ᵐ ω ∂P, ω ∈ A →
-      P⁻[V.indicator (fun _ => 1) | m] ω ≤ ENNReal.ofReal (C * r ^ p) *
-        ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-          (γ ω) (G ω) z u)) :
-    P E ≤ ENNReal.ofReal (C * r ^ p) * ENNReal.ofReal (initialGreen κ z) *
-      @finiteCenteredLaw Ω mΩ P κ B γ G z u A := by
-  let : MeasurableSpace Ω := mΩ
-  have h := measure_le_setLIntegral_of_conditional_approach P m hm hV hA hE
-    (fun ω => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z u)) (ENNReal.ofReal (C * r ^ p)) ENNReal.ofReal_ne_top happroach
-  rw [finiteCenteredLaw_event_identity P κ B γ G z hz u (hm A hA)] at h
-  simpa only [mul_assoc] using h
-
-/-- Any tilted-law failure estimate, in particular the stretched exponential
-bound from repeated adapted tests, gives the corresponding missed-cell bound. -/
-theorem finiteCenteredLaw_missed_cell_bound_of_failure_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ)
-    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ)
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    {E V A : Set Ω} (hV : MeasurableSet[mΩ] V) (hA : MeasurableSet[m] A)
-    (hE : E ⊆ V ∩ A) (C r p : ℝ) (ε : ℝ≥0∞)
-    (happroach : ∀ᵐ ω ∂P, ω ∈ A →
-      P⁻[V.indicator (fun _ => 1) | m] ω ≤ ENNReal.ofReal (C * r ^ p) *
-        ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-          (γ ω) (G ω) z u))
-    (hfailure : @finiteCenteredLaw Ω mΩ P κ B γ G z u A ≤ ε) :
-    P E ≤ ENNReal.ofReal (C * r ^ p) * ENNReal.ofReal (initialGreen κ z) * ε := by
-  let : MeasurableSpace Ω := mΩ
-  exact (finiteCenteredLaw_missed_cell_bound P κ B γ G z hz u m hm hV hA hE
-    C r p happroach).trans (mul_le_mul' le_rfl hfailure)
-
-/-- The failure probability is a product bound under the centered law; no
-independence assumption is used. -/
-theorem finiteCenteredLaw_adapted_failures_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
-    (filtration : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
-    (hmeas : ∀ n, MeasurableSet[filtration (n + 1)] (failure n))
-    (q : ℝ≥0∞)
-    (hstep : ∀ n (A : Set Ω), MeasurableSet[filtration n] A →
-      finiteCenteredLaw P κ B γ G z u (A ∩ failure n) ≤
-        q * finiteCenteredLaw P κ B γ G z u A)
-    (N : ℕ) {A : Set Ω} (hA : A ⊆ ⋂ n ∈ Finset.range N, failure n) :
-    finiteCenteredLaw P κ B γ G z u A ≤ q ^ N := by
-  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F hw z hz u hu
-  calc
-    finiteCenteredLaw P κ B γ G z u A ≤
-        finiteCenteredLaw P κ B γ G z u (⋂ n ∈ Finset.range N, failure n) :=
-      measure_mono hA
-    _ ≤ q ^ N * finiteCenteredLaw P κ B γ G z u univ :=
-      adapted_failures_bound _ filtration failure hmeas q hstep N
-    _ = q ^ N := by rw [measure_univ, mul_one]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.FiniteStoppedLaw
+import OAI.Probability.SLE.Gauge.DenseCoverCore
+
+namespace OAI
+
+/-!
+The finite-stop change of measure in the missed-cell estimate of
+`prop:dense-cover-upper`, *An explicit exact Hausdorff gauge for SLE*.
+
+A conditional approach estimate is the hypothesis for the conditioning
+and change-of-law steps.
+-/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- Restricting a normalized density to an event exactly recovers its
+unnormalized integral. No measurability of the density is needed here. -/
+theorem setLIntegral_eq_mul_normalizedDensity
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (w : Ω → ℝ) {a : ℝ} (ha : 0 < a) {A : Set Ω} (hA : MeasurableSet A) :
+    (∫⁻ ω in A, ENNReal.ofReal (w ω) ∂P) =
+      ENNReal.ofReal a *
+        P.withDensity (fun ω => ENNReal.ofReal (w ω / a)) A := by
+  rw [withDensity_apply _ hA,
+    ← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+  apply lintegral_congr
+  intro ω
+  rw [ENNReal.ofReal_div_of_pos ha,
+    ENNReal.mul_div_cancel (ENNReal.ofReal_ne_zero_iff.mpr ha) ENNReal.ofReal_ne_top]
+
+/-- The precise event identity for the finite centered law, at a fixed
+logarithmic conformal-radius stop. -/
+theorem finiteCenteredLaw_event_identity
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ)
+    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (u : ℝ) {A : Set Ω} (hA : MeasurableSet A) :
+    (∫⁻ ω in A, ENNReal.ofReal
+      (stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u) ∂P) =
+      ENNReal.ofReal (initialGreen κ z) * finiteCenteredLaw P κ B γ G z u A :=
+  setLIntegral_eq_mul_normalizedDensity P _ (initialGreen_pos κ z hz) hA
+
+/-- A conditional approach bound can be integrated on any event determined
+by the conditioning sigma algebra. The smaller event need not be measurable. -/
+theorem measure_le_setLIntegral_of_conditional_approach
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    {E V A : Set Ω} (hV : MeasurableSet[mΩ] V) (hA : MeasurableSet[m] A)
+    (hE : E ⊆ V ∩ A) (W : Ω → ℝ≥0∞) (K : ℝ≥0∞) (hK : K ≠ ∞)
+    (happroach : ∀ᵐ ω ∂P, ω ∈ A →
+      P⁻[V.indicator (fun _ => 1) | m] ω ≤ K * W ω) :
+    P E ≤ K * ∫⁻ ω in A, W ω ∂P := by
+  let : MeasurableSpace Ω := mΩ
+  calc
+    P E ≤ P (V ∩ A) := measure_mono hE
+    _ = ∫⁻ ω in A, V.indicator (fun _ => 1) ω ∂P := by
+      rw [setLIntegral_indicator hV, setLIntegral_const, one_mul]
+    _ = ∫⁻ ω in A, P⁻[V.indicator (fun _ => 1) | m] ω ∂P :=
+      (setLIntegral_condLExp hm P _ hA).symm
+    _ ≤ ∫⁻ ω in A, K * W ω ∂P :=
+      lintegral_mono_ae ((ae_restrict_iff' (hm A hA)).mpr happroach)
+    _ = K * ∫⁻ ω in A, W ω ∂P := lintegral_const_mul' K W hK
+
+/-- Transfer a visited cell with no candidate from the original law to a
+finite centered law. `A` is the adapted failure event and `V` the visit event.
+The displayed conditional approach bound is the only probabilistic estimate
+assumed by this reduction. -/
+theorem finiteCenteredLaw_missed_cell_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ)
+    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ)
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    {E V A : Set Ω} (hV : MeasurableSet[mΩ] V) (hA : MeasurableSet[m] A)
+    (hE : E ⊆ V ∩ A) (C r p : ℝ)
+    (happroach : ∀ᵐ ω ∂P, ω ∈ A →
+      P⁻[V.indicator (fun _ => 1) | m] ω ≤ ENNReal.ofReal (C * r ^ p) *
+        ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+          (γ ω) (G ω) z u)) :
+    P E ≤ ENNReal.ofReal (C * r ^ p) * ENNReal.ofReal (initialGreen κ z) *
+      @finiteCenteredLaw Ω mΩ P κ B γ G z u A := by
+  let : MeasurableSpace Ω := mΩ
+  have h := measure_le_setLIntegral_of_conditional_approach P m hm hV hA hE
+    (fun ω => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z u)) (ENNReal.ofReal (C * r ^ p)) ENNReal.ofReal_ne_top happroach
+  rw [finiteCenteredLaw_event_identity P κ B γ G z hz u (hm A hA)] at h
+  simpa only [mul_assoc] using h
+
+/-- Any tilted-law failure estimate, in particular the stretched exponential
+bound from repeated adapted tests, gives the corresponding missed-cell bound. -/
+theorem finiteCenteredLaw_missed_cell_bound_of_failure_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ)
+    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ)
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    {E V A : Set Ω} (hV : MeasurableSet[mΩ] V) (hA : MeasurableSet[m] A)
+    (hE : E ⊆ V ∩ A) (C r p : ℝ) (ε : ℝ≥0∞)
+    (happroach : ∀ᵐ ω ∂P, ω ∈ A →
+      P⁻[V.indicator (fun _ => 1) | m] ω ≤ ENNReal.ofReal (C * r ^ p) *
+        ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+          (γ ω) (G ω) z u))
+    (hfailure : @finiteCenteredLaw Ω mΩ P κ B γ G z u A ≤ ε) :
+    P E ≤ ENNReal.ofReal (C * r ^ p) * ENNReal.ofReal (initialGreen κ z) * ε := by
+  let : MeasurableSpace Ω := mΩ
+  exact (finiteCenteredLaw_missed_cell_bound P κ B γ G z hz u m hm hV hA hE
+    C r p happroach).trans (mul_le_mul' le_rfl hfailure)
+
+/-- The failure probability is a product bound under the centered law; no
+independence assumption is used. -/
+theorem finiteCenteredLaw_adapted_failures_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
+    (filtration : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
+    (hmeas : ∀ n, MeasurableSet[filtration (n + 1)] (failure n))
+    (q : ℝ≥0∞)
+    (hstep : ∀ n (A : Set Ω), MeasurableSet[filtration n] A →
+      finiteCenteredLaw P κ B γ G z u (A ∩ failure n) ≤
+        q * finiteCenteredLaw P κ B γ G z u A)
+    (N : ℕ) {A : Set Ω} (hA : A ⊆ ⋂ n ∈ Finset.range N, failure n) :
+    finiteCenteredLaw P κ B γ G z u A ≤ q ^ N := by
+  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F hw z hz u hu
+  calc
+    finiteCenteredLaw P κ B γ G z u A ≤
+        finiteCenteredLaw P κ B γ G z u (⋂ n ∈ Finset.range N, failure n) :=
+      measure_mono hA
+    _ ≤ q ^ N * finiteCenteredLaw P κ B γ G z u univ :=
+      adapted_failures_bound _ filtration failure hmeas q hstep N
+    _ = q ^ N := by rw [measure_univ, mul_one]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverEvents.lean b/lean/OAI/Probability/SLE/Gauge/CoverEvents.lean
index db788eb25de4adacad70732aa6825d9c320396d5..3470ea4176aaa52acef53f40757251018c56709f 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverEvents.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverEvents.lean
@@ -1,87 +1,87 @@
-import OAI.Probability.SLE.SausageCharge
-
-namespace OAI
-
-/-! Measurability of the actual all-time cell visits and terminal-sausage
-candidate tests used by the dense cover. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-local instance coverEventPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance coverEventPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-lemma isClosed_compact_time_hit (s t : ℝ≥0) {A : Set ℂ} (hA : IsClosed A) :
-    IsClosed {f : C(ℝ≥0, ℂ) | ∃ u ∈ Icc s t, f u ∈ A} := by
-  have hf : Continuous (fun p : C(ℝ≥0, ℂ) × Icc s t => p.1 p.2) :=
-    continuous_eval.comp (continuous_fst.prodMk
-      (continuous_subtype_val.comp continuous_snd))
-  have hc := isClosedMap_fst_of_compactSpace _ (hA.preimage hf)
-  convert hc using 1
-  ext f
-  simp only [mem_ofPred_eq, mem_image, Prod.exists]
-  constructor
-  · rintro ⟨u, hu, hfu⟩
-    exact ⟨f, ⟨u, hu⟩, hfu, rfl⟩
-  · rintro ⟨g, u, hgu, rfl⟩
-    exact ⟨u, u.property, hgu⟩
-
-lemma measurableSet_whole_time_hit {A : Set ℂ} (hA : IsClosed A) :
-    MeasurableSet {f : C(ℝ≥0, ℂ) | ∃ u : ℝ≥0, f u ∈ A} := by
-  have he : {f : C(ℝ≥0, ℂ) | ∃ u : ℝ≥0, f u ∈ A} =
-      ⋃ n : ℕ, {f | ∃ u ∈ Icc 0 (n : ℝ≥0), f u ∈ A} := by
-    ext f
-    simp only [mem_ofPred_eq, mem_iUnion]
-    constructor
-    · rintro ⟨u, hu⟩
-      obtain ⟨n, hn⟩ := exists_nat_ge u
-      exact ⟨n, u, ⟨zero_le, hn⟩, hu⟩
-    · rintro ⟨n, u, _, hu⟩
-      exact ⟨u, hu⟩
-  rw [he]
-  exact MeasurableSet.iUnion (fun n => (isClosed_compact_time_hit 0 n hA).measurableSet)
-
-lemma measurableSet_trace_hits_closed
-    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    {A : Set ℂ} (hA : IsClosed A) :
-    MeasurableSet {ω | ∃ u : ℝ≥0, f ω u ∈ A} :=
-  hf (measurableSet_whole_time_hit hA)
-
-lemma measurableSet_sausage_candidate
-    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (κ v b : ℝ) (D : Set ℂ) :
-    MeasurableSet {ω | ENNReal.ofReal b ≤ wholeSausageCharge κ (f ω) v D} :=
-  measurableSet_le measurable_const ((wholeSausageCharge_measurable κ v D).comp hf)
-
-/-- Both visit and no-candidate information is terminal information. The
-finite change of law is applied later only to an adapted containing event. -/
-lemma measurableSet_visited_without_candidate
-    {Ω ι : Type*} [MeasurableSpace Ω]
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (κ v : ℝ) (I : Finset ι) (D : ι → Set ℂ) (b : ι → ℝ)
-    {Q : Set ℂ} (hQ : IsClosed Q) :
-    MeasurableSet {ω | (∃ u : ℝ≥0, f ω u ∈ Q) ∧
-      ∀ i ∈ I, wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)} := by
-  apply (measurableSet_trace_hits_closed f hf hQ).inter
-  have hm : MeasurableSet (⋂ i ∈ I,
-      {ω | wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)}) :=
-    I.measurableSet_biInter (fun i _ =>
-      measurableSet_lt ((wholeSausageCharge_measurable κ v (D i)).comp hf) measurable_const)
-  have he : {ω | ∀ i ∈ I,
-      wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)} =
-      ⋂ i ∈ I, {ω | wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)} := by
-    ext ω
-    simp only [mem_iInter, mem_ofPred_eq]
-  change MeasurableSet {ω | ∀ i ∈ I,
-    wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)}
-  rw [he]
-  exact hm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.SausageCharge
+
+namespace OAI
+
+/-! Measurability of the actual all-time cell visits and terminal-sausage
+candidate tests used by the dense cover. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+local instance coverEventPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance coverEventPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+lemma isClosed_compact_time_hit (s t : ℝ≥0) {A : Set ℂ} (hA : IsClosed A) :
+    IsClosed {f : C(ℝ≥0, ℂ) | ∃ u ∈ Icc s t, f u ∈ A} := by
+  have hf : Continuous (fun p : C(ℝ≥0, ℂ) × Icc s t => p.1 p.2) :=
+    continuous_eval.comp (continuous_fst.prodMk
+      (continuous_subtype_val.comp continuous_snd))
+  have hc := isClosedMap_fst_of_compactSpace _ (hA.preimage hf)
+  convert hc using 1
+  ext f
+  simp only [mem_ofPred_eq, mem_image, Prod.exists]
+  constructor
+  · rintro ⟨u, hu, hfu⟩
+    exact ⟨f, ⟨u, hu⟩, hfu, rfl⟩
+  · rintro ⟨g, u, hgu, rfl⟩
+    exact ⟨u, u.property, hgu⟩
+
+lemma measurableSet_whole_time_hit {A : Set ℂ} (hA : IsClosed A) :
+    MeasurableSet {f : C(ℝ≥0, ℂ) | ∃ u : ℝ≥0, f u ∈ A} := by
+  have he : {f : C(ℝ≥0, ℂ) | ∃ u : ℝ≥0, f u ∈ A} =
+      ⋃ n : ℕ, {f | ∃ u ∈ Icc 0 (n : ℝ≥0), f u ∈ A} := by
+    ext f
+    simp only [mem_ofPred_eq, mem_iUnion]
+    constructor
+    · rintro ⟨u, hu⟩
+      obtain ⟨n, hn⟩ := exists_nat_ge u
+      exact ⟨n, u, ⟨zero_le, hn⟩, hu⟩
+    · rintro ⟨n, u, _, hu⟩
+      exact ⟨u, hu⟩
+  rw [he]
+  exact MeasurableSet.iUnion (fun n => (isClosed_compact_time_hit 0 n hA).measurableSet)
+
+lemma measurableSet_trace_hits_closed
+    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    {A : Set ℂ} (hA : IsClosed A) :
+    MeasurableSet {ω | ∃ u : ℝ≥0, f ω u ∈ A} :=
+  hf (measurableSet_whole_time_hit hA)
+
+lemma measurableSet_sausage_candidate
+    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (κ v b : ℝ) (D : Set ℂ) :
+    MeasurableSet {ω | ENNReal.ofReal b ≤ wholeSausageCharge κ (f ω) v D} :=
+  measurableSet_le measurable_const ((wholeSausageCharge_measurable κ v D).comp hf)
+
+/-- Both visit and no-candidate information is terminal information. The
+finite change of law is applied later only to an adapted containing event. -/
+lemma measurableSet_visited_without_candidate
+    {Ω ι : Type*} [MeasurableSpace Ω]
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (κ v : ℝ) (I : Finset ι) (D : ι → Set ℂ) (b : ι → ℝ)
+    {Q : Set ℂ} (hQ : IsClosed Q) :
+    MeasurableSet {ω | (∃ u : ℝ≥0, f ω u ∈ Q) ∧
+      ∀ i ∈ I, wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)} := by
+  apply (measurableSet_trace_hits_closed f hf hQ).inter
+  have hm : MeasurableSet (⋂ i ∈ I,
+      {ω | wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)}) :=
+    I.measurableSet_biInter (fun i _ =>
+      measurableSet_lt ((wholeSausageCharge_measurable κ v (D i)).comp hf) measurable_const)
+  have he : {ω | ∀ i ∈ I,
+      wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)} =
+      ⋂ i ∈ I, {ω | wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)} := by
+    ext ω
+    simp only [mem_iInter, mem_ofPred_eq]
+  change MeasurableSet {ω | ∀ i ∈ I,
+    wholeSausageCharge κ (f ω) v (D i) < ENNReal.ofReal (b i)}
+  rw [he]
+  exact hm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverGaugeBounds.lean b/lean/OAI/Probability/SLE/Gauge/CoverGaugeBounds.lean
index c25a09b7703e6b18b846b304c932a789923992aa..12e16f57e4db95ab9145f0a94f3be086604d308f 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverGaugeBounds.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverGaugeBounds.lean
@@ -1,156 +1,156 @@
-import OAI.Probability.SLE.Gauge.GaugeCompletion
-
-namespace OAI
-
-/-!
-The uniform gauge cost estimate in the finite-cover step of
-`prop:dense-cover-upper`, *An explicit exact Hausdorff gauge for SLE*.
-The window starts at `floor (n / 3)`; the intermediate estimate uses
-`n / 4` so it also covers that endpoint once `n` is large.
--/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-namespace CoverGaugeBounds
-
-def radius (T : ℝ) (j : ℕ) : ℝ := Real.exp (-T * (j : ℝ))
-
-lemma radius_pos (T : ℝ) (j : ℕ) : 0 < radius T j := Real.exp_pos _
-
-lemma exp_scale_tendsto_zero {T : ℝ} (hT : 0 < T) :
-    Tendsto (fun x : ℝ => Real.exp (-T * x)) atTop (𝓝 0) := by
-  simpa only [Real.rpow_zero, one_mul] using
-    tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 0 T hT
-
-lemma radius_tendsto_zero {T : ℝ} (hT : 0 < T) :
-    Tendsto (radius T) atTop (𝓝 0) :=
-  (exp_scale_tendsto_zero hT).comp tendsto_natCast_atTop_atTop
-
-lemma scaled_quarter_radius_tendsto_zero {C T : ℝ} (hT : 0 < T) :
-    Tendsto (fun n : ℕ => C * Real.exp (-T * ((n : ℝ) / 4))) atTop (𝓝 0) := by
-  convert ((exp_scale_tendsto_zero (div_pos hT (by norm_num : (0 : ℝ) < 4))).comp
-    tendsto_natCast_atTop_atTop).const_mul C using 1
-  · funext n
-    dsimp only [Function.comp_def]
-    congr 2
-    ring
-  · simp only [mul_zero]
-
-lemma logLog_scaled_radius {C T j : ℝ} (hC : 0 < C) :
-    Real.log (Real.log (1 / (C * Real.exp (-T * j)))) =
-      Real.log (T * j - Real.log C) := by
-  congr 1
-  rw [one_div, Real.log_inv, Real.log_mul hC.ne' (Real.exp_ne_zero _), Real.log_exp]
-  ring
-
-/-- Once the clock argument is at least `e`, its log-log factor is bounded by
-`(2 log n)^q` uniformly for every index below `n`. -/
-lemma formula_bound {κ C T n j : ℝ} (hκ8 : κ < 8) (hC : 0 < C)
-    (hT : 0 < T) (hn : 1 ≤ n) (hN : T + |Real.log C| ≤ n)
-    (hj : j ≤ n) (harg : Real.exp 1 ≤ T * j - Real.log C) :
-    hFormula κ (C * Real.exp (-T * j)) ≤
-      (C ^ dimension κ * (2 : ℝ) ^ (codimension κ / 2)) *
-        Real.exp (-T * j) ^ dimension κ * (Real.log n) ^ (codimension κ / 2) := by
-  have hn0 : 0 < n := zero_lt_one.trans_le hn
-  have hq : 0 ≤ codimension κ / 2 := by dsimp [codimension, dimension]; linarith
-  have harg0 : 0 < T * j - Real.log C := (Real.exp_pos 1).trans_le harg
-  have hlog0 : 0 ≤ Real.log (T * j - Real.log C) := by
-    have he : 1 ≤ T * j - Real.log C :=
-      (Real.one_lt_exp_iff.mpr zero_lt_one).le.trans harg
-    exact Real.log_nonneg he
-  have hupper : T * j - Real.log C ≤ n ^ 2 := by
-    calc
-      _ ≤ T * n + |Real.log C| := by
-        have hmul := mul_le_mul_of_nonneg_left hj hT.le
-        linarith [neg_le_abs (Real.log C)]
-      _ ≤ (T + |Real.log C|) * n := by
-        nlinarith [abs_nonneg (Real.log C)]
-      _ ≤ n ^ 2 := by
-        have hh := mul_le_mul_of_nonneg_right hN hn0.le
-        nlinarith
-  have hlog : Real.log (T * j - Real.log C) ≤ 2 * Real.log n := by
-    have hh := Real.log_le_log harg0 hupper
-    simpa only [Real.log_pow, Nat.cast_ofNat] using hh
-  unfold hFormula
-  rw [logLog_scaled_radius hC, Real.mul_rpow hC.le (Real.exp_pos _).le]
-  calc
-    _ ≤ (C ^ dimension κ * Real.exp (-T * j) ^ dimension κ) *
-        (2 * Real.log n) ^ (codimension κ / 2) :=
-      mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hlog0 hlog hq) (by positivity)
-    _ = _ := by
-      rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 2) (Real.log_nonneg hn)]
-      ring
-
-/-- The selected disks' diameter costs have one uniform constant across the
-whole linear window of clock indices. -/
-theorem eventual_window_bound {κ C T : ℝ} (hκ8 : κ < 8) (hC : 0 < C) (hT : 0 < T)
-    {h : ℝ → ℝ} (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∃ A : ℝ, 0 < A ∧ ∀ᶠ n : ℕ in atTop, ∀ j : ℕ,
-      (n : ℝ) / 4 ≤ (j : ℝ) → j ≤ n →
-      h (C * radius T j) ≤ A * radius T j ^ dimension κ *
-        (Real.log (n : ℝ)) ^ (codimension κ / 2) := by
-  obtain ⟨δ, hδ, heq⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hf
-  refine ⟨C ^ dimension κ * (2 : ℝ) ^ (codimension κ / 2), by positivity, ?_⟩
-  have hsmall := (scaled_quarter_radius_tendsto_zero (C := C) hT).eventually (gt_mem_nhds hδ)
-  have hlarge : ∀ᶠ n : ℕ in atTop,
-      max 1 (max (T + |Real.log C|) (4 * (Real.exp 1 + |Real.log C|) / T)) ≤ (n : ℝ) :=
-    tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _)
-  filter_upwards [hsmall, hlarge] with n hnsmall hnlarge
-  intro j hjlow hjhigh
-  have hn1 : 1 ≤ (n : ℝ) := (le_max_left _ _).trans hnlarge
-  have hN : T + |Real.log C| ≤ (n : ℝ) :=
-    (le_max_left _ _).trans ((le_max_right _ _).trans hnlarge)
-  have hnclock : 4 * (Real.exp 1 + |Real.log C|) / T ≤ (n : ℝ) :=
-    (le_max_right _ _).trans ((le_max_right _ _).trans hnlarge)
-  have harg : Real.exp 1 ≤ T * (j : ℝ) - Real.log C := by
-    have hb := (div_le_iff₀ hT).mp hnclock
-    have hj := mul_le_mul_of_nonneg_left hjlow hT.le
-    linarith [le_abs_self (Real.log C)]
-  have hrad : C * radius T j < δ := by
-    apply lt_of_le_of_lt _ hnsmall
-    apply mul_le_mul_of_nonneg_left _ hC.le
-    apply Real.exp_le_exp.mpr
-    exact mul_le_mul_of_nonpos_left hjlow (neg_nonpos.mpr hT.le)
-  rw [heq ⟨mul_pos hC (radius_pos T j), hrad⟩]
-  exact formula_bound hκ8 hC hT hn1 hN (by exact_mod_cast hjhigh) harg
-
-lemma quarter_le_floor_third {n : ℕ} (hn : 12 ≤ n) :
-    (n : ℝ) / 4 ≤ (⌊(n : ℝ) / 3⌋₊ : ℝ) := by
-  have hf := Nat.lt_floor_add_one ((n : ℝ) / 3)
-  have hnreal : (12 : ℝ) ≤ n := by exact_mod_cast hn
-  linarith
-
-/-- The source uses `J₀ = floor (n/3)`, so the endpoint is included in the
-uniform estimate rather than silently replaced by `n/3`. -/
-theorem eventual_floor_window_bound {κ C T : ℝ} (hκ8 : κ < 8) (hC : 0 < C) (hT : 0 < T)
-    {h : ℝ → ℝ} (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∃ A : ℝ, 0 < A ∧ ∀ᶠ n : ℕ in atTop, ∀ j : ℕ,
-      ⌊(n : ℝ) / 3⌋₊ ≤ j → j ≤ n →
-      h (C * radius T j) ≤ A * radius T j ^ dimension κ *
-        (Real.log (n : ℝ)) ^ (codimension κ / 2) := by
-  obtain ⟨A, hA, hbound⟩ := eventual_window_bound hκ8 hC hT hf
-  refine ⟨A, hA, ?_⟩
-  filter_upwards [hbound, eventually_ge_atTop 12] with n hn hn12
-  intro j hj hjn
-  exact hn j ((quarter_le_floor_third hn12).trans (by exact_mod_cast hj)) hjn
-
-/-- In particular the largest selected diameter, at the first tested clock
-index, tends to zero. -/
-theorem floor_window_diameter_tendsto_zero {C T : ℝ} (hT : 0 < T) :
-    Tendsto (fun n : ℕ => C * radius T ⌊(n : ℝ) / 3⌋₊) atTop (𝓝 0) := by
-  have hindex : Tendsto (fun n : ℕ => ⌊(n : ℝ) / 3⌋₊) atTop atTop := by
-    apply tendsto_nat_floor_atTop.comp
-    exact (tendsto_natCast_atTop_atTop.atTop_div_const (by norm_num : (0 : ℝ) < 3))
-  simpa only [Function.comp_def, mul_zero] using ((radius_tendsto_zero hT).comp hindex).const_mul C
-
-end CoverGaugeBounds
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.GaugeCompletion
+
+namespace OAI
+
+/-!
+The uniform gauge cost estimate in the finite-cover step of
+`prop:dense-cover-upper`, *An explicit exact Hausdorff gauge for SLE*.
+The window starts at `floor (n / 3)`; the intermediate estimate uses
+`n / 4` so it also covers that endpoint once `n` is large.
+-/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+namespace CoverGaugeBounds
+
+def radius (T : ℝ) (j : ℕ) : ℝ := Real.exp (-T * (j : ℝ))
+
+lemma radius_pos (T : ℝ) (j : ℕ) : 0 < radius T j := Real.exp_pos _
+
+lemma exp_scale_tendsto_zero {T : ℝ} (hT : 0 < T) :
+    Tendsto (fun x : ℝ => Real.exp (-T * x)) atTop (𝓝 0) := by
+  simpa only [Real.rpow_zero, one_mul] using
+    tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 0 T hT
+
+lemma radius_tendsto_zero {T : ℝ} (hT : 0 < T) :
+    Tendsto (radius T) atTop (𝓝 0) :=
+  (exp_scale_tendsto_zero hT).comp tendsto_natCast_atTop_atTop
+
+lemma scaled_quarter_radius_tendsto_zero {C T : ℝ} (hT : 0 < T) :
+    Tendsto (fun n : ℕ => C * Real.exp (-T * ((n : ℝ) / 4))) atTop (𝓝 0) := by
+  convert ((exp_scale_tendsto_zero (div_pos hT (by norm_num : (0 : ℝ) < 4))).comp
+    tendsto_natCast_atTop_atTop).const_mul C using 1
+  · funext n
+    dsimp only [Function.comp_def]
+    congr 2
+    ring
+  · simp only [mul_zero]
+
+lemma logLog_scaled_radius {C T j : ℝ} (hC : 0 < C) :
+    Real.log (Real.log (1 / (C * Real.exp (-T * j)))) =
+      Real.log (T * j - Real.log C) := by
+  congr 1
+  rw [one_div, Real.log_inv, Real.log_mul hC.ne' (Real.exp_ne_zero _), Real.log_exp]
+  ring
+
+/-- Once the clock argument is at least `e`, its log-log factor is bounded by
+`(2 log n)^q` uniformly for every index below `n`. -/
+lemma formula_bound {κ C T n j : ℝ} (hκ8 : κ < 8) (hC : 0 < C)
+    (hT : 0 < T) (hn : 1 ≤ n) (hN : T + |Real.log C| ≤ n)
+    (hj : j ≤ n) (harg : Real.exp 1 ≤ T * j - Real.log C) :
+    hFormula κ (C * Real.exp (-T * j)) ≤
+      (C ^ dimension κ * (2 : ℝ) ^ (codimension κ / 2)) *
+        Real.exp (-T * j) ^ dimension κ * (Real.log n) ^ (codimension κ / 2) := by
+  have hn0 : 0 < n := zero_lt_one.trans_le hn
+  have hq : 0 ≤ codimension κ / 2 := by dsimp [codimension, dimension]; linarith
+  have harg0 : 0 < T * j - Real.log C := (Real.exp_pos 1).trans_le harg
+  have hlog0 : 0 ≤ Real.log (T * j - Real.log C) := by
+    have he : 1 ≤ T * j - Real.log C :=
+      (Real.one_lt_exp_iff.mpr zero_lt_one).le.trans harg
+    exact Real.log_nonneg he
+  have hupper : T * j - Real.log C ≤ n ^ 2 := by
+    calc
+      _ ≤ T * n + |Real.log C| := by
+        have hmul := mul_le_mul_of_nonneg_left hj hT.le
+        linarith [neg_le_abs (Real.log C)]
+      _ ≤ (T + |Real.log C|) * n := by
+        nlinarith [abs_nonneg (Real.log C)]
+      _ ≤ n ^ 2 := by
+        have hh := mul_le_mul_of_nonneg_right hN hn0.le
+        nlinarith
+  have hlog : Real.log (T * j - Real.log C) ≤ 2 * Real.log n := by
+    have hh := Real.log_le_log harg0 hupper
+    simpa only [Real.log_pow, Nat.cast_ofNat] using hh
+  unfold hFormula
+  rw [logLog_scaled_radius hC, Real.mul_rpow hC.le (Real.exp_pos _).le]
+  calc
+    _ ≤ (C ^ dimension κ * Real.exp (-T * j) ^ dimension κ) *
+        (2 * Real.log n) ^ (codimension κ / 2) :=
+      mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hlog0 hlog hq) (by positivity)
+    _ = _ := by
+      rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 2) (Real.log_nonneg hn)]
+      ring
+
+/-- The selected disks' diameter costs have one uniform constant across the
+whole linear window of clock indices. -/
+theorem eventual_window_bound {κ C T : ℝ} (hκ8 : κ < 8) (hC : 0 < C) (hT : 0 < T)
+    {h : ℝ → ℝ} (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∃ A : ℝ, 0 < A ∧ ∀ᶠ n : ℕ in atTop, ∀ j : ℕ,
+      (n : ℝ) / 4 ≤ (j : ℝ) → j ≤ n →
+      h (C * radius T j) ≤ A * radius T j ^ dimension κ *
+        (Real.log (n : ℝ)) ^ (codimension κ / 2) := by
+  obtain ⟨δ, hδ, heq⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hf
+  refine ⟨C ^ dimension κ * (2 : ℝ) ^ (codimension κ / 2), by positivity, ?_⟩
+  have hsmall := (scaled_quarter_radius_tendsto_zero (C := C) hT).eventually (gt_mem_nhds hδ)
+  have hlarge : ∀ᶠ n : ℕ in atTop,
+      max 1 (max (T + |Real.log C|) (4 * (Real.exp 1 + |Real.log C|) / T)) ≤ (n : ℝ) :=
+    tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _)
+  filter_upwards [hsmall, hlarge] with n hnsmall hnlarge
+  intro j hjlow hjhigh
+  have hn1 : 1 ≤ (n : ℝ) := (le_max_left _ _).trans hnlarge
+  have hN : T + |Real.log C| ≤ (n : ℝ) :=
+    (le_max_left _ _).trans ((le_max_right _ _).trans hnlarge)
+  have hnclock : 4 * (Real.exp 1 + |Real.log C|) / T ≤ (n : ℝ) :=
+    (le_max_right _ _).trans ((le_max_right _ _).trans hnlarge)
+  have harg : Real.exp 1 ≤ T * (j : ℝ) - Real.log C := by
+    have hb := (div_le_iff₀ hT).mp hnclock
+    have hj := mul_le_mul_of_nonneg_left hjlow hT.le
+    linarith [le_abs_self (Real.log C)]
+  have hrad : C * radius T j < δ := by
+    apply lt_of_le_of_lt _ hnsmall
+    apply mul_le_mul_of_nonneg_left _ hC.le
+    apply Real.exp_le_exp.mpr
+    exact mul_le_mul_of_nonpos_left hjlow (neg_nonpos.mpr hT.le)
+  rw [heq ⟨mul_pos hC (radius_pos T j), hrad⟩]
+  exact formula_bound hκ8 hC hT hn1 hN (by exact_mod_cast hjhigh) harg
+
+lemma quarter_le_floor_third {n : ℕ} (hn : 12 ≤ n) :
+    (n : ℝ) / 4 ≤ (⌊(n : ℝ) / 3⌋₊ : ℝ) := by
+  have hf := Nat.lt_floor_add_one ((n : ℝ) / 3)
+  have hnreal : (12 : ℝ) ≤ n := by exact_mod_cast hn
+  linarith
+
+/-- The source uses `J₀ = floor (n/3)`, so the endpoint is included in the
+uniform estimate rather than silently replaced by `n/3`. -/
+theorem eventual_floor_window_bound {κ C T : ℝ} (hκ8 : κ < 8) (hC : 0 < C) (hT : 0 < T)
+    {h : ℝ → ℝ} (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∃ A : ℝ, 0 < A ∧ ∀ᶠ n : ℕ in atTop, ∀ j : ℕ,
+      ⌊(n : ℝ) / 3⌋₊ ≤ j → j ≤ n →
+      h (C * radius T j) ≤ A * radius T j ^ dimension κ *
+        (Real.log (n : ℝ)) ^ (codimension κ / 2) := by
+  obtain ⟨A, hA, hbound⟩ := eventual_window_bound hκ8 hC hT hf
+  refine ⟨A, hA, ?_⟩
+  filter_upwards [hbound, eventually_ge_atTop 12] with n hn hn12
+  intro j hj hjn
+  exact hn j ((quarter_le_floor_third hn12).trans (by exact_mod_cast hj)) hjn
+
+/-- In particular the largest selected diameter, at the first tested clock
+index, tends to zero. -/
+theorem floor_window_diameter_tendsto_zero {C T : ℝ} (hT : 0 < T) :
+    Tendsto (fun n : ℕ => C * radius T ⌊(n : ℝ) / 3⌋₊) atTop (𝓝 0) := by
+  have hindex : Tendsto (fun n : ℕ => ⌊(n : ℝ) / 3⌋₊) atTop atTop := by
+    apply tendsto_nat_floor_atTop.comp
+    exact (tendsto_natCast_atTop_atTop.atTop_div_const (by norm_num : (0 : ℝ) < 3))
+  simpa only [Function.comp_def, mul_zero] using ((radius_tendsto_zero hT).comp hindex).const_mul C
+
+end CoverGaugeBounds
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverGrid.lean b/lean/OAI/Probability/SLE/Gauge/CoverGrid.lean
index a54acd5e0001dd0fedbece4a12ae9337bdfcbf2a..e472d7c1727119a37ddc00595f35324c7884a2f2 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverGrid.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverGrid.lean
@@ -1,167 +1,167 @@
-import OAI.Probability.SLE.Gauge.CoverEvents
-
-namespace OAI
-
-/-! A finite physical grid in a bounded disk, with the planar cell count
-needed by the missed-cell estimate. Closed disks are used as fine cells. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-def coverGridCenter (r : ℝ) (ij : ℤ × ℤ) : ℂ :=
-  (r * (ij.1 : ℝ) : ℝ) + (r * (ij.2 : ℝ) : ℝ) * Complex.I
-
-def coverGridIndex (R r : ℝ) : Finset (ℤ × ℤ) :=
-  let N : ℤ := ⌈R / r⌉ + 1
-  (Finset.Icc (-N) N) ×ˢ (Finset.Icc (-N) N)
-
-def retainedCoverGridIndex (R r : ℝ) (K : Set ℂ) : Finset (ℤ × ℤ) := by
-  classical
-  exact (coverGridIndex R r).filter (fun ij =>
-    (K ∩ closedBall (coverGridCenter r ij) (2 * r)).Nonempty)
-
-lemma coverGrid_coordinate {R r x : ℝ} (_hR : 0 ≤ R) (hr : 0 < r)
-    (hx : |x| ≤ R) :
-    -(⌈R / r⌉ + 1) ≤ ⌊x / r⌋ ∧ ⌊x / r⌋ ≤ ⌈R / r⌉ + 1 := by
-  have hl := Int.sub_one_lt_floor (x / r)
-  have hu := Int.floor_le (x / r)
-  have hc := Int.le_ceil (R / r)
-  have hxl : -R / r ≤ x / r := div_le_div_of_nonneg_right (abs_le.mp hx).1 hr.le
-  have hxl' : -(R / r) ≤ x / r := by simpa only [neg_div] using hxl
-  have hxu : x / r ≤ R / r := div_le_div_of_nonneg_right (abs_le.mp hx).2 hr.le
-  have hlow : ((-(⌈R / r⌉ + 1) : ℤ) : ℝ) ≤ (⌊x / r⌋ : ℝ) := by
-    push_cast
-    linarith
-  have hhigh : (⌊x / r⌋ : ℝ) ≤ ((⌈R / r⌉ + 1 : ℤ) : ℝ) := by
-    push_cast
-    linarith
-  exact ⟨by exact_mod_cast hlow, by exact_mod_cast hhigh⟩
-
-lemma coverGrid_floor_distance {r : ℝ} (hr : 0 < r) (z : ℂ) :
-    dist z (coverGridCenter r (⌊z.re / r⌋, ⌊z.im / r⌋)) ≤ 2 * r := by
-  have hre0 : 0 ≤ z.re - r * (⌊z.re / r⌋ : ℝ) := by
-    have hh := (Int.floor_le (z.re / r))
-    have hm := (mul_le_mul_of_nonneg_left hh hr.le)
-    rw [mul_div_cancel₀ _ hr.ne'] at hm
-    linarith
-  have hre1 : z.re - r * (⌊z.re / r⌋ : ℝ) ≤ r := by
-    have hh := mul_lt_mul_of_pos_left (Int.lt_floor_add_one (z.re / r)) hr
-    rw [mul_div_cancel₀ _ hr.ne'] at hh
-    nlinarith
-  have him0 : 0 ≤ z.im - r * (⌊z.im / r⌋ : ℝ) := by
-    have hh := mul_le_mul_of_nonneg_left (Int.floor_le (z.im / r)) hr.le
-    rw [mul_div_cancel₀ _ hr.ne'] at hh
-    linarith
-  have him1 : z.im - r * (⌊z.im / r⌋ : ℝ) ≤ r := by
-    have hh := mul_lt_mul_of_pos_left (Int.lt_floor_add_one (z.im / r)) hr
-    rw [mul_div_cancel₀ _ hr.ne'] at hh
-    nlinarith
-  rw [dist_eq_norm]
-  apply (Complex.norm_le_abs_re_add_abs_im _).trans
-  simp only [coverGridCenter, Complex.sub_re, Complex.sub_im, Complex.add_re,
-    Complex.add_im, Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re,
-    Complex.mul_im, Complex.I_re, Complex.I_im, mul_zero, mul_one,
-    sub_zero, add_zero, zero_add, abs_of_nonneg hre0, abs_of_nonneg him0]
-  linarith
-
-theorem coverGrid_covers_disk {R r : ℝ} (hR : 0 ≤ R) (hr : 0 < r) :
-    closedBall (0 : ℂ) R ⊆
-      ⋃ ij ∈ coverGridIndex R r, closedBall (coverGridCenter r ij) (2 * r) := by
-  intro z hz
-  have hzn : ‖z‖ ≤ R := by simpa only [mem_closedBall, dist_zero_right] using hz
-  have hre := coverGrid_coordinate hR hr ((Complex.abs_re_le_norm z).trans hzn)
-  have him := coverGrid_coordinate hR hr ((Complex.abs_im_le_norm z).trans hzn)
-  apply mem_iUnion₂.mpr
-  refine ⟨(⌊z.re / r⌋, ⌊z.im / r⌋), ?_, coverGrid_floor_distance hr z⟩
-  simpa only [coverGridIndex, Finset.mem_product, Finset.mem_Icc] using And.intro hre him
-
-/-- The constant depends on the bounded disk, and not on the small mesh. -/
-theorem coverGrid_card_bound {R r : ℝ} (hR : 0 ≤ R) (hr : 0 < r) (hr1 : r ≤ 1) :
-    ((coverGridIndex R r).card : ℝ) ≤ (2 * R + 5) ^ 2 * r ^ (-2 : ℝ) := by
-  let N : ℤ := ⌈R / r⌉ + 1
-  have hN : 0 ≤ N := by
-    have hh : (0 : ℝ) ≤ (⌈R / r⌉ : ℝ) := (div_nonneg hR hr.le).trans (Int.le_ceil _)
-    have hh' : (0 : ℤ) ≤ ⌈R / r⌉ := by exact_mod_cast hh
-    dsimp [N]
-    omega
-  have hcard : ((coverGridIndex R r).card : ℝ) = (2 * (N : ℝ) + 1) ^ 2 := by
-    have hh := Int.card_Icc_of_le (a := -N) (b := N) (by omega)
-    have hhR : ((Finset.Icc (-N) N).card : ℝ) = 2 * (N : ℝ) + 1 := by
-      have hh' : ((Finset.Icc (-N) N).card : ℝ) = (N : ℝ) + 1 - -(N : ℝ) := by
-        exact_mod_cast hh
-      linarith
-    change (((Finset.Icc (-N) N) ×ˢ (Finset.Icc (-N) N)).card : ℝ) = _
-    rw [Finset.card_product, Nat.cast_mul, hhR]
-    ring
-  have hNbound : (N : ℝ) ≤ R / r + 2 := by
-    have hh := Int.ceil_lt_add_one (R / r)
-    dsimp [N]
-    push_cast
-    linarith
-  have hside : 2 * (N : ℝ) + 1 ≤ (2 * R + 5) / r := by
-    apply (le_div_iff₀ hr).mpr
-    have hh := mul_le_mul_of_nonneg_right hNbound hr.le
-    rw [add_mul, div_mul_cancel₀ _ hr.ne'] at hh
-    nlinarith
-  rw [hcard, Real.rpow_neg hr.le, Real.rpow_two]
-  calc
-    _ ≤ ((2 * R + 5) / r) ^ 2 := pow_le_pow_left₀ (by exact_mod_cast (show (0 : ℤ) ≤ 2 * N + 1 by omega)) hside 2
-    _ = _ := by rw [div_pow, div_eq_mul_inv]
-
-theorem retainedCoverGrid_covers {R r : ℝ} (hR : 0 ≤ R) (hr : 0 < r)
-    {K : Set ℂ} (hK : K ⊆ closedBall 0 R) :
-    K ⊆ ⋃ ij ∈ retainedCoverGridIndex R r K,
-      closedBall (coverGridCenter r ij) (2 * r) := by
-  classical
-  intro z hz
-  obtain ⟨ij, hij, hzcell⟩ := mem_iUnion₂.mp (coverGrid_covers_disk hR hr (hK hz))
-  exact mem_iUnion₂.mpr ⟨ij, Finset.mem_filter.mpr ⟨hij, ⟨z, hz, hzcell⟩⟩, hzcell⟩
-
-theorem retainedCoverGrid_card_bound {R r : ℝ} (hR : 0 ≤ R)
-    (hr : 0 < r) (hr1 : r ≤ 1) (K : Set ℂ) :
-    ((retainedCoverGridIndex R r K).card : ℝ) ≤ (2 * R + 5) ^ 2 * r ^ (-2 : ℝ) := by
-  classical
-  exact (Nat.cast_le.mpr (Finset.card_filter_le _ _)).trans (coverGrid_card_bound hR hr hr1)
-
-/-- Retaining only cells meeting the interior cutoff keeps all their centers
-away from the real boundary and in one fixed enlarged bounded disk. -/
-theorem retainedCoverGrid_center_bounds {R r η : ℝ} {K : Set ℂ}
-    (hK : K ⊆ closedBall 0 R) (hheight : ∀ z ∈ K, η ≤ z.im)
-    {ij : ℤ × ℤ} (hij : ij ∈ retainedCoverGridIndex R r K) :
-    ‖coverGridCenter r ij‖ ≤ R + 2 * r ∧
-      η - 2 * r ≤ (coverGridCenter r ij).im := by
-  classical
-  obtain ⟨z, hzK, hzcell⟩ := (Finset.mem_filter.mp hij).2
-  have hdist : dist z (coverGridCenter r ij) ≤ 2 * r := hzcell
-  have hzn : ‖z‖ ≤ R := by simpa only [mem_closedBall, dist_zero_right] using hK hzK
-  have hnorm : ‖coverGridCenter r ij‖ ≤ ‖z‖ + dist z (coverGridCenter r ij) := by
-    have hh := dist_triangle (coverGridCenter r ij) z 0
-    simpa only [dist_zero_right, dist_comm (coverGridCenter r ij) z, add_comm] using hh
-  have him := (Complex.abs_im_le_norm (z - coverGridCenter r ij)).trans
-    (by simpa only [dist_eq_norm] using hdist)
-  rw [Complex.sub_im] at him
-  constructor
-  · linarith
-  · have hh := (abs_le.mp him).2
-    have hz := hheight z hzK
-    linarith
-
-lemma coverGrid_cell_diameter {r : ℝ} (_hr : 0 < r) (ij : ℤ × ℤ) :
-    ediam (closedBall (coverGridCenter r ij) (2 * r)) ≤ ENNReal.ofReal (4 * r) := by
-  apply ediam_le_of_forall_dist_le
-  intro x hx y hy
-  have hxy := dist_triangle x (coverGridCenter r ij) y
-  have hx' : dist x (coverGridCenter r ij) ≤ 2 * r := hx
-  have hy' : dist (coverGridCenter r ij) y ≤ 2 * r := by
-    have hh : dist y (coverGridCenter r ij) ≤ 2 * r := hy
-    simpa only [dist_comm] using hh
-  linarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverEvents
+
+namespace OAI
+
+/-! A finite physical grid in a bounded disk, with the planar cell count
+needed by the missed-cell estimate. Closed disks are used as fine cells. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+def coverGridCenter (r : ℝ) (ij : ℤ × ℤ) : ℂ :=
+  (r * (ij.1 : ℝ) : ℝ) + (r * (ij.2 : ℝ) : ℝ) * Complex.I
+
+def coverGridIndex (R r : ℝ) : Finset (ℤ × ℤ) :=
+  let N : ℤ := ⌈R / r⌉ + 1
+  (Finset.Icc (-N) N) ×ˢ (Finset.Icc (-N) N)
+
+def retainedCoverGridIndex (R r : ℝ) (K : Set ℂ) : Finset (ℤ × ℤ) := by
+  classical
+  exact (coverGridIndex R r).filter (fun ij =>
+    (K ∩ closedBall (coverGridCenter r ij) (2 * r)).Nonempty)
+
+lemma coverGrid_coordinate {R r x : ℝ} (_hR : 0 ≤ R) (hr : 0 < r)
+    (hx : |x| ≤ R) :
+    -(⌈R / r⌉ + 1) ≤ ⌊x / r⌋ ∧ ⌊x / r⌋ ≤ ⌈R / r⌉ + 1 := by
+  have hl := Int.sub_one_lt_floor (x / r)
+  have hu := Int.floor_le (x / r)
+  have hc := Int.le_ceil (R / r)
+  have hxl : -R / r ≤ x / r := div_le_div_of_nonneg_right (abs_le.mp hx).1 hr.le
+  have hxl' : -(R / r) ≤ x / r := by simpa only [neg_div] using hxl
+  have hxu : x / r ≤ R / r := div_le_div_of_nonneg_right (abs_le.mp hx).2 hr.le
+  have hlow : ((-(⌈R / r⌉ + 1) : ℤ) : ℝ) ≤ (⌊x / r⌋ : ℝ) := by
+    push_cast
+    linarith
+  have hhigh : (⌊x / r⌋ : ℝ) ≤ ((⌈R / r⌉ + 1 : ℤ) : ℝ) := by
+    push_cast
+    linarith
+  exact ⟨by exact_mod_cast hlow, by exact_mod_cast hhigh⟩
+
+lemma coverGrid_floor_distance {r : ℝ} (hr : 0 < r) (z : ℂ) :
+    dist z (coverGridCenter r (⌊z.re / r⌋, ⌊z.im / r⌋)) ≤ 2 * r := by
+  have hre0 : 0 ≤ z.re - r * (⌊z.re / r⌋ : ℝ) := by
+    have hh := (Int.floor_le (z.re / r))
+    have hm := (mul_le_mul_of_nonneg_left hh hr.le)
+    rw [mul_div_cancel₀ _ hr.ne'] at hm
+    linarith
+  have hre1 : z.re - r * (⌊z.re / r⌋ : ℝ) ≤ r := by
+    have hh := mul_lt_mul_of_pos_left (Int.lt_floor_add_one (z.re / r)) hr
+    rw [mul_div_cancel₀ _ hr.ne'] at hh
+    nlinarith
+  have him0 : 0 ≤ z.im - r * (⌊z.im / r⌋ : ℝ) := by
+    have hh := mul_le_mul_of_nonneg_left (Int.floor_le (z.im / r)) hr.le
+    rw [mul_div_cancel₀ _ hr.ne'] at hh
+    linarith
+  have him1 : z.im - r * (⌊z.im / r⌋ : ℝ) ≤ r := by
+    have hh := mul_lt_mul_of_pos_left (Int.lt_floor_add_one (z.im / r)) hr
+    rw [mul_div_cancel₀ _ hr.ne'] at hh
+    nlinarith
+  rw [dist_eq_norm]
+  apply (Complex.norm_le_abs_re_add_abs_im _).trans
+  simp only [coverGridCenter, Complex.sub_re, Complex.sub_im, Complex.add_re,
+    Complex.add_im, Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re,
+    Complex.mul_im, Complex.I_re, Complex.I_im, mul_zero, mul_one,
+    sub_zero, add_zero, zero_add, abs_of_nonneg hre0, abs_of_nonneg him0]
+  linarith
+
+theorem coverGrid_covers_disk {R r : ℝ} (hR : 0 ≤ R) (hr : 0 < r) :
+    closedBall (0 : ℂ) R ⊆
+      ⋃ ij ∈ coverGridIndex R r, closedBall (coverGridCenter r ij) (2 * r) := by
+  intro z hz
+  have hzn : ‖z‖ ≤ R := by simpa only [mem_closedBall, dist_zero_right] using hz
+  have hre := coverGrid_coordinate hR hr ((Complex.abs_re_le_norm z).trans hzn)
+  have him := coverGrid_coordinate hR hr ((Complex.abs_im_le_norm z).trans hzn)
+  apply mem_iUnion₂.mpr
+  refine ⟨(⌊z.re / r⌋, ⌊z.im / r⌋), ?_, coverGrid_floor_distance hr z⟩
+  simpa only [coverGridIndex, Finset.mem_product, Finset.mem_Icc] using And.intro hre him
+
+/-- The constant depends on the bounded disk, and not on the small mesh. -/
+theorem coverGrid_card_bound {R r : ℝ} (hR : 0 ≤ R) (hr : 0 < r) (hr1 : r ≤ 1) :
+    ((coverGridIndex R r).card : ℝ) ≤ (2 * R + 5) ^ 2 * r ^ (-2 : ℝ) := by
+  let N : ℤ := ⌈R / r⌉ + 1
+  have hN : 0 ≤ N := by
+    have hh : (0 : ℝ) ≤ (⌈R / r⌉ : ℝ) := (div_nonneg hR hr.le).trans (Int.le_ceil _)
+    have hh' : (0 : ℤ) ≤ ⌈R / r⌉ := by exact_mod_cast hh
+    dsimp [N]
+    omega
+  have hcard : ((coverGridIndex R r).card : ℝ) = (2 * (N : ℝ) + 1) ^ 2 := by
+    have hh := Int.card_Icc_of_le (a := -N) (b := N) (by omega)
+    have hhR : ((Finset.Icc (-N) N).card : ℝ) = 2 * (N : ℝ) + 1 := by
+      have hh' : ((Finset.Icc (-N) N).card : ℝ) = (N : ℝ) + 1 - -(N : ℝ) := by
+        exact_mod_cast hh
+      linarith
+    change (((Finset.Icc (-N) N) ×ˢ (Finset.Icc (-N) N)).card : ℝ) = _
+    rw [Finset.card_product, Nat.cast_mul, hhR]
+    ring
+  have hNbound : (N : ℝ) ≤ R / r + 2 := by
+    have hh := Int.ceil_lt_add_one (R / r)
+    dsimp [N]
+    push_cast
+    linarith
+  have hside : 2 * (N : ℝ) + 1 ≤ (2 * R + 5) / r := by
+    apply (le_div_iff₀ hr).mpr
+    have hh := mul_le_mul_of_nonneg_right hNbound hr.le
+    rw [add_mul, div_mul_cancel₀ _ hr.ne'] at hh
+    nlinarith
+  rw [hcard, Real.rpow_neg hr.le, Real.rpow_two]
+  calc
+    _ ≤ ((2 * R + 5) / r) ^ 2 := pow_le_pow_left₀ (by exact_mod_cast (show (0 : ℤ) ≤ 2 * N + 1 by omega)) hside 2
+    _ = _ := by rw [div_pow, div_eq_mul_inv]
+
+theorem retainedCoverGrid_covers {R r : ℝ} (hR : 0 ≤ R) (hr : 0 < r)
+    {K : Set ℂ} (hK : K ⊆ closedBall 0 R) :
+    K ⊆ ⋃ ij ∈ retainedCoverGridIndex R r K,
+      closedBall (coverGridCenter r ij) (2 * r) := by
+  classical
+  intro z hz
+  obtain ⟨ij, hij, hzcell⟩ := mem_iUnion₂.mp (coverGrid_covers_disk hR hr (hK hz))
+  exact mem_iUnion₂.mpr ⟨ij, Finset.mem_filter.mpr ⟨hij, ⟨z, hz, hzcell⟩⟩, hzcell⟩
+
+theorem retainedCoverGrid_card_bound {R r : ℝ} (hR : 0 ≤ R)
+    (hr : 0 < r) (hr1 : r ≤ 1) (K : Set ℂ) :
+    ((retainedCoverGridIndex R r K).card : ℝ) ≤ (2 * R + 5) ^ 2 * r ^ (-2 : ℝ) := by
+  classical
+  exact (Nat.cast_le.mpr (Finset.card_filter_le _ _)).trans (coverGrid_card_bound hR hr hr1)
+
+/-- Retaining only cells meeting the interior cutoff keeps all their centers
+away from the real boundary and in one fixed enlarged bounded disk. -/
+theorem retainedCoverGrid_center_bounds {R r η : ℝ} {K : Set ℂ}
+    (hK : K ⊆ closedBall 0 R) (hheight : ∀ z ∈ K, η ≤ z.im)
+    {ij : ℤ × ℤ} (hij : ij ∈ retainedCoverGridIndex R r K) :
+    ‖coverGridCenter r ij‖ ≤ R + 2 * r ∧
+      η - 2 * r ≤ (coverGridCenter r ij).im := by
+  classical
+  obtain ⟨z, hzK, hzcell⟩ := (Finset.mem_filter.mp hij).2
+  have hdist : dist z (coverGridCenter r ij) ≤ 2 * r := hzcell
+  have hzn : ‖z‖ ≤ R := by simpa only [mem_closedBall, dist_zero_right] using hK hzK
+  have hnorm : ‖coverGridCenter r ij‖ ≤ ‖z‖ + dist z (coverGridCenter r ij) := by
+    have hh := dist_triangle (coverGridCenter r ij) z 0
+    simpa only [dist_zero_right, dist_comm (coverGridCenter r ij) z, add_comm] using hh
+  have him := (Complex.abs_im_le_norm (z - coverGridCenter r ij)).trans
+    (by simpa only [dist_eq_norm] using hdist)
+  rw [Complex.sub_im] at him
+  constructor
+  · linarith
+  · have hh := (abs_le.mp him).2
+    have hz := hheight z hzK
+    linarith
+
+lemma coverGrid_cell_diameter {r : ℝ} (_hr : 0 < r) (ij : ℤ × ℤ) :
+    ediam (closedBall (coverGridCenter r ij) (2 * r)) ≤ ENNReal.ofReal (4 * r) := by
+  apply ediam_le_of_forall_dist_le
+  intro x hx y hy
+  have hxy := dist_triangle x (coverGridCenter r ij) y
+  have hx' : dist x (coverGridCenter r ij) ≤ 2 * r := hx
+  have hy' : dist (coverGridCenter r ij) y ≤ 2 * r := by
+    have hh : dist y (coverGridCenter r ij) ≤ 2 * r := hy
+    simpa only [dist_comm] using hh
+  linarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverLimits.lean b/lean/OAI/Probability/SLE/Gauge/CoverLimits.lean
index 988eb73f41740cfd1cd514e9bbc9d4496b7b27a4..3a996c1deba6de9f3e6cd5110085f40b08a84436 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverLimits.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverLimits.lean
@@ -1,111 +1,111 @@
-import OAI.Probability.SLE.MainReduction
-
-namespace OAI
-
-/-!
-The deterministic and integration steps in the upper-cover arguments of
-the two exact-gauge manuscripts. These lemmas do not assume a Hausdorff
-upper bound: they pass from actual shrinking covers to that bound.
--/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- Cover costs control the Hausdorff limit even when they do not converge. -/
-theorem hausdorff_le_liminf_cover_cost
-    (h : ℝ → ℝ) (hh : IsGauge h) (E : Set ℂ)
-    (U : ℕ → ℕ → Set ℂ) (δ : ℕ → ℝ≥0∞)
-    (hδ : Tendsto δ atTop (𝓝 0))
-    (hcover : ∀ n, E ⊆ ⋃ j, U n j)
-    (hdiam : ∀ n j, ediam (U n j) ≤ δ n) :
-    hausdorffGauge h E ≤
-      liminf (fun n => ∑' j, extendedGauge h (ediam (U n j))) atTop := by
-  rw [source_gauge_cover_formula h hh]
-  refine iSup_le fun r => iSup_le fun hr => ?_
-  apply le_liminf_of_le (by isBoundedDefault)
-  filter_upwards [hδ.eventually (gt_mem_nhds hr)] with n hn
-  exact iInf_le_of_le (U n) (iInf_le_of_le (hcover n)
-    (iInf_le_of_le (fun j => (hdiam n j).trans hn.le) le_rfl))
-
-/-- Fatou's lemma for measurable costs of actual covers, with a limiting
-expectation bound rather than an assumption about the target measure. -/
-theorem expected_hausdorff_le_of_shrinking_covers
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (h : ℝ → ℝ) (hh : IsGauge h) (E : Ω → Set ℂ)
-    (U : ℕ → Ω → ℕ → Set ℂ) (δ : ℕ → ℝ≥0∞)
-    (hδ : Tendsto δ atTop (𝓝 0))
-    (hcover : ∀ᵐ ω ∂P, ∀ n, E ω ⊆ ⋃ j, U n ω j)
-    (hdiam : ∀ᵐ ω ∂P, ∀ n j, ediam (U n ω j) ≤ δ n)
-    (hmeas : ∀ n, AEMeasurable
-      (fun ω => ∑' j, extendedGauge h (ediam (U n ω j))) P)
-    (C : ℝ≥0∞)
-    (hcost : liminf (fun n => ∫⁻ ω,
-      ∑' j, extendedGauge h (ediam (U n ω j)) ∂P) atTop ≤ C) :
-    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤ C := by
-  calc
-    _ ≤ ∫⁻ ω, liminf (fun n => ∑' j,
-        extendedGauge h (ediam (U n ω j))) atTop ∂P := by
-      apply lintegral_mono_ae
-      filter_upwards [hcover, hdiam] with ω hc hd
-      exact hausdorff_le_liminf_cover_cost h hh (E ω)
-        (fun n => U n ω) δ hδ hc hd
-    _ ≤ liminf (fun n => ∫⁻ ω,
-        ∑' j, extendedGauge h (ediam (U n ω j)) ∂P) atTop :=
-      lintegral_liminf_le' hmeas
-    _ ≤ C := hcost
-
-/-- Disjoint candidate sets are all paid for by the same spatial measure. -/
-theorem disjoint_cover_charge {ι : Type*} (I : Finset ι)
-    (ν : Measure ℂ) (D : ι → Set ℂ) (A : Set ℂ)
-    (hdisj : (I : Set ι).PairwiseDisjoint D)
-    (hmeas : ∀ i ∈ I, MeasurableSet (D i))
-    (hsub : ∀ i ∈ I, D i ⊆ A)
-    (cost : ι → ℝ≥0∞) (C : ℝ≥0∞)
-    (hcost : ∀ i ∈ I, cost i ≤ C * ν (D i)) :
-    ∑ i ∈ I, cost i ≤ C * ν A := by
-  calc
-    _ ≤ ∑ i ∈ I, C * ν (D i) := Finset.sum_le_sum hcost
-    _ = C * ∑ i ∈ I, ν (D i) := (Finset.mul_sum I _ C).symm
-    _ = C * ν (⋃ i ∈ I, D i) := by
-      rw [measure_biUnion_finset hdisj hmeas]
-    _ ≤ C * ν A := mul_le_mul' le_rfl (measure_mono (iUnion₂_subset hsub))
-
-/-- Continuity from below retains a uniform expectation bound as both the
-time cutoff and the distance from the real boundary are removed. -/
-theorem expected_measure_iUnion_le
-    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (P : Measure Ω) (ν : Measure X) (E : ℕ → Ω → Set X)
-    (hmono : ∀ ω, Monotone (fun n => E n ω))
-    (hmeas : ∀ n, AEMeasurable (fun ω => ν (E n ω)) P)
-    (C : ℝ≥0∞) (hbound : ∀ n, (∫⁻ ω, ν (E n ω) ∂P) ≤ C) :
-    (∫⁻ ω, ν (⋃ n, E n ω) ∂P) ≤ C := by
-  simp_rw [Monotone.measure_iUnion (hmono _)]
-  rw [lintegral_iSup' hmeas (Eventually.of_forall fun ω i j hij =>
-    measure_mono (hmono ω hij))]
-  exact iSup_le hbound
-
-/-- Adding a set contained in a fixed null boundary costs no measure. -/
-theorem measure_inter_eq_interior_of_null_boundary
-    {X : Type*} [MeasurableSpace X] (ν : Measure X)
-    (E A H : Set X)
-    (hboundary : ν (E ∩ A ∩ Hᶜ) = 0) :
-    ν (E ∩ A) = ν (E ∩ A ∩ H) := by
-  apply le_antisymm
-  · calc
-      ν (E ∩ A) = ν ((E ∩ A ∩ H) ∪ (E ∩ A ∩ Hᶜ)) := by
-        congr 1
-        ext x
-        simp only [mem_inter_iff, mem_union, mem_compl_iff]
-        tauto
-      _ ≤ ν (E ∩ A ∩ H) + ν (E ∩ A ∩ Hᶜ) := measure_union_le _ _
-      _ = ν (E ∩ A ∩ H) := by rw [hboundary, add_zero]
-  · exact measure_mono inter_subset_left
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.MainReduction
+
+namespace OAI
+
+/-!
+The deterministic and integration steps in the upper-cover arguments of
+the two exact-gauge manuscripts. These lemmas do not assume a Hausdorff
+upper bound: they pass from actual shrinking covers to that bound.
+-/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- Cover costs control the Hausdorff limit even when they do not converge. -/
+theorem hausdorff_le_liminf_cover_cost
+    (h : ℝ → ℝ) (hh : IsGauge h) (E : Set ℂ)
+    (U : ℕ → ℕ → Set ℂ) (δ : ℕ → ℝ≥0∞)
+    (hδ : Tendsto δ atTop (𝓝 0))
+    (hcover : ∀ n, E ⊆ ⋃ j, U n j)
+    (hdiam : ∀ n j, ediam (U n j) ≤ δ n) :
+    hausdorffGauge h E ≤
+      liminf (fun n => ∑' j, extendedGauge h (ediam (U n j))) atTop := by
+  rw [source_gauge_cover_formula h hh]
+  refine iSup_le fun r => iSup_le fun hr => ?_
+  apply le_liminf_of_le (by isBoundedDefault)
+  filter_upwards [hδ.eventually (gt_mem_nhds hr)] with n hn
+  exact iInf_le_of_le (U n) (iInf_le_of_le (hcover n)
+    (iInf_le_of_le (fun j => (hdiam n j).trans hn.le) le_rfl))
+
+/-- Fatou's lemma for measurable costs of actual covers, with a limiting
+expectation bound rather than an assumption about the target measure. -/
+theorem expected_hausdorff_le_of_shrinking_covers
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (h : ℝ → ℝ) (hh : IsGauge h) (E : Ω → Set ℂ)
+    (U : ℕ → Ω → ℕ → Set ℂ) (δ : ℕ → ℝ≥0∞)
+    (hδ : Tendsto δ atTop (𝓝 0))
+    (hcover : ∀ᵐ ω ∂P, ∀ n, E ω ⊆ ⋃ j, U n ω j)
+    (hdiam : ∀ᵐ ω ∂P, ∀ n j, ediam (U n ω j) ≤ δ n)
+    (hmeas : ∀ n, AEMeasurable
+      (fun ω => ∑' j, extendedGauge h (ediam (U n ω j))) P)
+    (C : ℝ≥0∞)
+    (hcost : liminf (fun n => ∫⁻ ω,
+      ∑' j, extendedGauge h (ediam (U n ω j)) ∂P) atTop ≤ C) :
+    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤ C := by
+  calc
+    _ ≤ ∫⁻ ω, liminf (fun n => ∑' j,
+        extendedGauge h (ediam (U n ω j))) atTop ∂P := by
+      apply lintegral_mono_ae
+      filter_upwards [hcover, hdiam] with ω hc hd
+      exact hausdorff_le_liminf_cover_cost h hh (E ω)
+        (fun n => U n ω) δ hδ hc hd
+    _ ≤ liminf (fun n => ∫⁻ ω,
+        ∑' j, extendedGauge h (ediam (U n ω j)) ∂P) atTop :=
+      lintegral_liminf_le' hmeas
+    _ ≤ C := hcost
+
+/-- Disjoint candidate sets are all paid for by the same spatial measure. -/
+theorem disjoint_cover_charge {ι : Type*} (I : Finset ι)
+    (ν : Measure ℂ) (D : ι → Set ℂ) (A : Set ℂ)
+    (hdisj : (I : Set ι).PairwiseDisjoint D)
+    (hmeas : ∀ i ∈ I, MeasurableSet (D i))
+    (hsub : ∀ i ∈ I, D i ⊆ A)
+    (cost : ι → ℝ≥0∞) (C : ℝ≥0∞)
+    (hcost : ∀ i ∈ I, cost i ≤ C * ν (D i)) :
+    ∑ i ∈ I, cost i ≤ C * ν A := by
+  calc
+    _ ≤ ∑ i ∈ I, C * ν (D i) := Finset.sum_le_sum hcost
+    _ = C * ∑ i ∈ I, ν (D i) := (Finset.mul_sum I _ C).symm
+    _ = C * ν (⋃ i ∈ I, D i) := by
+      rw [measure_biUnion_finset hdisj hmeas]
+    _ ≤ C * ν A := mul_le_mul' le_rfl (measure_mono (iUnion₂_subset hsub))
+
+/-- Continuity from below retains a uniform expectation bound as both the
+time cutoff and the distance from the real boundary are removed. -/
+theorem expected_measure_iUnion_le
+    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (P : Measure Ω) (ν : Measure X) (E : ℕ → Ω → Set X)
+    (hmono : ∀ ω, Monotone (fun n => E n ω))
+    (hmeas : ∀ n, AEMeasurable (fun ω => ν (E n ω)) P)
+    (C : ℝ≥0∞) (hbound : ∀ n, (∫⁻ ω, ν (E n ω) ∂P) ≤ C) :
+    (∫⁻ ω, ν (⋃ n, E n ω) ∂P) ≤ C := by
+  simp_rw [Monotone.measure_iUnion (hmono _)]
+  rw [lintegral_iSup' hmeas (Eventually.of_forall fun ω i j hij =>
+    measure_mono (hmono ω hij))]
+  exact iSup_le hbound
+
+/-- Adding a set contained in a fixed null boundary costs no measure. -/
+theorem measure_inter_eq_interior_of_null_boundary
+    {X : Type*} [MeasurableSpace X] (ν : Measure X)
+    (E A H : Set X)
+    (hboundary : ν (E ∩ A ∩ Hᶜ) = 0) :
+    ν (E ∩ A) = ν (E ∩ A ∩ H) := by
+  apply le_antisymm
+  · calc
+      ν (E ∩ A) = ν ((E ∩ A ∩ H) ∪ (E ∩ A ∩ Hᶜ)) := by
+        congr 1
+        ext x
+        simp only [mem_inter_iff, mem_union, mem_compl_iff]
+        tauto
+      _ ≤ ν (E ∩ A ∩ H) + ν (E ∩ A ∩ Hᶜ) := measure_union_le _ _
+      _ = ν (E ∩ A ∩ H) := by rw [hboundary, add_zero]
+  · exact measure_mono inter_subset_left
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverMesh.lean b/lean/OAI/Probability/SLE/Gauge/CoverMesh.lean
index 60f8fb7766e5775ab0c39ef04271ea572ede149d..d4465d2d45b931df8b0a36ac2616ed035602eac2 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverMesh.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverMesh.lean
@@ -1,103 +1,103 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-The deterministic common mesh in the dense-cover construction. At each
-stage there are only finitely many positive test cutoffs. The mesh satisfies
-every cutoff and decreases by at least a factor of two at each stage.
--/
-
-noncomputable section
-open Set Filter
-open scoped Topology
-
-namespace SLEExactGauge.CoverMesh
-
-def cutoff (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) : ℝ := by
-  classical
-  exact (insert 1 ((I n).image (ε n))).min' (by simp)
-
-lemma cutoff_pos (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
-    (hε : ∀ n i, i ∈ I n → 0 < ε n i) (n : ℕ) : 0 < cutoff I ε n := by
-  classical
-  unfold cutoff
-  have hmem := Finset.min'_mem (insert 1 ((I n).image (ε n))) (by simp)
-  rcases Finset.mem_insert.mp hmem with h | h
-  · simpa only [h] using (show (0 : ℝ) < 1 by norm_num)
-  · obtain ⟨i, hi, he⟩ := Finset.mem_image.mp h
-    exact he ▸ hε n i hi
-
-lemma cutoff_le_one (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
-    cutoff I ε n ≤ 1 := by
-  classical
-  exact Finset.min'_le _ _ (Finset.mem_insert_self _ _)
-
-lemma cutoff_le_test (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
-    (n i : ℕ) (hi : i ∈ I n) : cutoff I ε n ≤ ε n i := by
-  classical
-  exact Finset.min'_le _ _
-    (Finset.mem_insert_of_mem (Finset.mem_image.mpr ⟨i, hi, rfl⟩))
-
-def mesh (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) : ℕ → ℝ
-  | 0 => cutoff I ε 0
-  | n + 1 => min (mesh I ε n / 2) (cutoff I ε (n + 1))
-
-lemma mesh_pos (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
-    (hε : ∀ n i, i ∈ I n → 0 < ε n i) (n : ℕ) : 0 < mesh I ε n := by
-  induction n with
-  | zero => exact cutoff_pos I ε hε 0
-  | succ n ih =>
-      exact lt_min (div_pos ih (by norm_num)) (cutoff_pos I ε hε (n + 1))
-
-lemma mesh_half (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
-    mesh I ε (n + 1) ≤ mesh I ε n / 2 := min_le_left _ _
-
-lemma mesh_le_cutoff (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
-    mesh I ε n ≤ cutoff I ε n := by
-  cases n with
-  | zero => exact le_rfl
-  | succ n => exact min_le_right _ _
-
-lemma mesh_le_test (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
-    (n i : ℕ) (hi : i ∈ I n) : mesh I ε n ≤ ε n i :=
-  (mesh_le_cutoff I ε n).trans (cutoff_le_test I ε n i hi)
-
-lemma mesh_le_half_pow (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
-    mesh I ε n ≤ (1 / 2 : ℝ) ^ n := by
-  induction n with
-  | zero => simpa only [mesh, pow_zero] using cutoff_le_one I ε 0
-  | succ n ih =>
-      calc
-        mesh I ε (n + 1) ≤ mesh I ε n / 2 := mesh_half I ε n
-        _ ≤ (1 / 2 : ℝ) ^ n / 2 := div_le_div_of_nonneg_right ih (by norm_num)
-        _ = (1 / 2 : ℝ) ^ (n + 1) := by rw [pow_succ]; ring
-
-lemma mesh_le_dyadic (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
-    mesh I ε n ≤ (2 : ℝ) ^ (-(n : ℝ)) := by
-  convert mesh_le_half_pow I ε n using 1
-  rw [Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 2), Real.rpow_natCast]
-  simp only [one_div, inv_pow]
-
-lemma mesh_tendsto_zero (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
-    (hε : ∀ n i, i ∈ I n → 0 < ε n i) :
-    Tendsto (mesh I ε) atTop (𝓝 0) :=
-  squeeze_zero (fun n => (mesh_pos I ε hε n).le) (mesh_le_half_pow I ε)
-    (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num))
-
-/-- One positive deterministic mesh works for every finite test window,
-including empty windows, and tends to zero with geometric decay. -/
-theorem exists_common_mesh (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
-    (hε : ∀ n i, i ∈ I n → 0 < ε n i) :
-    ∃ v : ℕ → ℝ, (∀ n, 0 < v n) ∧
-      (∀ n, v n ≤ (2 : ℝ) ^ (-(n : ℝ))) ∧
-      (∀ n, v (n + 1) ≤ v n / 2) ∧
-      (∀ n i, i ∈ I n → v n ≤ ε n i) ∧ Tendsto v atTop (𝓝 0) :=
-  ⟨mesh I ε, mesh_pos I ε hε, mesh_le_dyadic I ε, mesh_half I ε,
-    mesh_le_test I ε, mesh_tendsto_zero I ε hε⟩
-
-end SLEExactGauge.CoverMesh
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+The deterministic common mesh in the dense-cover construction. At each
+stage there are only finitely many positive test cutoffs. The mesh satisfies
+every cutoff and decreases by at least a factor of two at each stage.
+-/
+
+noncomputable section
+open Set Filter
+open scoped Topology
+
+namespace SLEExactGauge.CoverMesh
+
+def cutoff (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) : ℝ := by
+  classical
+  exact (insert 1 ((I n).image (ε n))).min' (by simp)
+
+lemma cutoff_pos (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
+    (hε : ∀ n i, i ∈ I n → 0 < ε n i) (n : ℕ) : 0 < cutoff I ε n := by
+  classical
+  unfold cutoff
+  have hmem := Finset.min'_mem (insert 1 ((I n).image (ε n))) (by simp)
+  rcases Finset.mem_insert.mp hmem with h | h
+  · simpa only [h] using (show (0 : ℝ) < 1 by norm_num)
+  · obtain ⟨i, hi, he⟩ := Finset.mem_image.mp h
+    exact he ▸ hε n i hi
+
+lemma cutoff_le_one (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
+    cutoff I ε n ≤ 1 := by
+  classical
+  exact Finset.min'_le _ _ (Finset.mem_insert_self _ _)
+
+lemma cutoff_le_test (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
+    (n i : ℕ) (hi : i ∈ I n) : cutoff I ε n ≤ ε n i := by
+  classical
+  exact Finset.min'_le _ _
+    (Finset.mem_insert_of_mem (Finset.mem_image.mpr ⟨i, hi, rfl⟩))
+
+def mesh (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) : ℕ → ℝ
+  | 0 => cutoff I ε 0
+  | n + 1 => min (mesh I ε n / 2) (cutoff I ε (n + 1))
+
+lemma mesh_pos (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
+    (hε : ∀ n i, i ∈ I n → 0 < ε n i) (n : ℕ) : 0 < mesh I ε n := by
+  induction n with
+  | zero => exact cutoff_pos I ε hε 0
+  | succ n ih =>
+      exact lt_min (div_pos ih (by norm_num)) (cutoff_pos I ε hε (n + 1))
+
+lemma mesh_half (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
+    mesh I ε (n + 1) ≤ mesh I ε n / 2 := min_le_left _ _
+
+lemma mesh_le_cutoff (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
+    mesh I ε n ≤ cutoff I ε n := by
+  cases n with
+  | zero => exact le_rfl
+  | succ n => exact min_le_right _ _
+
+lemma mesh_le_test (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
+    (n i : ℕ) (hi : i ∈ I n) : mesh I ε n ≤ ε n i :=
+  (mesh_le_cutoff I ε n).trans (cutoff_le_test I ε n i hi)
+
+lemma mesh_le_half_pow (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
+    mesh I ε n ≤ (1 / 2 : ℝ) ^ n := by
+  induction n with
+  | zero => simpa only [mesh, pow_zero] using cutoff_le_one I ε 0
+  | succ n ih =>
+      calc
+        mesh I ε (n + 1) ≤ mesh I ε n / 2 := mesh_half I ε n
+        _ ≤ (1 / 2 : ℝ) ^ n / 2 := div_le_div_of_nonneg_right ih (by norm_num)
+        _ = (1 / 2 : ℝ) ^ (n + 1) := by rw [pow_succ]; ring
+
+lemma mesh_le_dyadic (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ) (n : ℕ) :
+    mesh I ε n ≤ (2 : ℝ) ^ (-(n : ℝ)) := by
+  convert mesh_le_half_pow I ε n using 1
+  rw [Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 2), Real.rpow_natCast]
+  simp only [one_div, inv_pow]
+
+lemma mesh_tendsto_zero (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
+    (hε : ∀ n i, i ∈ I n → 0 < ε n i) :
+    Tendsto (mesh I ε) atTop (𝓝 0) :=
+  squeeze_zero (fun n => (mesh_pos I ε hε n).le) (mesh_le_half_pow I ε)
+    (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num))
+
+/-- One positive deterministic mesh works for every finite test window,
+including empty windows, and tends to zero with geometric decay. -/
+theorem exists_common_mesh (I : ℕ → Finset ℕ) (ε : ℕ → ℕ → ℝ)
+    (hε : ∀ n i, i ∈ I n → 0 < ε n i) :
+    ∃ v : ℕ → ℝ, (∀ n, 0 < v n) ∧
+      (∀ n, v n ≤ (2 : ℝ) ^ (-(n : ℝ))) ∧
+      (∀ n, v (n + 1) ≤ v n / 2) ∧
+      (∀ n i, i ∈ I n → v n ≤ ε n i) ∧ Tendsto v atTop (𝓝 0) :=
+  ⟨mesh I ε, mesh_pos I ε hε, mesh_le_dyadic I ε, mesh_half I ε,
+    mesh_le_test I ε, mesh_tendsto_zero I ε hε⟩
+
+end SLEExactGauge.CoverMesh
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverNumerics.lean b/lean/OAI/Probability/SLE/Gauge/CoverNumerics.lean
index b80266e11b3cc93d93e35cb76bf4c804e4d1f110..51f3845cff9da16d99f4b37878c6910e8f89ff67 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverNumerics.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverNumerics.lean
@@ -1,102 +1,102 @@
-import OAI.Probability.SLE.Gauge.DenseCoverCore
-
-namespace OAI
-
-/-! The failure and residual-cover estimates used by the explicit gauge. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- The dimensions in the hit probability, covering cost, and grid count cancel. -/
-lemma dense_cover_power_cancellation (κ : ℝ) {r : ℝ} (hr : 0 < r) :
-    r ^ (-2 : ℝ) * r ^ dimension κ * r ^ codimension κ = 1 := by
-  rw [← Real.rpow_add hr, ← Real.rpow_add hr]
-  have he : (-2 : ℝ) + dimension κ + codimension κ = 0 := by
-    unfold codimension
-    ring
-  rw [he, Real.rpow_zero]
-
-lemma failure_product_le_exp {p : ℝ} (hp : p ≤ 1) (N : ℕ) :
-    (1 - p) ^ N ≤ Real.exp (-p * (N : ℝ)) := by
-  have hb : 1 - p ≤ Real.exp (-p) := by
-    simpa only [sub_eq_add_neg, add_comm] using Real.add_one_le_exp (-p)
-  have hpow := pow_le_pow_left₀ (sub_nonneg.mpr hp) hb N
-  simpa only [← Real.exp_nat_mul, mul_comm (N : ℝ) (-p)] using hpow
-
-/-- A linear number of tests, each succeeding with probability at least
-`n^(-1/2)`, has the stretched-exponential failure bound used in the paper. -/
-theorem failure_window_le_stretchedExp {n : ℝ} (hn : 1 ≤ n)
-    (N : ℕ) (hN : n / 4 ≤ (N : ℝ)) :
-    (1 - 1 / Real.sqrt n) ^ N ≤ Real.exp (-Real.sqrt n / 4) := by
-  have hn0 : 0 < n := zero_lt_one.trans_le hn
-  have hs : 0 < Real.sqrt n := Real.sqrt_pos.mpr hn0
-  have hs1 : 1 ≤ Real.sqrt n := by
-    simpa only [Real.sqrt_one] using Real.sqrt_le_sqrt hn
-  have hp : 1 / Real.sqrt n ≤ 1 := (div_le_one hs).mpr hs1
-  apply (failure_product_le_exp hp N).trans
-  apply Real.exp_le_exp.mpr
-  have hmul := mul_le_mul_of_nonneg_left hN (one_div_pos.mpr hs).le
-  have he : (1 / Real.sqrt n) * (n / 4) = Real.sqrt n / 4 := by
-    field_simp
-    nlinarith [Real.sq_sqrt hn0.le]
-  rw [he] at hmul
-  nlinarith
-
-/-- A logarithmic correction cannot offset the stretched-exponential failure
-probability obtained from the finite sequence of adapted tests. -/
-theorem logarithmic_cleanup_tendsto_zero {q c : ℝ}
-    (hq : q ≤ 1) (hc : 0 < c) :
-    Tendsto (fun x : ℝ => (Real.log x) ^ q * Real.exp (-c * Real.sqrt x))
-      atTop (𝓝 0) := by
-  have ht := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 2 c hc).comp
-    Real.tendsto_sqrt_atTop
-  have hu : Tendsto (fun x : ℝ => x * Real.exp (-c * Real.sqrt x))
-      atTop (𝓝 0) := by
-    apply ht.congr'
-    filter_upwards [eventually_ge_atTop (0 : ℝ)] with x hx
-    simp only [Function.comp_def, Real.rpow_two, Real.sq_sqrt hx]
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu
-  · filter_upwards [eventually_ge_atTop (Real.exp 1)] with x hx
-    have h1 : 1 ≤ x := (Real.one_lt_exp_iff.mpr zero_lt_one).le.trans hx
-    exact mul_nonneg (Real.rpow_nonneg (Real.log_nonneg h1) q) (Real.exp_pos _).le
-  · filter_upwards [eventually_ge_atTop (Real.exp 1)] with x hx
-    have hxp : 0 < x := (Real.exp_pos 1).trans_le hx
-    have hlog : 1 ≤ Real.log x := by
-      simpa using Real.log_le_log (Real.exp_pos 1) hx
-    have hp : (Real.log x) ^ q ≤ Real.log x := by
-      simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hlog hq
-    exact mul_le_mul_of_nonneg_right (hp.trans (Real.log_le_self hxp.le))
-      (Real.exp_pos _).le
-
-theorem explicit_cleanup_tendsto_zero {κ c : ℝ} (hκ : 0 < κ) (hc : 0 < c) :
-    Tendsto (fun n : ℕ => (Real.log (n : ℝ)) ^ (codimension κ / 2) *
-      Real.exp (-c * Real.sqrt (n : ℝ))) atTop (𝓝 0) := by
-  have hq : codimension κ / 2 ≤ 1 := by
-    dsimp [codimension, dimension]
-    linarith
-  exact (logarithmic_cleanup_tendsto_zero hq hc).comp tendsto_natCast_atTop_atTop
-
-/-- Finite expected residual cost is the sum of each missed-cell probability
-times its deterministic covering cost. -/
-theorem expected_missed_cell_cost
-    {Ω ι : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (I : Finset ι) (missed : ι → Set Ω)
-    (hm : ∀ i ∈ I, MeasurableSet (missed i))
-    (cost bound : ι → ℝ≥0∞)
-    (hb : ∀ i ∈ I, P (missed i) ≤ bound i) :
-    (∫⁻ ω, ∑ i ∈ I, (missed i).indicator (fun _ => cost i) ω ∂P) ≤
-      ∑ i ∈ I, cost i * bound i := by
-  rw [lintegral_finsetSum I (fun i hi => measurable_const.indicator (hm i hi))]
-  apply Finset.sum_le_sum
-  intro i hi
-  rw [lintegral_indicator (hm i hi), lintegral_const, Measure.restrict_apply_univ]
-  exact mul_le_mul' le_rfl (hb i hi)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.DenseCoverCore
+
+namespace OAI
+
+/-! The failure and residual-cover estimates used by the explicit gauge. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- The dimensions in the hit probability, covering cost, and grid count cancel. -/
+lemma dense_cover_power_cancellation (κ : ℝ) {r : ℝ} (hr : 0 < r) :
+    r ^ (-2 : ℝ) * r ^ dimension κ * r ^ codimension κ = 1 := by
+  rw [← Real.rpow_add hr, ← Real.rpow_add hr]
+  have he : (-2 : ℝ) + dimension κ + codimension κ = 0 := by
+    unfold codimension
+    ring
+  rw [he, Real.rpow_zero]
+
+lemma failure_product_le_exp {p : ℝ} (hp : p ≤ 1) (N : ℕ) :
+    (1 - p) ^ N ≤ Real.exp (-p * (N : ℝ)) := by
+  have hb : 1 - p ≤ Real.exp (-p) := by
+    simpa only [sub_eq_add_neg, add_comm] using Real.add_one_le_exp (-p)
+  have hpow := pow_le_pow_left₀ (sub_nonneg.mpr hp) hb N
+  simpa only [← Real.exp_nat_mul, mul_comm (N : ℝ) (-p)] using hpow
+
+/-- A linear number of tests, each succeeding with probability at least
+`n^(-1/2)`, has the stretched-exponential failure bound used in the paper. -/
+theorem failure_window_le_stretchedExp {n : ℝ} (hn : 1 ≤ n)
+    (N : ℕ) (hN : n / 4 ≤ (N : ℝ)) :
+    (1 - 1 / Real.sqrt n) ^ N ≤ Real.exp (-Real.sqrt n / 4) := by
+  have hn0 : 0 < n := zero_lt_one.trans_le hn
+  have hs : 0 < Real.sqrt n := Real.sqrt_pos.mpr hn0
+  have hs1 : 1 ≤ Real.sqrt n := by
+    simpa only [Real.sqrt_one] using Real.sqrt_le_sqrt hn
+  have hp : 1 / Real.sqrt n ≤ 1 := (div_le_one hs).mpr hs1
+  apply (failure_product_le_exp hp N).trans
+  apply Real.exp_le_exp.mpr
+  have hmul := mul_le_mul_of_nonneg_left hN (one_div_pos.mpr hs).le
+  have he : (1 / Real.sqrt n) * (n / 4) = Real.sqrt n / 4 := by
+    field_simp
+    nlinarith [Real.sq_sqrt hn0.le]
+  rw [he] at hmul
+  nlinarith
+
+/-- A logarithmic correction cannot offset the stretched-exponential failure
+probability obtained from the finite sequence of adapted tests. -/
+theorem logarithmic_cleanup_tendsto_zero {q c : ℝ}
+    (hq : q ≤ 1) (hc : 0 < c) :
+    Tendsto (fun x : ℝ => (Real.log x) ^ q * Real.exp (-c * Real.sqrt x))
+      atTop (𝓝 0) := by
+  have ht := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 2 c hc).comp
+    Real.tendsto_sqrt_atTop
+  have hu : Tendsto (fun x : ℝ => x * Real.exp (-c * Real.sqrt x))
+      atTop (𝓝 0) := by
+    apply ht.congr'
+    filter_upwards [eventually_ge_atTop (0 : ℝ)] with x hx
+    simp only [Function.comp_def, Real.rpow_two, Real.sq_sqrt hx]
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu
+  · filter_upwards [eventually_ge_atTop (Real.exp 1)] with x hx
+    have h1 : 1 ≤ x := (Real.one_lt_exp_iff.mpr zero_lt_one).le.trans hx
+    exact mul_nonneg (Real.rpow_nonneg (Real.log_nonneg h1) q) (Real.exp_pos _).le
+  · filter_upwards [eventually_ge_atTop (Real.exp 1)] with x hx
+    have hxp : 0 < x := (Real.exp_pos 1).trans_le hx
+    have hlog : 1 ≤ Real.log x := by
+      simpa using Real.log_le_log (Real.exp_pos 1) hx
+    have hp : (Real.log x) ^ q ≤ Real.log x := by
+      simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hlog hq
+    exact mul_le_mul_of_nonneg_right (hp.trans (Real.log_le_self hxp.le))
+      (Real.exp_pos _).le
+
+theorem explicit_cleanup_tendsto_zero {κ c : ℝ} (hκ : 0 < κ) (hc : 0 < c) :
+    Tendsto (fun n : ℕ => (Real.log (n : ℝ)) ^ (codimension κ / 2) *
+      Real.exp (-c * Real.sqrt (n : ℝ))) atTop (𝓝 0) := by
+  have hq : codimension κ / 2 ≤ 1 := by
+    dsimp [codimension, dimension]
+    linarith
+  exact (logarithmic_cleanup_tendsto_zero hq hc).comp tendsto_natCast_atTop_atTop
+
+/-- Finite expected residual cost is the sum of each missed-cell probability
+times its deterministic covering cost. -/
+theorem expected_missed_cell_cost
+    {Ω ι : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (I : Finset ι) (missed : ι → Set Ω)
+    (hm : ∀ i ∈ I, MeasurableSet (missed i))
+    (cost bound : ι → ℝ≥0∞)
+    (hb : ∀ i ∈ I, P (missed i) ≤ bound i) :
+    (∫⁻ ω, ∑ i ∈ I, (missed i).indicator (fun _ => cost i) ω ∂P) ≤
+      ∑ i ∈ I, cost i * bound i := by
+  rw [lintegral_finsetSum I (fun i hi => measurable_const.indicator (hm i hi))]
+  apply Finset.sum_le_sum
+  intro i hi
+  rw [lintegral_indicator (hm i hi), lintegral_const, Measure.restrict_apply_univ]
+  exact mul_le_mul' le_rfl (hb i hi)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverRadiusLevels.lean b/lean/OAI/Probability/SLE/Gauge/CoverRadiusLevels.lean
index 0cecf497fd3036c5ed2bc09eb61e2de5fa60ab26..58b64cfca73cf8898c0fea5d878254fdcb59c95f 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverRadiusLevels.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverRadiusLevels.lean
@@ -1,67 +1,67 @@
-import OAI.Probability.SLE.Gauge.CoverSchedule
-import OAI.Probability.SLE.Loewner.ActualRadiusStops
-
-namespace OAI
-
-/-! Absolute geometric radii are represented by nonnegative logarithmic
-clock levels. The initial truncation disappears simultaneously throughout
-every sufficiently late cover-test window. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def coverLogLevel (z : ℂ) (T : ℝ) (j : ℕ) : ℝ :=
-  max 0 (Real.log (2 * z.im) + T * (j : ℝ))
-
-lemma coverLogLevel_nonneg (z : ℂ) (T : ℝ) (j : ℕ) : 0 ≤ coverLogLevel z T j :=
-  le_max_left _ _
-
-lemma coverLogLevel_monotone (z : ℂ) {T : ℝ} (hT : 0 ≤ T) :
-    Monotone (coverLogLevel z T) := by
-  intro i j hij
-  change max 0 (Real.log (2 * z.im) + T * (i : ℝ)) ≤
-    max 0 (Real.log (2 * z.im) + T * (j : ℝ))
-  apply max_le_max le_rfl
-  have hijR : (i : ℝ) ≤ j := by exact_mod_cast hij
-  linarith [mul_le_mul_of_nonneg_left hijR hT]
-
-lemma eventually_cover_window_untruncated
-    (z : ℂ) {T : ℝ} (hT : 0 < T) :
-    ∀ᶠ n : ℕ in atTop, ∀ j ∈ coverTestWindow n,
-      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ) := by
-  obtain ⟨N, hN⟩ := exists_nat_gt (-Real.log (2 * z.im) / T)
-  have hNT : -Real.log (2 * z.im) < T * (N : ℝ) := by
-    simpa only [mul_comm] using (div_lt_iff₀ hT).mp hN
-  filter_upwards [eventually_ge_atTop (3 * N)] with n hn
-  intro j hj
-  have hjN : N ≤ j := by
-    have hh := (Finset.mem_Icc.mp hj).1
-    omega
-  have hNj : T * (N : ℝ) ≤ T * (j : ℝ) :=
-    mul_le_mul_of_nonneg_left (by exact_mod_cast hjN) hT.le
-  exact max_eq_right (by linarith)
-
-lemma radius_target_eq_coverRadius
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (T : ℝ) (j : ℕ)
-    (hlevel : coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) :
-    (2 * z.im) * Real.exp (-coverLogLevel z T j) = coverRadius T j := by
-  have hr : 0 < 2 * z.im := mul_pos (by norm_num) hz
-  rw [hlevel, neg_add, Real.exp_add, Real.exp_neg, Real.exp_log hr]
-  unfold coverRadius
-  field_simp [(show 0 < z.im from hz).ne']
-
-lemma eventually_cover_window_radius_target
-    (z : ℂ) (hz : z ∈ upperHalfPlane) {T : ℝ} (hT : 0 < T) :
-    ∀ᶠ n : ℕ in atTop, ∀ j ∈ coverTestWindow n,
-      (2 * z.im) * Real.exp (-coverLogLevel z T j) = coverRadius T j := by
-  filter_upwards [eventually_cover_window_untruncated z hT] with n hn
-  exact fun j hj => radius_target_eq_coverRadius z hz T j (hn j hj)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverSchedule
+import OAI.Probability.SLE.Loewner.ActualRadiusStops
+
+namespace OAI
+
+/-! Absolute geometric radii are represented by nonnegative logarithmic
+clock levels. The initial truncation disappears simultaneously throughout
+every sufficiently late cover-test window. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def coverLogLevel (z : ℂ) (T : ℝ) (j : ℕ) : ℝ :=
+  max 0 (Real.log (2 * z.im) + T * (j : ℝ))
+
+lemma coverLogLevel_nonneg (z : ℂ) (T : ℝ) (j : ℕ) : 0 ≤ coverLogLevel z T j :=
+  le_max_left _ _
+
+lemma coverLogLevel_monotone (z : ℂ) {T : ℝ} (hT : 0 ≤ T) :
+    Monotone (coverLogLevel z T) := by
+  intro i j hij
+  change max 0 (Real.log (2 * z.im) + T * (i : ℝ)) ≤
+    max 0 (Real.log (2 * z.im) + T * (j : ℝ))
+  apply max_le_max le_rfl
+  have hijR : (i : ℝ) ≤ j := by exact_mod_cast hij
+  linarith [mul_le_mul_of_nonneg_left hijR hT]
+
+lemma eventually_cover_window_untruncated
+    (z : ℂ) {T : ℝ} (hT : 0 < T) :
+    ∀ᶠ n : ℕ in atTop, ∀ j ∈ coverTestWindow n,
+      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ) := by
+  obtain ⟨N, hN⟩ := exists_nat_gt (-Real.log (2 * z.im) / T)
+  have hNT : -Real.log (2 * z.im) < T * (N : ℝ) := by
+    simpa only [mul_comm] using (div_lt_iff₀ hT).mp hN
+  filter_upwards [eventually_ge_atTop (3 * N)] with n hn
+  intro j hj
+  have hjN : N ≤ j := by
+    have hh := (Finset.mem_Icc.mp hj).1
+    omega
+  have hNj : T * (N : ℝ) ≤ T * (j : ℝ) :=
+    mul_le_mul_of_nonneg_left (by exact_mod_cast hjN) hT.le
+  exact max_eq_right (by linarith)
+
+lemma radius_target_eq_coverRadius
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (T : ℝ) (j : ℕ)
+    (hlevel : coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) :
+    (2 * z.im) * Real.exp (-coverLogLevel z T j) = coverRadius T j := by
+  have hr : 0 < 2 * z.im := mul_pos (by norm_num) hz
+  rw [hlevel, neg_add, Real.exp_add, Real.exp_neg, Real.exp_log hr]
+  unfold coverRadius
+  field_simp [(show 0 < z.im from hz).ne']
+
+lemma eventually_cover_window_radius_target
+    (z : ℂ) (hz : z ∈ upperHalfPlane) {T : ℝ} (hT : 0 < T) :
+    ∀ᶠ n : ℕ in atTop, ∀ j ∈ coverTestWindow n,
+      (2 * z.im) * Real.exp (-coverLogLevel z T j) = coverRadius T j := by
+  filter_upwards [eventually_cover_window_untruncated z hT] with n hn
+  exact fun j hj => radius_target_eq_coverRadius z hz T j (hn j hj)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/CoverSchedule.lean b/lean/OAI/Probability/SLE/Gauge/CoverSchedule.lean
index 36a6bf937b7a996873484e364d128f15a47f77fa..913b8d140619779fbfc5e4c3f8a003e74ea1e26a 100644
--- a/lean/OAI/Probability/SLE/Gauge/CoverSchedule.lean
+++ b/lean/OAI/Probability/SLE/Gauge/CoverSchedule.lean
@@ -1,99 +1,99 @@
-import OAI.Probability.SLE.Gauge.DenseVisitNumerics
-
-namespace OAI
-
-/-! The common deterministic physical mesh and geometric clock radii in
-the explicit-gauge cover. The cutoff depends on the multiplicity and
-radius, never on the stopped state or spatial center. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology ENNReal
-
-namespace SLEExactGauge
-
-def coverRadius (T : ℝ) (j : ℕ) : ℝ := Real.exp (-T * (j : ℝ))
-
-theorem coverRadius_pos (T : ℝ) (j : ℕ) : 0 < coverRadius T j := Real.exp_pos _
-
-theorem coverRadius_antitone {T : ℝ} (hT : 0 ≤ T) : Antitone (coverRadius T) := by
-  intro i j hij
-  apply Real.exp_le_exp.mpr
-  have hh : (i : ℝ) ≤ j := by exact_mod_cast hij
-  nlinarith
-
-theorem coverRadius_tendsto_zero {T : ℝ} (hT : 0 < T) :
-    Tendsto (coverRadius T) atTop (𝓝 0) := by
-  have he : Real.exp (-T) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
-  convert tendsto_pow_atTop_nhds_zero_of_lt_one (Real.exp_pos (-T)).le he using 1
-  funext j
-  simp only [coverRadius, ← Real.exp_nat_mul]
-  congr 1
-  ring
-
-theorem coverRadius_third_tendsto_zero {T : ℝ} (hT : 0 < T) :
-    Tendsto (fun n : ℕ => coverRadius T (n / 3)) atTop (𝓝 0) := by
-  apply (coverRadius_tendsto_zero hT).comp
-  apply Filter.tendsto_atTop.mpr
-  intro N
-  filter_upwards [eventually_ge_atTop (3 * N)] with n hn
-  omega
-
-theorem coverRadius_window_upper {T : ℝ} (hT : 0 ≤ T) {n j : ℕ}
-    (hj : j ∈ coverTestWindow n) : coverRadius T j ≤ coverRadius T (n / 3) :=
-  coverRadius_antitone hT (Finset.mem_Icc.mp hj).1
-
-/-- The one-unit angle test and the dense experiment both finish before
-the next scheduled radius. -/
-theorem coverRadius_next_below_dense_cutoff
-    {T C₁ : ℝ} (hC₁ : 0 < C₁) (hgap : 1 + Real.log C₁ < T) (j : ℕ) :
-    coverRadius T (j + 1) < Real.exp (-1) * coverRadius T j / C₁ := by
-  have he : coverRadius T (j + 1) = coverRadius T j * Real.exp (-T) := by
-    simp only [coverRadius, Nat.cast_add, Nat.cast_one, ← Real.exp_add]
-    congr 1
-    ring
-  rw [he, mul_comm (Real.exp (-1)), mul_div_assoc]
-  apply mul_lt_mul_of_pos_left _ (coverRadius_pos T j)
-  rw [← Real.exp_log hC₁, ← Real.exp_sub]
-  exact Real.exp_lt_exp.mpr (by linarith)
-
-/-- One sequence works at every test and every center. It has strict slack
-below all dense-visit cutoffs, even though those cutoffs vary with `m_n`. -/
-theorem exists_cover_physical_mesh
-    (T c : ℝ) (δ : ℕ → ℝ) (hδ : ∀ m, 0 < δ m) :
-    ∃ v : ℕ → ℝ, (∀ n, 0 < v n) ∧
-      Tendsto v atTop (𝓝 0) ∧ (∀ n, v (n + 1) ≤ v n / 2) ∧
-      (∀ n, v n ≤ (2 : ℝ) ^ (-(n : ℝ))) ∧
-      ∀ n j, j ∈ coverTestWindow n →
-        v n < Real.exp (-1) * coverRadius T j * δ (coverMultiplicity c n) := by
-  let cutoff := fun n j => Real.exp (-1) * coverRadius T j * δ (coverMultiplicity c n) / 2
-  have hcut : ∀ n j, j ∈ coverTestWindow n → 0 < cutoff n j := by
-    intro n j _
-    dsimp only [cutoff]
-    exact div_pos (mul_pos (mul_pos (Real.exp_pos _) (coverRadius_pos T j)) (hδ _))
-      (by norm_num)
-  obtain ⟨v, hv, hdyadic, hhalf, htest, hlim⟩ :=
-    CoverMesh.exists_common_mesh coverTestWindow cutoff hcut
-  refine ⟨v, hv, hlim, hhalf, hdyadic, ?_⟩
-  intro n j hj
-  refine (htest n j hj).trans_lt ?_
-  have hp : 0 < Real.exp (-1) * coverRadius T j * δ (coverMultiplicity c n) :=
-    mul_pos (mul_pos (Real.exp_pos _) (coverRadius_pos T j)) (hδ _)
-  dsimp only [cutoff]
-  linarith
-
-/-- A common diameter bound for the selected enlarged disks and the fine
-grid cells tends to zero. -/
-theorem cover_diameter_tendsto_zero {T : ℝ} (hT : 0 < T) (A B : ℝ) :
-    Tendsto (fun n => ENNReal.ofReal
-      (max (A * coverRadius T (n / 3)) (B * coverRadius T n))) atTop (𝓝 0) := by
-  have ht := ((coverRadius_third_tendsto_zero hT).const_mul A).max
-    ((coverRadius_tendsto_zero hT).const_mul B)
-  simpa only [Function.comp_def, mul_zero, max_self, ENNReal.ofReal_zero] using
-    ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.DenseVisitNumerics
+
+namespace OAI
+
+/-! The common deterministic physical mesh and geometric clock radii in
+the explicit-gauge cover. The cutoff depends on the multiplicity and
+radius, never on the stopped state or spatial center. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology ENNReal
+
+namespace SLEExactGauge
+
+def coverRadius (T : ℝ) (j : ℕ) : ℝ := Real.exp (-T * (j : ℝ))
+
+theorem coverRadius_pos (T : ℝ) (j : ℕ) : 0 < coverRadius T j := Real.exp_pos _
+
+theorem coverRadius_antitone {T : ℝ} (hT : 0 ≤ T) : Antitone (coverRadius T) := by
+  intro i j hij
+  apply Real.exp_le_exp.mpr
+  have hh : (i : ℝ) ≤ j := by exact_mod_cast hij
+  nlinarith
+
+theorem coverRadius_tendsto_zero {T : ℝ} (hT : 0 < T) :
+    Tendsto (coverRadius T) atTop (𝓝 0) := by
+  have he : Real.exp (-T) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
+  convert tendsto_pow_atTop_nhds_zero_of_lt_one (Real.exp_pos (-T)).le he using 1
+  funext j
+  simp only [coverRadius, ← Real.exp_nat_mul]
+  congr 1
+  ring
+
+theorem coverRadius_third_tendsto_zero {T : ℝ} (hT : 0 < T) :
+    Tendsto (fun n : ℕ => coverRadius T (n / 3)) atTop (𝓝 0) := by
+  apply (coverRadius_tendsto_zero hT).comp
+  apply Filter.tendsto_atTop.mpr
+  intro N
+  filter_upwards [eventually_ge_atTop (3 * N)] with n hn
+  omega
+
+theorem coverRadius_window_upper {T : ℝ} (hT : 0 ≤ T) {n j : ℕ}
+    (hj : j ∈ coverTestWindow n) : coverRadius T j ≤ coverRadius T (n / 3) :=
+  coverRadius_antitone hT (Finset.mem_Icc.mp hj).1
+
+/-- The one-unit angle test and the dense experiment both finish before
+the next scheduled radius. -/
+theorem coverRadius_next_below_dense_cutoff
+    {T C₁ : ℝ} (hC₁ : 0 < C₁) (hgap : 1 + Real.log C₁ < T) (j : ℕ) :
+    coverRadius T (j + 1) < Real.exp (-1) * coverRadius T j / C₁ := by
+  have he : coverRadius T (j + 1) = coverRadius T j * Real.exp (-T) := by
+    simp only [coverRadius, Nat.cast_add, Nat.cast_one, ← Real.exp_add]
+    congr 1
+    ring
+  rw [he, mul_comm (Real.exp (-1)), mul_div_assoc]
+  apply mul_lt_mul_of_pos_left _ (coverRadius_pos T j)
+  rw [← Real.exp_log hC₁, ← Real.exp_sub]
+  exact Real.exp_lt_exp.mpr (by linarith)
+
+/-- One sequence works at every test and every center. It has strict slack
+below all dense-visit cutoffs, even though those cutoffs vary with `m_n`. -/
+theorem exists_cover_physical_mesh
+    (T c : ℝ) (δ : ℕ → ℝ) (hδ : ∀ m, 0 < δ m) :
+    ∃ v : ℕ → ℝ, (∀ n, 0 < v n) ∧
+      Tendsto v atTop (𝓝 0) ∧ (∀ n, v (n + 1) ≤ v n / 2) ∧
+      (∀ n, v n ≤ (2 : ℝ) ^ (-(n : ℝ))) ∧
+      ∀ n j, j ∈ coverTestWindow n →
+        v n < Real.exp (-1) * coverRadius T j * δ (coverMultiplicity c n) := by
+  let cutoff := fun n j => Real.exp (-1) * coverRadius T j * δ (coverMultiplicity c n) / 2
+  have hcut : ∀ n j, j ∈ coverTestWindow n → 0 < cutoff n j := by
+    intro n j _
+    dsimp only [cutoff]
+    exact div_pos (mul_pos (mul_pos (Real.exp_pos _) (coverRadius_pos T j)) (hδ _))
+      (by norm_num)
+  obtain ⟨v, hv, hdyadic, hhalf, htest, hlim⟩ :=
+    CoverMesh.exists_common_mesh coverTestWindow cutoff hcut
+  refine ⟨v, hv, hlim, hhalf, hdyadic, ?_⟩
+  intro n j hj
+  refine (htest n j hj).trans_lt ?_
+  have hp : 0 < Real.exp (-1) * coverRadius T j * δ (coverMultiplicity c n) :=
+    mul_pos (mul_pos (Real.exp_pos _) (coverRadius_pos T j)) (hδ _)
+  dsimp only [cutoff]
+  linarith
+
+/-- A common diameter bound for the selected enlarged disks and the fine
+grid cells tends to zero. -/
+theorem cover_diameter_tendsto_zero {T : ℝ} (hT : 0 < T) (A B : ℝ) :
+    Tendsto (fun n => ENNReal.ofReal
+      (max (A * coverRadius T (n / 3)) (B * coverRadius T n))) atTop (𝓝 0) := by
+  have ht := ((coverRadius_third_tendsto_zero hT).const_mul A).max
+    ((coverRadius_tendsto_zero hT).const_mul B)
+  simpa only [Function.comp_def, mul_zero, max_self, ENNReal.ofReal_zero] using
+    ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseCandidateCover.lean b/lean/OAI/Probability/SLE/Gauge/DenseCandidateCover.lean
index f8ec2016f16aba88441ce500d24371a31028b8c1..93234368efc797fa921017d63e37301818e524a1 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseCandidateCover.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseCandidateCover.lean
@@ -1,338 +1,338 @@
-import OAI.Probability.SLE.Gauge.CoverGrid
-import OAI.Probability.SLE.Gauge.CoverSchedule
-import OAI.Probability.SLE.Gauge.CoverGaugeBounds
-import OAI.Probability.SLE.Gauge.DenseCoverUpperReduction
-
-namespace OAI
-
-/-! The actual finite terminal-sausage candidates and missed cells in the
-explicit-gauge cover. All selections here are deterministic at one mesh. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-abbrev DenseCandidateIndex := (ℤ × ℤ) × ℕ
-
-def denseCandidateCenter (T : ℝ) (n : ℕ) (i : DenseCandidateIndex) : ℂ :=
-  coverGridCenter (coverRadius T n) i.1
-
-def denseCandidateRadius (T : ℝ) (i : DenseCandidateIndex) : ℝ :=
-  Real.exp (-1) * coverRadius T i.2
-
-def denseCandidateThreshold (κ T b c : ℝ) (n : ℕ) (i : DenseCandidateIndex) : ℝ :=
-  b * denseCandidateRadius T i ^ dimension κ * (coverMultiplicity c n : ℝ) ^ codimension κ
-
-def denseCandidateIndices (R T : ℝ) (K : Set ℂ) (κ b c v : ℝ)
-    (n : ℕ) (γ : ℝ≥0 → ℂ) : Finset DenseCandidateIndex := by
-  classical
-  exact ((retainedCoverGridIndex R (coverRadius T n) K) ×ˢ coverTestWindow n).filter
-    (fun i => ENNReal.ofReal (denseCandidateThreshold κ T b c n i) ≤
-      wholeSausageCharge κ γ v (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)))
-
-def denseFineCell (T : ℝ) (n : ℕ) (ij : ℤ × ℤ) : Set ℂ :=
-  closedBall (coverGridCenter (coverRadius T n) ij) (2 * coverRadius T n)
-
-def denseMissedCell (κ T b c v : ℝ) (n : ℕ) (ij : ℤ × ℤ)
-    (γ : ℝ≥0 → ℂ) : Prop :=
-  (range γ ∩ denseFineCell T n ij).Nonempty ∧
-    ∀ j ∈ coverTestWindow n,
-      wholeSausageCharge κ γ v
-        (closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j))) <
-          ENNReal.ofReal (denseCandidateThreshold κ T b c n (ij, j))
-
-lemma denseCandidate_mem {R T κ b c v : ℝ} {K : Set ℂ} {n : ℕ}
-    {γ : ℝ≥0 → ℂ} {i : DenseCandidateIndex}
-    (hi : i ∈ denseCandidateIndices R T K κ b c v n γ) :
-    i.1 ∈ retainedCoverGridIndex R (coverRadius T n) K ∧ i.2 ∈ coverTestWindow n ∧
-      ENNReal.ofReal (denseCandidateThreshold κ T b c n i) ≤
-        wholeSausageCharge κ γ v (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) := by
-  classical
-  exact ⟨(Finset.mem_product.mp (Finset.mem_filter.mp hi).1).1,
-    (Finset.mem_product.mp (Finset.mem_filter.mp hi).1).2, (Finset.mem_filter.mp hi).2⟩
-
-lemma denseCandidateRadius_pos (T : ℝ) (i : DenseCandidateIndex) :
-    0 < denseCandidateRadius T i := mul_pos (Real.exp_pos _) (coverRadius_pos _ _)
-
-lemma denseCandidateRadius_le_one {T : ℝ} (hT : 0 ≤ T) (i : DenseCandidateIndex) :
-    denseCandidateRadius T i ≤ 1 := by
-  have hr : coverRadius T i.2 ≤ 1 := by
-    apply Real.exp_le_one_iff.mpr
-    exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hT) (Nat.cast_nonneg _)
-  have he : Real.exp (-1) ≤ 1 := Real.exp_le_one_iff.mpr (by norm_num)
-  calc
-    _ ≤ Real.exp (-1) * 1 := mul_le_mul_of_nonneg_left hr (Real.exp_pos _).le
-    _ = Real.exp (-1) := mul_one _
-    _ ≤ 1 := he
-
-/-- Every fine cell is eventually contained in every candidate disk at its
-center throughout the exact window `floor(n/3)..floor(2n/3)`. -/
-theorem denseFineCell_eventually_subset_candidate {T : ℝ} (hT : 0 < T) :
-    ∀ᶠ n : ℕ in atTop, ∀ j ∈ coverTestWindow n, ∀ ij : ℤ × ℤ,
-      denseFineCell T n ij ⊆
-        closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j)) := by
-  have hlarge : ∀ᶠ n : ℕ in atTop, 3 * (1 + Real.log 2) / T ≤ (n : ℝ) :=
-    tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _)
-  filter_upwards [hlarge] with n hn
-  intro j hj ij z hz
-  have hjn : 3 * j ≤ 2 * n := by
-    have hh := (Finset.mem_Icc.mp hj).2
-    omega
-  have hjR : (3 : ℝ) * j ≤ 2 * n := by exact_mod_cast hjn
-  have hnT := (div_le_iff₀ hT).mp hn
-  have he : 2 * coverRadius T n ≤ denseCandidateRadius T (ij, j) := by
-    dsimp only [coverRadius, denseCandidateRadius]
-    rw [← Real.exp_log (by norm_num : (0 : ℝ) < 2), ← Real.exp_add, ← Real.exp_add]
-    apply Real.exp_le_exp.mpr
-    nlinarith
-  exact hz.trans he
-
-/-- The missed cells are exactly the visited retained cells with no terminal
-candidate; this yields a pointwise cover without a random selection rule. -/
-theorem dense_candidates_and_missed_cover
-    {R T : ℝ} (hR : 0 ≤ R) (hT : 0 < T) {K : Set ℂ} (hK : K ⊆ closedBall 0 R)
-    (κ b c : ℝ) :
-    ∀ᶠ n : ℕ in atTop, ∀ (v : ℝ) (γ : ℝ≥0 → ℂ) z, z ∈ K ∩ range γ →
-      (∃ i ∈ denseCandidateIndices R T K κ b c v n γ,
-        z ∈ closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) ∨
-      ∃ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-        denseMissedCell κ T b c v n ij γ ∧ z ∈ denseFineCell T n ij := by
-  classical
-  filter_upwards [denseFineCell_eventually_subset_candidate hT] with n hn
-  intro v γ z hz
-  obtain ⟨ij, hij, hzcell⟩ := mem_iUnion₂.mp
-    (retainedCoverGrid_covers hR (coverRadius_pos T n) hK hz.1)
-  by_cases hs : ∃ j ∈ coverTestWindow n,
-      ENNReal.ofReal (denseCandidateThreshold κ T b c n (ij, j)) ≤
-        wholeSausageCharge κ γ v
-          (closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j)))
-  · obtain ⟨j, hj, hmass⟩ := hs
-    exact Or.inl ⟨(ij, j), Finset.mem_filter.mpr
-      ⟨Finset.mem_product.mpr ⟨hij, hj⟩, hmass⟩, hn j hj ij hzcell⟩
-  · refine Or.inr ⟨ij, hij, ⟨⟨z, hz.2, hzcell⟩, ?_⟩, hzcell⟩
-    intro j hj
-    exact lt_of_not_ge (fun hmass => hs ⟨j, hj, hmass⟩)
-
-lemma denseCandidate_disk_subset {R T κ b c v : ℝ} (hT : 0 ≤ T) {K : Set ℂ}
-    (hK : K ⊆ closedBall 0 R) {n : ℕ} {γ : ℝ≥0 → ℂ} {i : DenseCandidateIndex}
-    (hi : i ∈ denseCandidateIndices R T K κ b c v n γ) :
-    closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i) ⊆ closedBall 0 (R + 3) := by
-  have hheight : ∀ z ∈ K, -R ≤ z.im := by
-    intro z hz
-    have hn : ‖z‖ ≤ R := by simpa only [mem_closedBall, dist_zero_right] using hK hz
-    have him := (Complex.abs_im_le_norm z).trans hn
-    exact (abs_le.mp him).1
-  have hcenter := (retainedCoverGrid_center_bounds hK hheight (denseCandidate_mem hi).1).1
-  have hrn : coverRadius T n ≤ 1 := Real.exp_le_one_iff.mpr (by
-    have hp := mul_nonneg hT (Nat.cast_nonneg n)
-    linarith)
-  intro z hz
-  have ht := dist_triangle z (denseCandidateCenter T n i) 0
-  have hr := denseCandidateRadius_le_one hT i
-  change dist z 0 ≤ R + 3
-  change dist z (denseCandidateCenter T n i) ≤ denseCandidateRadius T i at hz
-  change ‖denseCandidateCenter T n i‖ ≤ R + 2 * coverRadius T n at hcenter
-  simp only [dist_zero_right] at ht ⊢
-  linarith
-
-def denseCoverDiameter (T : ℝ) (n : ℕ) : ℝ≥0∞ :=
-  ENNReal.ofReal (max (8 * Real.exp (-1) * coverRadius T (n / 3)) (4 * coverRadius T n))
-
-lemma denseCoverDiameter_tendsto_zero {T : ℝ} (hT : 0 < T) :
-    Tendsto (denseCoverDiameter T) atTop (𝓝 0) :=
-  cover_diameter_tendsto_zero hT (8 * Real.exp (-1)) 4
-
-lemma denseCandidate_enlarged_diameter {R T κ b c v : ℝ} (hT : 0 ≤ T) {K : Set ℂ}
-    {n : ℕ} {γ : ℝ≥0 → ℂ} {i : DenseCandidateIndex}
-    (hi : i ∈ denseCandidateIndices R T K κ b c v n γ) :
-    ediam (closedBall (denseCandidateCenter T n i) (4 * denseCandidateRadius T i)) ≤
-      denseCoverDiameter T n := by
-  apply (ediam_le_of_forall_dist_le ?_).trans (ENNReal.ofReal_le_ofReal (le_max_left _ _))
-  intro x hx y hy
-  have ht := dist_triangle x (denseCandidateCenter T n i) y
-  have hx' : dist x (denseCandidateCenter T n i) ≤ 4 * denseCandidateRadius T i := hx
-  have hy' : dist (denseCandidateCenter T n i) y ≤ 4 * denseCandidateRadius T i := by
-    change dist y (denseCandidateCenter T n i) ≤ 4 * denseCandidateRadius T i at hy
-    simpa only [dist_comm] using hy
-  have hr := coverRadius_window_upper hT (denseCandidate_mem hi).2.1
-  have he := Real.exp_pos (-1)
-  dsimp only [denseCandidateRadius] at hx' hy'
-  nlinarith
-
-lemma denseFineCell_diameter (T : ℝ) (n : ℕ) (ij : ℤ × ℤ) :
-    ediam (denseFineCell T n ij) ≤ denseCoverDiameter T n :=
-  (coverGrid_cell_diameter (coverRadius_pos T n) ij).trans
-    (ENNReal.ofReal_le_ofReal (le_max_right _ _))
-
-
-local instance denseCandidatePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance denseCandidatePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem denseMissedCell_measurable
-    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (κ T b c v : ℝ) (n : ℕ) (ij : ℤ × ℤ) :
-    MeasurableSet {ω | denseMissedCell κ T b c v n ij (f ω)} := by
-  have hm := measurableSet_visited_without_candidate f hf κ v (coverTestWindow n)
-    (fun j => closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j)))
-    (fun j => denseCandidateThreshold κ T b c n (ij, j))
-    (Q := denseFineCell T n ij) isClosed_closedBall
-  convert hm using 1
-  ext ω
-  simp only [denseMissedCell, mem_ofPred_eq, Set.Nonempty, mem_inter_iff, mem_range]
-  constructor
-  · rintro ⟨⟨z, ⟨t, rfl⟩, hz⟩, hh⟩
-    exact ⟨⟨t, hz⟩, hh⟩
-  · rintro ⟨⟨t, ht⟩, hh⟩
-    exact ⟨⟨f ω t, ⟨t, rfl⟩, ht⟩, hh⟩
-
-lemma extendedGauge_closedBall_le {h : ℝ → ℝ} (hh : IsGauge h)
-    (z : ℂ) {r : ℝ} (hr : 0 ≤ r) :
-    extendedGauge h (ediam (closedBall z r)) ≤ ENNReal.ofReal (h (2 * r)) := by
-  have hd : ediam (closedBall z r) ≤ ENNReal.ofReal (2 * r) := by
-    apply ediam_le_of_forall_dist_le
-    intro x hx y hy
-    have hx' : dist x z ≤ r := hx
-    have hy' : dist z y ≤ r := by
-      change dist y z ≤ r at hy
-      simpa only [dist_comm] using hy
-    exact (dist_triangle x z y).trans (by linarith)
-  have hdR : (ediam (closedBall z r)).toReal ≤ 2 * r := by
-    exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hd).trans_eq
-      (ENNReal.toReal_ofReal (by positivity))
-  exact ENNReal.ofReal_le_ofReal
-    (hh.monotoneOn ENNReal.toReal_nonneg (show 0 ≤ 2 * r by positivity) hdR)
-
-/-- The prescribed integer multiplicity pays the logarithmic correction of
-all selected disks, uniformly throughout the window. -/
-theorem denseCandidate_eventual_gauge_charge
-    {κ T b c : ℝ} (hκ8 : κ < 8) (hT : 0 < T) (hb : 0 < b) (hc : 0 < c)
-    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∃ C : ℝ≥0, 0 < C ∧ ∀ᶠ n : ℕ in atTop,
-      ∀ (R : ℝ) (K : Set ℂ) (v : ℝ) (γ : ℝ≥0 → ℂ) i,
-        i ∈ denseCandidateIndices R T K κ b c v n γ →
-        extendedGauge h (ediam (closedBall (denseCandidateCenter T n i)
-          (4 * denseCandidateRadius T i))) ≤
-            (C : ℝ≥0∞) * wholeSausageCharge κ γ v
-              (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) := by
-  have hp : 0 ≤ codimension κ := by dsimp [codimension, dimension]; linarith
-  obtain ⟨A, hA, hbound⟩ := CoverGaugeBounds.eventual_window_bound hκ8
-    (mul_pos (by norm_num : (0 : ℝ) < 8) (Real.exp_pos (-1))) hT hf
-  let L := b * (Real.exp (-1)) ^ dimension κ * (c / 2) ^ codimension κ
-  have hL : 0 < L := by dsimp only [L]; positivity
-  let C : ℝ≥0 := ⟨A / L, (div_pos hA hL).le⟩
-  refine ⟨C, div_pos hA hL, ?_⟩
-  filter_upwards [hbound, coverMultiplicity_mass_lower hc hp,
-    eventually_ge_atTop (12 : ℕ)] with n hn hmass hn12
-  intro R K v γ i hi
-  have hj := (denseCandidate_mem hi).2.1
-  have hjlow : (n : ℝ) / 4 ≤ (i.2 : ℝ) := by
-    have hnI : n / 3 ≤ i.2 := (Finset.mem_Icc.mp hj).1
-    have hfour : n ≤ 4 * i.2 := by omega
-    have hfourR : (n : ℝ) ≤ 4 * i.2 := by exact_mod_cast hfour
-    linarith
-  have hjhigh : i.2 ≤ n := by
-    have hnI := (Finset.mem_Icc.mp hj).2
-    omega
-  have hg := hn i.2 hjlow hjhigh
-  change h (8 * Real.exp (-1) * coverRadius T i.2) ≤
-    A * coverRadius T i.2 ^ dimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2) at hg
-  have hscale : L * (coverRadius T i.2 ^ dimension κ *
-      (Real.log (n : ℝ)) ^ (codimension κ / 2)) ≤ denseCandidateThreshold κ T b c n i := by
-    unfold denseCandidateThreshold denseCandidateRadius
-    rw [Real.mul_rpow (Real.exp_pos _).le (coverRadius_pos _ _).le]
-    calc
-      _ = (b * Real.exp (-1) ^ dimension κ * coverRadius T i.2 ^ dimension κ) *
-          ((c / 2) ^ codimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2)) := by
-        dsimp only [L]
-        ring
-      _ ≤ (b * Real.exp (-1) ^ dimension κ * coverRadius T i.2 ^ dimension κ) *
-          (coverMultiplicity c n : ℝ) ^ codimension κ :=
-        mul_le_mul_of_nonneg_left hmass
-          (mul_nonneg (mul_nonneg hb.le (Real.rpow_nonneg (Real.exp_pos _).le _))
-            (Real.rpow_nonneg (coverRadius_pos _ _).le _))
-      _ = _ := by ring
-  have hreal : h (8 * denseCandidateRadius T i) ≤ (C : ℝ) * denseCandidateThreshold κ T b c n i := by
-    calc
-      _ = h (8 * Real.exp (-1) * coverRadius T i.2) := by congr 1; simp only [denseCandidateRadius]; ring
-      _ ≤ A * coverRadius T i.2 ^ dimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2) := hg
-      _ = (C : ℝ) * (L * (coverRadius T i.2 ^ dimension κ *
-          (Real.log (n : ℝ)) ^ (codimension κ / 2))) := by
-        change _ = A / L * _
-        field_simp
-      _ ≤ _ := mul_le_mul_of_nonneg_left hscale C.2
-  calc
-    _ ≤ ENNReal.ofReal (h (8 * denseCandidateRadius T i)) := by
-      have he : 2 * (4 * denseCandidateRadius T i) = 8 * denseCandidateRadius T i := by ring
-      simpa only [he] using extendedGauge_closedBall_le hh (denseCandidateCenter T n i)
-        (r := 4 * denseCandidateRadius T i) (by positivity [denseCandidateRadius_pos T i])
-    _ ≤ ENNReal.ofReal ((C : ℝ) * denseCandidateThreshold κ T b c n i) := ENNReal.ofReal_le_ofReal hreal
-    _ = (C : ℝ≥0∞) * ENNReal.ofReal (denseCandidateThreshold κ T b c n i) := by
-      have hC : ENNReal.ofReal (C : ℝ) = (C : ℝ≥0∞) := ENNReal.ofReal_coe_nnreal
-      exact (ENNReal.ofReal_mul C.2).trans (congrArg
-        (fun t : ℝ≥0∞ => t * ENNReal.ofReal (denseCandidateThreshold κ T b c n i)) hC)
-    _ ≤ _ := mul_le_mul' le_rfl (denseCandidate_mem hi).2.2
-
-/-- The cost of each fine cleanup cell has the matching explicit-gauge
-factor, independent of the cell center. -/
-theorem denseFineCell_eventual_gauge_bound
-    {κ T : ℝ} (hκ8 : κ < 8) (hT : 0 < T)
-    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∃ A : ℝ, 0 < A ∧ ∀ᶠ n : ℕ in atTop, ∀ ij : ℤ × ℤ,
-      extendedGauge h (ediam (denseFineCell T n ij)) ≤
-        ENNReal.ofReal (A * coverRadius T n ^ dimension κ *
-          (Real.log (n : ℝ)) ^ (codimension κ / 2)) := by
-  obtain ⟨A, hA, hbound⟩ := CoverGaugeBounds.eventual_window_bound hκ8 (by norm_num : (0 : ℝ) < 4) hT hf
-  refine ⟨A, hA, ?_⟩
-  filter_upwards [hbound] with n hn
-  intro ij
-  have hg := hn n (by have hn0 : (0 : ℝ) ≤ n := Nat.cast_nonneg n; linarith) le_rfl
-  change h (4 * coverRadius T n) ≤ _ at hg
-  have he : 2 * (2 * coverRadius T n) = 4 * coverRadius T n := by ring
-  have hd : extendedGauge h (ediam (denseFineCell T n ij)) ≤
-      ENNReal.ofReal (h (4 * coverRadius T n)) := by
-    simpa only [denseFineCell, he] using
-      (extendedGauge_closedBall_le hh (coverGridCenter (coverRadius T n) ij)
-        (r := 2 * coverRadius T n) (by positivity [coverRadius_pos T n]))
-  exact hd.trans (ENNReal.ofReal_le_ofReal hg)
-
-/-- The six concrete geometric hypotheses of the expected-cover reduction,
-with one constant and one sufficiently late cutoff for every path and mesh. -/
-theorem dense_candidate_cover_geometry
-    {κ R T b c : ℝ} (hκ8 : κ < 8) (hR : 0 ≤ R) (hT : 0 < T)
-    (hb : 0 < b) (hc : 0 < c) {K : Set ℂ} (hK : K ⊆ closedBall 0 R)
-    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∃ C : ℝ≥0, 0 < C ∧ ∀ᶠ n : ℕ in atTop, ∀ (v : ℝ) (γ : ℝ≥0 → ℂ),
-      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ, denseCandidateRadius T i ≤ R + 3) ∧
-      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ,
-        closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i) ⊆ closedBall 0 (R + 3)) ∧
-      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ,
-        extendedGauge h (ediam (closedBall (denseCandidateCenter T n i) (4 * denseCandidateRadius T i))) ≤
-          (C : ℝ≥0∞) * wholeSausageCharge κ γ v
-            (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i))) ∧
-      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ,
-        ediam (closedBall (denseCandidateCenter T n i) (4 * denseCandidateRadius T i)) ≤ denseCoverDiameter T n) ∧
-      (∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-        ediam (denseFineCell T n ij) ≤ denseCoverDiameter T n) ∧
-      (∀ z ∈ K ∩ range γ,
-        (∃ i ∈ denseCandidateIndices R T K κ b c v n γ,
-          z ∈ closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) ∨
-        ∃ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-          denseMissedCell κ T b c v n ij γ ∧ z ∈ denseFineCell T n ij) := by
-  obtain ⟨C, hC, hcharge⟩ := denseCandidate_eventual_gauge_charge hκ8 hT hb hc hh hf
-  refine ⟨C, hC, ?_⟩
-  filter_upwards [hcharge, dense_candidates_and_missed_cover hR hT hK κ b c] with n hn hcover
-  intro v γ
-  refine ⟨?_, ?_, hn R K v γ, ?_, ?_, hcover v γ⟩
-  · intro i _
-    exact (denseCandidateRadius_le_one hT.le i).trans (by linarith)
-  · exact fun _ hi => denseCandidate_disk_subset hT.le hK hi
-  · exact fun _ hi => denseCandidate_enlarged_diameter hT.le hi
-  · exact fun ij _ => denseFineCell_diameter T n ij
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverGrid
+import OAI.Probability.SLE.Gauge.CoverSchedule
+import OAI.Probability.SLE.Gauge.CoverGaugeBounds
+import OAI.Probability.SLE.Gauge.DenseCoverUpperReduction
+
+namespace OAI
+
+/-! The actual finite terminal-sausage candidates and missed cells in the
+explicit-gauge cover. All selections here are deterministic at one mesh. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+abbrev DenseCandidateIndex := (ℤ × ℤ) × ℕ
+
+def denseCandidateCenter (T : ℝ) (n : ℕ) (i : DenseCandidateIndex) : ℂ :=
+  coverGridCenter (coverRadius T n) i.1
+
+def denseCandidateRadius (T : ℝ) (i : DenseCandidateIndex) : ℝ :=
+  Real.exp (-1) * coverRadius T i.2
+
+def denseCandidateThreshold (κ T b c : ℝ) (n : ℕ) (i : DenseCandidateIndex) : ℝ :=
+  b * denseCandidateRadius T i ^ dimension κ * (coverMultiplicity c n : ℝ) ^ codimension κ
+
+def denseCandidateIndices (R T : ℝ) (K : Set ℂ) (κ b c v : ℝ)
+    (n : ℕ) (γ : ℝ≥0 → ℂ) : Finset DenseCandidateIndex := by
+  classical
+  exact ((retainedCoverGridIndex R (coverRadius T n) K) ×ˢ coverTestWindow n).filter
+    (fun i => ENNReal.ofReal (denseCandidateThreshold κ T b c n i) ≤
+      wholeSausageCharge κ γ v (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)))
+
+def denseFineCell (T : ℝ) (n : ℕ) (ij : ℤ × ℤ) : Set ℂ :=
+  closedBall (coverGridCenter (coverRadius T n) ij) (2 * coverRadius T n)
+
+def denseMissedCell (κ T b c v : ℝ) (n : ℕ) (ij : ℤ × ℤ)
+    (γ : ℝ≥0 → ℂ) : Prop :=
+  (range γ ∩ denseFineCell T n ij).Nonempty ∧
+    ∀ j ∈ coverTestWindow n,
+      wholeSausageCharge κ γ v
+        (closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j))) <
+          ENNReal.ofReal (denseCandidateThreshold κ T b c n (ij, j))
+
+lemma denseCandidate_mem {R T κ b c v : ℝ} {K : Set ℂ} {n : ℕ}
+    {γ : ℝ≥0 → ℂ} {i : DenseCandidateIndex}
+    (hi : i ∈ denseCandidateIndices R T K κ b c v n γ) :
+    i.1 ∈ retainedCoverGridIndex R (coverRadius T n) K ∧ i.2 ∈ coverTestWindow n ∧
+      ENNReal.ofReal (denseCandidateThreshold κ T b c n i) ≤
+        wholeSausageCharge κ γ v (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) := by
+  classical
+  exact ⟨(Finset.mem_product.mp (Finset.mem_filter.mp hi).1).1,
+    (Finset.mem_product.mp (Finset.mem_filter.mp hi).1).2, (Finset.mem_filter.mp hi).2⟩
+
+lemma denseCandidateRadius_pos (T : ℝ) (i : DenseCandidateIndex) :
+    0 < denseCandidateRadius T i := mul_pos (Real.exp_pos _) (coverRadius_pos _ _)
+
+lemma denseCandidateRadius_le_one {T : ℝ} (hT : 0 ≤ T) (i : DenseCandidateIndex) :
+    denseCandidateRadius T i ≤ 1 := by
+  have hr : coverRadius T i.2 ≤ 1 := by
+    apply Real.exp_le_one_iff.mpr
+    exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hT) (Nat.cast_nonneg _)
+  have he : Real.exp (-1) ≤ 1 := Real.exp_le_one_iff.mpr (by norm_num)
+  calc
+    _ ≤ Real.exp (-1) * 1 := mul_le_mul_of_nonneg_left hr (Real.exp_pos _).le
+    _ = Real.exp (-1) := mul_one _
+    _ ≤ 1 := he
+
+/-- Every fine cell is eventually contained in every candidate disk at its
+center throughout the exact window `floor(n/3)..floor(2n/3)`. -/
+theorem denseFineCell_eventually_subset_candidate {T : ℝ} (hT : 0 < T) :
+    ∀ᶠ n : ℕ in atTop, ∀ j ∈ coverTestWindow n, ∀ ij : ℤ × ℤ,
+      denseFineCell T n ij ⊆
+        closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j)) := by
+  have hlarge : ∀ᶠ n : ℕ in atTop, 3 * (1 + Real.log 2) / T ≤ (n : ℝ) :=
+    tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _)
+  filter_upwards [hlarge] with n hn
+  intro j hj ij z hz
+  have hjn : 3 * j ≤ 2 * n := by
+    have hh := (Finset.mem_Icc.mp hj).2
+    omega
+  have hjR : (3 : ℝ) * j ≤ 2 * n := by exact_mod_cast hjn
+  have hnT := (div_le_iff₀ hT).mp hn
+  have he : 2 * coverRadius T n ≤ denseCandidateRadius T (ij, j) := by
+    dsimp only [coverRadius, denseCandidateRadius]
+    rw [← Real.exp_log (by norm_num : (0 : ℝ) < 2), ← Real.exp_add, ← Real.exp_add]
+    apply Real.exp_le_exp.mpr
+    nlinarith
+  exact hz.trans he
+
+/-- The missed cells are exactly the visited retained cells with no terminal
+candidate; this yields a pointwise cover without a random selection rule. -/
+theorem dense_candidates_and_missed_cover
+    {R T : ℝ} (hR : 0 ≤ R) (hT : 0 < T) {K : Set ℂ} (hK : K ⊆ closedBall 0 R)
+    (κ b c : ℝ) :
+    ∀ᶠ n : ℕ in atTop, ∀ (v : ℝ) (γ : ℝ≥0 → ℂ) z, z ∈ K ∩ range γ →
+      (∃ i ∈ denseCandidateIndices R T K κ b c v n γ,
+        z ∈ closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) ∨
+      ∃ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+        denseMissedCell κ T b c v n ij γ ∧ z ∈ denseFineCell T n ij := by
+  classical
+  filter_upwards [denseFineCell_eventually_subset_candidate hT] with n hn
+  intro v γ z hz
+  obtain ⟨ij, hij, hzcell⟩ := mem_iUnion₂.mp
+    (retainedCoverGrid_covers hR (coverRadius_pos T n) hK hz.1)
+  by_cases hs : ∃ j ∈ coverTestWindow n,
+      ENNReal.ofReal (denseCandidateThreshold κ T b c n (ij, j)) ≤
+        wholeSausageCharge κ γ v
+          (closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j)))
+  · obtain ⟨j, hj, hmass⟩ := hs
+    exact Or.inl ⟨(ij, j), Finset.mem_filter.mpr
+      ⟨Finset.mem_product.mpr ⟨hij, hj⟩, hmass⟩, hn j hj ij hzcell⟩
+  · refine Or.inr ⟨ij, hij, ⟨⟨z, hz.2, hzcell⟩, ?_⟩, hzcell⟩
+    intro j hj
+    exact lt_of_not_ge (fun hmass => hs ⟨j, hj, hmass⟩)
+
+lemma denseCandidate_disk_subset {R T κ b c v : ℝ} (hT : 0 ≤ T) {K : Set ℂ}
+    (hK : K ⊆ closedBall 0 R) {n : ℕ} {γ : ℝ≥0 → ℂ} {i : DenseCandidateIndex}
+    (hi : i ∈ denseCandidateIndices R T K κ b c v n γ) :
+    closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i) ⊆ closedBall 0 (R + 3) := by
+  have hheight : ∀ z ∈ K, -R ≤ z.im := by
+    intro z hz
+    have hn : ‖z‖ ≤ R := by simpa only [mem_closedBall, dist_zero_right] using hK hz
+    have him := (Complex.abs_im_le_norm z).trans hn
+    exact (abs_le.mp him).1
+  have hcenter := (retainedCoverGrid_center_bounds hK hheight (denseCandidate_mem hi).1).1
+  have hrn : coverRadius T n ≤ 1 := Real.exp_le_one_iff.mpr (by
+    have hp := mul_nonneg hT (Nat.cast_nonneg n)
+    linarith)
+  intro z hz
+  have ht := dist_triangle z (denseCandidateCenter T n i) 0
+  have hr := denseCandidateRadius_le_one hT i
+  change dist z 0 ≤ R + 3
+  change dist z (denseCandidateCenter T n i) ≤ denseCandidateRadius T i at hz
+  change ‖denseCandidateCenter T n i‖ ≤ R + 2 * coverRadius T n at hcenter
+  simp only [dist_zero_right] at ht ⊢
+  linarith
+
+def denseCoverDiameter (T : ℝ) (n : ℕ) : ℝ≥0∞ :=
+  ENNReal.ofReal (max (8 * Real.exp (-1) * coverRadius T (n / 3)) (4 * coverRadius T n))
+
+lemma denseCoverDiameter_tendsto_zero {T : ℝ} (hT : 0 < T) :
+    Tendsto (denseCoverDiameter T) atTop (𝓝 0) :=
+  cover_diameter_tendsto_zero hT (8 * Real.exp (-1)) 4
+
+lemma denseCandidate_enlarged_diameter {R T κ b c v : ℝ} (hT : 0 ≤ T) {K : Set ℂ}
+    {n : ℕ} {γ : ℝ≥0 → ℂ} {i : DenseCandidateIndex}
+    (hi : i ∈ denseCandidateIndices R T K κ b c v n γ) :
+    ediam (closedBall (denseCandidateCenter T n i) (4 * denseCandidateRadius T i)) ≤
+      denseCoverDiameter T n := by
+  apply (ediam_le_of_forall_dist_le ?_).trans (ENNReal.ofReal_le_ofReal (le_max_left _ _))
+  intro x hx y hy
+  have ht := dist_triangle x (denseCandidateCenter T n i) y
+  have hx' : dist x (denseCandidateCenter T n i) ≤ 4 * denseCandidateRadius T i := hx
+  have hy' : dist (denseCandidateCenter T n i) y ≤ 4 * denseCandidateRadius T i := by
+    change dist y (denseCandidateCenter T n i) ≤ 4 * denseCandidateRadius T i at hy
+    simpa only [dist_comm] using hy
+  have hr := coverRadius_window_upper hT (denseCandidate_mem hi).2.1
+  have he := Real.exp_pos (-1)
+  dsimp only [denseCandidateRadius] at hx' hy'
+  nlinarith
+
+lemma denseFineCell_diameter (T : ℝ) (n : ℕ) (ij : ℤ × ℤ) :
+    ediam (denseFineCell T n ij) ≤ denseCoverDiameter T n :=
+  (coverGrid_cell_diameter (coverRadius_pos T n) ij).trans
+    (ENNReal.ofReal_le_ofReal (le_max_right _ _))
+
+
+local instance denseCandidatePathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance denseCandidatePathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem denseMissedCell_measurable
+    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (κ T b c v : ℝ) (n : ℕ) (ij : ℤ × ℤ) :
+    MeasurableSet {ω | denseMissedCell κ T b c v n ij (f ω)} := by
+  have hm := measurableSet_visited_without_candidate f hf κ v (coverTestWindow n)
+    (fun j => closedBall (denseCandidateCenter T n (ij, j)) (denseCandidateRadius T (ij, j)))
+    (fun j => denseCandidateThreshold κ T b c n (ij, j))
+    (Q := denseFineCell T n ij) isClosed_closedBall
+  convert hm using 1
+  ext ω
+  simp only [denseMissedCell, mem_ofPred_eq, Set.Nonempty, mem_inter_iff, mem_range]
+  constructor
+  · rintro ⟨⟨z, ⟨t, rfl⟩, hz⟩, hh⟩
+    exact ⟨⟨t, hz⟩, hh⟩
+  · rintro ⟨⟨t, ht⟩, hh⟩
+    exact ⟨⟨f ω t, ⟨t, rfl⟩, ht⟩, hh⟩
+
+lemma extendedGauge_closedBall_le {h : ℝ → ℝ} (hh : IsGauge h)
+    (z : ℂ) {r : ℝ} (hr : 0 ≤ r) :
+    extendedGauge h (ediam (closedBall z r)) ≤ ENNReal.ofReal (h (2 * r)) := by
+  have hd : ediam (closedBall z r) ≤ ENNReal.ofReal (2 * r) := by
+    apply ediam_le_of_forall_dist_le
+    intro x hx y hy
+    have hx' : dist x z ≤ r := hx
+    have hy' : dist z y ≤ r := by
+      change dist y z ≤ r at hy
+      simpa only [dist_comm] using hy
+    exact (dist_triangle x z y).trans (by linarith)
+  have hdR : (ediam (closedBall z r)).toReal ≤ 2 * r := by
+    exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hd).trans_eq
+      (ENNReal.toReal_ofReal (by positivity))
+  exact ENNReal.ofReal_le_ofReal
+    (hh.monotoneOn ENNReal.toReal_nonneg (show 0 ≤ 2 * r by positivity) hdR)
+
+/-- The prescribed integer multiplicity pays the logarithmic correction of
+all selected disks, uniformly throughout the window. -/
+theorem denseCandidate_eventual_gauge_charge
+    {κ T b c : ℝ} (hκ8 : κ < 8) (hT : 0 < T) (hb : 0 < b) (hc : 0 < c)
+    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∃ C : ℝ≥0, 0 < C ∧ ∀ᶠ n : ℕ in atTop,
+      ∀ (R : ℝ) (K : Set ℂ) (v : ℝ) (γ : ℝ≥0 → ℂ) i,
+        i ∈ denseCandidateIndices R T K κ b c v n γ →
+        extendedGauge h (ediam (closedBall (denseCandidateCenter T n i)
+          (4 * denseCandidateRadius T i))) ≤
+            (C : ℝ≥0∞) * wholeSausageCharge κ γ v
+              (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) := by
+  have hp : 0 ≤ codimension κ := by dsimp [codimension, dimension]; linarith
+  obtain ⟨A, hA, hbound⟩ := CoverGaugeBounds.eventual_window_bound hκ8
+    (mul_pos (by norm_num : (0 : ℝ) < 8) (Real.exp_pos (-1))) hT hf
+  let L := b * (Real.exp (-1)) ^ dimension κ * (c / 2) ^ codimension κ
+  have hL : 0 < L := by dsimp only [L]; positivity
+  let C : ℝ≥0 := ⟨A / L, (div_pos hA hL).le⟩
+  refine ⟨C, div_pos hA hL, ?_⟩
+  filter_upwards [hbound, coverMultiplicity_mass_lower hc hp,
+    eventually_ge_atTop (12 : ℕ)] with n hn hmass hn12
+  intro R K v γ i hi
+  have hj := (denseCandidate_mem hi).2.1
+  have hjlow : (n : ℝ) / 4 ≤ (i.2 : ℝ) := by
+    have hnI : n / 3 ≤ i.2 := (Finset.mem_Icc.mp hj).1
+    have hfour : n ≤ 4 * i.2 := by omega
+    have hfourR : (n : ℝ) ≤ 4 * i.2 := by exact_mod_cast hfour
+    linarith
+  have hjhigh : i.2 ≤ n := by
+    have hnI := (Finset.mem_Icc.mp hj).2
+    omega
+  have hg := hn i.2 hjlow hjhigh
+  change h (8 * Real.exp (-1) * coverRadius T i.2) ≤
+    A * coverRadius T i.2 ^ dimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2) at hg
+  have hscale : L * (coverRadius T i.2 ^ dimension κ *
+      (Real.log (n : ℝ)) ^ (codimension κ / 2)) ≤ denseCandidateThreshold κ T b c n i := by
+    unfold denseCandidateThreshold denseCandidateRadius
+    rw [Real.mul_rpow (Real.exp_pos _).le (coverRadius_pos _ _).le]
+    calc
+      _ = (b * Real.exp (-1) ^ dimension κ * coverRadius T i.2 ^ dimension κ) *
+          ((c / 2) ^ codimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2)) := by
+        dsimp only [L]
+        ring
+      _ ≤ (b * Real.exp (-1) ^ dimension κ * coverRadius T i.2 ^ dimension κ) *
+          (coverMultiplicity c n : ℝ) ^ codimension κ :=
+        mul_le_mul_of_nonneg_left hmass
+          (mul_nonneg (mul_nonneg hb.le (Real.rpow_nonneg (Real.exp_pos _).le _))
+            (Real.rpow_nonneg (coverRadius_pos _ _).le _))
+      _ = _ := by ring
+  have hreal : h (8 * denseCandidateRadius T i) ≤ (C : ℝ) * denseCandidateThreshold κ T b c n i := by
+    calc
+      _ = h (8 * Real.exp (-1) * coverRadius T i.2) := by congr 1; simp only [denseCandidateRadius]; ring
+      _ ≤ A * coverRadius T i.2 ^ dimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2) := hg
+      _ = (C : ℝ) * (L * (coverRadius T i.2 ^ dimension κ *
+          (Real.log (n : ℝ)) ^ (codimension κ / 2))) := by
+        change _ = A / L * _
+        field_simp
+      _ ≤ _ := mul_le_mul_of_nonneg_left hscale C.2
+  calc
+    _ ≤ ENNReal.ofReal (h (8 * denseCandidateRadius T i)) := by
+      have he : 2 * (4 * denseCandidateRadius T i) = 8 * denseCandidateRadius T i := by ring
+      simpa only [he] using extendedGauge_closedBall_le hh (denseCandidateCenter T n i)
+        (r := 4 * denseCandidateRadius T i) (by positivity [denseCandidateRadius_pos T i])
+    _ ≤ ENNReal.ofReal ((C : ℝ) * denseCandidateThreshold κ T b c n i) := ENNReal.ofReal_le_ofReal hreal
+    _ = (C : ℝ≥0∞) * ENNReal.ofReal (denseCandidateThreshold κ T b c n i) := by
+      have hC : ENNReal.ofReal (C : ℝ) = (C : ℝ≥0∞) := ENNReal.ofReal_coe_nnreal
+      exact (ENNReal.ofReal_mul C.2).trans (congrArg
+        (fun t : ℝ≥0∞ => t * ENNReal.ofReal (denseCandidateThreshold κ T b c n i)) hC)
+    _ ≤ _ := mul_le_mul' le_rfl (denseCandidate_mem hi).2.2
+
+/-- The cost of each fine cleanup cell has the matching explicit-gauge
+factor, independent of the cell center. -/
+theorem denseFineCell_eventual_gauge_bound
+    {κ T : ℝ} (hκ8 : κ < 8) (hT : 0 < T)
+    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∃ A : ℝ, 0 < A ∧ ∀ᶠ n : ℕ in atTop, ∀ ij : ℤ × ℤ,
+      extendedGauge h (ediam (denseFineCell T n ij)) ≤
+        ENNReal.ofReal (A * coverRadius T n ^ dimension κ *
+          (Real.log (n : ℝ)) ^ (codimension κ / 2)) := by
+  obtain ⟨A, hA, hbound⟩ := CoverGaugeBounds.eventual_window_bound hκ8 (by norm_num : (0 : ℝ) < 4) hT hf
+  refine ⟨A, hA, ?_⟩
+  filter_upwards [hbound] with n hn
+  intro ij
+  have hg := hn n (by have hn0 : (0 : ℝ) ≤ n := Nat.cast_nonneg n; linarith) le_rfl
+  change h (4 * coverRadius T n) ≤ _ at hg
+  have he : 2 * (2 * coverRadius T n) = 4 * coverRadius T n := by ring
+  have hd : extendedGauge h (ediam (denseFineCell T n ij)) ≤
+      ENNReal.ofReal (h (4 * coverRadius T n)) := by
+    simpa only [denseFineCell, he] using
+      (extendedGauge_closedBall_le hh (coverGridCenter (coverRadius T n) ij)
+        (r := 2 * coverRadius T n) (by positivity [coverRadius_pos T n]))
+  exact hd.trans (ENNReal.ofReal_le_ofReal hg)
+
+/-- The six concrete geometric hypotheses of the expected-cover reduction,
+with one constant and one sufficiently late cutoff for every path and mesh. -/
+theorem dense_candidate_cover_geometry
+    {κ R T b c : ℝ} (hκ8 : κ < 8) (hR : 0 ≤ R) (hT : 0 < T)
+    (hb : 0 < b) (hc : 0 < c) {K : Set ℂ} (hK : K ⊆ closedBall 0 R)
+    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∃ C : ℝ≥0, 0 < C ∧ ∀ᶠ n : ℕ in atTop, ∀ (v : ℝ) (γ : ℝ≥0 → ℂ),
+      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ, denseCandidateRadius T i ≤ R + 3) ∧
+      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ,
+        closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i) ⊆ closedBall 0 (R + 3)) ∧
+      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ,
+        extendedGauge h (ediam (closedBall (denseCandidateCenter T n i) (4 * denseCandidateRadius T i))) ≤
+          (C : ℝ≥0∞) * wholeSausageCharge κ γ v
+            (closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i))) ∧
+      (∀ i ∈ denseCandidateIndices R T K κ b c v n γ,
+        ediam (closedBall (denseCandidateCenter T n i) (4 * denseCandidateRadius T i)) ≤ denseCoverDiameter T n) ∧
+      (∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+        ediam (denseFineCell T n ij) ≤ denseCoverDiameter T n) ∧
+      (∀ z ∈ K ∩ range γ,
+        (∃ i ∈ denseCandidateIndices R T K κ b c v n γ,
+          z ∈ closedBall (denseCandidateCenter T n i) (denseCandidateRadius T i)) ∨
+        ∃ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+          denseMissedCell κ T b c v n ij γ ∧ z ∈ denseFineCell T n ij) := by
+  obtain ⟨C, hC, hcharge⟩ := denseCandidate_eventual_gauge_charge hκ8 hT hb hc hh hf
+  refine ⟨C, hC, ?_⟩
+  filter_upwards [hcharge, dense_candidates_and_missed_cover hR hT hK κ b c] with n hn hcover
+  intro v γ
+  refine ⟨?_, ?_, hn R K v γ, ?_, ?_, hcover v γ⟩
+  · intro i _
+    exact (denseCandidateRadius_le_one hT.le i).trans (by linarith)
+  · exact fun _ hi => denseCandidate_disk_subset hT.le hK hi
+  · exact fun _ hi => denseCandidate_enlarged_diameter hT.le hi
+  · exact fun ij _ => denseFineCell_diameter T n ij
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseCoverAdaptedTests.lean b/lean/OAI/Probability/SLE/Gauge/DenseCoverAdaptedTests.lean
index 7c7b572d54b6c1322e63781bf3c9a4475a6c7593..9774bcedd68efa5795c4930877433a6079ee9cc4 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseCoverAdaptedTests.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseCoverAdaptedTests.lean
@@ -1,112 +1,112 @@
-import OAI.Probability.SLE.Gauge.DenseVisitNumerics
-
-namespace OAI
-
-/-! Adapted dense-visit tests at the manuscript's actual logarithmic
-multiplicity and linear index window.  This proves the failure calculation
-without independence between the tests. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem complement_inter_bound_of_success_lower
-    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
-    {B E : Set Ω} (hE : MeasurableSet E) {p : ℝ} (hp : 0 ≤ p) (hp1 : p ≤ 1)
-    (hsuccess : ENNReal.ofReal p * μ B ≤ μ (B ∩ E)) :
-    μ (B ∩ Eᶜ) ≤ ENNReal.ofReal (1 - p) * μ B := by
-  have he := congrArg ENNReal.toReal (measure_sdiff_add_inter (μ := μ) B hE)
-  rw [ENNReal.toReal_add (measure_ne_top _ _) (measure_ne_top _ _)] at he
-  have hs := ENNReal.toReal_mono (measure_ne_top _ _) hsuccess
-  rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal hp] at hs
-  apply (ENNReal.toReal_le_toReal (measure_ne_top _ _)
-    (ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _))).mp
-  rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal (sub_nonneg.mpr hp1)]
-  rw [Set.sdiff_eq] at he
-  nlinarith
-
-/-- The exact finite window can start at any filtration index. -/
-theorem adapted_failures_Icc_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω)
-    (F : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
-    (hmeas : ∀ j, MeasurableSet[F (j + 1)] (failure j))
-    (q : ℝ≥0∞)
-    (hstep : ∀ j (B : Set Ω), MeasurableSet[F j] B →
-      μ (B ∩ failure j) ≤ q * μ B)
-    (a b : ℕ) (hab : a ≤ b) :
-    μ (⋂ j ∈ Finset.Icc a b, failure j) ≤ q ^ (b + 1 - a) * μ univ := by
-  let F' : Filtration ℕ mΩ :=
-    ⟨fun j => F (a + j), fun _ _ h => F.mono (Nat.add_le_add_left h a),
-      fun j => F.le (a + j)⟩
-  have hh := adapted_failures_bound μ F' (fun j => failure (a + j))
-    (fun j => by simpa only [F', Nat.add_assoc] using hmeas (a + j)) q
-    (fun j B hB => hstep (a + j) B hB) (b + 1 - a)
-  have he : (⋂ j ∈ Finset.range (b + 1 - a), failure (a + j)) =
-      ⋂ j ∈ Finset.Icc a b, failure j := by
-    ext ω
-    simp only [mem_iInter, Finset.mem_range, Finset.mem_Icc]
-    constructor
-    · intro h j hj
-      have heq : a + (j - a) = j := by omega
-      simpa only [heq] using h (j - a) (by omega)
-    · intro h j hj
-      exact h (a + j) ⟨by omega, by omega⟩
-  rwa [he] at hh
-
-/-- Conditional success bounds at the actual scale yield the stretched
-exponential bound for failure of every test in the cover window. -/
-theorem cover_window_failure_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
-    (F : Filtration ℕ mΩ) (success : ℕ → Set Ω)
-    (hmeas : ∀ j, MeasurableSet[F (j + 1)] (success j))
-    (n : ℕ) (hn : 1 ≤ n)
-    (hstep : ∀ j (B : Set Ω), MeasurableSet[F j] B →
-      ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ μ (B ∩ success j)) :
-    μ (⋂ j ∈ coverTestWindow n, (success j)ᶜ) ≤
-      ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
-  have hnR : (1 : ℝ) ≤ n := by exact_mod_cast hn
-  have hs1 : 1 ≤ Real.sqrt (n : ℝ) := by
-    simpa only [Real.sqrt_one] using Real.sqrt_le_sqrt hnR
-  have hspos : 0 < Real.sqrt (n : ℝ) := zero_lt_one.trans_le hs1
-  have hp : 1 / Real.sqrt (n : ℝ) ≤ 1 := (div_le_one hspos).mpr hs1
-  have hb := adapted_failures_Icc_bound μ F (fun j => (success j)ᶜ)
-    (fun j => (hmeas j).compl) (ENNReal.ofReal (1 - 1 / Real.sqrt (n : ℝ)))
-    (fun j B hB => complement_inter_bound_of_success_lower μ
-      (F.le (j + 1) _ (hmeas j)) (one_div_nonneg.mpr hspos.le) hp (hstep j B hB))
-    (n / 3) (2 * n / 3) (by omega)
-  rw [measure_univ, mul_one] at hb
-  change μ (⋂ j ∈ coverTestWindow n, (success j)ᶜ) ≤ _ at hb
-  have hcard : (coverTestWindow n).card = 2 * n / 3 + 1 - n / 3 := by
-    simp only [coverTestWindow, Nat.card_Icc]
-  rw [← hcard] at hb
-  refine hb.trans ?_
-  rw [← ENNReal.ofReal_pow (sub_nonneg.mpr hp)]
-  exact ENNReal.ofReal_le_ofReal
-    (failure_window_le_stretchedExp hnR (coverTestWindow n).card
-      (coverTestWindow_card_real_lower n))
-
-theorem denseVisit_cover_window_failure
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
-    (F : ℕ → Filtration ℕ mΩ) (success : ℕ → ℕ → Set Ω)
-    (hmeas : ∀ n j, MeasurableSet[F n (j + 1)] (success n j))
-    {a C c : ℝ} (ha : 0 < a) (hC : 0 ≤ C) (hc : 0 < c)
-    (hsmall : C * c ^ 2 ≤ 1 / 4)
-    (hstep : ∀ n j (B : Set Ω), MeasurableSet[F n j] B →
-      ENNReal.ofReal (a * Real.exp (-C * (coverMultiplicity c n : ℝ) ^ 2)) * μ B ≤
-        μ (B ∩ success n j)) :
-    ∀ᶠ n : ℕ in atTop,
-      μ (⋂ j ∈ coverTestWindow n, (success n j)ᶜ) ≤
-        ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
-  filter_upwards [denseVisit_probability_eventually_lower ha hC hc hsmall,
-    eventually_ge_atTop (1 : ℕ)] with n hn hn1
-  apply cover_window_failure_bound μ (F n) (success n) (hmeas n) n hn1
-  intro j B hB
-  exact (mul_le_mul' (ENNReal.ofReal_le_ofReal hn) le_rfl).trans (hstep n j B hB)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.DenseVisitNumerics
+
+namespace OAI
+
+/-! Adapted dense-visit tests at the manuscript's actual logarithmic
+multiplicity and linear index window.  This proves the failure calculation
+without independence between the tests. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem complement_inter_bound_of_success_lower
+    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
+    {B E : Set Ω} (hE : MeasurableSet E) {p : ℝ} (hp : 0 ≤ p) (hp1 : p ≤ 1)
+    (hsuccess : ENNReal.ofReal p * μ B ≤ μ (B ∩ E)) :
+    μ (B ∩ Eᶜ) ≤ ENNReal.ofReal (1 - p) * μ B := by
+  have he := congrArg ENNReal.toReal (measure_sdiff_add_inter (μ := μ) B hE)
+  rw [ENNReal.toReal_add (measure_ne_top _ _) (measure_ne_top _ _)] at he
+  have hs := ENNReal.toReal_mono (measure_ne_top _ _) hsuccess
+  rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal hp] at hs
+  apply (ENNReal.toReal_le_toReal (measure_ne_top _ _)
+    (ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _))).mp
+  rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal (sub_nonneg.mpr hp1)]
+  rw [Set.sdiff_eq] at he
+  nlinarith
+
+/-- The exact finite window can start at any filtration index. -/
+theorem adapted_failures_Icc_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω)
+    (F : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
+    (hmeas : ∀ j, MeasurableSet[F (j + 1)] (failure j))
+    (q : ℝ≥0∞)
+    (hstep : ∀ j (B : Set Ω), MeasurableSet[F j] B →
+      μ (B ∩ failure j) ≤ q * μ B)
+    (a b : ℕ) (hab : a ≤ b) :
+    μ (⋂ j ∈ Finset.Icc a b, failure j) ≤ q ^ (b + 1 - a) * μ univ := by
+  let F' : Filtration ℕ mΩ :=
+    ⟨fun j => F (a + j), fun _ _ h => F.mono (Nat.add_le_add_left h a),
+      fun j => F.le (a + j)⟩
+  have hh := adapted_failures_bound μ F' (fun j => failure (a + j))
+    (fun j => by simpa only [F', Nat.add_assoc] using hmeas (a + j)) q
+    (fun j B hB => hstep (a + j) B hB) (b + 1 - a)
+  have he : (⋂ j ∈ Finset.range (b + 1 - a), failure (a + j)) =
+      ⋂ j ∈ Finset.Icc a b, failure j := by
+    ext ω
+    simp only [mem_iInter, Finset.mem_range, Finset.mem_Icc]
+    constructor
+    · intro h j hj
+      have heq : a + (j - a) = j := by omega
+      simpa only [heq] using h (j - a) (by omega)
+    · intro h j hj
+      exact h (a + j) ⟨by omega, by omega⟩
+  rwa [he] at hh
+
+/-- Conditional success bounds at the actual scale yield the stretched
+exponential bound for failure of every test in the cover window. -/
+theorem cover_window_failure_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
+    (F : Filtration ℕ mΩ) (success : ℕ → Set Ω)
+    (hmeas : ∀ j, MeasurableSet[F (j + 1)] (success j))
+    (n : ℕ) (hn : 1 ≤ n)
+    (hstep : ∀ j (B : Set Ω), MeasurableSet[F j] B →
+      ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ μ (B ∩ success j)) :
+    μ (⋂ j ∈ coverTestWindow n, (success j)ᶜ) ≤
+      ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
+  have hnR : (1 : ℝ) ≤ n := by exact_mod_cast hn
+  have hs1 : 1 ≤ Real.sqrt (n : ℝ) := by
+    simpa only [Real.sqrt_one] using Real.sqrt_le_sqrt hnR
+  have hspos : 0 < Real.sqrt (n : ℝ) := zero_lt_one.trans_le hs1
+  have hp : 1 / Real.sqrt (n : ℝ) ≤ 1 := (div_le_one hspos).mpr hs1
+  have hb := adapted_failures_Icc_bound μ F (fun j => (success j)ᶜ)
+    (fun j => (hmeas j).compl) (ENNReal.ofReal (1 - 1 / Real.sqrt (n : ℝ)))
+    (fun j B hB => complement_inter_bound_of_success_lower μ
+      (F.le (j + 1) _ (hmeas j)) (one_div_nonneg.mpr hspos.le) hp (hstep j B hB))
+    (n / 3) (2 * n / 3) (by omega)
+  rw [measure_univ, mul_one] at hb
+  change μ (⋂ j ∈ coverTestWindow n, (success j)ᶜ) ≤ _ at hb
+  have hcard : (coverTestWindow n).card = 2 * n / 3 + 1 - n / 3 := by
+    simp only [coverTestWindow, Nat.card_Icc]
+  rw [← hcard] at hb
+  refine hb.trans ?_
+  rw [← ENNReal.ofReal_pow (sub_nonneg.mpr hp)]
+  exact ENNReal.ofReal_le_ofReal
+    (failure_window_le_stretchedExp hnR (coverTestWindow n).card
+      (coverTestWindow_card_real_lower n))
+
+theorem denseVisit_cover_window_failure
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
+    (F : ℕ → Filtration ℕ mΩ) (success : ℕ → ℕ → Set Ω)
+    (hmeas : ∀ n j, MeasurableSet[F n (j + 1)] (success n j))
+    {a C c : ℝ} (ha : 0 < a) (hC : 0 ≤ C) (hc : 0 < c)
+    (hsmall : C * c ^ 2 ≤ 1 / 4)
+    (hstep : ∀ n j (B : Set Ω), MeasurableSet[F n j] B →
+      ENNReal.ofReal (a * Real.exp (-C * (coverMultiplicity c n : ℝ) ^ 2)) * μ B ≤
+        μ (B ∩ success n j)) :
+    ∀ᶠ n : ℕ in atTop,
+      μ (⋂ j ∈ coverTestWindow n, (success n j)ᶜ) ≤
+        ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by
+  filter_upwards [denseVisit_probability_eventually_lower ha hC hc hsmall,
+    eventually_ge_atTop (1 : ℕ)] with n hn hn1
+  apply cover_window_failure_bound μ (F n) (success n) (hmeas n) n hn1
+  intro j B hB
+  exact (mul_le_mul' (ENNReal.ofReal_le_ofReal hn) le_rfl).trans (hstep n j B hB)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseCoverCore.lean b/lean/OAI/Probability/SLE/Gauge/DenseCoverCore.lean
index f86398ba622ad1fad20c43135596826a17797c8a..1e6b88afb2b76f97149ea960c23a685bc52561e0 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseCoverCore.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseCoverCore.lean
@@ -1,103 +1,103 @@
-import OAI.Probability.SLE.Gauge.CoverLimits
-
-namespace OAI
-
-/-! Finite spatial selection and adapted failure accounting for the dense cover. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- Extract disjoint candidate disks whose fourfold enlargements cover all
-candidates, and charge their complete covering cost to one measure. -/
-theorem select_disjoint_candidates_and_charge
-    {ι : Type*} (I : Finset ι) (x : ι → ℂ) (r : ι → ℝ)
-    (R : ℝ) (hr : ∀ i ∈ I, r i ≤ R)
-    (ν : Measure ℂ) (A : Set ℂ)
-    (hA : ∀ i ∈ I, closedBall (x i) (r i) ⊆ A)
-    (cost : ι → ℝ≥0∞) (C : ℝ≥0∞)
-    (hcost : ∀ i ∈ I, cost i ≤ C * ν (closedBall (x i) (r i))) :
-    ∃ J : Finset ι, J ⊆ I ∧
-      (J : Set ι).PairwiseDisjoint (fun i => closedBall (x i) (r i)) ∧
-      (∀ i ∈ I, ∃ j ∈ J,
-        closedBall (x i) (r i) ⊆ closedBall (x j) (4 * r j)) ∧
-      ∑ j ∈ J, cost j ≤ C * ν A := by
-  classical
-  obtain ⟨u, hu, hdisj, hcover⟩ :=
-    Vitali.exists_disjoint_subfamily_covering_enlargement_closedBall
-      (I : Set ι) x r R hr 4 (by norm_num)
-  have hfin : u.Finite := I.finite_toSet.subset hu
-  refine ⟨hfin.toFinset, ?_, ?_, ?_, ?_⟩
-  · intro j hj
-    exact hu (hfin.mem_toFinset.mp hj)
-  · simpa only [hfin.coe_toFinset] using hdisj
-  · intro i hi
-    obtain ⟨j, hj, hij⟩ := hcover i hi
-    exact ⟨j, hfin.mem_toFinset.mpr hj, hij⟩
-  · apply disjoint_cover_charge hfin.toFinset ν
-      (fun i => closedBall (x i) (r i)) A
-      (by simpa only [hfin.coe_toFinset] using hdisj)
-      (fun _ _ => isClosed_closedBall.measurableSet)
-    · intro j hj
-      exact hA j (hu (hfin.mem_toFinset.mp hj))
-    · intro j hj
-      exact hcost j (hu (hfin.mem_toFinset.mp hj))
-
-/-- Candidate-containing cells and the remaining visited cells together form
-an actual cover after disjoint spatial selection. -/
-theorem selected_and_missed_cover
-    {ι η : Type*} (I J : Finset ι) (M : Finset η)
-    (D V : ι → Set ℂ) (Q : η → Set ℂ) (E : Set ℂ)
-    (hselect : ∀ i ∈ I, ∃ j ∈ J, D i ⊆ V j)
-    (hcell : ∀ z ∈ E, (∃ i ∈ I, z ∈ D i) ∨ ∃ k ∈ M, z ∈ Q k) :
-    E ⊆ (⋃ j ∈ J, V j) ∪ ⋃ k ∈ M, Q k := by
-  intro z hz
-  rcases hcell z hz with ⟨i, hi, hzi⟩ | ⟨k, hk, hzk⟩
-  · obtain ⟨j, hj, hij⟩ := hselect i hi
-    exact Or.inl (mem_iUnion₂.mpr ⟨j, hj, hij hzi⟩)
-  · exact Or.inr (mem_iUnion₂.mpr ⟨k, hk, hzk⟩)
-
-/-- Repeated adapted tests require conditional success bounds, with no
-independence assumption. All previous failures form a measurable past event. -/
-theorem adapted_failures_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω)
-    (F : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
-    (hmeas : ∀ n, MeasurableSet[F (n + 1)] (failure n))
-    (q : ℝ≥0∞)
-    (hstep : ∀ n (B : Set Ω), MeasurableSet[F n] B →
-      P (B ∩ failure n) ≤ q * P B) :
-    ∀ N : ℕ, P (⋂ n ∈ Finset.range N, failure n) ≤ q ^ N * P univ := by
-  intro N
-  induction N with
-  | zero => simp
-  | succ N ih =>
-    let B : Set Ω := ⋂ n ∈ Finset.range N, failure n
-    have hB : MeasurableSet[F N] B := by
-      apply MeasurableSet.iInter
-      intro n
-      apply MeasurableSet.iInter
-      intro hn
-      exact (F.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n)
-    have he : (⋂ n ∈ Finset.range (N + 1), failure n) = B ∩ failure N := by
-      ext ω
-      simp only [mem_iInter, Finset.mem_range, mem_inter_iff, B]
-      constructor
-      · intro hω
-        exact ⟨fun n hn => hω n (Nat.lt_succ_of_lt hn), hω N (Nat.lt_succ_self N)⟩
-      · rintro ⟨hω, hN⟩ n hn
-        rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl
-        · exact hω n hn
-        · exact hN
-    rw [he, pow_succ]
-    calc
-      P (B ∩ failure N) ≤ q * P B := hstep N B hB
-      _ ≤ q * (q ^ N * P univ) := mul_le_mul' le_rfl ih
-      _ = q ^ N * q * P univ := by ac_rfl
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverLimits
+
+namespace OAI
+
+/-! Finite spatial selection and adapted failure accounting for the dense cover. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- Extract disjoint candidate disks whose fourfold enlargements cover all
+candidates, and charge their complete covering cost to one measure. -/
+theorem select_disjoint_candidates_and_charge
+    {ι : Type*} (I : Finset ι) (x : ι → ℂ) (r : ι → ℝ)
+    (R : ℝ) (hr : ∀ i ∈ I, r i ≤ R)
+    (ν : Measure ℂ) (A : Set ℂ)
+    (hA : ∀ i ∈ I, closedBall (x i) (r i) ⊆ A)
+    (cost : ι → ℝ≥0∞) (C : ℝ≥0∞)
+    (hcost : ∀ i ∈ I, cost i ≤ C * ν (closedBall (x i) (r i))) :
+    ∃ J : Finset ι, J ⊆ I ∧
+      (J : Set ι).PairwiseDisjoint (fun i => closedBall (x i) (r i)) ∧
+      (∀ i ∈ I, ∃ j ∈ J,
+        closedBall (x i) (r i) ⊆ closedBall (x j) (4 * r j)) ∧
+      ∑ j ∈ J, cost j ≤ C * ν A := by
+  classical
+  obtain ⟨u, hu, hdisj, hcover⟩ :=
+    Vitali.exists_disjoint_subfamily_covering_enlargement_closedBall
+      (I : Set ι) x r R hr 4 (by norm_num)
+  have hfin : u.Finite := I.finite_toSet.subset hu
+  refine ⟨hfin.toFinset, ?_, ?_, ?_, ?_⟩
+  · intro j hj
+    exact hu (hfin.mem_toFinset.mp hj)
+  · simpa only [hfin.coe_toFinset] using hdisj
+  · intro i hi
+    obtain ⟨j, hj, hij⟩ := hcover i hi
+    exact ⟨j, hfin.mem_toFinset.mpr hj, hij⟩
+  · apply disjoint_cover_charge hfin.toFinset ν
+      (fun i => closedBall (x i) (r i)) A
+      (by simpa only [hfin.coe_toFinset] using hdisj)
+      (fun _ _ => isClosed_closedBall.measurableSet)
+    · intro j hj
+      exact hA j (hu (hfin.mem_toFinset.mp hj))
+    · intro j hj
+      exact hcost j (hu (hfin.mem_toFinset.mp hj))
+
+/-- Candidate-containing cells and the remaining visited cells together form
+an actual cover after disjoint spatial selection. -/
+theorem selected_and_missed_cover
+    {ι η : Type*} (I J : Finset ι) (M : Finset η)
+    (D V : ι → Set ℂ) (Q : η → Set ℂ) (E : Set ℂ)
+    (hselect : ∀ i ∈ I, ∃ j ∈ J, D i ⊆ V j)
+    (hcell : ∀ z ∈ E, (∃ i ∈ I, z ∈ D i) ∨ ∃ k ∈ M, z ∈ Q k) :
+    E ⊆ (⋃ j ∈ J, V j) ∪ ⋃ k ∈ M, Q k := by
+  intro z hz
+  rcases hcell z hz with ⟨i, hi, hzi⟩ | ⟨k, hk, hzk⟩
+  · obtain ⟨j, hj, hij⟩ := hselect i hi
+    exact Or.inl (mem_iUnion₂.mpr ⟨j, hj, hij hzi⟩)
+  · exact Or.inr (mem_iUnion₂.mpr ⟨k, hk, hzk⟩)
+
+/-- Repeated adapted tests require conditional success bounds, with no
+independence assumption. All previous failures form a measurable past event. -/
+theorem adapted_failures_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω)
+    (F : Filtration ℕ mΩ) (failure : ℕ → Set Ω)
+    (hmeas : ∀ n, MeasurableSet[F (n + 1)] (failure n))
+    (q : ℝ≥0∞)
+    (hstep : ∀ n (B : Set Ω), MeasurableSet[F n] B →
+      P (B ∩ failure n) ≤ q * P B) :
+    ∀ N : ℕ, P (⋂ n ∈ Finset.range N, failure n) ≤ q ^ N * P univ := by
+  intro N
+  induction N with
+  | zero => simp
+  | succ N ih =>
+    let B : Set Ω := ⋂ n ∈ Finset.range N, failure n
+    have hB : MeasurableSet[F N] B := by
+      apply MeasurableSet.iInter
+      intro n
+      apply MeasurableSet.iInter
+      intro hn
+      exact (F.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hn))) _ (hmeas n)
+    have he : (⋂ n ∈ Finset.range (N + 1), failure n) = B ∩ failure N := by
+      ext ω
+      simp only [mem_iInter, Finset.mem_range, mem_inter_iff, B]
+      constructor
+      · intro hω
+        exact ⟨fun n hn => hω n (Nat.lt_succ_of_lt hn), hω N (Nat.lt_succ_self N)⟩
+      · rintro ⟨hω, hN⟩ n hn
+        rcases Nat.lt_succ_iff_lt_or_eq.mp hn with hn | rfl
+        · exact hω n hn
+        · exact hN
+    rw [he, pow_succ]
+    calc
+      P (B ∩ failure N) ≤ q * P B := hstep N B hB
+      _ ≤ q * (q ^ N * P univ) := mul_le_mul' le_rfl ih
+      _ = q ^ N * q * P univ := by ac_rfl
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseCoverExpectation.lean b/lean/OAI/Probability/SLE/Gauge/DenseCoverExpectation.lean
index b3454f875c54bda8cc14936f2dc5213b2638ad95..f6a65e69dd33698aed37af1c31120ee774ac57ae 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseCoverExpectation.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseCoverExpectation.lean
@@ -1,116 +1,116 @@
-import OAI.Probability.SLE.Gauge.DenseCandidateCover
-import OAI.Probability.SLE.Gauge.MissedCellCleanup
-
-namespace OAI
-
-/-! The expected explicit-gauge cost of the actual finite candidate cover.
-The leading constant is chosen before the compact interior cutoff; only the
-vanishing missed-cell prefactor is allowed to depend on that cutoff. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance denseExpectationPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance denseExpectationPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-def denseMissedProbabilityBound (κ T D : ℝ) (n : ℕ) : ℝ≥0∞ :=
-  ENNReal.ofReal (D * coverRadius T n ^ codimension κ * Real.exp (-Real.sqrt (n : ℝ) / 4))
-
-theorem dense_missed_cost_tendsto_zero
-    {κ T R D : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hT : 0 < T)
-    (hR : 0 ≤ R) (hD : 0 ≤ D) (K : Set ℂ)
-    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    Tendsto (fun n => ∑ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-      extendedGauge h (ediam (denseFineCell T n ij)) * denseMissedProbabilityBound κ T D n)
-      atTop (𝓝 0) := by
-  obtain ⟨A, hA, hcost⟩ := denseFineCell_eventual_gauge_bound hκ8 hT hh hf
-  have hlimR : Tendsto (fun n : ℕ => (2 * R + 5) ^ 2 * A * D *
-      (Real.log (n : ℝ)) ^ (codimension κ / 2) * Real.exp (-Real.sqrt (n : ℝ) / 4))
-      atTop (𝓝 0) := by
-    have ht := (explicit_cleanup_tendsto_zero (κ := κ) (c := 1 / 4) hκ (by norm_num)).const_mul
-      ((2 * R + 5) ^ 2 * A * D)
-    convert ht using 1
-    · funext n
-      have he : -Real.sqrt (n : ℝ) / 4 = -(1 / 4 : ℝ) * Real.sqrt (n : ℝ) := by ring
-      rw [he]
-      ring
-    · simp only [mul_zero]
-  have hlim : Tendsto (fun n : ℕ => ENNReal.ofReal ((2 * R + 5) ^ 2 * A * D *
-      (Real.log (n : ℝ)) ^ (codimension κ / 2) * Real.exp (-Real.sqrt (n : ℝ) / 4)))
-      atTop (𝓝 0) := by
-    simpa only [Function.comp_def, ENNReal.ofReal_zero] using
-      ENNReal.continuous_ofReal.continuousAt.tendsto.comp hlimR
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
-    (Eventually.of_forall fun _ => bot_le)
-  filter_upwards [hcost] with n hn
-  have hr1 : coverRadius T n ≤ 1 := Real.exp_le_one_iff.mpr
-    (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hT.le) (Nat.cast_nonneg _))
-  have he : -(1 / 4 : ℝ) * Real.sqrt (n : ℝ) = -Real.sqrt (n : ℝ) / 4 := by ring
-  have hb := missed_cell_sum_bound (retainedCoverGridIndex R (coverRadius T n) K)
-    (fun ij => extendedGauge h (ediam (denseFineCell T n ij)))
-    (fun _ => denseMissedProbabilityBound κ T D n) κ (coverRadius T n) n (1 / 4)
-    ((2 * R + 5) ^ 2) A D (coverRadius_pos T n) (Real.log_natCast_nonneg n)
-    (sq_nonneg _) hA.le hD
-    (retainedCoverGrid_card_bound hR (coverRadius_pos T n) hr1 K)
-    (fun ij _ => hn ij) (fun _ _ => by simp only [denseMissedProbabilityBound, he]; exact le_rfl)
-  simpa only [denseMissedProbabilityBound, he] using hb
-
-/-- A uniform candidate charge and the actual stretched-exponential missed
-bound give the upper expectation. `C` is fixed before `R`, `K`, and `D`. -/
-theorem expected_hausdorff_of_dense_candidates
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hγ : ∀ ω, Continuous (γ ω))
-    {T b c : ℝ} (hT : 0 < T) (hb : 0 < b) (hc : 0 < c)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∃ C : ℝ≥0, 0 < C ∧ ∀ (R : ℝ), 0 ≤ R → ∀ K : Set ℂ,
-      K ⊆ closedBall 0 R → ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) →
-      ∀ (D : ℝ), 0 ≤ D →
-      (∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-        P {ω | denseMissedCell κ T b c (v n) n ij (γ ω)} ≤ denseMissedProbabilityBound κ T D n) →
-      (∫⁻ ω, hausdorffGauge h (K ∩ range (γ ω)) ∂P) ≤
-        (C : ℝ≥0∞) * wholeSausageChargeBound κ (R + 3) := by
-  classical
-  obtain ⟨C, hC, hcharge⟩ := denseCandidate_eventual_gauge_charge hκ8 hT hb hc hh hf
-  refine ⟨C, hC, ?_⟩
-  intro R hR K hK v hv D hD hprob
-  let f : Ω → C(ℝ≥0, ℂ) := fun ω => ⟨γ ω, hγ ω⟩
-  have hfmeas : Measurable f := source_whole_path_measurable f hSLE.1
-  have hcover := dense_candidates_and_missed_cover hR hT hK κ b c
-  have hall := hcharge.and (hcover.and hprob)
-  obtain ⟨N, hN⟩ := eventually_atTop.mp hall
-  let k : ℕ → ℕ := fun n => n + N
-  have hk (n : ℕ) : N ≤ k n := Nat.le_add_left N n
-  have hkTop : Tendsto k atTop atTop := tendsto_add_atTop_nat N
-  apply expected_hausdorff_le_of_candidate_disks P κ hκ hκ8 γ hSLE h
-    (fun ω => K ∩ range (γ ω))
-    (fun n ω => denseCandidateIndices R T K κ b c (v (k n)) (k n) (γ ω))
-    (fun n => retainedCoverGridIndex R (coverRadius T (k n)) K)
-    (fun n => denseCandidateCenter T (k n)) (fun _ => denseCandidateRadius T)
-    (fun n => denseFineCell T (k n))
-    (fun n ij => {ω | denseMissedCell κ T b c (v (k n)) (k n) ij (γ ω)})
-    (fun n ij _ => denseMissedCell_measurable f hfmeas κ T b c (v (k n)) (k n) ij)
-    (fun n => v (k n)) (fun n => hv (k n)) (R + 3) C
-    (fun n => denseCoverDiameter T (k n)) ((denseCoverDiameter_tendsto_zero hT).comp hkTop)
-    (fun n _ => denseMissedProbabilityBound κ T D (k n))
-    (fun n ij hij => (hN (k n) (hk n)).2.2 ij hij)
-    ((dense_missed_cost_tendsto_zero hκ hκ8 hT hR hD K hh hf).comp hkTop)
-  apply Eventually.of_forall
-  intro ω n
-  have hn := hN (k n) (hk n)
-  refine ⟨?_, ?_, hn.1 R K (v (k n)) (γ ω), ?_, ?_, hn.2.1 (v (k n)) (γ ω)⟩
-  · intro i _
-    exact (denseCandidateRadius_le_one hT.le i).trans (by linarith)
-  · exact fun _ hi => denseCandidate_disk_subset hT.le hK hi
-  · exact fun _ hi => denseCandidate_enlarged_diameter hT.le hi
-  · exact fun ij _ => denseFineCell_diameter T (k n) ij
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.DenseCandidateCover
+import OAI.Probability.SLE.Gauge.MissedCellCleanup
+
+namespace OAI
+
+/-! The expected explicit-gauge cost of the actual finite candidate cover.
+The leading constant is chosen before the compact interior cutoff; only the
+vanishing missed-cell prefactor is allowed to depend on that cutoff. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance denseExpectationPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance denseExpectationPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+def denseMissedProbabilityBound (κ T D : ℝ) (n : ℕ) : ℝ≥0∞ :=
+  ENNReal.ofReal (D * coverRadius T n ^ codimension κ * Real.exp (-Real.sqrt (n : ℝ) / 4))
+
+theorem dense_missed_cost_tendsto_zero
+    {κ T R D : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hT : 0 < T)
+    (hR : 0 ≤ R) (hD : 0 ≤ D) (K : Set ℂ)
+    {h : ℝ → ℝ} (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    Tendsto (fun n => ∑ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+      extendedGauge h (ediam (denseFineCell T n ij)) * denseMissedProbabilityBound κ T D n)
+      atTop (𝓝 0) := by
+  obtain ⟨A, hA, hcost⟩ := denseFineCell_eventual_gauge_bound hκ8 hT hh hf
+  have hlimR : Tendsto (fun n : ℕ => (2 * R + 5) ^ 2 * A * D *
+      (Real.log (n : ℝ)) ^ (codimension κ / 2) * Real.exp (-Real.sqrt (n : ℝ) / 4))
+      atTop (𝓝 0) := by
+    have ht := (explicit_cleanup_tendsto_zero (κ := κ) (c := 1 / 4) hκ (by norm_num)).const_mul
+      ((2 * R + 5) ^ 2 * A * D)
+    convert ht using 1
+    · funext n
+      have he : -Real.sqrt (n : ℝ) / 4 = -(1 / 4 : ℝ) * Real.sqrt (n : ℝ) := by ring
+      rw [he]
+      ring
+    · simp only [mul_zero]
+  have hlim : Tendsto (fun n : ℕ => ENNReal.ofReal ((2 * R + 5) ^ 2 * A * D *
+      (Real.log (n : ℝ)) ^ (codimension κ / 2) * Real.exp (-Real.sqrt (n : ℝ) / 4)))
+      atTop (𝓝 0) := by
+    simpa only [Function.comp_def, ENNReal.ofReal_zero] using
+      ENNReal.continuous_ofReal.continuousAt.tendsto.comp hlimR
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
+    (Eventually.of_forall fun _ => bot_le)
+  filter_upwards [hcost] with n hn
+  have hr1 : coverRadius T n ≤ 1 := Real.exp_le_one_iff.mpr
+    (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hT.le) (Nat.cast_nonneg _))
+  have he : -(1 / 4 : ℝ) * Real.sqrt (n : ℝ) = -Real.sqrt (n : ℝ) / 4 := by ring
+  have hb := missed_cell_sum_bound (retainedCoverGridIndex R (coverRadius T n) K)
+    (fun ij => extendedGauge h (ediam (denseFineCell T n ij)))
+    (fun _ => denseMissedProbabilityBound κ T D n) κ (coverRadius T n) n (1 / 4)
+    ((2 * R + 5) ^ 2) A D (coverRadius_pos T n) (Real.log_natCast_nonneg n)
+    (sq_nonneg _) hA.le hD
+    (retainedCoverGrid_card_bound hR (coverRadius_pos T n) hr1 K)
+    (fun ij _ => hn ij) (fun _ _ => by simp only [denseMissedProbabilityBound, he]; exact le_rfl)
+  simpa only [denseMissedProbabilityBound, he] using hb
+
+/-- A uniform candidate charge and the actual stretched-exponential missed
+bound give the upper expectation. `C` is fixed before `R`, `K`, and `D`. -/
+theorem expected_hausdorff_of_dense_candidates
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hγ : ∀ ω, Continuous (γ ω))
+    {T b c : ℝ} (hT : 0 < T) (hb : 0 < b) (hc : 0 < c)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∃ C : ℝ≥0, 0 < C ∧ ∀ (R : ℝ), 0 ≤ R → ∀ K : Set ℂ,
+      K ⊆ closedBall 0 R → ∀ (v : ℕ → ℝ), (∀ n, 0 < v n) →
+      ∀ (D : ℝ), 0 ≤ D →
+      (∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+        P {ω | denseMissedCell κ T b c (v n) n ij (γ ω)} ≤ denseMissedProbabilityBound κ T D n) →
+      (∫⁻ ω, hausdorffGauge h (K ∩ range (γ ω)) ∂P) ≤
+        (C : ℝ≥0∞) * wholeSausageChargeBound κ (R + 3) := by
+  classical
+  obtain ⟨C, hC, hcharge⟩ := denseCandidate_eventual_gauge_charge hκ8 hT hb hc hh hf
+  refine ⟨C, hC, ?_⟩
+  intro R hR K hK v hv D hD hprob
+  let f : Ω → C(ℝ≥0, ℂ) := fun ω => ⟨γ ω, hγ ω⟩
+  have hfmeas : Measurable f := source_whole_path_measurable f hSLE.1
+  have hcover := dense_candidates_and_missed_cover hR hT hK κ b c
+  have hall := hcharge.and (hcover.and hprob)
+  obtain ⟨N, hN⟩ := eventually_atTop.mp hall
+  let k : ℕ → ℕ := fun n => n + N
+  have hk (n : ℕ) : N ≤ k n := Nat.le_add_left N n
+  have hkTop : Tendsto k atTop atTop := tendsto_add_atTop_nat N
+  apply expected_hausdorff_le_of_candidate_disks P κ hκ hκ8 γ hSLE h
+    (fun ω => K ∩ range (γ ω))
+    (fun n ω => denseCandidateIndices R T K κ b c (v (k n)) (k n) (γ ω))
+    (fun n => retainedCoverGridIndex R (coverRadius T (k n)) K)
+    (fun n => denseCandidateCenter T (k n)) (fun _ => denseCandidateRadius T)
+    (fun n => denseFineCell T (k n))
+    (fun n ij => {ω | denseMissedCell κ T b c (v (k n)) (k n) ij (γ ω)})
+    (fun n ij _ => denseMissedCell_measurable f hfmeas κ T b c (v (k n)) (k n) ij)
+    (fun n => v (k n)) (fun n => hv (k n)) (R + 3) C
+    (fun n => denseCoverDiameter T (k n)) ((denseCoverDiameter_tendsto_zero hT).comp hkTop)
+    (fun n _ => denseMissedProbabilityBound κ T D (k n))
+    (fun n ij hij => (hN (k n) (hk n)).2.2 ij hij)
+    ((dense_missed_cost_tendsto_zero hκ hκ8 hT hR hD K hh hf).comp hkTop)
+  apply Eventually.of_forall
+  intro ω n
+  have hn := hN (k n) (hk n)
+  refine ⟨?_, ?_, hn.1 R K (v (k n)) (γ ω), ?_, ?_, hn.2.1 (v (k n)) (γ ω)⟩
+  · intro i _
+    exact (denseCandidateRadius_le_one hT.le i).trans (by linarith)
+  · exact fun _ hi => denseCandidate_disk_subset hT.le hK hi
+  · exact fun _ hi => denseCandidate_enlarged_diameter hT.le hi
+  · exact fun ij _ => denseFineCell_diameter T (k n) ij
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseCoverUpperReduction.lean b/lean/OAI/Probability/SLE/Gauge/DenseCoverUpperReduction.lean
index c0215ea7b5b1ea2c60865b7993677ab33300d1d3..00fcd780ee45ace5f481a09c569444a2f799ac2b 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseCoverUpperReduction.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseCoverUpperReduction.lean
@@ -1,88 +1,88 @@
-import OAI.Probability.SLE.Gauge.FiniteCoverAssembly
-import OAI.Probability.SLE.Gauge.CoverNumerics
-
-namespace OAI
-
-/-! Candidate and missed-cell tests imply the dense-cover upper bound. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- Spatial selection, missed-cell probability bounds, and one-mesh charging
-produce the upper Hausdorff expectation. No measurable choice of disks is needed. -/
-theorem expected_hausdorff_le_of_candidate_disks
-    {Ω ι η : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (h : ℝ → ℝ) (E : Ω → Set ℂ)
-    (I : ℕ → Ω → Finset ι) (M : ℕ → Finset η)
-    (x : ℕ → ι → ℂ) (r : ℕ → ι → ℝ) (Q : ℕ → η → Set ℂ)
-    (missed : ℕ → η → Set Ω)
-    (hmissmeas : ∀ n k, k ∈ M n → MeasurableSet (missed n k))
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (R : ℝ) (C : ℝ≥0)
-    (δ : ℕ → ℝ≥0∞) (hδ : Tendsto δ atTop (𝓝 0))
-    (probBound : ℕ → η → ℝ≥0∞)
-    (hprob : ∀ n k, k ∈ M n → P (missed n k) ≤ probBound n k)
-    (hcleanup : Tendsto (fun n => ∑ k ∈ M n,
-      extendedGauge h (ediam (Q n k)) * probBound n k) atTop (𝓝 0))
-    (hgeometry : ∀ᵐ ω ∂P, ∀ n,
-      (∀ i ∈ I n ω, r n i ≤ R) ∧
-      (∀ i ∈ I n ω, closedBall (x n i) (r n i) ⊆ closedBall 0 R) ∧
-      (∀ i ∈ I n ω,
-        extendedGauge h (ediam (closedBall (x n i) (4 * r n i))) ≤
-          (C : ℝ≥0∞) * wholeSausageCharge κ (γ ω) (v n)
-            (closedBall (x n i) (r n i))) ∧
-      (∀ i ∈ I n ω, ediam (closedBall (x n i) (4 * r n i)) ≤ δ n) ∧
-      (∀ k ∈ M n, ediam (Q n k) ≤ δ n) ∧
-      (∀ z ∈ E ω, (∃ i ∈ I n ω, z ∈ closedBall (x n i) (r n i)) ∨
-        ∃ k ∈ M n, ω ∈ missed n k ∧ z ∈ Q n k)) :
-    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤
-      (C : ℝ≥0∞) * wholeSausageChargeBound κ R := by
-  classical
-  let error (n : ℕ) (ω : Ω) : ℝ≥0∞ :=
-    ∑ k ∈ M n, (missed n k).indicator
-      (fun _ => extendedGauge h (ediam (Q n k))) ω
-  have herr (n : ℕ) : AEMeasurable (error n) P := by
-    apply Measurable.aemeasurable
-    exact Finset.measurable_sum (M n) (fun k hk =>
-      measurable_const.indicator (hmissmeas n k hk))
-  have herrlim : Tendsto (fun n => ∫⁻ ω, error n ω ∂P) atTop (𝓝 0) := by
-    apply tendsto_of_tendsto_of_tendsto_of_le_of_le
-      tendsto_const_nhds hcleanup (fun _ => bot_le) (fun n => ?_)
-    exact expected_missed_cell_cost P (M n) (missed n)
-      (hmissmeas n) (fun k => extendedGauge h (ediam (Q n k)))
-      (probBound n) (hprob n)
-  apply expected_hausdorff_le_of_finite_sausage_covers P κ hκ hκ8 γ hSLE
-    h E δ hδ v hv R C error herr herrlim
-  filter_upwards [hgeometry] with ω hω
-  intro n
-  obtain ⟨hr, hA, hcost, hdisk, hmiss, hcell⟩ := hω n
-  let missedIndices := (M n).filter (fun k => ω ∈ missed n k)
-  have he : (∑ k ∈ missedIndices, extendedGauge h (ediam (Q n k))) = error n ω := by
-    simp [missedIndices, error, Finset.sum_filter, Set.indicator]
-  have hc : ∀ z ∈ E ω,
-      (∃ i ∈ I n ω, z ∈ closedBall (x n i) (r n i)) ∨
-        ∃ k ∈ missedIndices, z ∈ Q n k := by
-    intro z hz
-    rcases hcell z hz with hz | ⟨k, hk, hkm, hkz⟩
-    · exact Or.inl hz
-    · exact Or.inr ⟨k, Finset.mem_filter.mpr ⟨hk, hkm⟩, hkz⟩
-  obtain ⟨N, U, hU, hUd, hUc⟩ := exists_finite_selected_missed_cover
-    (I n ω) missedIndices (x n) (r n) (Q n) (E ω) (closedBall 0 R)
-    (wholeSausageMeasure κ (γ ω) (v n)) h R hr (δ n) C hA
-    (by simpa only [wholeSausageMeasure_apply _ _ _ isClosed_closedBall.measurableSet]
-      using hcost)
-    hdisk (fun k hk => hmiss k (Finset.mem_filter.mp hk).1) hc
-  refine ⟨N, U, hU, hUd, ?_⟩
-  simpa only [wholeSausageMeasure_apply _ _ _ isClosed_closedBall.measurableSet, he]
-    using hUc
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.FiniteCoverAssembly
+import OAI.Probability.SLE.Gauge.CoverNumerics
+
+namespace OAI
+
+/-! Candidate and missed-cell tests imply the dense-cover upper bound. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- Spatial selection, missed-cell probability bounds, and one-mesh charging
+produce the upper Hausdorff expectation. No measurable choice of disks is needed. -/
+theorem expected_hausdorff_le_of_candidate_disks
+    {Ω ι η : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (h : ℝ → ℝ) (E : Ω → Set ℂ)
+    (I : ℕ → Ω → Finset ι) (M : ℕ → Finset η)
+    (x : ℕ → ι → ℂ) (r : ℕ → ι → ℝ) (Q : ℕ → η → Set ℂ)
+    (missed : ℕ → η → Set Ω)
+    (hmissmeas : ∀ n k, k ∈ M n → MeasurableSet (missed n k))
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (R : ℝ) (C : ℝ≥0)
+    (δ : ℕ → ℝ≥0∞) (hδ : Tendsto δ atTop (𝓝 0))
+    (probBound : ℕ → η → ℝ≥0∞)
+    (hprob : ∀ n k, k ∈ M n → P (missed n k) ≤ probBound n k)
+    (hcleanup : Tendsto (fun n => ∑ k ∈ M n,
+      extendedGauge h (ediam (Q n k)) * probBound n k) atTop (𝓝 0))
+    (hgeometry : ∀ᵐ ω ∂P, ∀ n,
+      (∀ i ∈ I n ω, r n i ≤ R) ∧
+      (∀ i ∈ I n ω, closedBall (x n i) (r n i) ⊆ closedBall 0 R) ∧
+      (∀ i ∈ I n ω,
+        extendedGauge h (ediam (closedBall (x n i) (4 * r n i))) ≤
+          (C : ℝ≥0∞) * wholeSausageCharge κ (γ ω) (v n)
+            (closedBall (x n i) (r n i))) ∧
+      (∀ i ∈ I n ω, ediam (closedBall (x n i) (4 * r n i)) ≤ δ n) ∧
+      (∀ k ∈ M n, ediam (Q n k) ≤ δ n) ∧
+      (∀ z ∈ E ω, (∃ i ∈ I n ω, z ∈ closedBall (x n i) (r n i)) ∨
+        ∃ k ∈ M n, ω ∈ missed n k ∧ z ∈ Q n k)) :
+    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤
+      (C : ℝ≥0∞) * wholeSausageChargeBound κ R := by
+  classical
+  let error (n : ℕ) (ω : Ω) : ℝ≥0∞ :=
+    ∑ k ∈ M n, (missed n k).indicator
+      (fun _ => extendedGauge h (ediam (Q n k))) ω
+  have herr (n : ℕ) : AEMeasurable (error n) P := by
+    apply Measurable.aemeasurable
+    exact Finset.measurable_sum (M n) (fun k hk =>
+      measurable_const.indicator (hmissmeas n k hk))
+  have herrlim : Tendsto (fun n => ∫⁻ ω, error n ω ∂P) atTop (𝓝 0) := by
+    apply tendsto_of_tendsto_of_tendsto_of_le_of_le
+      tendsto_const_nhds hcleanup (fun _ => bot_le) (fun n => ?_)
+    exact expected_missed_cell_cost P (M n) (missed n)
+      (hmissmeas n) (fun k => extendedGauge h (ediam (Q n k)))
+      (probBound n) (hprob n)
+  apply expected_hausdorff_le_of_finite_sausage_covers P κ hκ hκ8 γ hSLE
+    h E δ hδ v hv R C error herr herrlim
+  filter_upwards [hgeometry] with ω hω
+  intro n
+  obtain ⟨hr, hA, hcost, hdisk, hmiss, hcell⟩ := hω n
+  let missedIndices := (M n).filter (fun k => ω ∈ missed n k)
+  have he : (∑ k ∈ missedIndices, extendedGauge h (ediam (Q n k))) = error n ω := by
+    simp [missedIndices, error, Finset.sum_filter, Set.indicator]
+  have hc : ∀ z ∈ E ω,
+      (∃ i ∈ I n ω, z ∈ closedBall (x n i) (r n i)) ∨
+        ∃ k ∈ missedIndices, z ∈ Q n k := by
+    intro z hz
+    rcases hcell z hz with hz | ⟨k, hk, hkm, hkz⟩
+    · exact Or.inl hz
+    · exact Or.inr ⟨k, Finset.mem_filter.mpr ⟨hk, hkm⟩, hkz⟩
+  obtain ⟨N, U, hU, hUd, hUc⟩ := exists_finite_selected_missed_cover
+    (I n ω) missedIndices (x n) (r n) (Q n) (E ω) (closedBall 0 R)
+    (wholeSausageMeasure κ (γ ω) (v n)) h R hr (δ n) C hA
+    (by simpa only [wholeSausageMeasure_apply _ _ _ isClosed_closedBall.measurableSet]
+      using hcost)
+    hdisk (fun k hk => hmiss k (Finset.mem_filter.mp hk).1) hc
+  refine ⟨N, U, hU, hUd, ?_⟩
+  simpa only [wholeSausageMeasure_apply _ _ _ isClosed_closedBall.measurableSet, he]
+    using hUc
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseInfinityCriterion.lean b/lean/OAI/Probability/SLE/Gauge/DenseInfinityCriterion.lean
index 6823d4d746f0b8126994de00762234550eed6c5b..eaa34130c1add058a61a5049388e86caba3342f4 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseInfinityCriterion.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseInfinityCriterion.lean
@@ -1,63 +1,63 @@
-import Mathlib
-
-namespace OAI
-
-/-! A countable test for a continuous function's limit at infinity on an
-open planar domain. Only the values on one fixed dense sequence are used. -/
-noncomputable section
-open Set Filter Metric TopologicalSpace
-open scoped Topology
-
-namespace SLEExactGauge
-
-def DenseInfinityCriterion (f : ℂ → ℂ) (D : Set ℂ) (c : ℂ) : Prop :=
-  ∀ k : ℕ, ∃ N : ℕ, ∀ n : ℕ,
-    (N:ℝ) < ‖denseSeq ℂ n‖ → denseSeq ℂ n ∈ D →
-      ‖f (denseSeq ℂ n)-c‖ ≤ 1/((k:ℝ)+1)
-
-lemma denseInfinityCriterion_of_tendsto {f : ℂ → ℂ} {D : Set ℂ} {c : ℂ}
-    (h : Tendsto f (cocompact ℂ ⊓ 𝓟 D) (𝓝 c)) : DenseInfinityCriterion f D c := by
-  intro k
-  have he := (Metric.tendsto_nhds.mp h) (1/((k:ℝ)+1)) (by positivity)
-  rw [eventually_inf_principal,← Metric.cobounded_eq_cocompact] at he
-  obtain ⟨R,_,hR⟩ := Filter.hasBasis_cobounded_norm.eventually_iff.mp he
-  obtain ⟨N,hN⟩ := exists_nat_gt R
-  refine ⟨N,fun n hn hD => ?_⟩
-  simpa only [dist_eq_norm] using (hR ((hN.trans hn).le) hD).le
-
-lemma DenseInfinityCriterion.bound {f : ℂ → ℂ} {D : Set ℂ} {c : ℂ}
-    (hD : IsOpen D) (hf : ContinuousOn f D) (h : DenseInfinityCriterion f D c)
-    (k : ℕ) : ∃ N : ℕ, ∀ z ∈ D, (N:ℝ)<‖z‖ → ‖f z-c‖ ≤ 1/((k:ℝ)+1) := by
-  obtain ⟨N,hN⟩ := h k
-  refine ⟨N,fun z hz hNz => ?_⟩
-  by_contra hn
-  have hbad : 1/((k:ℝ)+1)<‖f z-c‖ := lt_of_not_ge hn
-  have hc : ContinuousAt (fun w => ‖f w-c‖) z :=
-    (((hf z hz).continuousAt (hD.mem_nhds hz)).sub_const c).norm
-  have hb : {w : ℂ | 1/((k:ℝ)+1)<‖f w-c‖} ∈ 𝓝 z :=
-    hc.preimage_mem_nhds (Ioi_mem_nhds hbad)
-  have hr : {w : ℂ | (N:ℝ)<‖w‖} ∈ 𝓝 z :=
-    continuous_norm.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hNz)
-  obtain ⟨n,hnD,hnR,hnB⟩ := (denseRange_denseSeq ℂ).mem_nhds
-    (inter_mem (hD.mem_nhds hz) (inter_mem hr hb))
-  exact (not_lt_of_ge (hN n hnR hnD)) hnB
-
-lemma DenseInfinityCriterion.tendsto {f : ℂ → ℂ} {D : Set ℂ} {c : ℂ}
-    (hD : IsOpen D) (hf : ContinuousOn f D) (h : DenseInfinityCriterion f D c) :
-    Tendsto f (cocompact ℂ ⊓ 𝓟 D) (𝓝 c) := by
-  apply Metric.tendsto_nhds.mpr
-  intro ε hε
-  obtain ⟨k,hk⟩ := ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
-    (gt_mem_nhds hε)).exists
-  obtain ⟨N,hN⟩ := h.bound hD hf k
-  rw [eventually_inf_principal]
-  have hn : ∀ᶠ z : ℂ in cocompact ℂ, (N:ℝ)<‖z‖ :=
-    tendsto_norm_cocompact_atTop.eventually (eventually_gt_atTop (N:ℝ))
-  filter_upwards [hn] with z hz hDz
-  simpa only [dist_eq_norm] using (hN z hDz hz).trans_lt hk
-
-end SLEExactGauge
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! A countable test for a continuous function's limit at infinity on an
+open planar domain. Only the values on one fixed dense sequence are used. -/
+noncomputable section
+open Set Filter Metric TopologicalSpace
+open scoped Topology
+
+namespace SLEExactGauge
+
+def DenseInfinityCriterion (f : ℂ → ℂ) (D : Set ℂ) (c : ℂ) : Prop :=
+  ∀ k : ℕ, ∃ N : ℕ, ∀ n : ℕ,
+    (N:ℝ) < ‖denseSeq ℂ n‖ → denseSeq ℂ n ∈ D →
+      ‖f (denseSeq ℂ n)-c‖ ≤ 1/((k:ℝ)+1)
+
+lemma denseInfinityCriterion_of_tendsto {f : ℂ → ℂ} {D : Set ℂ} {c : ℂ}
+    (h : Tendsto f (cocompact ℂ ⊓ 𝓟 D) (𝓝 c)) : DenseInfinityCriterion f D c := by
+  intro k
+  have he := (Metric.tendsto_nhds.mp h) (1/((k:ℝ)+1)) (by positivity)
+  rw [eventually_inf_principal,← Metric.cobounded_eq_cocompact] at he
+  obtain ⟨R,_,hR⟩ := Filter.hasBasis_cobounded_norm.eventually_iff.mp he
+  obtain ⟨N,hN⟩ := exists_nat_gt R
+  refine ⟨N,fun n hn hD => ?_⟩
+  simpa only [dist_eq_norm] using (hR ((hN.trans hn).le) hD).le
+
+lemma DenseInfinityCriterion.bound {f : ℂ → ℂ} {D : Set ℂ} {c : ℂ}
+    (hD : IsOpen D) (hf : ContinuousOn f D) (h : DenseInfinityCriterion f D c)
+    (k : ℕ) : ∃ N : ℕ, ∀ z ∈ D, (N:ℝ)<‖z‖ → ‖f z-c‖ ≤ 1/((k:ℝ)+1) := by
+  obtain ⟨N,hN⟩ := h k
+  refine ⟨N,fun z hz hNz => ?_⟩
+  by_contra hn
+  have hbad : 1/((k:ℝ)+1)<‖f z-c‖ := lt_of_not_ge hn
+  have hc : ContinuousAt (fun w => ‖f w-c‖) z :=
+    (((hf z hz).continuousAt (hD.mem_nhds hz)).sub_const c).norm
+  have hb : {w : ℂ | 1/((k:ℝ)+1)<‖f w-c‖} ∈ 𝓝 z :=
+    hc.preimage_mem_nhds (Ioi_mem_nhds hbad)
+  have hr : {w : ℂ | (N:ℝ)<‖w‖} ∈ 𝓝 z :=
+    continuous_norm.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hNz)
+  obtain ⟨n,hnD,hnR,hnB⟩ := (denseRange_denseSeq ℂ).mem_nhds
+    (inter_mem (hD.mem_nhds hz) (inter_mem hr hb))
+  exact (not_lt_of_ge (hN n hnR hnD)) hnB
+
+lemma DenseInfinityCriterion.tendsto {f : ℂ → ℂ} {D : Set ℂ} {c : ℂ}
+    (hD : IsOpen D) (hf : ContinuousOn f D) (h : DenseInfinityCriterion f D c) :
+    Tendsto f (cocompact ℂ ⊓ 𝓟 D) (𝓝 c) := by
+  apply Metric.tendsto_nhds.mpr
+  intro ε hε
+  obtain ⟨k,hk⟩ := ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
+    (gt_mem_nhds hε)).exists
+  obtain ⟨N,hN⟩ := h.bound hD hf k
+  rw [eventually_inf_principal]
+  have hn : ∀ᶠ z : ℂ in cocompact ℂ, (N:ℝ)<‖z‖ :=
+    tendsto_norm_cocompact_atTop.eventually (eventually_gt_atTop (N:ℝ))
+  filter_upwards [hn] with z hz hDz
+  simpa only [dist_eq_norm] using (hN z hDz hz).trans_lt hk
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseRadiusGap.lean b/lean/OAI/Probability/SLE/Gauge/DenseRadiusGap.lean
index 9656eb4724f83c1d9a04e6ccca762697908d1022..576a655fd454f463888549a1993b16634cad9062 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseRadiusGap.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseRadiusGap.lean
@@ -1,39 +1,39 @@
-import OAI.Probability.SLE.Loewner.ActualRadiusStops
-
-namespace OAI
-
-/-! A live current radius level is strictly before every deeper radius
-level, including the event that the deeper level is never attained. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma IsCapacityTwoWitness.before_deeper_radius_level
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (z : ℂ) (u q : ℝ) (hu : 0 ≤ u) (hq : 0 < q) (T : ℝ≥0)
-    (hz : z ∈ survivingDomain γ T)
-    (hR : fullRadius G T z = (2 * z.im) * Real.exp (-u)) :
-    (T : ℝ≥0∞) < radiusLevelTime γ G z (u + q) := by
-  have hmono := radiusLevelTime_mono hU hw z hu (show u ≤ u + q by linarith)
-  rw [radiusLevelTime_eq_of_live_level hU hw z u T hz hR] at hmono
-  refine lt_of_le_of_ne hmono ?_
-  intro he
-  have hfinite : radiusLevelTime γ G z (u + q) < ∞ := by rw [← he]; exact ENNReal.coe_lt_top
-  obtain ⟨t, ht, hRt⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z (u + q) hfinite
-  have hte := radiusLevelTime_eq_of_live_level hU hw z (u + q) t ht hRt
-  have hTt : T = t := by exact_mod_cast (he.trans hte)
-  rw [← hTt, hR] at hRt
-  have hy : 2 * z.im ≠ 0 := (mul_pos (by norm_num) hz.1.1).ne'
-  have hexp := (mul_left_cancel₀ hy hRt)
-  have hneg := Real.exp_injective hexp
-  linarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.ActualRadiusStops
+
+namespace OAI
+
+/-! A live current radius level is strictly before every deeper radius
+level, including the event that the deeper level is never attained. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma IsCapacityTwoWitness.before_deeper_radius_level
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (z : ℂ) (u q : ℝ) (hu : 0 ≤ u) (hq : 0 < q) (T : ℝ≥0)
+    (hz : z ∈ survivingDomain γ T)
+    (hR : fullRadius G T z = (2 * z.im) * Real.exp (-u)) :
+    (T : ℝ≥0∞) < radiusLevelTime γ G z (u + q) := by
+  have hmono := radiusLevelTime_mono hU hw z hu (show u ≤ u + q by linarith)
+  rw [radiusLevelTime_eq_of_live_level hU hw z u T hz hR] at hmono
+  refine lt_of_le_of_ne hmono ?_
+  intro he
+  have hfinite : radiusLevelTime γ G z (u + q) < ∞ := by rw [← he]; exact ENNReal.coe_lt_top
+  obtain ⟨t, ht, hRt⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z (u + q) hfinite
+  have hte := radiusLevelTime_eq_of_live_level hU hw z (u + q) t ht hRt
+  have hTt : T = t := by exact_mod_cast (he.trans hte)
+  rw [← hTt, hR] at hRt
+  have hy : 2 * z.im ≠ 0 := (mul_pos (by norm_num) hz.1.1).ne'
+  have hexp := (mul_left_cancel₀ hy hRt)
+  have hneg := Real.exp_injective hexp
+  linarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseRibbonNumerics.lean b/lean/OAI/Probability/SLE/Gauge/DenseRibbonNumerics.lean
index 6d6c3a51a7474a71169c6919dff7a82e195da093..9dd2249b63563e2b208e73e6a9cc78d1f3273bad 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseRibbonNumerics.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseRibbonNumerics.lean
@@ -1,47 +1,47 @@
-import OAI.Probability.SLE.Gauge.RibbonExperimentAssembly
-
-namespace OAI
-
-/-! Absorb the fixed initial passage probability into the quadratic
-experiment cost, and convert width-scaled unit-strip mass to m^(2-d). -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma ribbon_mass_after_width_scaling (κ c a : ℝ) (ha : 0 < a) (m : ℕ) (hm : 0 < m) :
-    (a / (m : ℝ)) ^ dimension κ * ((m : ℝ) ^ 2 * c) =
-      c * a ^ dimension κ * (m : ℝ) ^ codimension κ := by
-  have hmR : (0 : ℝ) < m := by exact_mod_cast hm
-  rw [div_eq_mul_one_div, Real.mul_rpow ha.le (one_div_nonneg.mpr hmR.le)]
-  calc
-    _ = c * a ^ dimension κ * ((m : ℝ) ^ 2 * (1 / (m : ℝ)) ^ dimension κ) := by ring
-    _ = _ := by rw [ribbon_mass_power_cancellation κ hmR]
-
-/-- One initial positive-probability event and quadratically many uniform
-steps have one stretched-exponential cost, uniformly in the mesh. -/
-theorem ribbon_initial_and_steps_quadratic_cost
-    {q p D : ℝ} (hq : 0 < q) (hq1 : q ≤ 1)
-    (hp : 0 < p) (hD : 0 ≤ D) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (m N : ℕ), 0 < m → (N : ℝ) ≤ D * (m : ℝ) ^ 2 →
-      Real.exp (-C * (m : ℝ) ^ 2) ≤ q ^ N * p := by
-  obtain ⟨C, hC, hcost⟩ := ribbon_probability_of_quadratic_completion
-    (lt_min hq hp) ((min_le_left q p).trans hq1) (show 0 ≤ D + 1 by linarith)
-  refine ⟨C, hC, fun m N hm hN => ?_⟩
-  have hmR : (1 : ℝ) ≤ m := by exact_mod_cast hm
-  have hcount : ((N + 1 : ℕ) : ℝ) ≤ (D + 1) * (m : ℝ) ^ 2 := by
-    rw [Nat.cast_add, Nat.cast_one]
-    nlinarith
-  calc
-    _ ≤ min q p ^ (N + 1) := hcost m (N + 1) hcount
-    _ = min q p ^ N * min q p := pow_succ _ _
-    _ ≤ q ^ N * p := mul_le_mul (pow_le_pow_left₀ (le_of_lt (lt_min hq hp))
-      (min_le_left q p) N) (min_le_right q p) (le_of_lt (lt_min hq hp)) (pow_nonneg hq.le N)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RibbonExperimentAssembly
+
+namespace OAI
+
+/-! Absorb the fixed initial passage probability into the quadratic
+experiment cost, and convert width-scaled unit-strip mass to m^(2-d). -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma ribbon_mass_after_width_scaling (κ c a : ℝ) (ha : 0 < a) (m : ℕ) (hm : 0 < m) :
+    (a / (m : ℝ)) ^ dimension κ * ((m : ℝ) ^ 2 * c) =
+      c * a ^ dimension κ * (m : ℝ) ^ codimension κ := by
+  have hmR : (0 : ℝ) < m := by exact_mod_cast hm
+  rw [div_eq_mul_one_div, Real.mul_rpow ha.le (one_div_nonneg.mpr hmR.le)]
+  calc
+    _ = c * a ^ dimension κ * ((m : ℝ) ^ 2 * (1 / (m : ℝ)) ^ dimension κ) := by ring
+    _ = _ := by rw [ribbon_mass_power_cancellation κ hmR]
+
+/-- One initial positive-probability event and quadratically many uniform
+steps have one stretched-exponential cost, uniformly in the mesh. -/
+theorem ribbon_initial_and_steps_quadratic_cost
+    {q p D : ℝ} (hq : 0 < q) (hq1 : q ≤ 1)
+    (hp : 0 < p) (hD : 0 ≤ D) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (m N : ℕ), 0 < m → (N : ℝ) ≤ D * (m : ℝ) ^ 2 →
+      Real.exp (-C * (m : ℝ) ^ 2) ≤ q ^ N * p := by
+  obtain ⟨C, hC, hcost⟩ := ribbon_probability_of_quadratic_completion
+    (lt_min hq hp) ((min_le_left q p).trans hq1) (show 0 ≤ D + 1 by linarith)
+  refine ⟨C, hC, fun m N hm hN => ?_⟩
+  have hmR : (1 : ℝ) ≤ m := by exact_mod_cast hm
+  have hcount : ((N + 1 : ℕ) : ℝ) ≤ (D + 1) * (m : ℝ) ^ 2 := by
+    rw [Nat.cast_add, Nat.cast_one]
+    nlinarith
+  calc
+    _ ≤ min q p ^ (N + 1) := hcost m (N + 1) hcount
+    _ = min q p ^ N * min q p := pow_succ _ _
+    _ ≤ q ^ N * p := mul_le_mul (pow_le_pow_left₀ (le_of_lt (lt_min hq hp))
+      (min_le_left q p) N) (min_le_right q p) (le_of_lt (lt_min hq hp)) (pow_nonneg hq.le N)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DenseVisitNumerics.lean b/lean/OAI/Probability/SLE/Gauge/DenseVisitNumerics.lean
index 649229e1fc8dfcbaf27923616343620a794de238..6c3100cf4db4293b921dbd69b19b25be5bbed9c5 100644
--- a/lean/OAI/Probability/SLE/Gauge/DenseVisitNumerics.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DenseVisitNumerics.lean
@@ -1,110 +1,110 @@
-import OAI.Probability.SLE.Gauge.CoverNumerics
-import OAI.Probability.SLE.Gauge.CoverMesh
-
-namespace OAI
-
-/-! The actual logarithmic choice of dense-visit multiplicity and the
-linear test window used in the explicit gauge cover. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def coverMultiplicity (c : ℝ) (n : ℕ) : ℕ :=
-  ⌊c * Real.sqrt (Real.log (n : ℝ))⌋₊
-
-def coverTestWindow (n : ℕ) : Finset ℕ := Finset.Icc (n / 3) (2 * n / 3)
-
-theorem coverTestWindow_card_lower (n : ℕ) : n / 4 ≤ (coverTestWindow n).card := by
-  simp only [coverTestWindow, Nat.card_Icc]
-  omega
-
-theorem coverTestWindow_card_real_lower (n : ℕ) :
-    (n : ℝ) / 4 ≤ ((coverTestWindow n).card : ℝ) := by
-  have hh : n ≤ 4 * (coverTestWindow n).card := by
-    simp only [coverTestWindow, Nat.card_Icc]
-    omega
-  have hh' : (n : ℝ) ≤ 4 * ((coverTestWindow n).card : ℝ) := by exact_mod_cast hh
-  linarith
-
-theorem coverMultiplicity_upper {c : ℝ} (hc : 0 ≤ c) (n : ℕ) :
-    (coverMultiplicity c n : ℝ) ≤ c * Real.sqrt (Real.log (n : ℝ)) :=
-  Nat.floor_le (mul_nonneg hc (Real.sqrt_nonneg _))
-
-theorem coverMultiplicity_eventually_lower {c : ℝ} (hc : 0 < c) :
-    ∀ᶠ n : ℕ in atTop,
-      c / 2 * Real.sqrt (Real.log (n : ℝ)) ≤ (coverMultiplicity c n : ℝ) := by
-  have hlim : Tendsto (fun n : ℕ => c * Real.sqrt (Real.log (n : ℝ))) atTop atTop :=
-    (Real.tendsto_sqrt_atTop.comp (Real.tendsto_log_atTop.comp
-      tendsto_natCast_atTop_atTop)).const_mul_atTop hc
-  filter_upwards [hlim.eventually_ge_atTop 2] with n hn
-  have hh := Nat.lt_floor_add_one (c * Real.sqrt (Real.log (n : ℝ)))
-  change _ < (coverMultiplicity c n : ℝ) + 1 at hh
-  linarith
-
-theorem coverMultiplicity_eventually_ge {c : ℝ} (hc : 0 < c) (m₀ : ℕ) :
-    ∀ᶠ n : ℕ in atTop, m₀ ≤ coverMultiplicity c n := by
-  have hlim : Tendsto (fun n : ℕ => c * Real.sqrt (Real.log (n : ℝ))) atTop atTop :=
-    (Real.tendsto_sqrt_atTop.comp (Real.tendsto_log_atTop.comp
-      tendsto_natCast_atTop_atTop)).const_mul_atTop hc
-  filter_upwards [hlim.eventually_ge_atTop (m₀ : ℝ)] with n hn
-  exact Nat.le_floor hn
-
-/-- The mass created by the prescribed integer multiplicity is at least
-the logarithmic correction of the explicit gauge. -/
-theorem coverMultiplicity_mass_lower {c p : ℝ} (hc : 0 < c) (hp : 0 ≤ p) :
-    ∀ᶠ n : ℕ in atTop,
-      (c / 2) ^ p * (Real.log (n : ℝ)) ^ (p / 2) ≤
-        (coverMultiplicity c n : ℝ) ^ p := by
-  filter_upwards [coverMultiplicity_eventually_lower hc,
-    eventually_ge_atTop (1 : ℕ)] with n hn hn1
-  have hlog : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn1)
-  calc
-    _ = (c / 2 * Real.sqrt (Real.log (n : ℝ))) ^ p := by
-      rw [Real.mul_rpow (by positivity) (Real.sqrt_nonneg _), Real.sqrt_eq_rpow,
-        ← Real.rpow_mul hlog]
-      congr 2
-      ring
-    _ ≤ _ := Real.rpow_le_rpow (by positivity) hn hp
-
-/-- A fixed dense-visit probability becomes at least `n^(-1/2)` at the
-logarithmic multiplicity. The prefactor is retained, rather than discarded. -/
-theorem denseVisit_probability_eventually_lower
-    {a C c : ℝ} (ha : 0 < a) (hC : 0 ≤ C) (hc : 0 < c)
-    (hsmall : C * c ^ 2 ≤ 1 / 4) :
-    ∀ᶠ n : ℕ in atTop,
-      1 / Real.sqrt (n : ℝ) ≤
-        a * Real.exp (-C * (coverMultiplicity c n : ℝ) ^ 2) := by
-  have hloglim : Tendsto (fun n : ℕ => Real.log (n : ℝ)) atTop atTop :=
-    Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop
-  filter_upwards [eventually_ge_atTop (1 : ℕ),
-    hloglim.eventually_ge_atTop (-4 * Real.log a)] with n hn hlarge
-  have hnpos : (0 : ℝ) < n := by exact_mod_cast (zero_lt_one.trans_le hn)
-  have hlog : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
-  have hup := coverMultiplicity_upper hc.le n
-  have hsq : (coverMultiplicity c n : ℝ) ^ 2 ≤ c ^ 2 * Real.log (n : ℝ) := by
-    calc
-      _ ≤ (c * Real.sqrt (Real.log (n : ℝ))) ^ 2 :=
-        pow_le_pow_left₀ (Nat.cast_nonneg _) hup 2
-      _ = _ := by rw [mul_pow, Real.sq_sqrt hlog]
-  have hCbound : C * (coverMultiplicity c n : ℝ) ^ 2 ≤ Real.log (n : ℝ) / 4 := by
-    calc
-      _ ≤ C * (c ^ 2 * Real.log (n : ℝ)) := mul_le_mul_of_nonneg_left hsq hC
-      _ = (C * c ^ 2) * Real.log (n : ℝ) := by ring
-      _ ≤ (1 / 4) * Real.log (n : ℝ) := mul_le_mul_of_nonneg_right hsmall hlog
-      _ = _ := by ring
-  have hsqrt : 1 / Real.sqrt (n : ℝ) = Real.exp (-(Real.log (n : ℝ) / 2)) := by
-    rw [Real.sqrt_eq_rpow, Real.rpow_def_of_pos hnpos, one_div, ← Real.exp_neg]
-    congr 1
-    ring
-  rw [hsqrt, ← Real.exp_log ha, ← Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  linarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverNumerics
+import OAI.Probability.SLE.Gauge.CoverMesh
+
+namespace OAI
+
+/-! The actual logarithmic choice of dense-visit multiplicity and the
+linear test window used in the explicit gauge cover. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def coverMultiplicity (c : ℝ) (n : ℕ) : ℕ :=
+  ⌊c * Real.sqrt (Real.log (n : ℝ))⌋₊
+
+def coverTestWindow (n : ℕ) : Finset ℕ := Finset.Icc (n / 3) (2 * n / 3)
+
+theorem coverTestWindow_card_lower (n : ℕ) : n / 4 ≤ (coverTestWindow n).card := by
+  simp only [coverTestWindow, Nat.card_Icc]
+  omega
+
+theorem coverTestWindow_card_real_lower (n : ℕ) :
+    (n : ℝ) / 4 ≤ ((coverTestWindow n).card : ℝ) := by
+  have hh : n ≤ 4 * (coverTestWindow n).card := by
+    simp only [coverTestWindow, Nat.card_Icc]
+    omega
+  have hh' : (n : ℝ) ≤ 4 * ((coverTestWindow n).card : ℝ) := by exact_mod_cast hh
+  linarith
+
+theorem coverMultiplicity_upper {c : ℝ} (hc : 0 ≤ c) (n : ℕ) :
+    (coverMultiplicity c n : ℝ) ≤ c * Real.sqrt (Real.log (n : ℝ)) :=
+  Nat.floor_le (mul_nonneg hc (Real.sqrt_nonneg _))
+
+theorem coverMultiplicity_eventually_lower {c : ℝ} (hc : 0 < c) :
+    ∀ᶠ n : ℕ in atTop,
+      c / 2 * Real.sqrt (Real.log (n : ℝ)) ≤ (coverMultiplicity c n : ℝ) := by
+  have hlim : Tendsto (fun n : ℕ => c * Real.sqrt (Real.log (n : ℝ))) atTop atTop :=
+    (Real.tendsto_sqrt_atTop.comp (Real.tendsto_log_atTop.comp
+      tendsto_natCast_atTop_atTop)).const_mul_atTop hc
+  filter_upwards [hlim.eventually_ge_atTop 2] with n hn
+  have hh := Nat.lt_floor_add_one (c * Real.sqrt (Real.log (n : ℝ)))
+  change _ < (coverMultiplicity c n : ℝ) + 1 at hh
+  linarith
+
+theorem coverMultiplicity_eventually_ge {c : ℝ} (hc : 0 < c) (m₀ : ℕ) :
+    ∀ᶠ n : ℕ in atTop, m₀ ≤ coverMultiplicity c n := by
+  have hlim : Tendsto (fun n : ℕ => c * Real.sqrt (Real.log (n : ℝ))) atTop atTop :=
+    (Real.tendsto_sqrt_atTop.comp (Real.tendsto_log_atTop.comp
+      tendsto_natCast_atTop_atTop)).const_mul_atTop hc
+  filter_upwards [hlim.eventually_ge_atTop (m₀ : ℝ)] with n hn
+  exact Nat.le_floor hn
+
+/-- The mass created by the prescribed integer multiplicity is at least
+the logarithmic correction of the explicit gauge. -/
+theorem coverMultiplicity_mass_lower {c p : ℝ} (hc : 0 < c) (hp : 0 ≤ p) :
+    ∀ᶠ n : ℕ in atTop,
+      (c / 2) ^ p * (Real.log (n : ℝ)) ^ (p / 2) ≤
+        (coverMultiplicity c n : ℝ) ^ p := by
+  filter_upwards [coverMultiplicity_eventually_lower hc,
+    eventually_ge_atTop (1 : ℕ)] with n hn hn1
+  have hlog : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn1)
+  calc
+    _ = (c / 2 * Real.sqrt (Real.log (n : ℝ))) ^ p := by
+      rw [Real.mul_rpow (by positivity) (Real.sqrt_nonneg _), Real.sqrt_eq_rpow,
+        ← Real.rpow_mul hlog]
+      congr 2
+      ring
+    _ ≤ _ := Real.rpow_le_rpow (by positivity) hn hp
+
+/-- A fixed dense-visit probability becomes at least `n^(-1/2)` at the
+logarithmic multiplicity. The prefactor is retained, rather than discarded. -/
+theorem denseVisit_probability_eventually_lower
+    {a C c : ℝ} (ha : 0 < a) (hC : 0 ≤ C) (hc : 0 < c)
+    (hsmall : C * c ^ 2 ≤ 1 / 4) :
+    ∀ᶠ n : ℕ in atTop,
+      1 / Real.sqrt (n : ℝ) ≤
+        a * Real.exp (-C * (coverMultiplicity c n : ℝ) ^ 2) := by
+  have hloglim : Tendsto (fun n : ℕ => Real.log (n : ℝ)) atTop atTop :=
+    Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop
+  filter_upwards [eventually_ge_atTop (1 : ℕ),
+    hloglim.eventually_ge_atTop (-4 * Real.log a)] with n hn hlarge
+  have hnpos : (0 : ℝ) < n := by exact_mod_cast (zero_lt_one.trans_le hn)
+  have hlog : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
+  have hup := coverMultiplicity_upper hc.le n
+  have hsq : (coverMultiplicity c n : ℝ) ^ 2 ≤ c ^ 2 * Real.log (n : ℝ) := by
+    calc
+      _ ≤ (c * Real.sqrt (Real.log (n : ℝ))) ^ 2 :=
+        pow_le_pow_left₀ (Nat.cast_nonneg _) hup 2
+      _ = _ := by rw [mul_pow, Real.sq_sqrt hlog]
+  have hCbound : C * (coverMultiplicity c n : ℝ) ^ 2 ≤ Real.log (n : ℝ) / 4 := by
+    calc
+      _ ≤ C * (c ^ 2 * Real.log (n : ℝ)) := mul_le_mul_of_nonneg_left hsq hC
+      _ = (C * c ^ 2) * Real.log (n : ℝ) := by ring
+      _ ≤ (1 / 4) * Real.log (n : ℝ) := mul_le_mul_of_nonneg_right hsmall hlog
+      _ = _ := by ring
+  have hsqrt : 1 / Real.sqrt (n : ℝ) = Real.exp (-(Real.log (n : ℝ) / 2)) := by
+    rw [Real.sqrt_eq_rpow, Real.rpow_def_of_pos hnpos, one_div, ← Real.exp_neg]
+    congr 1
+    ring
+  rw [hsqrt, ← Real.exp_log ha, ← Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  linarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DensityWeightedMoments.lean b/lean/OAI/Probability/SLE/Gauge/DensityWeightedMoments.lean
index e92dc005d17fc6f231f42bdcb0bbdf262bfbb93b..d4f944f93bc08662400f4a4de85df0e4365c25ee 100644
--- a/lean/OAI/Probability/SLE/Gauge/DensityWeightedMoments.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DensityWeightedMoments.lean
@@ -1,85 +1,85 @@
-import OAI.Probability.SLE.Loewner.TargetDensityBound
-import OAI.Probability.SLE.Angle.AngleMomentWeight
-
-namespace OAI
-
-/-! The center-hit decay cancels the finite change-of-law density.
-Consequently every terminal disk moment has one integrable angular bound. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-theorem uniform_density_disk_terminal_moments
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ) :
-    ∃ M : ℝ≥0∞, M<∞ ∧ ∀ (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ), 0≤ℓ →
-      (∫⁻ ω, ENNReal.ofReal (Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ *
-        (C.diskTerminalScalar θ hθ ω)^m) ∂P) ≤
-          M*ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ) := by
-  obtain ⟨M,hM,hbound⟩ := C.uniform_diskTerminalScalar_centered_moments hB hc h0 hκ hκ8 F hw hgrowth m
-  refine ⟨M,hM,?_⟩
-  intro θ hθ ℓ hℓ
-  let u := 4*ℓ/κ
-  let e := Real.exp (-u)
-  let c := Real.exp (u*codimension κ)/(Real.sin θ)^greenExponent κ
-  let E := diskCenterHit γ G θ e
-  let Z := C.diskTerminalScalar θ hθ
-  have hs : 0<Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  have hc0 : 0≤c := div_nonneg (Real.exp_pos _).le (Real.rpow_nonneg hs.le _)
-  have hu : 0≤u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
-  have he : 0<e := Real.exp_pos _
-  have he1 : e≤1 := Real.exp_le_one_iff.mpr (neg_nonpos.mpr hu)
-  have hE := measurableSet_diskCenterHit
-    (fun t => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t))) hc
-    κ θ hθ γ G F hw e he he1
-  have hZ (ω) : 0≤Z ω := compactWeightedMass_nonneg _ _ _
-  have hd (ω) : Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ ≤
-      E.indicator (fun _ => c) ω := by
-    exact Target.levelDensity_le_centerHit κ hκ hκ8 _ _ ((hc ω).const_mul _)
-      (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ
-  have hp : (∫⁻ ω, ENNReal.ofReal (Target.levelDensity κ θ
-        (Target.normalizedDriver κ B hc h0 ω) ℓ * (Z ω)^m) ∂P) ≤
-      ENNReal.ofReal c * ∫⁻ ω in E, (ENNReal.ofReal (Z ω))^m ∂P := by
-    rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,← lintegral_indicator hE]
-    apply lintegral_mono
-    intro ω
-    by_cases hω : ω∈E
-    · rw [indicator_of_mem hω]
-      rw [← ENNReal.ofReal_pow (hZ ω),← ENNReal.ofReal_mul hc0]
-      exact ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right
-        (by simpa only [indicator_of_mem hω] using hd ω) (pow_nonneg (hZ ω) m))
-    · rw [indicator_of_notMem hω]
-      apply le_of_eq
-      apply ENNReal.ofReal_eq_zero.mpr
-      have hh : Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ ≤ 0 := by
-        simpa only [indicator_of_notMem hω] using hd ω
-      exact mul_nonpos_of_nonpos_of_nonneg hh (pow_nonneg (hZ ω) m)
-  have hec : c * e^codimension κ=1/(Real.sin θ)^greenExponent κ := by
-    dsimp only [c,e]
-    rw [← Real.exp_mul]
-    field_simp [(Real.rpow_pos_of_pos hs (greenExponent κ)).ne']
-    rw [← Real.exp_add]
-    ring_nf
-    exact Real.exp_zero
-  calc
-    _ ≤ ENNReal.ofReal c * ∫⁻ ω in E, (ENNReal.ofReal (Z ω))^m ∂P := hp
-    _ ≤ ENNReal.ofReal c * (M*ENNReal.ofReal (e^codimension κ)) :=
-      mul_le_mul' le_rfl (hbound θ hθ e he he1)
-    _ = _ := by rw [mul_left_comm,← ENNReal.ofReal_mul hc0,hec]
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.TargetDensityBound
+import OAI.Probability.SLE.Angle.AngleMomentWeight
+
+namespace OAI
+
+/-! The center-hit decay cancels the finite change-of-law density.
+Consequently every terminal disk moment has one integrable angular bound. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+theorem uniform_density_disk_terminal_moments
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ) :
+    ∃ M : ℝ≥0∞, M<∞ ∧ ∀ (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ), 0≤ℓ →
+      (∫⁻ ω, ENNReal.ofReal (Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ *
+        (C.diskTerminalScalar θ hθ ω)^m) ∂P) ≤
+          M*ENNReal.ofReal (1/(Real.sin θ)^greenExponent κ) := by
+  obtain ⟨M,hM,hbound⟩ := C.uniform_diskTerminalScalar_centered_moments hB hc h0 hκ hκ8 F hw hgrowth m
+  refine ⟨M,hM,?_⟩
+  intro θ hθ ℓ hℓ
+  let u := 4*ℓ/κ
+  let e := Real.exp (-u)
+  let c := Real.exp (u*codimension κ)/(Real.sin θ)^greenExponent κ
+  let E := diskCenterHit γ G θ e
+  let Z := C.diskTerminalScalar θ hθ
+  have hs : 0<Real.sin θ := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  have hc0 : 0≤c := div_nonneg (Real.exp_pos _).le (Real.rpow_nonneg hs.le _)
+  have hu : 0≤u := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
+  have he : 0<e := Real.exp_pos _
+  have he1 : e≤1 := Real.exp_le_one_iff.mpr (neg_nonpos.mpr hu)
+  have hE := measurableSet_diskCenterHit
+    (fun t => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t))) hc
+    κ θ hθ γ G F hw e he he1
+  have hZ (ω) : 0≤Z ω := compactWeightedMass_nonneg _ _ _
+  have hd (ω) : Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ ≤
+      E.indicator (fun _ => c) ω := by
+    exact Target.levelDensity_le_centerHit κ hκ hκ8 _ _ ((hc ω).const_mul _)
+      (γ ω) (G ω) (F ω) (hw ω) (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ
+  have hp : (∫⁻ ω, ENNReal.ofReal (Target.levelDensity κ θ
+        (Target.normalizedDriver κ B hc h0 ω) ℓ * (Z ω)^m) ∂P) ≤
+      ENNReal.ofReal c * ∫⁻ ω in E, (ENNReal.ofReal (Z ω))^m ∂P := by
+    rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,← lintegral_indicator hE]
+    apply lintegral_mono
+    intro ω
+    by_cases hω : ω∈E
+    · rw [indicator_of_mem hω]
+      rw [← ENNReal.ofReal_pow (hZ ω),← ENNReal.ofReal_mul hc0]
+      exact ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right
+        (by simpa only [indicator_of_mem hω] using hd ω) (pow_nonneg (hZ ω) m))
+    · rw [indicator_of_notMem hω]
+      apply le_of_eq
+      apply ENNReal.ofReal_eq_zero.mpr
+      have hh : Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ ≤ 0 := by
+        simpa only [indicator_of_notMem hω] using hd ω
+      exact mul_nonpos_of_nonpos_of_nonneg hh (pow_nonneg (hZ ω) m)
+  have hec : c * e^codimension κ=1/(Real.sin θ)^greenExponent κ := by
+    dsimp only [c,e]
+    rw [← Real.exp_mul]
+    field_simp [(Real.rpow_pos_of_pos hs (greenExponent κ)).ne']
+    rw [← Real.exp_add]
+    ring_nf
+    exact Real.exp_zero
+  calc
+    _ ≤ ENNReal.ofReal c * ∫⁻ ω in E, (ENNReal.ofReal (Z ω))^m ∂P := hp
+    _ ≤ ENNReal.ofReal c * (M*ENNReal.ofReal (e^codimension κ)) :=
+      mul_le_mul' le_rfl (hbound θ hθ e he he1)
+    _ = _ := by rw [mul_left_comm,← ENNReal.ofReal_mul hc0,hec]
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DiskCanonicalMoments.lean b/lean/OAI/Probability/SLE/Gauge/DiskCanonicalMoments.lean
index af870c47106a898525f4a097af6170680b5dab24..37d4d102f7009d5dbb0af4ca10a54a2343c7f8d7 100644
--- a/lean/OAI/Probability/SLE/Gauge/DiskCanonicalMoments.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DiskCanonicalMoments.lean
@@ -1,57 +1,57 @@
-import OAI.Probability.SLE.Mass.DiskLevelTests
-
-namespace OAI
-
-/-! The centered physical-disk estimate survives the canonical convex
-limit on the same event and with its original target-radius factor. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped NNReal ENNReal Topology CompactlySupported
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-theorem uniform_disk_terminal_test_centered_moments
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (K : Set ℂ) (hK : IsCompact K) (hKd : K ⊆ ball 0 1)
-    (b : ℝ) (hb : 0<b) (hb1 : b≤1) (hbK : ∀ x∈K, b≤1-‖x‖^2) (m : ℕ) :
-    ∃ M : ℝ≥0∞, M<∞ ∧ ∀ (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi),
-      ∀ f : C_c((ball (0:ℂ) 1),ℝ), (∀ x, 0≤f x) → (∀ x, f x≤1) →
-        (∀ x : (ball (0:ℂ) 1), (x:ℂ)∉K → f x=0) → ∀ e : ℝ, 0<e → e≤1 →
-      (∫⁻ ω in diskCenterHit γ G θ e,
-        ENNReal.ofReal (∫ x, f x ∂conformalMass (Target.diskHomeomorph θ hθ)
-          (Target.diskMassWeight κ θ hθ) (C.rational.terminal ω)) ^ m ∂P) ≤
-            M*ENNReal.ofReal (e^codimension κ) := by
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  obtain ⟨M,hM,hbound⟩ := uniform_diskLevelMass_centered_moments hB hm hBm hc h0 hκ hκ8
-    γ G F hw hgrowth K hK hKd b hb hb1 hbK m
-  refine ⟨M,hM,?_⟩
-  intro θ hθ f hf hf1 hfs e he he1
-  let g := conformalMassTest (Target.diskHomeomorph θ hθ) (Target.diskMassWeight κ θ hθ) f
-  have hg : ∀ z, 0≤g z := fun z => mul_nonneg (NNReal.coe_nonneg _) (hf _)
-  simp_rw [conformalMass_integral]
-  apply level_radon_limit_test_moment_le hm hc hκ hκ8 γ G F hw
-    (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n)
-    C.rational.supports C.rational.weights C.rational.weights_nonneg C.rational.weights_sum
-    ∞ C.rational.terminal (ae_restrict_of_ae C.rational.terminal_limit) g hg m
-    (M*ENNReal.ofReal (e^codimension κ))
-  intro n
-  apply (lintegral_mono (fun ω => pow_le_pow_left' ?_ m)).trans (hbound θ hθ e he he1 n (Nat.cast_nonneg n))
-  simpa only [g,conformalMass_integral] using
-    ofReal_disk_level_test_le hm hc hκ hκ8 γ G F hw θ hθ K hK hKd f hf hf1 hfs
-      n (Nat.cast_nonneg n) ∞ ω
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.DiskLevelTests
+
+namespace OAI
+
+/-! The centered physical-disk estimate survives the canonical convex
+limit on the same event and with its original target-radius factor. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped NNReal ENNReal Topology CompactlySupported
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+theorem uniform_disk_terminal_test_centered_moments
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (K : Set ℂ) (hK : IsCompact K) (hKd : K ⊆ ball 0 1)
+    (b : ℝ) (hb : 0<b) (hb1 : b≤1) (hbK : ∀ x∈K, b≤1-‖x‖^2) (m : ℕ) :
+    ∃ M : ℝ≥0∞, M<∞ ∧ ∀ (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi),
+      ∀ f : C_c((ball (0:ℂ) 1),ℝ), (∀ x, 0≤f x) → (∀ x, f x≤1) →
+        (∀ x : (ball (0:ℂ) 1), (x:ℂ)∉K → f x=0) → ∀ e : ℝ, 0<e → e≤1 →
+      (∫⁻ ω in diskCenterHit γ G θ e,
+        ENNReal.ofReal (∫ x, f x ∂conformalMass (Target.diskHomeomorph θ hθ)
+          (Target.diskMassWeight κ θ hθ) (C.rational.terminal ω)) ^ m ∂P) ≤
+            M*ENNReal.ofReal (e^codimension κ) := by
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  obtain ⟨M,hM,hbound⟩ := uniform_diskLevelMass_centered_moments hB hm hBm hc h0 hκ hκ8
+    γ G F hw hgrowth K hK hKd b hb hb1 hbK m
+  refine ⟨M,hM,?_⟩
+  intro θ hθ f hf hf1 hfs e he he1
+  let g := conformalMassTest (Target.diskHomeomorph θ hθ) (Target.diskMassWeight κ θ hθ) f
+  have hg : ∀ z, 0≤g z := fun z => mul_nonneg (NNReal.coe_nonneg _) (hf _)
+  simp_rw [conformalMass_integral]
+  apply level_radon_limit_test_moment_le hm hc hκ hκ8 γ G F hw
+    (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n)
+    C.rational.supports C.rational.weights C.rational.weights_nonneg C.rational.weights_sum
+    ∞ C.rational.terminal (ae_restrict_of_ae C.rational.terminal_limit) g hg m
+    (M*ENNReal.ofReal (e^codimension κ))
+  intro n
+  apply (lintegral_mono (fun ω => pow_le_pow_left' ?_ m)).trans (hbound θ hθ e he he1 n (Nat.cast_nonneg n))
+  simpa only [g,conformalMass_integral] using
+    ofReal_disk_level_test_le hm hc hκ hκ8 γ G F hw θ hθ K hK hKd f hf hf1 hfs
+      n (Nat.cast_nonneg n) ∞ ω
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/DiskIntegratedMoments.lean b/lean/OAI/Probability/SLE/Gauge/DiskIntegratedMoments.lean
index 5ab8189e5d8e71268f43a60c4125fe6493b7772e..68ac9ce0a5b58c25092777bf935aea87d77bf7c0 100644
--- a/lean/OAI/Probability/SLE/Gauge/DiskIntegratedMoments.lean
+++ b/lean/OAI/Probability/SLE/Gauge/DiskIntegratedMoments.lean
@@ -1,94 +1,94 @@
-import OAI.Probability.SLE.Mass.DiskLevelMoments
-
-namespace OAI
-
-/-! Uniform centered moments of the physical disk level masses. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem diskLevelMass_centered_moment_bound
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKd : K ⊆ ball 0 1)
-    (b : ℝ) (hb : 0<b) (hb1 : b≤1) (hbK : ∀ x∈K, b≤1-‖x‖^2)
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (m : ℕ)
-    (e : ℝ) (he : 0<e) (he1 : e≤1) (u : ℝ) (hu : 0≤u) :
-    (∫⁻ ω in diskCenterHit γ G θ e, diskLevelMass κ θ B γ G u K ω ^ m ∂P) ≤
-      ENNReal.ofReal (e^codimension κ)*centeredLevelMomentBound κ b 0 K m := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hE := measurableSet_diskCenterHit hm hc κ θ hθ γ G F hw e he he1
-  have hW := (measurable_diskStoppedGreen hm hc κ θ u hu γ G F hw).ennreal_ofReal
-  change (∫⁻ ω in diskCenterHit γ G θ e,
-    (∫⁻ x in K, ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω x))^m ∂P) ≤ _
-  rw [lintegral_mass_pow_eq (P.restrict (diskCenterHit γ G θ e)) (volume.restrict K)
-    (fun ω x => ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω x)) hW m]
-  have hk : Measurable (fun x : Fin m → ℂ =>
-      cappedDistanceKernel (codimension κ) b (Fin.cons 0 x)) := by
-    apply (measurable_cappedDistanceKernel _ _).comp
-    apply Measurable.of_eval
-    intro i
-    exact Fin.cases measurable_const (fun j => measurable_pi_apply j) i
-  have hbnd : (∫⁻ x : Fin m → ℂ, ∫⁻ ω in diskCenterHit γ G θ e,
-      ∏ i, ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω (x i)) ∂P
-      ∂Measure.pi (fun _ : Fin m => volume.restrict K)) ≤
-      ENNReal.ofReal (multipointConstant κ (m+1)*e^codimension κ)*
-        ∫⁻ x : Fin m → ℂ, cappedDistanceKernel (codimension κ) b (Fin.cons 0 x)
-          ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
-    rw [← lintegral_const_mul _ hk]
-    apply lintegral_mono_ae
-    filter_upwards [ae_tuple_mem (ι:=Fin m) K hK,ae_fin_cons_injective m K 0] with x hx hi
-    have h := diskStoppedGreen_centered_product_bound hB hm hBm hc h0 hκ hκ8
-      γ G F hw hgrowth θ hθ m x (fun i => hKd (hx i)) hi b hb hb1
-      (fun i => hbK _ (hx i)) e he u
-    rw [lintegral_indicator hE] at h
-    simpa only [ENNReal.ofReal_prod_of_nonneg
-      (fun i _ => diskStoppedGreen_nonneg hw u _ (x i))] using h
-  apply hbnd.trans_eq
-  rw [ENNReal.ofReal_mul (multipointConstant_pos _ _).le]
-  unfold centeredLevelMomentBound
-  ring
-
-theorem uniform_diskLevelMass_centered_moments
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (K : Set ℂ) (hK : IsCompact K) (hKd : K ⊆ ball 0 1)
-    (b : ℝ) (hb : 0<b) (hb1 : b≤1) (hbK : ∀ x∈K, b≤1-‖x‖^2) (m : ℕ) :
-    ∃ M : ℝ≥0∞, M<∞ ∧ ∀ θ∈Ioo (0:ℝ) Real.pi, ∀ e : ℝ, 0<e → e≤1 →
-      ∀ u : ℝ, 0≤u →
-        (∫⁻ ω in diskCenterHit γ G θ e, diskLevelMass κ θ B γ G u K ω ^ m ∂P) ≤
-          M*ENNReal.ofReal (e^codimension κ) := by
-  obtain ⟨D,hD,hbound⟩ := cappedDistanceKernel_fixed_center_bound m
-    (codimension_pos κ hκ8) (by unfold codimension dimension; linarith) b K hK.isBounded
-  refine ⟨ENNReal.ofReal (multipointConstant κ (m+1))*D,
-    ENNReal.mul_lt_top ENNReal.ofReal_lt_top hD,?_⟩
-  intro θ hθ e he he1 u hu
-  apply (diskLevelMass_centered_moment_bound hB hm hBm hc h0 hκ hκ8
-    γ G F hw hgrowth K hK.measurableSet hKd b hb hb1 hbK θ hθ m e he he1 u hu).trans
-  unfold centeredLevelMomentBound
-  calc
-    _ ≤ ENNReal.ofReal (e^codimension κ)*(ENNReal.ofReal (multipointConstant κ (m+1))*D) := by
-      gcongr
-      exact hbound 0
-    _ = _ := mul_comm _ _
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.DiskLevelMoments
+
+namespace OAI
+
+/-! Uniform centered moments of the physical disk level masses. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem diskLevelMass_centered_moment_bound
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKd : K ⊆ ball 0 1)
+    (b : ℝ) (hb : 0<b) (hb1 : b≤1) (hbK : ∀ x∈K, b≤1-‖x‖^2)
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (m : ℕ)
+    (e : ℝ) (he : 0<e) (he1 : e≤1) (u : ℝ) (hu : 0≤u) :
+    (∫⁻ ω in diskCenterHit γ G θ e, diskLevelMass κ θ B γ G u K ω ^ m ∂P) ≤
+      ENNReal.ofReal (e^codimension κ)*centeredLevelMomentBound κ b 0 K m := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hE := measurableSet_diskCenterHit hm hc κ θ hθ γ G F hw e he he1
+  have hW := (measurable_diskStoppedGreen hm hc κ θ u hu γ G F hw).ennreal_ofReal
+  change (∫⁻ ω in diskCenterHit γ G θ e,
+    (∫⁻ x in K, ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω x))^m ∂P) ≤ _
+  rw [lintegral_mass_pow_eq (P.restrict (diskCenterHit γ G θ e)) (volume.restrict K)
+    (fun ω x => ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω x)) hW m]
+  have hk : Measurable (fun x : Fin m → ℂ =>
+      cappedDistanceKernel (codimension κ) b (Fin.cons 0 x)) := by
+    apply (measurable_cappedDistanceKernel _ _).comp
+    apply Measurable.of_eval
+    intro i
+    exact Fin.cases measurable_const (fun j => measurable_pi_apply j) i
+  have hbnd : (∫⁻ x : Fin m → ℂ, ∫⁻ ω in diskCenterHit γ G θ e,
+      ∏ i, ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω (x i)) ∂P
+      ∂Measure.pi (fun _ : Fin m => volume.restrict K)) ≤
+      ENNReal.ofReal (multipointConstant κ (m+1)*e^codimension κ)*
+        ∫⁻ x : Fin m → ℂ, cappedDistanceKernel (codimension κ) b (Fin.cons 0 x)
+          ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
+    rw [← lintegral_const_mul _ hk]
+    apply lintegral_mono_ae
+    filter_upwards [ae_tuple_mem (ι:=Fin m) K hK,ae_fin_cons_injective m K 0] with x hx hi
+    have h := diskStoppedGreen_centered_product_bound hB hm hBm hc h0 hκ hκ8
+      γ G F hw hgrowth θ hθ m x (fun i => hKd (hx i)) hi b hb hb1
+      (fun i => hbK _ (hx i)) e he u
+    rw [lintegral_indicator hE] at h
+    simpa only [ENNReal.ofReal_prod_of_nonneg
+      (fun i _ => diskStoppedGreen_nonneg hw u _ (x i))] using h
+  apply hbnd.trans_eq
+  rw [ENNReal.ofReal_mul (multipointConstant_pos _ _).le]
+  unfold centeredLevelMomentBound
+  ring
+
+theorem uniform_diskLevelMass_centered_moments
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (K : Set ℂ) (hK : IsCompact K) (hKd : K ⊆ ball 0 1)
+    (b : ℝ) (hb : 0<b) (hb1 : b≤1) (hbK : ∀ x∈K, b≤1-‖x‖^2) (m : ℕ) :
+    ∃ M : ℝ≥0∞, M<∞ ∧ ∀ θ∈Ioo (0:ℝ) Real.pi, ∀ e : ℝ, 0<e → e≤1 →
+      ∀ u : ℝ, 0≤u →
+        (∫⁻ ω in diskCenterHit γ G θ e, diskLevelMass κ θ B γ G u K ω ^ m ∂P) ≤
+          M*ENNReal.ofReal (e^codimension κ) := by
+  obtain ⟨D,hD,hbound⟩ := cappedDistanceKernel_fixed_center_bound m
+    (codimension_pos κ hκ8) (by unfold codimension dimension; linarith) b K hK.isBounded
+  refine ⟨ENNReal.ofReal (multipointConstant κ (m+1))*D,
+    ENNReal.mul_lt_top ENNReal.ofReal_lt_top hD,?_⟩
+  intro θ hθ e he he1 u hu
+  apply (diskLevelMass_centered_moment_bound hB hm hBm hc h0 hκ hκ8
+    γ G F hw hgrowth K hK.measurableSet hKd b hb hb1 hbK θ hθ m e he he1 u hu).trans
+  unfold centeredLevelMomentBound
+  calc
+    _ ≤ ENNReal.ofReal (e^codimension κ)*(ENNReal.ofReal (multipointConstant κ (m+1))*D) := by
+      gcongr
+      exact hbound 0
+    _ = _ := mul_comm _ _
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/EarlyPassageLocalization.lean b/lean/OAI/Probability/SLE/Gauge/EarlyPassageLocalization.lean
index 2e98e030c6db1f79a173f9f206e6283e0f263f82..2e41980e1efeabe3daa38ad901677b2861bcbc97 100644
--- a/lean/OAI/Probability/SLE/Gauge/EarlyPassageLocalization.lean
+++ b/lean/OAI/Probability/SLE/Gauge/EarlyPassageLocalization.lean
@@ -1,274 +1,274 @@
-import OAI.Probability.SLE.Geometry.FirstHeightBounded
-import OAI.Probability.SLE.Geometry.RandomBoundaryTubes
-import OAI.Probability.SLE.StripProbabilities
-
-namespace OAI
-
-/-! Early physical containment of the narrow first-height past. The event
-is measurable in the stopped past, and failure is an actual conformal
-localization escape event. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.EarlyPassageLocalization
-
-local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-local instance : LocallyCompactSpace closedHalfPlane :=
-  isClosed_closedHalfPlane.locallyCompactSpace
-
-open FirstHeightSupport
-
-/-- Physical containment of the entire trace segment through a finite stop. -/
-def stoppedImageEvent {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (σ : Ω → ℝ≥0)
-    (Fbar : ℂ → OnePoint ℂ) (a : ℂ) (r : ℝ) : Set Ω :=
-  {ω | ∀ t ∈ Icc 0 (σ ω), Fbar (γ ω t) ∈
-    (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r}
-
-theorem stoppedImageEvent_measurable
-    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
-    (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t))
-    (hc : ∀ ω, Continuous (γ ω)) (hH : ∀ ω t, 0 ≤ (γ ω t).im)
-    (σ : Ω → ℝ≥0) (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hT : ∀ ω, σ ω ≤ T)
-    (Fbar : ℂ → OnePoint ℂ) (hFbar : ContinuousOn Fbar closedHalfPlane)
-    (a : ℂ) (r : ℝ) :
-    MeasurableSet[hσ.measurableSpace] (stoppedImageEvent γ σ Fbar a r) := by
-  let f : C(closedHalfPlane, OnePoint ℂ) := ⟨fun z => Fbar z, hFbar.domRestrict⟩
-  let p := stoppedTracePath γ hc σ
-  have hp : Measurable[hσ.measurableSpace] p :=
-    stoppedTracePath_measurable ℱ γ ha hc σ hσ
-  have hpm : Measurable[hσ.measurableSpace] (fun ω => f.comp (liftClosedPath (p ω))) := by
-    exact (ContinuousMap.continuous_postcomp f).measurable.comp
-      (measurable_liftClosedPath.comp hp)
-  have hO : IsOpen ((fun z : ℂ => (z : OnePoint ℂ)) '' ball a r) :=
-    OnePoint.isOpenEmbedding_coe.isOpenMap _ isOpen_ball
-  have he : stoppedImageEvent γ σ Fbar a r =
-      {ω | MapsTo (f.comp (liftClosedPath (p ω))) (Icc 0 T)
-        ((fun z : ℂ => (z : OnePoint ℂ)) '' ball a r)} := by
-    ext ω
-    have hpc : ClosedPathProperty (p ω) := fun t => hH ω (min t (σ ω))
-    have heval (t : ℝ≥0) : f.comp (liftClosedPath (p ω)) t =
-        Fbar (γ ω (min t (σ ω))) := by
-      change Fbar ((liftClosedPath (p ω) t : closedHalfPlane) : ℂ) = _
-      rw [liftClosedPath_apply _ hpc]
-      rfl
-    constructor
-    · intro h t ht
-      rw [heval]
-      exact h _ ⟨bot_le, min_le_right _ _⟩
-    · intro h t ht
-      have hh := h ⟨bot_le, ht.2.trans (hT ω)⟩
-      simpa only [heval, min_eq_left ht.2] using hh
-  rw [he]
-  exact (ContinuousMap.isOpen_setOfPred_mapsTo isCompact_Icc hO).measurableSet.preimage hpm
-
-/-- Containment failure before a bounded parameter-radius stop is exactly
-an escape covered by conformal localization. -/
-theorem stoppedImage_failure_subset_escape
-    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (σ : Ω → ℝ≥0)
-    (E : Set Ω) (Fbar : ℂ → OnePoint ℂ) (a : ℂ) (r A : ℝ)
-    (hbound : ∀ ω ∈ E, ∀ t ∈ Icc 0 (σ ω), ‖γ ω t‖ < A) :
-    E \ stoppedImageEvent γ σ Fbar a r ⊆
-      {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) A t ∧
-        Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r} := by
-  rintro ω ⟨hE, hbad⟩
-  change ¬ ∀ t ∈ Icc 0 (σ ω), Fbar (γ ω t) ∈
-    (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r at hbad
-  push Not at hbad
-  obtain ⟨t, ht, hbad⟩ := hbad
-  refine ⟨t, ?_, hbad⟩
-  intro s hs
-  exact hbound ω hE s ⟨bot_le, hs.le.trans ht.2⟩
-
-/-- A bounded first-height certificate stays strictly inside the ball of
-half the rectangle scale. -/
-theorem bounded_firstHeight_norm
-    (p : C(ℝ≥0, ℂ)) (hp0 : p 0 = 0) (hpH : ∀ t, 0 ≤ (p t).im)
-    {B : ℝ} (hB : 0 < B) (T : ℝ≥0)
-    (hcert : let σ := boundedFirstHeightTime p B T
-      0 < σ ∧ (p σ).im = B / 4 ∧
-        MapsTo (fun t => p (min t σ)) (Icc 0 T) (narrowStrip B)) :
-    ∀ t ∈ Icc 0 (boundedFirstHeightTime p B T), ‖p t‖ < B / 2 := by
-  obtain ⟨σ, htime, _, _, _, _, hpast⟩ :=
-    bounded_certificate_gives_first_height p hp0 hpH hB T hcert
-  have hσ : σ = boundedFirstHeightTime p B T := by
-    have hle : σ ≤ T := by
-      by_contra hn
-      have hσmin := boundedFirstHeightTime_le p B T
-      obtain ⟨_, hhit, _⟩ := hcert
-      have hfirstle : firstHeightTime p (B / 4) ≤
-          (boundedFirstHeightTime p B T : WithTop ℝ≥0) := by
-        apply hittingAfter_le_of_mem (show (0 : ℝ≥0) ≤ boundedFirstHeightTime p B T from bot_le)
-        change (p (boundedFirstHeightTime p B T)).im - B / 4 = 0
-        rw [hhit, sub_self]
-      rw [htime, WithTop.coe_le_coe] at hfirstle
-      exact (not_le_of_gt (lt_of_not_ge hn)) (hfirstle.trans hσmin)
-    apply WithTop.coe_injective
-    rw [boundedFirstHeightTime_coe, htime, min_eq_left (WithTop.coe_le_coe.mpr hle)]
-  intro t ht
-  rw [← hσ] at ht
-  exact mem_ball_zero_iff.mp
-    (GuideSplice.closedInitialRectangle_subset_smallBall hB (hpast (mem_image_of_mem p ht)))
-
-/-- Conformal localization controls a stopped containment failure whenever
-the past remains below its parameter-radius threshold. -/
-theorem unit_guide_localization
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∃ b C c L₀ : ℝ, 0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L₀ ∧
-    ∀ (Ω : Type*) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
-      DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane →
-      ContinuousOn Fbar closedHalfPlane →
-      (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) →
-    ∀ (a : ℂ) (r : ℝ), 0 < r →
-    ∀ (η : ℝ → ℂ) (S : ℝ), 0 < S → ContinuousOn η (Icc 0 S) →
-      η 0 = 0 → ‖η S‖ = 1 → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) →
-      (∀ t ∈ Icc 0 S, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-a‖ ≤ r) →
-    ∀ (L : ℝ), L₀ ≤ L → ∀ (σ : Ω → ℝ≥0) (E : Set Ω),
-      (∀ ω ∈ E, ∀ t ∈ Icc 0 (σ ω), ‖γ ω t‖ < L ^ b) →
-      P (E \ stoppedImageEvent γ σ Fbar a (L * r)) ≤ ENNReal.ofReal (C * L ^ (-c)) := by
-  obtain ⟨b, C, c, L₀, hb, hC, hc, hL₀, hloc⟩ :=
-    sourceUniformConformalLocalization κ hκ hκ8
-  refine ⟨b, C, c, L₀, hb, hC, hc, hL₀, ?_⟩
-  intro Ω m P hP γ hSLE F Fbar hF hinj hbar hext a r hr η S hS hη hη0 hηS hH him L hL σ E hbound
-  exact (measure_mono (stoppedImage_failure_subset_escape γ σ E Fbar a (L * r) (L ^ b) hbound)).trans
-    (hloc Ω m P hP γ hSLE F Fbar hF hinj hbar hext a r hr η S hS hη hη0 hηS hH him L hL)
-
-/-- Actual capacity scaling transports an escape event, including its
-strict before-first-modulus condition. -/
-theorem escape_under_scaling
-    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (Fbar : ℂ → OnePoint ℂ)
-    (c : ℝ≥0) (hc : 0 < c) (A r : ℝ) (a : ℂ) :
-    {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) A t ∧
-      Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r} ⊆
-    {ω | ∃ t : ℝ≥0, BeforeFirstModulus (sourceScaleTrace c (γ ω)) ((c : ℝ) * A) t ∧
-      Fbar (sourceScaleTrace c (γ ω) t / (c : ℂ)) ∉
-        (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r} := by
-  have hc2 : 0 < c ^ 2 := pow_pos hc 2
-  have hcC : (c : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (ne_of_gt hc)
-  rintro ω ⟨t, ht, hbad⟩
-  refine ⟨t * c ^ 2, ?_, ?_⟩
-  · intro u hu
-    have hu' : sourceScaleTime c u < t := (div_lt_iff₀ hc2).mpr hu
-    have hn : ‖sourceScaleTrace c (γ ω) u‖ = (c : ℝ) * ‖γ ω (sourceScaleTime c u)‖ := by
-      simp only [sourceScaleTrace, norm_mul, Complex.norm_real, Real.norm_eq_abs,
-        abs_of_nonneg c.coe_nonneg]
-    rw [hn]
-    exact mul_lt_mul_of_pos_left (ht _ hu') hc
-  · simpa only [sourceScaleTrace, sourceScaleTime,
-      mul_div_cancel_right₀ _ (ne_of_gt hc2), mul_div_cancel_left₀ _ hcC] using hbad
-
-/-- The localization estimate with a guide ending at an arbitrary positive
-parameter radius, derived by capacity scaling of ordinary SLE. -/
-theorem localization_at_guide_radius
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∃ b C c L₀ : ℝ, 0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L₀ ∧
-    ∀ (Ω : Type*) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
-      DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane →
-      ContinuousOn Fbar closedHalfPlane →
-      (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) →
-    ∀ (a : ℂ) (r : ℝ), 0 < r →
-    ∀ (η : ℝ → ℂ) (S b₀ : ℝ), 0 < S → 0 < b₀ → ContinuousOn η (Icc 0 S) →
-      η 0 = 0 → ‖η S‖ = b₀ → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) →
-      (∀ t ∈ Icc 0 S, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-a‖ ≤ r) →
-    ∀ (L : ℝ), L₀ ≤ L →
-      P {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) (b₀ * L ^ b) t ∧
-        Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a (L * r)} ≤
-      ENNReal.ofReal (C * L ^ (-c)) := by
-  obtain ⟨b, C, c, L₀, hb, hC, hc, hL₀, hloc⟩ :=
-    sourceUniformConformalLocalization κ hκ hκ8
-  refine ⟨b, C, c, L₀, hb, hC, hc, hL₀, ?_⟩
-  intro Ω m P hP γ hSLE F Fbar hF hinj hbar hext a r hr η S b₀ hS hb₀ hη hη0 hηS hH him L hL
-  let d : ℝ≥0 := ⟨b₀, hb₀.le⟩
-  let k := d⁻¹
-  have hk : 0 < k := inv_pos.mpr hb₀
-  have hkR : 0 < (k : ℝ) := hk
-  have hkC : (k : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hkR.ne'
-  have hkd : (k : ℝ) * b₀ = 1 := by
-    change b₀⁻¹ * b₀ = 1
-    exact inv_mul_cancel₀ hb₀.ne'
-  let Fs := fun z : ℂ => F (z / (k : ℂ))
-  let Fbars := fun z : ℂ => Fbar (z / (k : ℂ))
-  let ηs := fun t : ℝ => (k : ℂ) * η t
-  have hmap (z : ℂ) (hz : z ∈ upperHalfPlane) : z / (k : ℂ) ∈ upperHalfPlane := by
-    change 0 < (z / (k : ℂ)).im
-    rw [Complex.div_ofReal_im]
-    exact div_pos hz hkR
-  have hFs : DifferentiableOn ℂ Fs upperHalfPlane :=
-    hF.comp (differentiable_id.div_const (k : ℂ)).differentiableOn hmap
-  have hFsi : InjOn Fs upperHalfPlane := by
-    intro z hz w hw he
-    have hh := hinj (hmap z hz) (hmap w hw) he
-    exact (div_left_inj' hkC).mp hh
-  have hbars : ContinuousOn Fbars closedHalfPlane :=
-    hbar.comp (continuous_id.div_const (k : ℂ)).continuousOn (by
-      intro z hz
-      change 0 ≤ (z / (k : ℂ)).im
-      rw [Complex.div_ofReal_im]
-      exact div_nonneg hz hkR.le)
-  have hexts (z : ℂ) (hz : z ∈ upperHalfPlane) : Fbars z = (Fs z : OnePoint ℂ) :=
-    hext _ (hmap z hz)
-  have hηsc : ContinuousOn ηs (Icc 0 S) := continuousOn_const.mul hη
-  have hηs0 : ηs 0 = 0 := by simp only [ηs, hη0, mul_zero]
-  have hηsS : ‖ηs S‖ = 1 := by
-    simpa only [ηs, norm_mul, Complex.norm_real, Real.norm_eq_abs,
-      abs_of_nonneg k.coe_nonneg, hηS] using hkd
-  have hηsH (t : ℝ) (ht : t ∈ Icc 0 S) : 0 ≤ (ηs t).im := by
-    change 0 ≤ ((k : ℂ) * η t).im
-    simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero]
-    exact mul_nonneg hkR.le (hH t ht)
-  have hims (t : ℝ) (ht : t ∈ Icc 0 S) :
-      ∃ w : ℂ, Fbars (ηs t) = (w : OnePoint ℂ) ∧ ‖w - a‖ ≤ r := by
-    simpa only [Fbars, ηs, mul_div_cancel_left₀ _ hkC] using him t ht
-  have hscaled := sourceCapacityScaling.2 κ hκ hκ8 γ hSLE k hk
-  have hbound := hloc Ω m P hP (fun ω => sourceScaleTrace k (γ ω)) hscaled
-    Fs Fbars hFs hFsi hbars hexts a r hr ηs S hS hηsc hηs0 hηsS hηsH hims L hL
-  have hsub := escape_under_scaling γ Fbar k hk (b₀ * L ^ b) (L * r) a
-  have hfactor : (k : ℝ) * (b₀ * L ^ b) = L ^ b := by rw [← mul_assoc, hkd, one_mul]
-  rw [hfactor] at hsub
-  exact (measure_mono hsub).trans hbound
-
-/-- The precise early-passage estimate: the bounded narrow first-height
-past fails physical containment with probability at most the localization
-power tail. -/
-theorem bounded_firstHeight_localization
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∃ b C c L₀ : ℝ, 0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L₀ ∧
-    ∀ (Ω : Type*) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-      (∀ ω, Continuous (γ ω)) → (∀ ω, γ ω 0 = 0) → (∀ ω t, 0 ≤ (γ ω t).im) →
-    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
-      DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane →
-      ContinuousOn Fbar closedHalfPlane →
-      (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) →
-    ∀ (a : ℂ) (r : ℝ), 0 < r →
-    ∀ (η : ℝ → ℂ) (S b₀ : ℝ), 0 < S → 0 < b₀ → ContinuousOn η (Icc 0 S) →
-      η 0 = 0 → ‖η S‖ = b₀ → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) →
-      (∀ t ∈ Icc 0 S, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-a‖ ≤ r) →
-    ∀ (L : ℝ), L₀ ≤ L → ∀ (B : ℝ), 0 < B → B / 2 ≤ b₀ * L ^ b → ∀ T : ℝ≥0,
-      let σ := fun ω => boundedFirstHeightTime (γ ω) B T
-      P (stoppedNarrowCertificate γ σ B T \ stoppedImageEvent γ σ Fbar a (L * r)) ≤
-        ENNReal.ofReal (C * L ^ (-c)) := by
-  obtain ⟨b, C, c, L₀, hb, hC, hc, hL₀, hloc⟩ := localization_at_guide_radius κ hκ hκ8
-  refine ⟨b, C, c, L₀, hb, hC, hc, hL₀, ?_⟩
-  intro Ω m P hP γ hSLE hγc hγ0 hγH F Fbar hF hinj hbar hext a r hr η S b₀ hS hb₀ hη hη0 hηS hH him L hL B hB hBL T
-  let σ := fun ω => boundedFirstHeightTime (γ ω) B T
-  have hnorm : ∀ ω ∈ stoppedNarrowCertificate γ σ B T,
-      ∀ t ∈ Icc 0 (σ ω), ‖γ ω t‖ < b₀ * L ^ b := by
-    intro ω hω t ht
-    exact (bounded_firstHeight_norm ⟨γ ω, hγc ω⟩ (hγ0 ω) (hγH ω) hB T hω t ht).trans_le hBL
-  exact (measure_mono (stoppedImage_failure_subset_escape γ σ
-    (stoppedNarrowCertificate γ σ B T) Fbar a (L * r) (b₀ * L ^ b) hnorm)).trans
-      (hloc Ω P γ hSLE F Fbar hF hinj hbar hext a r hr η S b₀ hS hb₀ hη hη0 hηS hH him L hL)
-
-end SLEExactGauge.EarlyPassageLocalization
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.FirstHeightBounded
+import OAI.Probability.SLE.Geometry.RandomBoundaryTubes
+import OAI.Probability.SLE.StripProbabilities
+
+namespace OAI
+
+/-! Early physical containment of the narrow first-height past. The event
+is measurable in the stopped past, and failure is an actual conformal
+localization escape event. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.EarlyPassageLocalization
+
+local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+local instance : LocallyCompactSpace closedHalfPlane :=
+  isClosed_closedHalfPlane.locallyCompactSpace
+
+open FirstHeightSupport
+
+/-- Physical containment of the entire trace segment through a finite stop. -/
+def stoppedImageEvent {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (σ : Ω → ℝ≥0)
+    (Fbar : ℂ → OnePoint ℂ) (a : ℂ) (r : ℝ) : Set Ω :=
+  {ω | ∀ t ∈ Icc 0 (σ ω), Fbar (γ ω t) ∈
+    (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r}
+
+theorem stoppedImageEvent_measurable
+    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
+    (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t))
+    (hc : ∀ ω, Continuous (γ ω)) (hH : ∀ ω t, 0 ≤ (γ ω t).im)
+    (σ : Ω → ℝ≥0) (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hT : ∀ ω, σ ω ≤ T)
+    (Fbar : ℂ → OnePoint ℂ) (hFbar : ContinuousOn Fbar closedHalfPlane)
+    (a : ℂ) (r : ℝ) :
+    MeasurableSet[hσ.measurableSpace] (stoppedImageEvent γ σ Fbar a r) := by
+  let f : C(closedHalfPlane, OnePoint ℂ) := ⟨fun z => Fbar z, hFbar.domRestrict⟩
+  let p := stoppedTracePath γ hc σ
+  have hp : Measurable[hσ.measurableSpace] p :=
+    stoppedTracePath_measurable ℱ γ ha hc σ hσ
+  have hpm : Measurable[hσ.measurableSpace] (fun ω => f.comp (liftClosedPath (p ω))) := by
+    exact (ContinuousMap.continuous_postcomp f).measurable.comp
+      (measurable_liftClosedPath.comp hp)
+  have hO : IsOpen ((fun z : ℂ => (z : OnePoint ℂ)) '' ball a r) :=
+    OnePoint.isOpenEmbedding_coe.isOpenMap _ isOpen_ball
+  have he : stoppedImageEvent γ σ Fbar a r =
+      {ω | MapsTo (f.comp (liftClosedPath (p ω))) (Icc 0 T)
+        ((fun z : ℂ => (z : OnePoint ℂ)) '' ball a r)} := by
+    ext ω
+    have hpc : ClosedPathProperty (p ω) := fun t => hH ω (min t (σ ω))
+    have heval (t : ℝ≥0) : f.comp (liftClosedPath (p ω)) t =
+        Fbar (γ ω (min t (σ ω))) := by
+      change Fbar ((liftClosedPath (p ω) t : closedHalfPlane) : ℂ) = _
+      rw [liftClosedPath_apply _ hpc]
+      rfl
+    constructor
+    · intro h t ht
+      rw [heval]
+      exact h _ ⟨bot_le, min_le_right _ _⟩
+    · intro h t ht
+      have hh := h ⟨bot_le, ht.2.trans (hT ω)⟩
+      simpa only [heval, min_eq_left ht.2] using hh
+  rw [he]
+  exact (ContinuousMap.isOpen_setOfPred_mapsTo isCompact_Icc hO).measurableSet.preimage hpm
+
+/-- Containment failure before a bounded parameter-radius stop is exactly
+an escape covered by conformal localization. -/
+theorem stoppedImage_failure_subset_escape
+    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (σ : Ω → ℝ≥0)
+    (E : Set Ω) (Fbar : ℂ → OnePoint ℂ) (a : ℂ) (r A : ℝ)
+    (hbound : ∀ ω ∈ E, ∀ t ∈ Icc 0 (σ ω), ‖γ ω t‖ < A) :
+    E \ stoppedImageEvent γ σ Fbar a r ⊆
+      {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) A t ∧
+        Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r} := by
+  rintro ω ⟨hE, hbad⟩
+  change ¬ ∀ t ∈ Icc 0 (σ ω), Fbar (γ ω t) ∈
+    (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r at hbad
+  push Not at hbad
+  obtain ⟨t, ht, hbad⟩ := hbad
+  refine ⟨t, ?_, hbad⟩
+  intro s hs
+  exact hbound ω hE s ⟨bot_le, hs.le.trans ht.2⟩
+
+/-- A bounded first-height certificate stays strictly inside the ball of
+half the rectangle scale. -/
+theorem bounded_firstHeight_norm
+    (p : C(ℝ≥0, ℂ)) (hp0 : p 0 = 0) (hpH : ∀ t, 0 ≤ (p t).im)
+    {B : ℝ} (hB : 0 < B) (T : ℝ≥0)
+    (hcert : let σ := boundedFirstHeightTime p B T
+      0 < σ ∧ (p σ).im = B / 4 ∧
+        MapsTo (fun t => p (min t σ)) (Icc 0 T) (narrowStrip B)) :
+    ∀ t ∈ Icc 0 (boundedFirstHeightTime p B T), ‖p t‖ < B / 2 := by
+  obtain ⟨σ, htime, _, _, _, _, hpast⟩ :=
+    bounded_certificate_gives_first_height p hp0 hpH hB T hcert
+  have hσ : σ = boundedFirstHeightTime p B T := by
+    have hle : σ ≤ T := by
+      by_contra hn
+      have hσmin := boundedFirstHeightTime_le p B T
+      obtain ⟨_, hhit, _⟩ := hcert
+      have hfirstle : firstHeightTime p (B / 4) ≤
+          (boundedFirstHeightTime p B T : WithTop ℝ≥0) := by
+        apply hittingAfter_le_of_mem (show (0 : ℝ≥0) ≤ boundedFirstHeightTime p B T from bot_le)
+        change (p (boundedFirstHeightTime p B T)).im - B / 4 = 0
+        rw [hhit, sub_self]
+      rw [htime, WithTop.coe_le_coe] at hfirstle
+      exact (not_le_of_gt (lt_of_not_ge hn)) (hfirstle.trans hσmin)
+    apply WithTop.coe_injective
+    rw [boundedFirstHeightTime_coe, htime, min_eq_left (WithTop.coe_le_coe.mpr hle)]
+  intro t ht
+  rw [← hσ] at ht
+  exact mem_ball_zero_iff.mp
+    (GuideSplice.closedInitialRectangle_subset_smallBall hB (hpast (mem_image_of_mem p ht)))
+
+/-- Conformal localization controls a stopped containment failure whenever
+the past remains below its parameter-radius threshold. -/
+theorem unit_guide_localization
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∃ b C c L₀ : ℝ, 0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L₀ ∧
+    ∀ (Ω : Type*) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
+      DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane →
+      ContinuousOn Fbar closedHalfPlane →
+      (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) →
+    ∀ (a : ℂ) (r : ℝ), 0 < r →
+    ∀ (η : ℝ → ℂ) (S : ℝ), 0 < S → ContinuousOn η (Icc 0 S) →
+      η 0 = 0 → ‖η S‖ = 1 → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) →
+      (∀ t ∈ Icc 0 S, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-a‖ ≤ r) →
+    ∀ (L : ℝ), L₀ ≤ L → ∀ (σ : Ω → ℝ≥0) (E : Set Ω),
+      (∀ ω ∈ E, ∀ t ∈ Icc 0 (σ ω), ‖γ ω t‖ < L ^ b) →
+      P (E \ stoppedImageEvent γ σ Fbar a (L * r)) ≤ ENNReal.ofReal (C * L ^ (-c)) := by
+  obtain ⟨b, C, c, L₀, hb, hC, hc, hL₀, hloc⟩ :=
+    sourceUniformConformalLocalization κ hκ hκ8
+  refine ⟨b, C, c, L₀, hb, hC, hc, hL₀, ?_⟩
+  intro Ω m P hP γ hSLE F Fbar hF hinj hbar hext a r hr η S hS hη hη0 hηS hH him L hL σ E hbound
+  exact (measure_mono (stoppedImage_failure_subset_escape γ σ E Fbar a (L * r) (L ^ b) hbound)).trans
+    (hloc Ω m P hP γ hSLE F Fbar hF hinj hbar hext a r hr η S hS hη hη0 hηS hH him L hL)
+
+/-- Actual capacity scaling transports an escape event, including its
+strict before-first-modulus condition. -/
+theorem escape_under_scaling
+    {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) (Fbar : ℂ → OnePoint ℂ)
+    (c : ℝ≥0) (hc : 0 < c) (A r : ℝ) (a : ℂ) :
+    {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) A t ∧
+      Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r} ⊆
+    {ω | ∃ t : ℝ≥0, BeforeFirstModulus (sourceScaleTrace c (γ ω)) ((c : ℝ) * A) t ∧
+      Fbar (sourceScaleTrace c (γ ω) t / (c : ℂ)) ∉
+        (fun z : ℂ => (z : OnePoint ℂ)) '' ball a r} := by
+  have hc2 : 0 < c ^ 2 := pow_pos hc 2
+  have hcC : (c : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (ne_of_gt hc)
+  rintro ω ⟨t, ht, hbad⟩
+  refine ⟨t * c ^ 2, ?_, ?_⟩
+  · intro u hu
+    have hu' : sourceScaleTime c u < t := (div_lt_iff₀ hc2).mpr hu
+    have hn : ‖sourceScaleTrace c (γ ω) u‖ = (c : ℝ) * ‖γ ω (sourceScaleTime c u)‖ := by
+      simp only [sourceScaleTrace, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+        abs_of_nonneg c.coe_nonneg]
+    rw [hn]
+    exact mul_lt_mul_of_pos_left (ht _ hu') hc
+  · simpa only [sourceScaleTrace, sourceScaleTime,
+      mul_div_cancel_right₀ _ (ne_of_gt hc2), mul_div_cancel_left₀ _ hcC] using hbad
+
+/-- The localization estimate with a guide ending at an arbitrary positive
+parameter radius, derived by capacity scaling of ordinary SLE. -/
+theorem localization_at_guide_radius
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∃ b C c L₀ : ℝ, 0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L₀ ∧
+    ∀ (Ω : Type*) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
+      DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane →
+      ContinuousOn Fbar closedHalfPlane →
+      (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) →
+    ∀ (a : ℂ) (r : ℝ), 0 < r →
+    ∀ (η : ℝ → ℂ) (S b₀ : ℝ), 0 < S → 0 < b₀ → ContinuousOn η (Icc 0 S) →
+      η 0 = 0 → ‖η S‖ = b₀ → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) →
+      (∀ t ∈ Icc 0 S, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-a‖ ≤ r) →
+    ∀ (L : ℝ), L₀ ≤ L →
+      P {ω | ∃ t : ℝ≥0, BeforeFirstModulus (γ ω) (b₀ * L ^ b) t ∧
+        Fbar (γ ω t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' ball a (L * r)} ≤
+      ENNReal.ofReal (C * L ^ (-c)) := by
+  obtain ⟨b, C, c, L₀, hb, hC, hc, hL₀, hloc⟩ :=
+    sourceUniformConformalLocalization κ hκ hκ8
+  refine ⟨b, C, c, L₀, hb, hC, hc, hL₀, ?_⟩
+  intro Ω m P hP γ hSLE F Fbar hF hinj hbar hext a r hr η S b₀ hS hb₀ hη hη0 hηS hH him L hL
+  let d : ℝ≥0 := ⟨b₀, hb₀.le⟩
+  let k := d⁻¹
+  have hk : 0 < k := inv_pos.mpr hb₀
+  have hkR : 0 < (k : ℝ) := hk
+  have hkC : (k : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hkR.ne'
+  have hkd : (k : ℝ) * b₀ = 1 := by
+    change b₀⁻¹ * b₀ = 1
+    exact inv_mul_cancel₀ hb₀.ne'
+  let Fs := fun z : ℂ => F (z / (k : ℂ))
+  let Fbars := fun z : ℂ => Fbar (z / (k : ℂ))
+  let ηs := fun t : ℝ => (k : ℂ) * η t
+  have hmap (z : ℂ) (hz : z ∈ upperHalfPlane) : z / (k : ℂ) ∈ upperHalfPlane := by
+    change 0 < (z / (k : ℂ)).im
+    rw [Complex.div_ofReal_im]
+    exact div_pos hz hkR
+  have hFs : DifferentiableOn ℂ Fs upperHalfPlane :=
+    hF.comp (differentiable_id.div_const (k : ℂ)).differentiableOn hmap
+  have hFsi : InjOn Fs upperHalfPlane := by
+    intro z hz w hw he
+    have hh := hinj (hmap z hz) (hmap w hw) he
+    exact (div_left_inj' hkC).mp hh
+  have hbars : ContinuousOn Fbars closedHalfPlane :=
+    hbar.comp (continuous_id.div_const (k : ℂ)).continuousOn (by
+      intro z hz
+      change 0 ≤ (z / (k : ℂ)).im
+      rw [Complex.div_ofReal_im]
+      exact div_nonneg hz hkR.le)
+  have hexts (z : ℂ) (hz : z ∈ upperHalfPlane) : Fbars z = (Fs z : OnePoint ℂ) :=
+    hext _ (hmap z hz)
+  have hηsc : ContinuousOn ηs (Icc 0 S) := continuousOn_const.mul hη
+  have hηs0 : ηs 0 = 0 := by simp only [ηs, hη0, mul_zero]
+  have hηsS : ‖ηs S‖ = 1 := by
+    simpa only [ηs, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+      abs_of_nonneg k.coe_nonneg, hηS] using hkd
+  have hηsH (t : ℝ) (ht : t ∈ Icc 0 S) : 0 ≤ (ηs t).im := by
+    change 0 ≤ ((k : ℂ) * η t).im
+    simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero]
+    exact mul_nonneg hkR.le (hH t ht)
+  have hims (t : ℝ) (ht : t ∈ Icc 0 S) :
+      ∃ w : ℂ, Fbars (ηs t) = (w : OnePoint ℂ) ∧ ‖w - a‖ ≤ r := by
+    simpa only [Fbars, ηs, mul_div_cancel_left₀ _ hkC] using him t ht
+  have hscaled := sourceCapacityScaling.2 κ hκ hκ8 γ hSLE k hk
+  have hbound := hloc Ω m P hP (fun ω => sourceScaleTrace k (γ ω)) hscaled
+    Fs Fbars hFs hFsi hbars hexts a r hr ηs S hS hηsc hηs0 hηsS hηsH hims L hL
+  have hsub := escape_under_scaling γ Fbar k hk (b₀ * L ^ b) (L * r) a
+  have hfactor : (k : ℝ) * (b₀ * L ^ b) = L ^ b := by rw [← mul_assoc, hkd, one_mul]
+  rw [hfactor] at hsub
+  exact (measure_mono hsub).trans hbound
+
+/-- The precise early-passage estimate: the bounded narrow first-height
+past fails physical containment with probability at most the localization
+power tail. -/
+theorem bounded_firstHeight_localization
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∃ b C c L₀ : ℝ, 0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L₀ ∧
+    ∀ (Ω : Type*) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+      (∀ ω, Continuous (γ ω)) → (∀ ω, γ ω 0 = 0) → (∀ ω t, 0 ≤ (γ ω t).im) →
+    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
+      DifferentiableOn ℂ F upperHalfPlane → InjOn F upperHalfPlane →
+      ContinuousOn Fbar closedHalfPlane →
+      (∀ z ∈ upperHalfPlane, Fbar z = (F z : OnePoint ℂ)) →
+    ∀ (a : ℂ) (r : ℝ), 0 < r →
+    ∀ (η : ℝ → ℂ) (S b₀ : ℝ), 0 < S → 0 < b₀ → ContinuousOn η (Icc 0 S) →
+      η 0 = 0 → ‖η S‖ = b₀ → (∀ t ∈ Icc 0 S, 0 ≤ (η t).im) →
+      (∀ t ∈ Icc 0 S, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-a‖ ≤ r) →
+    ∀ (L : ℝ), L₀ ≤ L → ∀ (B : ℝ), 0 < B → B / 2 ≤ b₀ * L ^ b → ∀ T : ℝ≥0,
+      let σ := fun ω => boundedFirstHeightTime (γ ω) B T
+      P (stoppedNarrowCertificate γ σ B T \ stoppedImageEvent γ σ Fbar a (L * r)) ≤
+        ENNReal.ofReal (C * L ^ (-c)) := by
+  obtain ⟨b, C, c, L₀, hb, hC, hc, hL₀, hloc⟩ := localization_at_guide_radius κ hκ hκ8
+  refine ⟨b, C, c, L₀, hb, hC, hc, hL₀, ?_⟩
+  intro Ω m P hP γ hSLE hγc hγ0 hγH F Fbar hF hinj hbar hext a r hr η S b₀ hS hb₀ hη hη0 hηS hH him L hL B hB hBL T
+  let σ := fun ω => boundedFirstHeightTime (γ ω) B T
+  have hnorm : ∀ ω ∈ stoppedNarrowCertificate γ σ B T,
+      ∀ t ∈ Icc 0 (σ ω), ‖γ ω t‖ < b₀ * L ^ b := by
+    intro ω hω t ht
+    exact (bounded_firstHeight_norm ⟨γ ω, hγc ω⟩ (hγ0 ω) (hγH ω) hB T hω t ht).trans_le hBL
+  exact (measure_mono (stoppedImage_failure_subset_escape γ σ
+    (stoppedNarrowCertificate γ σ B T) Fbar a (L * r) (b₀ * L ^ b) hnorm)).trans
+      (hloc Ω P γ hSLE F Fbar hF hinj hbar hext a r hr η S b₀ hS hb₀ hη hη0 hηS hH him L hL)
+
+end SLEExactGauge.EarlyPassageLocalization
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExactGaugeComparison.lean b/lean/OAI/Probability/SLE/Gauge/ExactGaugeComparison.lean
index 2b72783876bfbbef919c55cca2c6614fa9135ebc..0fd6bb165d7be346ce3ef3454d82db08db419f0a 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExactGaugeComparison.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExactGaugeComparison.lean
@@ -1,141 +1,141 @@
-import OAI.Probability.SLE.HausdorffLimits
-
-namespace OAI
-
-/-!
-The deterministic comparison behind Corollary `cor:larger-gauge` in
-*An explicit exact Hausdorff gauge for SLE* (26 September 2026).
-
-The Hausdorff measures here are the existing `Measure.mkMetric` constructions, so
-the comparison applies to arbitrary countable spatial covers. No SLE estimate is
-assumed in the comparison theorem: positive measure for the smaller gauge is the
-sole input to the non-sigma-finiteness conclusion.
--/
-
-noncomputable section
-
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-lemma logLog_inv_tendsto_atTop :
-    Tendsto (fun r : ℝ => Real.log (Real.log (1 / r))) (𝓝[>] 0) atTop := by
-  simpa only [one_div, Function.comp_def] using
-    Real.tendsto_log_atTop.comp
-      (Real.tendsto_log_atTop.comp tendsto_inv_nhdsGT_zero)
-
-/-- The regular exact gauge is asymptotically smaller than Schramm's proposed
-exponent-one log-log gauge. -/
-theorem hFormula_div_gFormula_tendsto_zero {κ : ℝ} (hκ : 0 < κ) :
-    Tendsto (fun r => hFormula κ r / gFormula κ r) (𝓝[>] 0) (𝓝 0) := by
-  have hq : 0 < 1 - codimension κ / 2 := by
-    dsimp [codimension, dimension]
-    linarith
-  have ht := (tendsto_rpow_neg_atTop hq).comp logLog_inv_tendsto_atTop
-  apply ht.congr'
-  filter_upwards [self_mem_nhdsWithin,
-    logLog_inv_tendsto_atTop.eventually (eventually_gt_atTop 0)] with r hr hl
-  have hrp : 0 < r := hr
-  have hrpow : r ^ dimension κ ≠ 0 := (Real.rpow_pos_of_pos hrp _).ne'
-  dsimp only [Function.comp_def, hFormula, gFormula]
-  rw [mul_div_mul_left _ _ hrpow, ← Real.rpow_sub_one hl.ne']
-  congr 1
-  ring
-
-theorem gauge_ratio_tendsto_zero {κ : ℝ} (hκ : 0 < κ)
-    {h g : ℝ → ℝ} (hf : HasSmallRadiusFormula h (hFormula κ))
-    (gf : HasSmallRadiusFormula g (gFormula κ)) :
-    Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0) := by
-  apply (hFormula_div_gFormula_tendsto_zero hκ).congr'
-  exact (hf.div gf).symm
-
-/-- A small-radius ratio estimate gives a comparison of the associated
-Hausdorff measures, independently of their values at large radii. -/
-theorem hausdorffGauge_le_smul_of_ratio_tendsto_zero
-    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
-    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
-    {c : ℝ} (hc : 0 < c) :
-    hausdorffGauge h ≤ ENNReal.ofReal c • hausdorffGauge g := by
-  have hsmall : ∀ᶠ r in 𝓝[>] (0 : ℝ), h r / g r < c :=
-    hratio.eventually (gt_mem_nhds hc)
-  obtain ⟨δ, hδ, hbound⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hsmall
-  have htoReal : Tendsto ENNReal.toReal (𝓝 (0 : ℝ≥0∞)) (𝓝 (0 : ℝ)) := by
-    simpa using ENNReal.tendsto_toReal ENNReal.zero_ne_top
-  have hevent : ∀ᶠ r : ℝ≥0∞ in 𝓝 (0 : ℝ≥0∞),
-      extendedGauge h r ≤ ENNReal.ofReal c * extendedGauge g r := by
-    filter_upwards [htoReal.eventually (gt_mem_nhds hδ)] with r hr
-    by_cases hz : r.toReal = 0
-    · simp [extendedGauge, hz, hh.zero, hg.zero]
-    · have hp : 0 < r.toReal := lt_of_le_of_ne ENNReal.toReal_nonneg (Ne.symm hz)
-      have hs := (div_lt_iff₀ (hg.positive _ hp)).mp (hbound ⟨hp, hr⟩)
-      dsimp only [extendedGauge]
-      rw [← ENNReal.ofReal_mul hc.le]
-      exact ENNReal.ofReal_le_ofReal hs.le
-  exact Measure.mkMetric_mono_smul ENNReal.ofReal_ne_top
-    (ENNReal.ofReal_ne_zero_iff.mpr hc) (hevent.filter_mono nhdsWithin_le_nhds)
-
-/-- Finite measure for a larger gauge forces zero measure for any gauge with
-vanishing ratio. This statement does not require the set to be measurable. -/
-theorem hausdorffGauge_eq_zero_of_larger_finite
-    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
-    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
-    {E : Set ℂ} (hfinite : hausdorffGauge g E < ∞) :
-    hausdorffGauge h E = 0 := by
-  have ht : Tendsto (fun c : ℝ => ENNReal.ofReal c * hausdorffGauge g E)
-      (𝓝[>] 0) (𝓝 0) := by
-    have hid : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) :=
-      tendsto_id.mono_left nhdsWithin_le_nhds
-    simpa using ENNReal.Tendsto.mul_const (ENNReal.tendsto_ofReal hid)
-      (Or.inr hfinite.ne)
-  apply le_antisymm _ bot_le
-  apply ge_of_tendsto ht
-  filter_upwards [self_mem_nhdsWithin] with c hc
-  exact hausdorffGauge_le_smul_of_ratio_tendsto_zero hh hg hratio hc E
-
-/-- Positive measure for the smaller gauge prevents sigma-finiteness of the
-larger Hausdorff measure restricted to that set. -/
-theorem not_sigmaFinite_larger_of_positive
-    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
-    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
-    {E : Set ℂ} (hpositive : 0 < hausdorffGauge h E) :
-    ¬ SigmaFinite ((hausdorffGauge g).restrict E) := by
-  intro hσ
-  let : SigmaFinite ((hausdorffGauge g).restrict E) := hσ
-  let A := spanningSets ((hausdorffGauge g).restrict E)
-  have hzero (n : ℕ) : hausdorffGauge h (A n ∩ E) = 0 := by
-    apply hausdorffGauge_eq_zero_of_larger_finite hh hg hratio
-    have hf := measure_spanningSets_lt_top ((hausdorffGauge g).restrict E) n
-    rwa [Measure.restrict_apply (measurableSet_spanningSets _ _)] at hf
-  have hcover : (⋃ n, A n ∩ E) = E := by
-    rw [← iUnion_inter]
-    simp only [A, iUnion_spanningSets, univ_inter]
-  have he : hausdorffGauge h E = 0 := by
-    rw [← hcover]
-    exact measure_iUnion_null hzero
-  exact hpositive.ne' he
-
-theorem hausdorffGauge_larger_eq_top_of_positive
-    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
-    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
-    {E : Set ℂ} (hpositive : 0 < hausdorffGauge h E) :
-    hausdorffGauge g E = ∞ := by
-  by_contra htop
-  exact hpositive.ne'
-    (hausdorffGauge_eq_zero_of_larger_finite hh hg hratio (lt_top_iff_ne_top.mpr htop))
-
-/-- Corollary `cor:larger-gauge`: once the explicit exact gauge has positive
-measure on a set, the exponent-one gauge is not sigma-finite there. -/
-theorem larger_logLog_gauge_not_sigmaFinite {κ : ℝ} (hκ : 0 < κ)
-    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
-    (hf : HasSmallRadiusFormula h (hFormula κ))
-    (gf : HasSmallRadiusFormula g (gFormula κ))
-    {E : Set ℂ} (hpositive : 0 < hausdorffGauge h E) :
-    ¬ SigmaFinite ((hausdorffGauge g).restrict E) :=
-  not_sigmaFinite_larger_of_positive hh hg (gauge_ratio_tendsto_zero hκ hf gf) hpositive
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.HausdorffLimits
+
+namespace OAI
+
+/-!
+The deterministic comparison behind Corollary `cor:larger-gauge` in
+*An explicit exact Hausdorff gauge for SLE* (26 September 2026).
+
+The Hausdorff measures here are the existing `Measure.mkMetric` constructions, so
+the comparison applies to arbitrary countable spatial covers. No SLE estimate is
+assumed in the comparison theorem: positive measure for the smaller gauge is the
+sole input to the non-sigma-finiteness conclusion.
+-/
+
+noncomputable section
+
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+lemma logLog_inv_tendsto_atTop :
+    Tendsto (fun r : ℝ => Real.log (Real.log (1 / r))) (𝓝[>] 0) atTop := by
+  simpa only [one_div, Function.comp_def] using
+    Real.tendsto_log_atTop.comp
+      (Real.tendsto_log_atTop.comp tendsto_inv_nhdsGT_zero)
+
+/-- The regular exact gauge is asymptotically smaller than Schramm's proposed
+exponent-one log-log gauge. -/
+theorem hFormula_div_gFormula_tendsto_zero {κ : ℝ} (hκ : 0 < κ) :
+    Tendsto (fun r => hFormula κ r / gFormula κ r) (𝓝[>] 0) (𝓝 0) := by
+  have hq : 0 < 1 - codimension κ / 2 := by
+    dsimp [codimension, dimension]
+    linarith
+  have ht := (tendsto_rpow_neg_atTop hq).comp logLog_inv_tendsto_atTop
+  apply ht.congr'
+  filter_upwards [self_mem_nhdsWithin,
+    logLog_inv_tendsto_atTop.eventually (eventually_gt_atTop 0)] with r hr hl
+  have hrp : 0 < r := hr
+  have hrpow : r ^ dimension κ ≠ 0 := (Real.rpow_pos_of_pos hrp _).ne'
+  dsimp only [Function.comp_def, hFormula, gFormula]
+  rw [mul_div_mul_left _ _ hrpow, ← Real.rpow_sub_one hl.ne']
+  congr 1
+  ring
+
+theorem gauge_ratio_tendsto_zero {κ : ℝ} (hκ : 0 < κ)
+    {h g : ℝ → ℝ} (hf : HasSmallRadiusFormula h (hFormula κ))
+    (gf : HasSmallRadiusFormula g (gFormula κ)) :
+    Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0) := by
+  apply (hFormula_div_gFormula_tendsto_zero hκ).congr'
+  exact (hf.div gf).symm
+
+/-- A small-radius ratio estimate gives a comparison of the associated
+Hausdorff measures, independently of their values at large radii. -/
+theorem hausdorffGauge_le_smul_of_ratio_tendsto_zero
+    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
+    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
+    {c : ℝ} (hc : 0 < c) :
+    hausdorffGauge h ≤ ENNReal.ofReal c • hausdorffGauge g := by
+  have hsmall : ∀ᶠ r in 𝓝[>] (0 : ℝ), h r / g r < c :=
+    hratio.eventually (gt_mem_nhds hc)
+  obtain ⟨δ, hδ, hbound⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hsmall
+  have htoReal : Tendsto ENNReal.toReal (𝓝 (0 : ℝ≥0∞)) (𝓝 (0 : ℝ)) := by
+    simpa using ENNReal.tendsto_toReal ENNReal.zero_ne_top
+  have hevent : ∀ᶠ r : ℝ≥0∞ in 𝓝 (0 : ℝ≥0∞),
+      extendedGauge h r ≤ ENNReal.ofReal c * extendedGauge g r := by
+    filter_upwards [htoReal.eventually (gt_mem_nhds hδ)] with r hr
+    by_cases hz : r.toReal = 0
+    · simp [extendedGauge, hz, hh.zero, hg.zero]
+    · have hp : 0 < r.toReal := lt_of_le_of_ne ENNReal.toReal_nonneg (Ne.symm hz)
+      have hs := (div_lt_iff₀ (hg.positive _ hp)).mp (hbound ⟨hp, hr⟩)
+      dsimp only [extendedGauge]
+      rw [← ENNReal.ofReal_mul hc.le]
+      exact ENNReal.ofReal_le_ofReal hs.le
+  exact Measure.mkMetric_mono_smul ENNReal.ofReal_ne_top
+    (ENNReal.ofReal_ne_zero_iff.mpr hc) (hevent.filter_mono nhdsWithin_le_nhds)
+
+/-- Finite measure for a larger gauge forces zero measure for any gauge with
+vanishing ratio. This statement does not require the set to be measurable. -/
+theorem hausdorffGauge_eq_zero_of_larger_finite
+    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
+    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
+    {E : Set ℂ} (hfinite : hausdorffGauge g E < ∞) :
+    hausdorffGauge h E = 0 := by
+  have ht : Tendsto (fun c : ℝ => ENNReal.ofReal c * hausdorffGauge g E)
+      (𝓝[>] 0) (𝓝 0) := by
+    have hid : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) :=
+      tendsto_id.mono_left nhdsWithin_le_nhds
+    simpa using ENNReal.Tendsto.mul_const (ENNReal.tendsto_ofReal hid)
+      (Or.inr hfinite.ne)
+  apply le_antisymm _ bot_le
+  apply ge_of_tendsto ht
+  filter_upwards [self_mem_nhdsWithin] with c hc
+  exact hausdorffGauge_le_smul_of_ratio_tendsto_zero hh hg hratio hc E
+
+/-- Positive measure for the smaller gauge prevents sigma-finiteness of the
+larger Hausdorff measure restricted to that set. -/
+theorem not_sigmaFinite_larger_of_positive
+    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
+    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
+    {E : Set ℂ} (hpositive : 0 < hausdorffGauge h E) :
+    ¬ SigmaFinite ((hausdorffGauge g).restrict E) := by
+  intro hσ
+  let : SigmaFinite ((hausdorffGauge g).restrict E) := hσ
+  let A := spanningSets ((hausdorffGauge g).restrict E)
+  have hzero (n : ℕ) : hausdorffGauge h (A n ∩ E) = 0 := by
+    apply hausdorffGauge_eq_zero_of_larger_finite hh hg hratio
+    have hf := measure_spanningSets_lt_top ((hausdorffGauge g).restrict E) n
+    rwa [Measure.restrict_apply (measurableSet_spanningSets _ _)] at hf
+  have hcover : (⋃ n, A n ∩ E) = E := by
+    rw [← iUnion_inter]
+    simp only [A, iUnion_spanningSets, univ_inter]
+  have he : hausdorffGauge h E = 0 := by
+    rw [← hcover]
+    exact measure_iUnion_null hzero
+  exact hpositive.ne' he
+
+theorem hausdorffGauge_larger_eq_top_of_positive
+    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
+    (hratio : Tendsto (fun r => h r / g r) (𝓝[>] 0) (𝓝 0))
+    {E : Set ℂ} (hpositive : 0 < hausdorffGauge h E) :
+    hausdorffGauge g E = ∞ := by
+  by_contra htop
+  exact hpositive.ne'
+    (hausdorffGauge_eq_zero_of_larger_finite hh hg hratio (lt_top_iff_ne_top.mpr htop))
+
+/-- Corollary `cor:larger-gauge`: once the explicit exact gauge has positive
+measure on a set, the exponent-one gauge is not sigma-finite there. -/
+theorem larger_logLog_gauge_not_sigmaFinite {κ : ℝ} (hκ : 0 < κ)
+    {h g : ℝ → ℝ} (hh : IsGauge h) (hg : IsGauge g)
+    (hf : HasSmallRadiusFormula h (hFormula κ))
+    (gf : HasSmallRadiusFormula g (gFormula κ))
+    {E : Set ℂ} (hpositive : 0 < hausdorffGauge h E) :
+    ¬ SigmaFinite ((hausdorffGauge g).restrict E) :=
+  not_sigmaFinite_larger_of_positive hh hg (gauge_ratio_tendsto_zero hκ hf gf) hpositive
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExactMomentGauge.lean b/lean/OAI/Probability/SLE/Gauge/ExactMomentGauge.lean
index eb2db1d9a578f59a075ccc2dad13c6dbbfc1d40c..50040c12e2bdcb43ae416e39802abd459eabbbe8 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExactMomentGauge.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExactMomentGauge.lean
@@ -1,82 +1,82 @@
-import OAI.Probability.SLE.Geometry.CanonicalDiskPositive
-import OAI.Probability.SLE.Mass.MomentLevels
-
-namespace OAI
-
-/-! Positive finite prescribed moments and the resulting exact quadratic gauge.
-All coefficients are the original nested Wiener integrals in `Target`. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-  (hB : IsBrownianReal B P)
-  (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-  (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-  (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-
-include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
-
-lemma finiteMoment_positive (ℓ : ℝ) (hℓ : 0≤ℓ) (hsmall : Real.exp (-(4*ℓ/κ))<1/8) :
-    0<Target.finiteMoment κ W 1 ℓ := by
-  let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
-  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1]
-  have hp : ∀ᵐ θ ∂Target.angleLaw κ, 0<Target.levelCoefficient κ W ℓ 1 θ := by
-    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-    exact C.levelCoefficient_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW θ hθ ℓ hℓ hsmall
-  apply (integral_pos_iff_support_of_nonneg_ae (hp.mono (fun _ h => h.le))
-    (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1).2).2
-  have he : (Target.angleLaw κ) (Function.support (Target.levelCoefficient κ W ℓ 1))=
-      (Target.angleLaw κ) univ := measure_congr (hp.mono (fun θ h => propext
-        (show Target.levelCoefficient κ W ℓ 1 θ≠0 ↔ True from iff_true_intro h.ne')))
-  rw [he,measure_univ]
-  exact zero_lt_one
-
-theorem moment_one_positive : 0<Target.moment κ W 1 := by
-  let ℓ : ℝ := κ*Real.log 16/4
-  have hℓ : 0≤ℓ := div_nonneg (mul_nonneg hκ.le (Real.log_nonneg (by norm_num))) (by norm_num)
-  have hlevel : 4*ℓ/κ=Real.log 16 := by dsimp only [ℓ]; field_simp
-  have hsmall : Real.exp (-(4*ℓ/κ))<1/8 := by
-    rw [hlevel,Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<16)]
-    norm_num
-  have hp := C.finiteMoment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW ℓ hℓ hsmall
-  apply hp.trans_le
-  obtain ⟨n,hn⟩ := exists_nat_ge ℓ
-  apply ge_of_tendsto (C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW 1)
-  filter_upwards [eventually_ge_atTop n] with j hj
-  exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW 1 hℓ
-    (by change (0:ℝ)≤(j:ℝ); exact Nat.cast_nonneg j) (hn.trans (Nat.cast_le.mpr hj))
-
-theorem moment_positive (n : ℕ) : 0<Target.moment κ W n :=
-  (pow_pos (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW) n).trans_le
-    (C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
-
-theorem prescribed_gauge_admissible : Target.IsHausdorffGauge (Target.gauge κ W) :=
-  Target.gauge_admissible_of_moment_inequalities hκ hκ8 W
-    (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
-    (fun n _ => C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
-
-theorem prescribed_batch_retains_moment (n : ℕ) :
-    1≤Target.batchLevel κ W n ∧ 3*Target.moment κ W n/4<
-      Target.finiteMoment κ W n (Target.batchLevel κ W n) := by
-  apply Target.batchLevel_retains_moment κ W n
-  · intro i j hij
-    exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW n
-      (by change (0:ℝ)≤(i:ℝ); exact Nat.cast_nonneg i)
-      (by change (0:ℝ)≤(j:ℝ); exact Nat.cast_nonneg j) (Nat.cast_le.mpr hij)
-  · exact C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW n
-  · exact C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW n
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CanonicalDiskPositive
+import OAI.Probability.SLE.Mass.MomentLevels
+
+namespace OAI
+
+/-! Positive finite prescribed moments and the resulting exact quadratic gauge.
+All coefficients are the original nested Wiener integrals in `Target`. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+  (hB : IsBrownianReal B P)
+  (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+  (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+  (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+  (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+
+include C hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
+
+lemma finiteMoment_positive (ℓ : ℝ) (hℓ : 0≤ℓ) (hsmall : Real.exp (-(4*ℓ/κ))<1/8) :
+    0<Target.finiteMoment κ W 1 ℓ := by
+  let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ
+  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1]
+  have hp : ∀ᵐ θ ∂Target.angleLaw κ, 0<Target.levelCoefficient κ W ℓ 1 θ := by
+    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+    exact C.levelCoefficient_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW θ hθ ℓ hℓ hsmall
+  apply (integral_pos_iff_support_of_nonneg_ae (hp.mono (fun _ h => h.le))
+    (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ 1).2).2
+  have he : (Target.angleLaw κ) (Function.support (Target.levelCoefficient κ W ℓ 1))=
+      (Target.angleLaw κ) univ := measure_congr (hp.mono (fun θ h => propext
+        (show Target.levelCoefficient κ W ℓ 1 θ≠0 ↔ True from iff_true_intro h.ne')))
+  rw [he,measure_univ]
+  exact zero_lt_one
+
+theorem moment_one_positive : 0<Target.moment κ W 1 := by
+  let ℓ : ℝ := κ*Real.log 16/4
+  have hℓ : 0≤ℓ := div_nonneg (mul_nonneg hκ.le (Real.log_nonneg (by norm_num))) (by norm_num)
+  have hlevel : 4*ℓ/κ=Real.log 16 := by dsimp only [ℓ]; field_simp
+  have hsmall : Real.exp (-(4*ℓ/κ))<1/8 := by
+    rw [hlevel,Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<16)]
+    norm_num
+  have hp := C.finiteMoment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW ℓ hℓ hsmall
+  apply hp.trans_le
+  obtain ⟨n,hn⟩ := exists_nat_ge ℓ
+  apply ge_of_tendsto (C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW 1)
+  filter_upwards [eventually_ge_atTop n] with j hj
+  exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW 1 hℓ
+    (by change (0:ℝ)≤(j:ℝ); exact Nat.cast_nonneg j) (hn.trans (Nat.cast_le.mpr hj))
+
+theorem moment_positive (n : ℕ) : 0<Target.moment κ W n :=
+  (pow_pos (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW) n).trans_le
+    (C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
+
+theorem prescribed_gauge_admissible : Target.IsHausdorffGauge (Target.gauge κ W) :=
+  Target.gauge_admissible_of_moment_inequalities hκ hκ8 W
+    (C.moment_one_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW)
+    (fun n _ => C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n)
+
+theorem prescribed_batch_retains_moment (n : ℕ) :
+    1≤Target.batchLevel κ W n ∧ 3*Target.moment κ W n/4<
+      Target.finiteMoment κ W n (Target.batchLevel κ W n) := by
+  apply Target.batchLevel_retains_moment κ W n
+  · intro i j hij
+    exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW n
+      (by change (0:ℝ)≤(i:ℝ); exact Nat.cast_nonneg i)
+      (by change (0:ℝ)≤(j:ℝ); exact Nat.cast_nonneg j) (Nat.cast_le.mpr hij)
+  · exact C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW n
+  · exact C.finiteMoment_tendsto_moment hB hc h0 hκ hκ8 F hw hgrowth W hW n
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExpectedBadGrid.lean b/lean/OAI/Probability/SLE/Gauge/ExpectedBadGrid.lean
index a778d906288ea2ec82fdef31179982f2c8a918a0..e59582000b481d4d0c528b7abf74d6319de91d8f 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExpectedBadGrid.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExpectedBadGrid.lean
@@ -1,67 +1,67 @@
-import OAI.Probability.SLE.Gauge.RandomBadGrid
-import OAI.Probability.SLE.Mass.GridMassOverlap
-
-namespace OAI
-
-/-! Individual cell charge estimates combine with fixed grid overlap and
-summable prescribed tails to give the complete lower-bound criterion. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma expected_badGrid_mass_le_overlap
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (μ : Ω → Measure ℂ) (hμ : Measurable μ) (η : Measure ℂ)
-    (r : ℝ) (hr : 0 < r) (H : ℝ) (p : ℝ≥0∞)
-    (hcell : ∀ i : ℤ×ℤ,
-      (∫⁻ ω, if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then
-        μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P) ≤
-      p*η (closedBall (coverGridCenter r i) (2*r))) :
-    (∫⁻ ω, μ ω (badGridCells (μ ω) r H) ∂P) ≤ 81*p*η univ := by
-  calc
-    _ ≤ ∑' i : ℤ×ℤ,
-        ∫⁻ ω, if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then
-          μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P :=
-      expected_badGrid_mass_le_sum P μ hμ r H
-    _ ≤ ∑' i : ℤ×ℤ, p*η (closedBall (coverGridCenter r i) (2*r)) := ENNReal.tsum_le_tsum hcell
-    _ = p*(∑' i : ℤ×ℤ, η (closedBall (coverGridCenter r i) (2*r))) := ENNReal.tsum_mul_left
-    _ ≤ p*(81*η univ) := mul_le_mul' le_rfl (grid_mass_sum_le η hr)
-    _ = _ := by ac_rfl
-
-lemma ae_positive_prescribed_hausdorff_of_cell_charges
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (μ : Ω → Measure ℂ) (hμ : Measurable μ) (hfin : ∀ ω, IsFiniteMeasure (μ ω))
-    (η : Measure ℂ) [IsFiniteMeasure η] (κ : ℝ) (W : Target.WienerLaw)
-    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
-    (hm1 : 0 < Target.moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
-    (D : ℝ) (hD : 0 ≤ D) (p : Target.GaugeIndex κ W → ℝ≥0∞)
-    (hp : (∑' e, p e) ≠ ∞)
-    (hcell : ∀ e : Target.GaugeIndex κ W, ∀ i : ℤ×ℤ,
-      (∫⁻ ω, if ENNReal.ofReal
-          (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ) <
-            μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) then
-          μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) else 0 ∂P) ≤
-        p e*η (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e))) :
-    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E → 0 < μ ω E →
-      0 < Target.hausdorff (Target.gauge κ W) E := by
-  apply ae_positive_prescribed_hausdorff_of_bad_mass P μ hμ hfin κ W hg hm1 hJ D hD
-  have hb : (∑' e : Target.GaugeIndex κ W,
-      ∫⁻ ω, μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≤
-      81*(∑' e, p e)*η univ := by
-    calc
-      _ ≤ ∑' e : Target.GaugeIndex κ W, 81*p e*η univ := ENNReal.tsum_le_tsum
-        (fun e => expected_badGrid_mass_le_overlap P μ hμ η _ (Target.radii_pos κ W e) _ (p e) (hcell e))
-      _ = _ := by rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
-  exact ne_top_of_le_ne_top
-    (ENNReal.mul_lt_top (ENNReal.mul_lt_top (by norm_num) (lt_top_iff_ne_top.mpr hp))
-      (measure_lt_top η univ)).ne hb
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RandomBadGrid
+import OAI.Probability.SLE.Mass.GridMassOverlap
+
+namespace OAI
+
+/-! Individual cell charge estimates combine with fixed grid overlap and
+summable prescribed tails to give the complete lower-bound criterion. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma expected_badGrid_mass_le_overlap
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (μ : Ω → Measure ℂ) (hμ : Measurable μ) (η : Measure ℂ)
+    (r : ℝ) (hr : 0 < r) (H : ℝ) (p : ℝ≥0∞)
+    (hcell : ∀ i : ℤ×ℤ,
+      (∫⁻ ω, if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then
+        μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P) ≤
+      p*η (closedBall (coverGridCenter r i) (2*r))) :
+    (∫⁻ ω, μ ω (badGridCells (μ ω) r H) ∂P) ≤ 81*p*η univ := by
+  calc
+    _ ≤ ∑' i : ℤ×ℤ,
+        ∫⁻ ω, if ENNReal.ofReal H < μ ω (closedBall (coverGridCenter r i) (2*r)) then
+          μ ω (closedBall (coverGridCenter r i) (2*r)) else 0 ∂P :=
+      expected_badGrid_mass_le_sum P μ hμ r H
+    _ ≤ ∑' i : ℤ×ℤ, p*η (closedBall (coverGridCenter r i) (2*r)) := ENNReal.tsum_le_tsum hcell
+    _ = p*(∑' i : ℤ×ℤ, η (closedBall (coverGridCenter r i) (2*r))) := ENNReal.tsum_mul_left
+    _ ≤ p*(81*η univ) := mul_le_mul' le_rfl (grid_mass_sum_le η hr)
+    _ = _ := by ac_rfl
+
+lemma ae_positive_prescribed_hausdorff_of_cell_charges
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (μ : Ω → Measure ℂ) (hμ : Measurable μ) (hfin : ∀ ω, IsFiniteMeasure (μ ω))
+    (η : Measure ℂ) [IsFiniteMeasure η] (κ : ℝ) (W : Target.WienerLaw)
+    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
+    (hm1 : 0 < Target.moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
+    (D : ℝ) (hD : 0 ≤ D) (p : Target.GaugeIndex κ W → ℝ≥0∞)
+    (hp : (∑' e, p e) ≠ ∞)
+    (hcell : ∀ e : Target.GaugeIndex κ W, ∀ i : ℤ×ℤ,
+      (∫⁻ ω, if ENNReal.ofReal
+          (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ) <
+            μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) then
+          μ ω (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e)) else 0 ∂P) ≤
+        p e*η (closedBall (coverGridCenter (Target.radii κ W e) i) (2*Target.radii κ W e))) :
+    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E → 0 < μ ω E →
+      0 < Target.hausdorff (Target.gauge κ W) E := by
+  apply ae_positive_prescribed_hausdorff_of_bad_mass P μ hμ hfin κ W hg hm1 hJ D hD
+  have hb : (∑' e : Target.GaugeIndex κ W,
+      ∫⁻ ω, μ ω (prescribedBadGrid (μ ω) κ W D e) ∂P) ≤
+      81*(∑' e, p e)*η univ := by
+    calc
+      _ ≤ ∑' e : Target.GaugeIndex κ W, 81*p e*η univ := ENNReal.tsum_le_tsum
+        (fun e => expected_badGrid_mass_le_overlap P μ hμ η _ (Target.radii_pos κ W e) _ (p e) (hcell e))
+      _ = _ := by rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
+  exact ne_top_of_le_ne_top
+    (ENNReal.mul_lt_top (ENNReal.mul_lt_top (by norm_num) (lt_top_iff_ne_top.mpr hp))
+      (measure_lt_top η univ)).ne hb
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExpectedCoverCharge.lean b/lean/OAI/Probability/SLE/Gauge/ExpectedCoverCharge.lean
index 2fa2c276085a0a79b448e7f419837b940704199c..2b7247a1a6426f360101c9fe28fdbacbd63d835f 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExpectedCoverCharge.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExpectedCoverCharge.lean
@@ -1,62 +1,62 @@
-import OAI.Probability.SLE.Geometry.SpatialExhaustion
-import OAI.Probability.SLE.SausageCharge
-
-namespace OAI
-
-/-! Turning one-mesh sausage charges into upper Hausdorff expectations. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- A shrinking cover whose selected part is charged to a single sausage
-mesh and whose expected residual cost vanishes has the desired uniform
-Hausdorff expectation bound. The hypothesis describes actual covers. -/
-theorem expected_hausdorff_le_of_sausage_charged_covers
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (h : ℝ → ℝ) (hh : IsGauge h) (E : Ω → Set ℂ)
-    (U : ℕ → Ω → ℕ → Set ℂ) (δ : ℕ → ℝ≥0∞)
-    (hδ : Tendsto δ atTop (𝓝 0))
-    (hcover : ∀ᵐ ω ∂P, ∀ n, E ω ⊆ ⋃ j, U n ω j)
-    (hdiam : ∀ᵐ ω ∂P, ∀ n j, ediam (U n ω j) ≤ δ n)
-    (hmeas : ∀ n, AEMeasurable
-      (fun ω => ∑' j, extendedGauge h (ediam (U n ω j))) P)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (R : ℝ) (C : ℝ≥0)
-    (error : ℕ → Ω → ℝ≥0∞) (herr : ∀ n, AEMeasurable (error n) P)
-    (herrlim : Tendsto (fun n => ∫⁻ ω, error n ω ∂P) atTop (𝓝 0))
-    (hcharge : ∀ n, ∀ᵐ ω ∂P,
-      (∑' j, extendedGauge h (ediam (U n ω j))) ≤
-        (C : ℝ≥0∞) * wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω) :
-    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤
-      (C : ℝ≥0∞) * wholeSausageChargeBound κ R := by
-  obtain ⟨_, hsausage⟩ := source_cover_charge P κ hκ hκ8 γ hSLE R
-  have hcost (n : ℕ) :
-      (∫⁻ ω, ∑' j, extendedGauge h (ediam (U n ω j)) ∂P) ≤
-        (C : ℝ≥0∞) * wholeSausageChargeBound κ R + ∫⁻ ω, error n ω ∂P := by
-    calc
-      _ ≤ ∫⁻ ω, (C : ℝ≥0∞) *
-          wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω ∂P :=
-        lintegral_mono_ae (hcharge n)
-      _ = (C : ℝ≥0∞) * (∫⁻ ω,
-          wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) ∂P) +
-          ∫⁻ ω, error n ω ∂P := by
-        rw [lintegral_add_right' _ (herr n),
-          lintegral_const_mul' _ _ ENNReal.coe_ne_top]
-      _ ≤ _ := add_le_add (mul_le_mul' le_rfl (hsausage (v n) (hv n)).2) le_rfl
-  apply expected_hausdorff_le_of_shrinking_covers P h hh E U δ hδ
-    hcover hdiam hmeas
-  have ht : Tendsto (fun n => (C : ℝ≥0∞) * wholeSausageChargeBound κ R +
-      ∫⁻ ω, error n ω ∂P) atTop
-      (𝓝 ((C : ℝ≥0∞) * wholeSausageChargeBound κ R)) := by
-    simpa only [add_zero] using tendsto_const_nhds.add herrlim
-  exact (liminf_le_liminf (Eventually.of_forall hcost)).trans_eq ht.liminf_eq
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.SpatialExhaustion
+import OAI.Probability.SLE.SausageCharge
+
+namespace OAI
+
+/-! Turning one-mesh sausage charges into upper Hausdorff expectations. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- A shrinking cover whose selected part is charged to a single sausage
+mesh and whose expected residual cost vanishes has the desired uniform
+Hausdorff expectation bound. The hypothesis describes actual covers. -/
+theorem expected_hausdorff_le_of_sausage_charged_covers
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (h : ℝ → ℝ) (hh : IsGauge h) (E : Ω → Set ℂ)
+    (U : ℕ → Ω → ℕ → Set ℂ) (δ : ℕ → ℝ≥0∞)
+    (hδ : Tendsto δ atTop (𝓝 0))
+    (hcover : ∀ᵐ ω ∂P, ∀ n, E ω ⊆ ⋃ j, U n ω j)
+    (hdiam : ∀ᵐ ω ∂P, ∀ n j, ediam (U n ω j) ≤ δ n)
+    (hmeas : ∀ n, AEMeasurable
+      (fun ω => ∑' j, extendedGauge h (ediam (U n ω j))) P)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (R : ℝ) (C : ℝ≥0)
+    (error : ℕ → Ω → ℝ≥0∞) (herr : ∀ n, AEMeasurable (error n) P)
+    (herrlim : Tendsto (fun n => ∫⁻ ω, error n ω ∂P) atTop (𝓝 0))
+    (hcharge : ∀ n, ∀ᵐ ω ∂P,
+      (∑' j, extendedGauge h (ediam (U n ω j))) ≤
+        (C : ℝ≥0∞) * wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω) :
+    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤
+      (C : ℝ≥0∞) * wholeSausageChargeBound κ R := by
+  obtain ⟨_, hsausage⟩ := source_cover_charge P κ hκ hκ8 γ hSLE R
+  have hcost (n : ℕ) :
+      (∫⁻ ω, ∑' j, extendedGauge h (ediam (U n ω j)) ∂P) ≤
+        (C : ℝ≥0∞) * wholeSausageChargeBound κ R + ∫⁻ ω, error n ω ∂P := by
+    calc
+      _ ≤ ∫⁻ ω, (C : ℝ≥0∞) *
+          wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω ∂P :=
+        lintegral_mono_ae (hcharge n)
+      _ = (C : ℝ≥0∞) * (∫⁻ ω,
+          wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) ∂P) +
+          ∫⁻ ω, error n ω ∂P := by
+        rw [lintegral_add_right' _ (herr n),
+          lintegral_const_mul' _ _ ENNReal.coe_ne_top]
+      _ ≤ _ := add_le_add (mul_le_mul' le_rfl (hsausage (v n) (hv n)).2) le_rfl
+  apply expected_hausdorff_le_of_shrinking_covers P h hh E U δ hδ
+    hcover hdiam hmeas
+  have ht : Tendsto (fun n => (C : ℝ≥0∞) * wholeSausageChargeBound κ R +
+      ∫⁻ ω, error n ω ∂P) atTop
+      (𝓝 ((C : ℝ≥0∞) * wholeSausageChargeBound κ R)) := by
+    simpa only [add_zero] using tendsto_const_nhds.add herrlim
+  exact (liminf_le_liminf (Eventually.of_forall hcost)).trans_eq ht.liminf_eq
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExperimentChangeOfLaw.lean b/lean/OAI/Probability/SLE/Gauge/ExperimentChangeOfLaw.lean
index 1194709e2f0ee5feb7a7ca2b1becc7467abbca81..5ffdd6e0a3b3a0666fa851c195c86c283c865051 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExperimentChangeOfLaw.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExperimentChangeOfLaw.lean
@@ -1,99 +1,99 @@
-import OAI.Probability.SLE.Gauge.RandomPassageBounds
-import OAI.Probability.SLE.InverseMoments
-
-namespace OAI
-
-/-! The finite centered law pays a uniform cost for an actual successful
-experiment whose marked-point angle stays positive. Its finite completion
-stop is used before the conformal-radius cutoff. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma greenDensity_lower_of_angle
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ) (T : ℝ≥0) (hT : z ∈ survivingDomain γ T)
-    {s : ℝ} (hs : 0 ≤ s)
-    (hangle : s ≤ Real.sin (Complex.arg (G T z - (U T : ℂ)))) :
-    s ^ greenExponent κ * initialGreen κ z ≤ greenDensity κ U γ G T z := by
-  have hR := hw.fullRadius_positive hU T z hT
-  have hR0 := hw.fullRadius_initial hU z (survivingDomain_antitone γ zero_le hT)
-  have hdec : fullRadius G T z ≤ 2 * z.im := by
-    rw [← hR0]
-    exact hw.radius_le_of_domain_subset hU T 0 (survivingDomain_antitone γ zero_le) z hT
-  have hinit : initialGreen κ z ≤ (2 * z.im) ^ (-codimension κ) := by
-    unfold initialGreen
-    exact mul_le_of_le_one_right (Real.rpow_nonneg (mul_nonneg (by norm_num) hT.1.1.le) _)
-      (Real.rpow_le_one (by
-        rw [Complex.sin_arg]
-        exact div_nonneg hT.1.1.le (norm_nonneg _))
-        (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
-  have hp : (2 * z.im) ^ (-codimension κ) ≤ (fullRadius G T z) ^ (-codimension κ) :=
-    Real.rpow_le_rpow_of_nonpos hR hdec (neg_nonpos.mpr (codimension_pos κ hκ8).le)
-  have ha : s ^ greenExponent κ ≤
-      (Real.sin (Complex.arg (G T z - (U T : ℂ)))) ^ greenExponent κ :=
-    Real.rpow_le_rpow hs hangle (greenExponent_pos κ hκ hκ8).le
-  rw [greenDensity, ite_eq_left hT]
-  calc
-    _ ≤ s ^ greenExponent κ * (2 * z.im) ^ (-codimension κ) :=
-      mul_le_mul_of_nonneg_left hinit (Real.rpow_nonneg hs _)
-    _ ≤ (Real.sin (Complex.arg (G T z - (U T : ℂ)))) ^ greenExponent κ *
-        (fullRadius G T z) ^ (-codimension κ) :=
-      mul_le_mul ha hp (Real.rpow_nonneg (mul_nonneg (by norm_num) hT.1.1.le) _) (Real.rpow_nonneg
-        (hs.trans hangle) _)
-    _ = _ := mul_comm _ _
-
-/-- An event observed at a finite experiment stop transfers to the finite
-radius law using its actual Green density there. -/
-theorem finiteCenteredLaw_experiment_lower_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
-    (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B
-      (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (E : Set Ω) (hE : MeasurableSet[hσ.measurableSpace] E)
-    (s : ℝ) (hs : 0 ≤ s)
-    (hsuccess : ∀ᵐ ω ∂P, ω ∈ E →
-      (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ∧
-      z ∈ survivingDomain (γ ω) (σ ω) ∧
-      s ≤ Real.sin (Complex.arg (G ω (σ ω) z - (Real.sqrt κ * B (σ ω) ω : ℂ)))) :
-    ENNReal.ofReal (s ^ greenExponent κ) * P E ≤ finiteCenteredLaw P κ B γ G z u E := by
-  obtain ⟨M, Z, _, hZi, _, hZb, _, hMstop, hsource, _⟩ :=
-    exists_physical_green_stopping_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu
-  have hbound : ∀ᵐ ω ∂P, ω ∈ E →
-      s ^ greenExponent κ * initialGreen κ z ≤ M (σ ω) ω := by
-    filter_upwards [hw, hsource, hsuccess] with ω hω hsrc hsucc hωE
-    obtain ⟨hbefore, hlive, hangle⟩ := hsucc hωE
-    rw [hsrc.1 (σ ω), physicalStoppedGreen, ite_eq_right (not_le.mpr hbefore)]
-    exact greenDensity_lower_of_angle ((hBc ω).const_mul _) hω κ hκ hκ8 z
-      (σ ω) hlive hs (by simpa only [Complex.ofReal_mul] using hangle)
-  have h := normalizedDensity_event_lower_bound_of_condExp P hσ.measurableSpace
-    hσ.measurableSpace_le Z (fun ω => M (σ ω) ω) hZi
-    (Eventually.of_forall (fun ω => (hZb ω).1)) (hMstop σ hσ)
-    (initialGreen κ z) (s ^ greenExponent κ) (initialGreen_pos κ z hz)
-    (Real.rpow_nonneg hs _) E hE hbound
-  have he : P.withDensity (fun ω => ENNReal.ofReal (Z ω / initialGreen κ z)) =
-      finiteCenteredLaw P κ B γ G z u := by
-    apply withDensity_congr_ae
-    filter_upwards [hsource] with ω hω
-    rw [hω.2]
-  exact he ▸ h
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RandomPassageBounds
+import OAI.Probability.SLE.InverseMoments
+
+namespace OAI
+
+/-! The finite centered law pays a uniform cost for an actual successful
+experiment whose marked-point angle stays positive. Its finite completion
+stop is used before the conformal-radius cutoff. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma greenDensity_lower_of_angle
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ) (T : ℝ≥0) (hT : z ∈ survivingDomain γ T)
+    {s : ℝ} (hs : 0 ≤ s)
+    (hangle : s ≤ Real.sin (Complex.arg (G T z - (U T : ℂ)))) :
+    s ^ greenExponent κ * initialGreen κ z ≤ greenDensity κ U γ G T z := by
+  have hR := hw.fullRadius_positive hU T z hT
+  have hR0 := hw.fullRadius_initial hU z (survivingDomain_antitone γ zero_le hT)
+  have hdec : fullRadius G T z ≤ 2 * z.im := by
+    rw [← hR0]
+    exact hw.radius_le_of_domain_subset hU T 0 (survivingDomain_antitone γ zero_le) z hT
+  have hinit : initialGreen κ z ≤ (2 * z.im) ^ (-codimension κ) := by
+    unfold initialGreen
+    exact mul_le_of_le_one_right (Real.rpow_nonneg (mul_nonneg (by norm_num) hT.1.1.le) _)
+      (Real.rpow_le_one (by
+        rw [Complex.sin_arg]
+        exact div_nonneg hT.1.1.le (norm_nonneg _))
+        (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
+  have hp : (2 * z.im) ^ (-codimension κ) ≤ (fullRadius G T z) ^ (-codimension κ) :=
+    Real.rpow_le_rpow_of_nonpos hR hdec (neg_nonpos.mpr (codimension_pos κ hκ8).le)
+  have ha : s ^ greenExponent κ ≤
+      (Real.sin (Complex.arg (G T z - (U T : ℂ)))) ^ greenExponent κ :=
+    Real.rpow_le_rpow hs hangle (greenExponent_pos κ hκ hκ8).le
+  rw [greenDensity, ite_eq_left hT]
+  calc
+    _ ≤ s ^ greenExponent κ * (2 * z.im) ^ (-codimension κ) :=
+      mul_le_mul_of_nonneg_left hinit (Real.rpow_nonneg hs _)
+    _ ≤ (Real.sin (Complex.arg (G T z - (U T : ℂ)))) ^ greenExponent κ *
+        (fullRadius G T z) ^ (-codimension κ) :=
+      mul_le_mul ha hp (Real.rpow_nonneg (mul_nonneg (by norm_num) hT.1.1.le) _) (Real.rpow_nonneg
+        (hs.trans hangle) _)
+    _ = _ := mul_comm _ _
+
+/-- An event observed at a finite experiment stop transfers to the finite
+radius law using its actual Green density there. -/
+theorem finiteCenteredLaw_experiment_lower_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
+    (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B
+      (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (E : Set Ω) (hE : MeasurableSet[hσ.measurableSpace] E)
+    (s : ℝ) (hs : 0 ≤ s)
+    (hsuccess : ∀ᵐ ω ∂P, ω ∈ E →
+      (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ∧
+      z ∈ survivingDomain (γ ω) (σ ω) ∧
+      s ≤ Real.sin (Complex.arg (G ω (σ ω) z - (Real.sqrt κ * B (σ ω) ω : ℂ)))) :
+    ENNReal.ofReal (s ^ greenExponent κ) * P E ≤ finiteCenteredLaw P κ B γ G z u E := by
+  obtain ⟨M, Z, _, hZi, _, hZb, _, hMstop, hsource, _⟩ :=
+    exists_physical_green_stopping_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu
+  have hbound : ∀ᵐ ω ∂P, ω ∈ E →
+      s ^ greenExponent κ * initialGreen κ z ≤ M (σ ω) ω := by
+    filter_upwards [hw, hsource, hsuccess] with ω hω hsrc hsucc hωE
+    obtain ⟨hbefore, hlive, hangle⟩ := hsucc hωE
+    rw [hsrc.1 (σ ω), physicalStoppedGreen, ite_eq_right (not_le.mpr hbefore)]
+    exact greenDensity_lower_of_angle ((hBc ω).const_mul _) hω κ hκ hκ8 z
+      (σ ω) hlive hs (by simpa only [Complex.ofReal_mul] using hangle)
+  have h := normalizedDensity_event_lower_bound_of_condExp P hσ.measurableSpace
+    hσ.measurableSpace_le Z (fun ω => M (σ ω) ω) hZi
+    (Eventually.of_forall (fun ω => (hZb ω).1)) (hMstop σ hσ)
+    (initialGreen κ z) (s ^ greenExponent κ) (initialGreen_pos κ z hz)
+    (Real.rpow_nonneg hs _) E hE hbound
+  have he : P.withDensity (fun ω => ENNReal.ofReal (Z ω / initialGreen κ z)) =
+      finiteCenteredLaw P κ B γ G z u := by
+    apply withDensity_congr_ae
+    filter_upwards [hsource] with ω hω
+    rw [hω.2]
+  exact he ▸ h
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExperimentClearance.lean b/lean/OAI/Probability/SLE/Gauge/ExperimentClearance.lean
index 68071658883123f13a8e4a80c9c238e61cb37bcb..6ffb7b5118e2a0654d85d8274f30737fcd49ccb7 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExperimentClearance.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExperimentClearance.lean
@@ -1,97 +1,97 @@
-import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry
-import OAI.Probability.SLE.Loewner.ActualRadiusStops
-
-namespace OAI
-
-/-! The dense experiment need not classify the filled hull of its obstacle.
-A free upward corridor puts the marked-point ball in the actual unbounded
-surviving component and gives its conformal-radius cutoff directly. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-/-- The upward shadow of a ball, with a slightly larger rectangular base. -/
-def upwardShadow (w : ℂ) (r : ℝ) : Set ℂ :=
-  {z | |z.re - w.re| < r ∧ w.im - r < z.im}
-
-lemma upwardRay_mem_shadow {w z : ℂ} {r : ℝ} (hz : z ∈ ball w r) (t : ℝ≥0) :
-    upwardRay z t ∈ upwardShadow w r := by
-  have hd : ‖z - w‖ < r := by simpa only [mem_ball, dist_eq_norm] using hz
-  have hre : |z.re - w.re| < r :=
-    (Complex.abs_re_le_norm (z - w)).trans_lt hd
-  have him : |z.im - w.im| < r :=
-    (Complex.abs_im_le_norm (z - w)).trans_lt hd
-  have hlow := (abs_lt.mp him).1
-  simp only [upwardRay, upwardShadow, mem_ofPred_eq, Complex.add_re, Complex.add_im,
-    Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im,
-    Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero]
-  exact ⟨hre, by linarith [t.coe_nonneg]⟩
-
-lemma marked_ball_survives_of_free_upwardShadow
-    (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (w : ℂ) {r : ℝ} (_hr : 0 < r)
-    (hheight : r ≤ w.im)
-    (hfree : Disjoint (γ '' Icc 0 T) (upwardShadow w r)) :
-    ball w r ⊆ survivingDomain γ T := by
-  intro z hz
-  have hshadow := upwardRay_mem_shadow hz
-  have hroute : range (upwardRay z) ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by
-    rintro u ⟨t, rfl⟩
-    refine ⟨?_, ?_⟩
-    · have hh := (hshadow t).2
-      change 0 < (upwardRay z t).im
-      linarith
-    · intro hpast
-      exact disjoint_left.mp hfree hpast (hshadow t)
-  have hsurv := survives_of_unbounded_free_curve γ T (upwardRay z)
-    (upwardRay_continuous z) hroute (upwardRay_unbounded z) (mem_range_self 0)
-  simpa only [upwardRay, NNReal.coe_zero, Complex.ofReal_zero, zero_mul, add_zero] using hsurv
-
-lemma fullRadius_lower_of_surviving_ball
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (T : ℝ≥0) (w : ℂ) {r : ℝ} (hr : 0 < r)
-    (hball : ball w r ⊆ survivingDomain γ T) : r ≤ fullRadius G T w := by
-  have hlive := hball (mem_ball_self hr)
-  apply radius_of_halfplane_map hr ((hw.2.2.2.1 T).mono hball)
-    (fun z hz => hw.2.2.2.2.2.1 T (hball hz))
-  exact norm_ne_zero_iff.mp (hw.derivative_norm_positive hU T w hlive).ne'
-
-/-- Uniform free clearance gives the actual radius bound throughout every
-successful prefix. Filled-hull containment is unnecessary. -/
-theorem experiment_radius_lower_from_shadow
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (T : ℝ≥0) (w : ℂ) {r : ℝ} (hr : 0 < r) (hheight : r ≤ w.im)
-    (hfree : Disjoint (γ '' Icc 0 T) (upwardShadow w r)) :
-    ∀ t : ℝ≥0, t ≤ T → w ∈ survivingDomain γ t ∧ r ≤ fullRadius G t w := by
-  intro t ht
-  have hball := marked_ball_survives_of_free_upwardShadow γ T w hr hheight hfree
-  have hballt : ball w r ⊆ survivingDomain γ t :=
-    hball.trans (survivingDomain_antitone γ ht)
-  exact ⟨hballt (mem_ball_self hr), fullRadius_lower_of_surviving_ball hU hw t w hr hballt⟩
-
-lemma completion_before_radiusLevelTime_of_prefix_lower
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (T : ℝ≥0) (w : ℂ) (u r : ℝ)
-    (hprefix : ∀ t : ℝ≥0, t ≤ T → w ∈ survivingDomain γ t ∧ r ≤ fullRadius G t w)
-    (hcutoff : (2 * w.im) * Real.exp (-u) < r) :
-    (T : ℝ≥0∞) < radiusLevelTime γ G w u := by
-  by_contra hn
-  have hle := le_of_not_gt hn
-  have hfinite : radiusLevelTime γ G w u < ∞ := hle.trans_lt ENNReal.coe_lt_top
-  obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G w u hfinite
-  rw [radiusLevelTime_eq_of_live_level hU hw w u t ht hR] at hle
-  have htT : t ≤ T := by exact_mod_cast hle
-  have hb := (hprefix t htT).2
-  rw [hR] at hb
-  exact (not_le_of_gt hcutoff) hb
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RibbonCoordinateGeometry
+import OAI.Probability.SLE.Loewner.ActualRadiusStops
+
+namespace OAI
+
+/-! The dense experiment need not classify the filled hull of its obstacle.
+A free upward corridor puts the marked-point ball in the actual unbounded
+surviving component and gives its conformal-radius cutoff directly. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+/-- The upward shadow of a ball, with a slightly larger rectangular base. -/
+def upwardShadow (w : ℂ) (r : ℝ) : Set ℂ :=
+  {z | |z.re - w.re| < r ∧ w.im - r < z.im}
+
+lemma upwardRay_mem_shadow {w z : ℂ} {r : ℝ} (hz : z ∈ ball w r) (t : ℝ≥0) :
+    upwardRay z t ∈ upwardShadow w r := by
+  have hd : ‖z - w‖ < r := by simpa only [mem_ball, dist_eq_norm] using hz
+  have hre : |z.re - w.re| < r :=
+    (Complex.abs_re_le_norm (z - w)).trans_lt hd
+  have him : |z.im - w.im| < r :=
+    (Complex.abs_im_le_norm (z - w)).trans_lt hd
+  have hlow := (abs_lt.mp him).1
+  simp only [upwardRay, upwardShadow, mem_ofPred_eq, Complex.add_re, Complex.add_im,
+    Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im,
+    Complex.I_re, Complex.I_im, mul_zero, mul_one, sub_zero, add_zero]
+  exact ⟨hre, by linarith [t.coe_nonneg]⟩
+
+lemma marked_ball_survives_of_free_upwardShadow
+    (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (w : ℂ) {r : ℝ} (_hr : 0 < r)
+    (hheight : r ≤ w.im)
+    (hfree : Disjoint (γ '' Icc 0 T) (upwardShadow w r)) :
+    ball w r ⊆ survivingDomain γ T := by
+  intro z hz
+  have hshadow := upwardRay_mem_shadow hz
+  have hroute : range (upwardRay z) ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by
+    rintro u ⟨t, rfl⟩
+    refine ⟨?_, ?_⟩
+    · have hh := (hshadow t).2
+      change 0 < (upwardRay z t).im
+      linarith
+    · intro hpast
+      exact disjoint_left.mp hfree hpast (hshadow t)
+  have hsurv := survives_of_unbounded_free_curve γ T (upwardRay z)
+    (upwardRay_continuous z) hroute (upwardRay_unbounded z) (mem_range_self 0)
+  simpa only [upwardRay, NNReal.coe_zero, Complex.ofReal_zero, zero_mul, add_zero] using hsurv
+
+lemma fullRadius_lower_of_surviving_ball
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (T : ℝ≥0) (w : ℂ) {r : ℝ} (hr : 0 < r)
+    (hball : ball w r ⊆ survivingDomain γ T) : r ≤ fullRadius G T w := by
+  have hlive := hball (mem_ball_self hr)
+  apply radius_of_halfplane_map hr ((hw.2.2.2.1 T).mono hball)
+    (fun z hz => hw.2.2.2.2.2.1 T (hball hz))
+  exact norm_ne_zero_iff.mp (hw.derivative_norm_positive hU T w hlive).ne'
+
+/-- Uniform free clearance gives the actual radius bound throughout every
+successful prefix. Filled-hull containment is unnecessary. -/
+theorem experiment_radius_lower_from_shadow
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (T : ℝ≥0) (w : ℂ) {r : ℝ} (hr : 0 < r) (hheight : r ≤ w.im)
+    (hfree : Disjoint (γ '' Icc 0 T) (upwardShadow w r)) :
+    ∀ t : ℝ≥0, t ≤ T → w ∈ survivingDomain γ t ∧ r ≤ fullRadius G t w := by
+  intro t ht
+  have hball := marked_ball_survives_of_free_upwardShadow γ T w hr hheight hfree
+  have hballt : ball w r ⊆ survivingDomain γ t :=
+    hball.trans (survivingDomain_antitone γ ht)
+  exact ⟨hballt (mem_ball_self hr), fullRadius_lower_of_surviving_ball hU hw t w hr hballt⟩
+
+lemma completion_before_radiusLevelTime_of_prefix_lower
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (T : ℝ≥0) (w : ℂ) (u r : ℝ)
+    (hprefix : ∀ t : ℝ≥0, t ≤ T → w ∈ survivingDomain γ t ∧ r ≤ fullRadius G t w)
+    (hcutoff : (2 * w.im) * Real.exp (-u) < r) :
+    (T : ℝ≥0∞) < radiusLevelTime γ G w u := by
+  by_contra hn
+  have hle := le_of_not_gt hn
+  have hfinite : radiusLevelTime γ G w u < ∞ := hle.trans_lt ENNReal.coe_lt_top
+  obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G w u hfinite
+  rw [radiusLevelTime_eq_of_live_level hU hw w u t ht hR] at hle
+  have htT : t ≤ T := by exact_mod_cast hle
+  have hb := (hprefix t htT).2
+  rw [hR] at hb
+  exact (not_le_of_gt hcutoff) hb
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExtendedExperimentStop.lean b/lean/OAI/Probability/SLE/Gauge/ExtendedExperimentStop.lean
index fe0bba823b3e688bf05c0e64189fff37954c0db0..8038d8027c4079a8804a59d5973c0dec27b9579a 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExtendedExperimentStop.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExtendedExperimentStop.lean
@@ -1,155 +1,155 @@
-import OAI.Probability.SLE.Gauge.ExperimentChangeOfLaw
-
-namespace OAI
-
-/-! A completion stop may be infinite on failure. Restrict success to
-finite deterministic horizons, apply the finite-stop density identity,
-and then exhaust the successful event. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def cutoffStop {Ω : Type*} (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) : ℝ≥0 :=
-  (min (τ ω) (H : WithTop ℝ≥0)).untopD 0
-
-lemma cutoffStop_coe {Ω : Type*} (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) :
-    (cutoffStop τ H ω : WithTop ℝ≥0) = min (τ ω) (H : WithTop ℝ≥0) := by
-  cases h : τ ω with
-  | top => simp only [cutoffStop, h, min_top_left, WithTop.untopD_coe]
-  | coe t => simp only [cutoffStop, h, ← WithTop.coe_min, WithTop.untopD_coe]
-
-lemma cutoffStop_stopping
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
-    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (H : ℝ≥0) :
-    IsStoppingTime ℱ (fun ω => (cutoffStop τ H ω : WithTop ℝ≥0)) := by
-  have he : (fun ω => (cutoffStop τ H ω : WithTop ℝ≥0)) =
-      fun ω => min (τ ω) (H : WithTop ℝ≥0) := funext (cutoffStop_coe τ H)
-  rw [he]
-  exact hτ.min_const H
-
-lemma cutoffStop_success_measurable
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
-    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (H : ℝ≥0)
-    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) :
-    MeasurableSet[(cutoffStop_stopping ℱ τ hτ H).measurableSpace]
-      (E ∩ {ω | τ ω ≤ (H : WithTop ℝ≥0)}) := by
-  have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ H) E hE
-  rw [IsStoppingTime.measurableSet] at hh ⊢
-  exact ⟨hh.1, fun t => by simpa only [cutoffStop_coe] using hh.2 t⟩
-
-lemma cutoffStop_eq_on_success {Ω : Type*}
-    (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) (hω : τ ω ≤ (H : WithTop ℝ≥0)) :
-    cutoffStop τ H ω = (τ ω).untopD 0 := by
-  simp only [cutoffStop, min_eq_left hω]
-
-/-- The normalized terminal density pays the same fixed cost when the
-completion stop is finite only on success. -/
-theorem normalizedDensity_completed_event_lower_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (ℱ : Filtration ℝ≥0 mΩ)
-    (Z : Ω → ℝ) (hZ : Integrable Z P) (hZpos : 0 ≤ᵐ[P] Z)
-    (M : ℝ≥0 → Ω → ℝ)
-    (hM : ∀ (σ : Ω → ℝ≥0) (hσ : IsStoppingTime ℱ
-        (fun ω => (σ ω : WithTop ℝ≥0))),
-      (fun ω => M (σ ω) ω) =ᵐ[P] P[Z | hσ.measurableSpace])
-    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ)
-    (a c : ℝ) (ha : 0 < a) (hc : 0 ≤ c)
-    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
-    (hfinite : ∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤)
-    (hbound : ∀ᵐ ω ∂P, ω ∈ E → c * a ≤ M ((τ ω).untopD 0) ω) :
-    ENNReal.ofReal c * P E ≤
-      P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E := by
-  let Eₙ (n : ℕ) := E ∩ {ω | τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)}
-  have hmono : Monotone Eₙ := by
-    intro n k hnk ω hω
-    refine ⟨hω.1, ?_⟩
-    change τ ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0)
-    exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))
-  have hcover : E =ᵐ[P] ⋃ n, Eₙ n := by
-    filter_upwards [hfinite] with ω hω
-    simp only [mem_iUnion]
-    apply propext
-    constructor
-    · intro hωE
-      lift τ ω to ℝ≥0 using hω hωE with t ht
-      obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ)
-      refine ⟨n, hωE, ?_⟩
-      change τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)
-      rw [← ht]
-      exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)
-    · rintro ⟨n, hn⟩
-      exact hn.1
-  have hlocal (n : ℕ) : ENNReal.ofReal c * P (Eₙ n) ≤
-      P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) (Eₙ n) := by
-    let σ := cutoffStop τ (n : ℝ≥0)
-    have hσ := cutoffStop_stopping ℱ τ hτ (n : ℝ≥0)
-    apply normalizedDensity_event_lower_bound_of_condExp P hσ.measurableSpace
-      hσ.measurableSpace_le Z (fun ω => M (σ ω) ω) hZ hZpos (hM σ hσ)
-      a c ha hc (Eₙ n) (cutoffStop_success_measurable ℱ τ hτ _ E hE)
-    filter_upwards [hbound] with ω hb hω
-    change c * a ≤ M (cutoffStop τ (n : ℝ≥0) ω) ω
-    rw [cutoffStop_eq_on_success τ _ ω hω.2]
-    exact hb hω.1
-  calc
-    _ = ENNReal.ofReal c * P (⋃ n, Eₙ n) := by rw [measure_congr hcover]
-    _ = ⨆ n, ENNReal.ofReal c * P (Eₙ n) := by
-      rw [hmono.measure_iUnion, ENNReal.mul_iSup]
-    _ ≤ _ := iSup_le fun n => (hlocal n).trans (measure_mono inter_subset_left)
-
-/-- The successful dense experiment need not finish on unsuccessful
-samples. Its actual finite completion and angle bound on success suffice
-for the centered-law lower bound. -/
-theorem finiteCenteredLaw_completed_experiment_lower_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
-    (τ : Ω → WithTop ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) τ)
-    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
-    (s : ℝ) (hs : 0 ≤ s)
-    (hsuccess : ∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤ ∧
-      τ ω < radiusLevelTime (γ ω) (G ω) z u ∧
-      z ∈ survivingDomain (γ ω) ((τ ω).untopD 0) ∧
-      s ≤ Real.sin (Complex.arg
-        (G ω ((τ ω).untopD 0) z - (Real.sqrt κ * B ((τ ω).untopD 0) ω : ℂ)))) :
-    ENNReal.ofReal (s ^ greenExponent κ) * P E ≤ finiteCenteredLaw P κ B γ G z u E := by
-  obtain ⟨M, Z, _, hZi, _, hZb, _, hMstop, hsource, _⟩ :=
-    exists_physical_green_stopping_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu
-  have hbound : ∀ᵐ ω ∂P, ω ∈ E →
-      s ^ greenExponent κ * initialGreen κ z ≤ M ((τ ω).untopD 0) ω := by
-    filter_upwards [hw, hsource, hsuccess] with ω hω hsrc hsucc hωE
-    obtain ⟨hfinite, hbefore, hlive, hangle⟩ := hsucc hωE
-    have he : (((τ ω).untopD 0 : ℝ≥0) : ℝ≥0∞) = τ ω := by
-      cases ht : τ ω with
-      | top => exact (hfinite ht).elim
-      | coe t => simp only [WithTop.untopD_coe]; rfl
-    rw [hsrc.1 ((τ ω).untopD 0), physicalStoppedGreen,
-      ite_eq_right (not_le.mpr (he.symm ▸ hbefore))]
-    exact greenDensity_lower_of_angle ((hBc ω).const_mul _) hω κ hκ hκ8 z
-      ((τ ω).untopD 0) hlive hs (by simpa only [Complex.ofReal_mul] using hangle)
-  have h := normalizedDensity_completed_event_lower_bound P _ Z hZi
-    (Eventually.of_forall (fun ω => (hZb ω).1)) M hMstop τ hτ
-    (initialGreen κ z) (s ^ greenExponent κ) (initialGreen_pos κ z hz)
-    (Real.rpow_nonneg hs _) E hE (hsuccess.mono fun _ hω hωE => (hω hωE).1) hbound
-  have he : P.withDensity (fun ω => ENNReal.ofReal (Z ω / initialGreen κ z)) =
-      finiteCenteredLaw P κ B γ G z u := by
-    apply withDensity_congr_ae
-    filter_upwards [hsource] with ω hω
-    rw [hω.2]
-  exact he ▸ h
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ExperimentChangeOfLaw
+
+namespace OAI
+
+/-! A completion stop may be infinite on failure. Restrict success to
+finite deterministic horizons, apply the finite-stop density identity,
+and then exhaust the successful event. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def cutoffStop {Ω : Type*} (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) : ℝ≥0 :=
+  (min (τ ω) (H : WithTop ℝ≥0)).untopD 0
+
+lemma cutoffStop_coe {Ω : Type*} (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) :
+    (cutoffStop τ H ω : WithTop ℝ≥0) = min (τ ω) (H : WithTop ℝ≥0) := by
+  cases h : τ ω with
+  | top => simp only [cutoffStop, h, min_top_left, WithTop.untopD_coe]
+  | coe t => simp only [cutoffStop, h, ← WithTop.coe_min, WithTop.untopD_coe]
+
+lemma cutoffStop_stopping
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
+    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (H : ℝ≥0) :
+    IsStoppingTime ℱ (fun ω => (cutoffStop τ H ω : WithTop ℝ≥0)) := by
+  have he : (fun ω => (cutoffStop τ H ω : WithTop ℝ≥0)) =
+      fun ω => min (τ ω) (H : WithTop ℝ≥0) := funext (cutoffStop_coe τ H)
+  rw [he]
+  exact hτ.min_const H
+
+lemma cutoffStop_success_measurable
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
+    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (H : ℝ≥0)
+    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) :
+    MeasurableSet[(cutoffStop_stopping ℱ τ hτ H).measurableSpace]
+      (E ∩ {ω | τ ω ≤ (H : WithTop ℝ≥0)}) := by
+  have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ H) E hE
+  rw [IsStoppingTime.measurableSet] at hh ⊢
+  exact ⟨hh.1, fun t => by simpa only [cutoffStop_coe] using hh.2 t⟩
+
+lemma cutoffStop_eq_on_success {Ω : Type*}
+    (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) (hω : τ ω ≤ (H : WithTop ℝ≥0)) :
+    cutoffStop τ H ω = (τ ω).untopD 0 := by
+  simp only [cutoffStop, min_eq_left hω]
+
+/-- The normalized terminal density pays the same fixed cost when the
+completion stop is finite only on success. -/
+theorem normalizedDensity_completed_event_lower_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (ℱ : Filtration ℝ≥0 mΩ)
+    (Z : Ω → ℝ) (hZ : Integrable Z P) (hZpos : 0 ≤ᵐ[P] Z)
+    (M : ℝ≥0 → Ω → ℝ)
+    (hM : ∀ (σ : Ω → ℝ≥0) (hσ : IsStoppingTime ℱ
+        (fun ω => (σ ω : WithTop ℝ≥0))),
+      (fun ω => M (σ ω) ω) =ᵐ[P] P[Z | hσ.measurableSpace])
+    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ)
+    (a c : ℝ) (ha : 0 < a) (hc : 0 ≤ c)
+    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
+    (hfinite : ∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤)
+    (hbound : ∀ᵐ ω ∂P, ω ∈ E → c * a ≤ M ((τ ω).untopD 0) ω) :
+    ENNReal.ofReal c * P E ≤
+      P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) E := by
+  let Eₙ (n : ℕ) := E ∩ {ω | τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)}
+  have hmono : Monotone Eₙ := by
+    intro n k hnk ω hω
+    refine ⟨hω.1, ?_⟩
+    change τ ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0)
+    exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))
+  have hcover : E =ᵐ[P] ⋃ n, Eₙ n := by
+    filter_upwards [hfinite] with ω hω
+    simp only [mem_iUnion]
+    apply propext
+    constructor
+    · intro hωE
+      lift τ ω to ℝ≥0 using hω hωE with t ht
+      obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ)
+      refine ⟨n, hωE, ?_⟩
+      change τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)
+      rw [← ht]
+      exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)
+    · rintro ⟨n, hn⟩
+      exact hn.1
+  have hlocal (n : ℕ) : ENNReal.ofReal c * P (Eₙ n) ≤
+      P.withDensity (fun ω => ENNReal.ofReal (Z ω / a)) (Eₙ n) := by
+    let σ := cutoffStop τ (n : ℝ≥0)
+    have hσ := cutoffStop_stopping ℱ τ hτ (n : ℝ≥0)
+    apply normalizedDensity_event_lower_bound_of_condExp P hσ.measurableSpace
+      hσ.measurableSpace_le Z (fun ω => M (σ ω) ω) hZ hZpos (hM σ hσ)
+      a c ha hc (Eₙ n) (cutoffStop_success_measurable ℱ τ hτ _ E hE)
+    filter_upwards [hbound] with ω hb hω
+    change c * a ≤ M (cutoffStop τ (n : ℝ≥0) ω) ω
+    rw [cutoffStop_eq_on_success τ _ ω hω.2]
+    exact hb hω.1
+  calc
+    _ = ENNReal.ofReal c * P (⋃ n, Eₙ n) := by rw [measure_congr hcover]
+    _ = ⨆ n, ENNReal.ofReal c * P (Eₙ n) := by
+      rw [hmono.measure_iUnion, ENNReal.mul_iSup]
+    _ ≤ _ := iSup_le fun n => (hlocal n).trans (measure_mono inter_subset_left)
+
+/-- The successful dense experiment need not finish on unsuccessful
+samples. Its actual finite completion and angle bound on success suffice
+for the centered-law lower bound. -/
+theorem finiteCenteredLaw_completed_experiment_lower_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
+    (τ : Ω → WithTop ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun _t => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) τ)
+    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
+    (s : ℝ) (hs : 0 ≤ s)
+    (hsuccess : ∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤ ∧
+      τ ω < radiusLevelTime (γ ω) (G ω) z u ∧
+      z ∈ survivingDomain (γ ω) ((τ ω).untopD 0) ∧
+      s ≤ Real.sin (Complex.arg
+        (G ω ((τ ω).untopD 0) z - (Real.sqrt κ * B ((τ ω).untopD 0) ω : ℂ)))) :
+    ENNReal.ofReal (s ^ greenExponent κ) * P E ≤ finiteCenteredLaw P κ B γ G z u E := by
+  obtain ⟨M, Z, _, hZi, _, hZb, _, hMstop, hsource, _⟩ :=
+    exists_physical_green_stopping_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu
+  have hbound : ∀ᵐ ω ∂P, ω ∈ E →
+      s ^ greenExponent κ * initialGreen κ z ≤ M ((τ ω).untopD 0) ω := by
+    filter_upwards [hw, hsource, hsuccess] with ω hω hsrc hsucc hωE
+    obtain ⟨hfinite, hbefore, hlive, hangle⟩ := hsucc hωE
+    have he : (((τ ω).untopD 0 : ℝ≥0) : ℝ≥0∞) = τ ω := by
+      cases ht : τ ω with
+      | top => exact (hfinite ht).elim
+      | coe t => simp only [WithTop.untopD_coe]; rfl
+    rw [hsrc.1 ((τ ω).untopD 0), physicalStoppedGreen,
+      ite_eq_right (not_le.mpr (he.symm ▸ hbefore))]
+    exact greenDensity_lower_of_angle ((hBc ω).const_mul _) hω κ hκ hκ8 z
+      ((τ ω).untopD 0) hlive hs (by simpa only [Complex.ofReal_mul] using hangle)
+  have h := normalizedDensity_completed_event_lower_bound P _ Z hZi
+    (Eventually.of_forall (fun ω => (hZb ω).1)) M hMstop τ hτ
+    (initialGreen κ z) (s ^ greenExponent κ) (initialGreen_pos κ z hz)
+    (Real.rpow_nonneg hs _) E hE (hsuccess.mono fun _ hω hωE => (hω hωE).1) hbound
+  have he : P.withDensity (fun ω => ENNReal.ofReal (Z ω / initialGreen κ z)) =
+      finiteCenteredLaw P κ B γ G z u := by
+    apply withDensity_congr_ae
+    filter_upwards [hsource] with ω hω
+    rw [hω.2]
+  exact he ▸ h
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/ExtendedRibbonStep.lean b/lean/OAI/Probability/SLE/Gauge/ExtendedRibbonStep.lean
index 43565336e58631280a5ddc75351b5c67b7d2d779..8b0f5cb5f104822ce092401bf00308be1ab1d061 100644
--- a/lean/OAI/Probability/SLE/Gauge/ExtendedRibbonStep.lean
+++ b/lean/OAI/Probability/SLE/Gauge/ExtendedRibbonStep.lean
@@ -1,141 +1,141 @@
-import OAI.Probability.SLE.Gauge.ConditionalRibbonStep
-import OAI.Probability.SLE.Gauge.ExtendedExperimentStop
-
-namespace OAI
-
-/-! First-front stops may be infinite on failed experiments. Exhaust the
-admissible event through finite horizons, so the actual finite-stop
-restart bound applies without requiring completion on failed samples. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open RibbonCoordinates BoundaryScaleLaw Target
-
-local instance extendedRibbonPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance extendedRibbonPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-def shiftedActualTrace {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
-    (hc : ∀ ω, Continuous (γ ω)) (ω : Ω) (t : ℝ≥0) : C(ℝ≥0, ℂ) :=
-  ⟨fun s => γ ω (t + s), (hc ω).comp (continuous_const.add continuous_id)⟩
-
-def ActualRibbonAdmissible {Ω : Type*} (U : Ω → ℝ≥0 → ℝ)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (C L : ℝ≥0) (ε κ v c : ℝ) (S : Set C(ℝ≥0, ℂ)) (ω : Ω) (t : ℝ≥0) : Prop :=
-  ∃ d : RibbonHistory C L ε, d.U = U ω ∧ d.trace = γ ω ∧
-    d.forward = G ω ∧ d.inverse = F ω ∧ d.time = t ∧ d.stepEvent κ v c = S
-
-/-- The quantitative actual ribbon restart bound extends to a stopping
-time finite on the admissible past event. All future traces use the
-original random completion time, rather than its horizon truncations. -/
-theorem actual_ribbon_restart_at_extended_stop
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω)
-    (ℱ : Filtration ℝ≥0 mΩ)
-    (U : Ω → ℝ≥0 → ℝ) (γ : Ω → ℝ≥0 → ℂ)
-    (hc : ∀ ω, Continuous (γ ω)) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (C L : ℝ≥0) (ε κ v c : ℝ) (q : ℝ≥0∞) (S : Set C(ℝ≥0, ℂ))
-    (hfiniteStep : ∀ (σ : Ω → ℝ≥0)
-      (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0))) (E : Set Ω),
-      (∀ᵐ ω ∂P, ω ∈ E → ActualRibbonAdmissible U γ G F C L ε κ v c S ω (σ ω)) →
-      ∀ A : Set Ω, MeasurableSet[hσ.measurableSpace] A → A ⊆ E →
-        q * P A ≤ P (A ∩ {ω | shiftedActualTrace γ hc ω (σ ω) ∈ S}))
-    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ)
-    (E : Set Ω) (hfinite : ∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤)
-    (hstate : ∀ᵐ ω ∂P, ω ∈ E →
-      ActualRibbonAdmissible U γ G F C L ε κ v c S ω ((τ ω).untopD 0))
-    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) (hAE : A ⊆ E) :
-    q * P A ≤ P (A ∩ {ω | shiftedActualTrace γ hc ω ((τ ω).untopD 0) ∈ S}) := by
-  let Aₙ (n : ℕ) := A ∩ {ω | τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)}
-  have hmono : Monotone Aₙ := by
-    intro n k hnk ω hω
-    refine ⟨hω.1, ?_⟩
-    change τ ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0)
-    exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))
-  have hcover : A =ᵐ[P] ⋃ n, Aₙ n := by
-    filter_upwards [hfinite] with ω hω
-    simp only [mem_iUnion]
-    apply propext
-    constructor
-    · intro hωA
-      lift τ ω to ℝ≥0 using hω (hAE hωA) with t ht
-      obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ)
-      refine ⟨n, hωA, ?_⟩
-      change τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)
-      rw [← ht]
-      exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)
-    · rintro ⟨n, hn⟩
-      exact hn.1
-  have hlocal (n : ℕ) : q * P (Aₙ n) ≤
-      P (A ∩ {ω | shiftedActualTrace γ hc ω ((τ ω).untopD 0) ∈ S}) := by
-    let σ := cutoffStop τ (n : ℝ≥0)
-    have hσ := cutoffStop_stopping ℱ τ hτ (n : ℝ≥0)
-    let Eₙ := E ∩ {ω | τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)}
-    have hstateₙ : ∀ᵐ ω ∂P, ω ∈ Eₙ →
-        ActualRibbonAdmissible U γ G F C L ε κ v c S ω (σ ω) := by
-      filter_upwards [hstate] with ω hω hωE
-      dsimp only [σ]
-      rw [cutoffStop_eq_on_success τ _ ω hωE.2]
-      exact hω hωE.1
-    have hh := hfiniteStep σ hσ Eₙ hstateₙ (Aₙ n)
-      (cutoffStop_success_measurable ℱ τ hτ _ A hA)
-      (fun ω hω => ⟨hAE hω.1, hω.2⟩)
-    apply hh.trans (measure_mono ?_)
-    intro ω hω
-    refine ⟨hω.1.1, ?_⟩
-    have he : σ ω = (τ ω).untopD 0 := cutoffStop_eq_on_success τ _ ω hω.1.2
-    simpa only [mem_ofPred_eq, he] using hω.2
-  calc
-    _ = q * P (⋃ n, Aₙ n) := by rw [measure_congr hcover]
-    _ = ⨆ n, q * P (Aₙ n) := by rw [hmono.measure_iUnion, ENNReal.mul_iSup]
-    _ ≤ _ := iSup_le hlocal
-
-/-- Actual adapted ribbon passages at the possibly infinite first-front
-stops used by the dense experiment. The constants are the same uniform
-constants obtained at finite stops from the published analytic inputs. -/
-theorem uniform_conditional_actual_ribbon_step_extended
-    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω))
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw)
-    (hwiener : PublishedPlanarWienerUnitSupport)
-    (W : WienerLaw) (hW : IsWiener W)
-    (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L)
-    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) :
-    ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧
-      ∀ (v : ℝ), 0 < v → v < v₀ → ∀ (τ : Ω → WithTop ℝ≥0)
-      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) τ),
-      ∀ (S : Set C(ℝ≥0, ℂ)), MeasurableSet S → ∀ E : Set Ω,
-      (∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤) →
-      (∀ᵐ ω ∂P, ω ∈ E → ActualRibbonAdmissible
-        (fun ω t => Real.sqrt κ * B t ω) γ G F C L ε κ v c S ω ((τ ω).untopD 0)) →
-      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → A ⊆ E →
-        ENNReal.ofReal q * P A ≤ P (A ∩ {ω |
-          shiftedActualTrace γ (fun ω => (hw ω).1) ω ((τ ω).untopD 0) ∈ S}) := by
-  obtain ⟨q, c, v₀, hq, hcm, hv₀, hstep⟩ :=
-    uniform_conditional_actual_ribbon_step P κ hκ hκ8 B hB hm hc γ G F hw
-      hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW
-      C L hC hL ε hε hεsmall
-  refine ⟨q, c, v₀, hq, hcm, hv₀, ?_⟩
-  intro v hv hvcut τ hτ S hS E hfinite hstate A hA hAE
-  apply actual_ribbon_restart_at_extended_stop P _
-    (fun ω t => Real.sqrt κ * B t ω) γ (fun ω => (hw ω).1) G F C L ε κ v c
-    (ENNReal.ofReal q) S ?_ τ hτ E hfinite hstate A hA hAE
-  intro σ hσ E' hstates A' hA' hAE'
-  exact hstep v hv hvcut σ hσ S hS E' hstates A' hA' hAE'
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ConditionalRibbonStep
+import OAI.Probability.SLE.Gauge.ExtendedExperimentStop
+
+namespace OAI
+
+/-! First-front stops may be infinite on failed experiments. Exhaust the
+admissible event through finite horizons, so the actual finite-stop
+restart bound applies without requiring completion on failed samples. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open RibbonCoordinates BoundaryScaleLaw Target
+
+local instance extendedRibbonPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance extendedRibbonPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+def shiftedActualTrace {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
+    (hc : ∀ ω, Continuous (γ ω)) (ω : Ω) (t : ℝ≥0) : C(ℝ≥0, ℂ) :=
+  ⟨fun s => γ ω (t + s), (hc ω).comp (continuous_const.add continuous_id)⟩
+
+def ActualRibbonAdmissible {Ω : Type*} (U : Ω → ℝ≥0 → ℝ)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (C L : ℝ≥0) (ε κ v c : ℝ) (S : Set C(ℝ≥0, ℂ)) (ω : Ω) (t : ℝ≥0) : Prop :=
+  ∃ d : RibbonHistory C L ε, d.U = U ω ∧ d.trace = γ ω ∧
+    d.forward = G ω ∧ d.inverse = F ω ∧ d.time = t ∧ d.stepEvent κ v c = S
+
+/-- The quantitative actual ribbon restart bound extends to a stopping
+time finite on the admissible past event. All future traces use the
+original random completion time, rather than its horizon truncations. -/
+theorem actual_ribbon_restart_at_extended_stop
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω)
+    (ℱ : Filtration ℝ≥0 mΩ)
+    (U : Ω → ℝ≥0 → ℝ) (γ : Ω → ℝ≥0 → ℂ)
+    (hc : ∀ ω, Continuous (γ ω)) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (C L : ℝ≥0) (ε κ v c : ℝ) (q : ℝ≥0∞) (S : Set C(ℝ≥0, ℂ))
+    (hfiniteStep : ∀ (σ : Ω → ℝ≥0)
+      (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0))) (E : Set Ω),
+      (∀ᵐ ω ∂P, ω ∈ E → ActualRibbonAdmissible U γ G F C L ε κ v c S ω (σ ω)) →
+      ∀ A : Set Ω, MeasurableSet[hσ.measurableSpace] A → A ⊆ E →
+        q * P A ≤ P (A ∩ {ω | shiftedActualTrace γ hc ω (σ ω) ∈ S}))
+    (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ)
+    (E : Set Ω) (hfinite : ∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤)
+    (hstate : ∀ᵐ ω ∂P, ω ∈ E →
+      ActualRibbonAdmissible U γ G F C L ε κ v c S ω ((τ ω).untopD 0))
+    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) (hAE : A ⊆ E) :
+    q * P A ≤ P (A ∩ {ω | shiftedActualTrace γ hc ω ((τ ω).untopD 0) ∈ S}) := by
+  let Aₙ (n : ℕ) := A ∩ {ω | τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)}
+  have hmono : Monotone Aₙ := by
+    intro n k hnk ω hω
+    refine ⟨hω.1, ?_⟩
+    change τ ω ≤ ((k : ℝ≥0) : WithTop ℝ≥0)
+    exact hω.2.trans (WithTop.coe_le_coe.mpr (by exact_mod_cast hnk))
+  have hcover : A =ᵐ[P] ⋃ n, Aₙ n := by
+    filter_upwards [hfinite] with ω hω
+    simp only [mem_iUnion]
+    apply propext
+    constructor
+    · intro hωA
+      lift τ ω to ℝ≥0 using hω (hAE hωA) with t ht
+      obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ)
+      refine ⟨n, hωA, ?_⟩
+      change τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)
+      rw [← ht]
+      exact WithTop.coe_le_coe.mpr (by exact_mod_cast hn)
+    · rintro ⟨n, hn⟩
+      exact hn.1
+  have hlocal (n : ℕ) : q * P (Aₙ n) ≤
+      P (A ∩ {ω | shiftedActualTrace γ hc ω ((τ ω).untopD 0) ∈ S}) := by
+    let σ := cutoffStop τ (n : ℝ≥0)
+    have hσ := cutoffStop_stopping ℱ τ hτ (n : ℝ≥0)
+    let Eₙ := E ∩ {ω | τ ω ≤ ((n : ℝ≥0) : WithTop ℝ≥0)}
+    have hstateₙ : ∀ᵐ ω ∂P, ω ∈ Eₙ →
+        ActualRibbonAdmissible U γ G F C L ε κ v c S ω (σ ω) := by
+      filter_upwards [hstate] with ω hω hωE
+      dsimp only [σ]
+      rw [cutoffStop_eq_on_success τ _ ω hωE.2]
+      exact hω hωE.1
+    have hh := hfiniteStep σ hσ Eₙ hstateₙ (Aₙ n)
+      (cutoffStop_success_measurable ℱ τ hτ _ A hA)
+      (fun ω hω => ⟨hAE hω.1, hω.2⟩)
+    apply hh.trans (measure_mono ?_)
+    intro ω hω
+    refine ⟨hω.1.1, ?_⟩
+    have he : σ ω = (τ ω).untopD 0 := cutoffStop_eq_on_success τ _ ω hω.1.2
+    simpa only [mem_ofPred_eq, he] using hω.2
+  calc
+    _ = q * P (⋃ n, Aₙ n) := by rw [measure_congr hcover]
+    _ = ⨆ n, q * P (Aₙ n) := by rw [hmono.measure_iUnion, ENNReal.mul_iSup]
+    _ ≤ _ := iSup_le hlocal
+
+/-- Actual adapted ribbon passages at the possibly infinite first-front
+stops used by the dense experiment. The constants are the same uniform
+constants obtained at finite stops from the published analytic inputs. -/
+theorem uniform_conditional_actual_ribbon_step_extended
+    {Ω : Type} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω))
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw)
+    (hwiener : PublishedPlanarWienerUnitSupport)
+    (W : WienerLaw) (hW : IsWiener W)
+    (C L : ℝ≥0) (hC : 0 < C) (hL : 0 < L)
+    (ε : ℝ) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8) :
+    ∃ q c v₀ : ℝ, 0 < q ∧ 0 < c ∧ 0 < v₀ ∧
+      ∀ (v : ℝ), 0 < v → v < v₀ → ∀ (τ : Ω → WithTop ℝ≥0)
+      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) τ),
+      ∀ (S : Set C(ℝ≥0, ℂ)), MeasurableSet S → ∀ E : Set Ω,
+      (∀ᵐ ω ∂P, ω ∈ E → τ ω ≠ ⊤) →
+      (∀ᵐ ω ∂P, ω ∈ E → ActualRibbonAdmissible
+        (fun ω t => Real.sqrt κ * B t ω) γ G F C L ε κ v c S ω ((τ ω).untopD 0)) →
+      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A → A ⊆ E →
+        ENNReal.ofReal q * P A ≤ P (A ∩ {ω |
+          shiftedActualTrace γ (fun ω => (hw ω).1) ω ((τ ω).untopD 0) ∈ S}) := by
+  obtain ⟨q, c, v₀, hq, hcm, hv₀, hstep⟩ :=
+    uniform_conditional_actual_ribbon_step P κ hκ hκ8 B hB hm hc γ G F hw
+      hcompact hboundary hsupport hplanar hstraight hexit hwiener W hW
+      C L hC hL ε hε hεsmall
+  refine ⟨q, c, v₀, hq, hcm, hv₀, ?_⟩
+  intro v hv hvcut τ hτ S hS E hfinite hstate A hA hAE
+  apply actual_ribbon_restart_at_extended_stop P _
+    (fun ω t => Real.sqrt κ * B t ω) γ (fun ω => (hw ω).1) G F C L ε κ v c
+    (ENNReal.ofReal q) S ?_ τ hτ E hfinite hstate A hA hAE
+  intro σ hσ E' hstates A' hA' hAE'
+  exact hstep v hv hvcut σ hσ S hS E' hstates A' hA' hAE'
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/FiniteCoverAssembly.lean b/lean/OAI/Probability/SLE/Gauge/FiniteCoverAssembly.lean
index e0656984cfa6596fc2d00d31d2b030d0d71c1534..009471d7b364d8e76d8733b776992438e92be2d0 100644
--- a/lean/OAI/Probability/SLE/Gauge/FiniteCoverAssembly.lean
+++ b/lean/OAI/Probability/SLE/Gauge/FiniteCoverAssembly.lean
@@ -1,148 +1,148 @@
-import OAI.Probability.SLE.Gauge.DenseCoverCore
-import OAI.Probability.SLE.Gauge.ExpectedCoverCharge
-
-namespace OAI
-
-/-! Finite selected/missed covers and measurable envelopes for their costs. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- The one-mesh charge is a measure, so disjoint candidates can be summed. -/
-def wholeSausageMeasure (κ : ℝ) (γ : ℝ≥0 → ℂ) (v : ℝ) : Measure ℂ :=
-  ENNReal.ofReal (v ^ (-codimension κ)) • volume.restrict (wholeSausageSet γ v)
-
-lemma wholeSausageMeasure_apply (κ : ℝ) (γ : ℝ≥0 → ℂ) (v : ℝ)
-    {A : Set ℂ} (hA : MeasurableSet A) :
-    wholeSausageMeasure κ γ v A = wholeSausageCharge κ γ v A := by
-  simp [wholeSausageMeasure, wholeSausageCharge, Measure.restrict_apply hA]
-
-/-- Pointwise existence of finite covers suffices. Their selection need not be
-measurable when a measurable envelope controls their costs. -/
-theorem hausdorff_le_liminf_finite_cover_envelope
-    (h : ℝ → ℝ) (E : Set ℂ) (δ : ℕ → ℝ≥0∞)
-    (hδ : Tendsto δ atTop (𝓝 0)) (Z : ℕ → ℝ≥0∞)
-    (hcovers : ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
-      E ⊆ ⋃ j, U j ∧ (∀ j, ediam (U j) ≤ δ n) ∧
-        ∑ j, extendedGauge h (ediam (U j)) ≤ Z n) :
-    hausdorffGauge h E ≤ liminf Z atTop := by
-  choose N U hc hd hz using hcovers
-  exact (Measure.mkMetric_le_liminf_sum E δ hδ U
-    (Eventually.of_forall hd) (Eventually.of_forall hc) (extendedGauge h)).trans
-      (liminf_le_liminf (Eventually.of_forall hz))
-
-/-- Fatou applied to the cost envelope, rather than to a chosen random cover. -/
-theorem expected_hausdorff_le_of_finite_cover_envelopes
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (h : ℝ → ℝ) (E : Ω → Set ℂ) (δ : ℕ → ℝ≥0∞)
-    (hδ : Tendsto δ atTop (𝓝 0)) (Z : ℕ → Ω → ℝ≥0∞)
-    (hZ : ∀ n, AEMeasurable (Z n) P)
-    (hcovers : ∀ᵐ ω ∂P, ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
-      E ω ⊆ ⋃ j, U j ∧ (∀ j, ediam (U j) ≤ δ n) ∧
-        ∑ j, extendedGauge h (ediam (U j)) ≤ Z n ω)
-    (C : ℝ≥0∞) (hcost : liminf (fun n => ∫⁻ ω, Z n ω ∂P) atTop ≤ C) :
-    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤ C := by
-  calc
-    _ ≤ ∫⁻ ω, liminf (fun n => Z n ω) atTop ∂P := by
-      apply lintegral_mono_ae
-      filter_upwards [hcovers] with ω hc
-      exact hausdorff_le_liminf_finite_cover_envelope h (E ω) δ hδ
-        (fun n => Z n ω) hc
-    _ ≤ liminf (fun n => ∫⁻ ω, Z n ω ∂P) atTop := lintegral_liminf_le' hZ
-    _ ≤ C := hcost
-
-/-- Disjoint candidate disks and all missed cells give a genuine finite cover
-with cost paid by one spatial measure and the missed-cell sum. -/
-theorem exists_finite_selected_missed_cover
-    {ι η : Type*} (I : Finset ι) (M : Finset η)
-    (x : ι → ℂ) (r : ι → ℝ) (Q : η → Set ℂ)
-    (E A : Set ℂ) (ν : Measure ℂ) (h : ℝ → ℝ)
-    (R : ℝ) (hr : ∀ i ∈ I, r i ≤ R) (δ C : ℝ≥0∞)
-    (hA : ∀ i ∈ I, closedBall (x i) (r i) ⊆ A)
-    (hcost : ∀ i ∈ I,
-      extendedGauge h (ediam (closedBall (x i) (4 * r i))) ≤
-        C * ν (closedBall (x i) (r i)))
-    (hdisk : ∀ i ∈ I, ediam (closedBall (x i) (4 * r i)) ≤ δ)
-    (hmiss : ∀ k ∈ M, ediam (Q k) ≤ δ)
-    (hcell : ∀ z ∈ E, (∃ i ∈ I, z ∈ closedBall (x i) (r i)) ∨
-      ∃ k ∈ M, z ∈ Q k) :
-    ∃ (N : ℕ) (U : Fin N → Set ℂ), E ⊆ ⋃ j, U j ∧
-      (∀ j, ediam (U j) ≤ δ) ∧
-      ∑ j, extendedGauge h (ediam (U j)) ≤
-        C * ν A + ∑ k ∈ M, extendedGauge h (ediam (Q k)) := by
-  classical
-  obtain ⟨J, hJI, _, hselect, hcharge⟩ :=
-    select_disjoint_candidates_and_charge I x r R hr ν A hA
-      (fun i => extendedGauge h (ediam (closedBall (x i) (4 * r i)))) C hcost
-  let W : ({i // i ∈ J} ⊕ {k // k ∈ M}) → Set ℂ :=
-    Sum.elim (fun i => closedBall (x i) (4 * r i)) (fun k => Q k)
-  let e := Fintype.equivFin ({i // i ∈ J} ⊕ {k // k ∈ M})
-  refine ⟨_, W ∘ e.symm, ?_, ?_, ?_⟩
-  · intro z hz
-    rcases selected_and_missed_cover I J M
-        (fun i => closedBall (x i) (r i))
-        (fun i => closedBall (x i) (4 * r i)) Q E hselect hcell hz with hz | hz
-    · obtain ⟨i, hi, hzi⟩ := mem_iUnion₂.mp hz
-      exact mem_iUnion.mpr ⟨e (Sum.inl ⟨i, hi⟩), by simpa [W] using hzi⟩
-    · obtain ⟨k, hk, hzk⟩ := mem_iUnion₂.mp hz
-      exact mem_iUnion.mpr ⟨e (Sum.inr ⟨k, hk⟩), by simpa [W] using hzk⟩
-  · intro j
-    change ediam (W (e.symm j)) ≤ δ
-    rcases e.symm j with i | k
-    · exact hdisk i (hJI i.property)
-    · exact hmiss k k.property
-  · calc
-      _ = (∑ i ∈ J, extendedGauge h (ediam (closedBall (x i) (4 * r i)))) +
-          ∑ k ∈ M, extendedGauge h (ediam (Q k)) := by
-        rw [← Equiv.sum_comp e]
-        simp only [Function.comp_apply, Equiv.symm_apply_apply, Fintype.sum_sum_type,
-          W, Sum.elim_inl, Sum.elim_inr]
-        exact congrArg₂ (· + ·)
-          (Finset.sum_coe_sort J
-            (fun i => extendedGauge h (ediam (closedBall (x i) (4 * r i)))))
-          (Finset.sum_coe_sort M (fun k => extendedGauge h (ediam (Q k))))
-      _ ≤ _ := add_le_add hcharge le_rfl
-
-/-- The finite cover can be chosen independently at every sample and scale.
-Only its common sausage charge and missed-cell envelope need measurability. -/
-theorem expected_hausdorff_le_of_finite_sausage_covers
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (h : ℝ → ℝ) (E : Ω → Set ℂ) (δ : ℕ → ℝ≥0∞)
-    (hδ : Tendsto δ atTop (𝓝 0))
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (R : ℝ) (C : ℝ≥0)
-    (error : ℕ → Ω → ℝ≥0∞) (herr : ∀ n, AEMeasurable (error n) P)
-    (herrlim : Tendsto (fun n => ∫⁻ ω, error n ω ∂P) atTop (𝓝 0))
-    (hcovers : ∀ᵐ ω ∂P, ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
-      E ω ⊆ ⋃ j, U j ∧ (∀ j, ediam (U j) ≤ δ n) ∧
-        ∑ j, extendedGauge h (ediam (U j)) ≤
-          (C : ℝ≥0∞) * wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω) :
-    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤
-      (C : ℝ≥0∞) * wholeSausageChargeBound κ R := by
-  obtain ⟨_, hsausage⟩ := source_cover_charge P κ hκ hκ8 γ hSLE R
-  let Z (n : ℕ) (ω : Ω) := (C : ℝ≥0∞) *
-    wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω
-  have hZ (n : ℕ) : AEMeasurable (Z n) P :=
-    ((hsausage (v n) (hv n)).1.const_mul _).add (herr n)
-  have hcost (n : ℕ) : (∫⁻ ω, Z n ω ∂P) ≤
-      (C : ℝ≥0∞) * wholeSausageChargeBound κ R + ∫⁻ ω, error n ω ∂P := by
-    dsimp [Z]
-    rw [lintegral_add_right' _ (herr n),
-      lintegral_const_mul' _ _ ENNReal.coe_ne_top]
-    exact add_le_add (mul_le_mul' le_rfl (hsausage (v n) (hv n)).2) le_rfl
-  apply expected_hausdorff_le_of_finite_cover_envelopes P h E δ hδ Z hZ hcovers
-  have ht : Tendsto (fun n => (C : ℝ≥0∞) * wholeSausageChargeBound κ R +
-      ∫⁻ ω, error n ω ∂P) atTop
-      (𝓝 ((C : ℝ≥0∞) * wholeSausageChargeBound κ R)) := by
-    simpa only [add_zero] using tendsto_const_nhds.add herrlim
-  exact (liminf_le_liminf (Eventually.of_forall hcost)).trans_eq ht.liminf_eq
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.DenseCoverCore
+import OAI.Probability.SLE.Gauge.ExpectedCoverCharge
+
+namespace OAI
+
+/-! Finite selected/missed covers and measurable envelopes for their costs. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- The one-mesh charge is a measure, so disjoint candidates can be summed. -/
+def wholeSausageMeasure (κ : ℝ) (γ : ℝ≥0 → ℂ) (v : ℝ) : Measure ℂ :=
+  ENNReal.ofReal (v ^ (-codimension κ)) • volume.restrict (wholeSausageSet γ v)
+
+lemma wholeSausageMeasure_apply (κ : ℝ) (γ : ℝ≥0 → ℂ) (v : ℝ)
+    {A : Set ℂ} (hA : MeasurableSet A) :
+    wholeSausageMeasure κ γ v A = wholeSausageCharge κ γ v A := by
+  simp [wholeSausageMeasure, wholeSausageCharge, Measure.restrict_apply hA]
+
+/-- Pointwise existence of finite covers suffices. Their selection need not be
+measurable when a measurable envelope controls their costs. -/
+theorem hausdorff_le_liminf_finite_cover_envelope
+    (h : ℝ → ℝ) (E : Set ℂ) (δ : ℕ → ℝ≥0∞)
+    (hδ : Tendsto δ atTop (𝓝 0)) (Z : ℕ → ℝ≥0∞)
+    (hcovers : ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
+      E ⊆ ⋃ j, U j ∧ (∀ j, ediam (U j) ≤ δ n) ∧
+        ∑ j, extendedGauge h (ediam (U j)) ≤ Z n) :
+    hausdorffGauge h E ≤ liminf Z atTop := by
+  choose N U hc hd hz using hcovers
+  exact (Measure.mkMetric_le_liminf_sum E δ hδ U
+    (Eventually.of_forall hd) (Eventually.of_forall hc) (extendedGauge h)).trans
+      (liminf_le_liminf (Eventually.of_forall hz))
+
+/-- Fatou applied to the cost envelope, rather than to a chosen random cover. -/
+theorem expected_hausdorff_le_of_finite_cover_envelopes
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (h : ℝ → ℝ) (E : Ω → Set ℂ) (δ : ℕ → ℝ≥0∞)
+    (hδ : Tendsto δ atTop (𝓝 0)) (Z : ℕ → Ω → ℝ≥0∞)
+    (hZ : ∀ n, AEMeasurable (Z n) P)
+    (hcovers : ∀ᵐ ω ∂P, ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
+      E ω ⊆ ⋃ j, U j ∧ (∀ j, ediam (U j) ≤ δ n) ∧
+        ∑ j, extendedGauge h (ediam (U j)) ≤ Z n ω)
+    (C : ℝ≥0∞) (hcost : liminf (fun n => ∫⁻ ω, Z n ω ∂P) atTop ≤ C) :
+    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤ C := by
+  calc
+    _ ≤ ∫⁻ ω, liminf (fun n => Z n ω) atTop ∂P := by
+      apply lintegral_mono_ae
+      filter_upwards [hcovers] with ω hc
+      exact hausdorff_le_liminf_finite_cover_envelope h (E ω) δ hδ
+        (fun n => Z n ω) hc
+    _ ≤ liminf (fun n => ∫⁻ ω, Z n ω ∂P) atTop := lintegral_liminf_le' hZ
+    _ ≤ C := hcost
+
+/-- Disjoint candidate disks and all missed cells give a genuine finite cover
+with cost paid by one spatial measure and the missed-cell sum. -/
+theorem exists_finite_selected_missed_cover
+    {ι η : Type*} (I : Finset ι) (M : Finset η)
+    (x : ι → ℂ) (r : ι → ℝ) (Q : η → Set ℂ)
+    (E A : Set ℂ) (ν : Measure ℂ) (h : ℝ → ℝ)
+    (R : ℝ) (hr : ∀ i ∈ I, r i ≤ R) (δ C : ℝ≥0∞)
+    (hA : ∀ i ∈ I, closedBall (x i) (r i) ⊆ A)
+    (hcost : ∀ i ∈ I,
+      extendedGauge h (ediam (closedBall (x i) (4 * r i))) ≤
+        C * ν (closedBall (x i) (r i)))
+    (hdisk : ∀ i ∈ I, ediam (closedBall (x i) (4 * r i)) ≤ δ)
+    (hmiss : ∀ k ∈ M, ediam (Q k) ≤ δ)
+    (hcell : ∀ z ∈ E, (∃ i ∈ I, z ∈ closedBall (x i) (r i)) ∨
+      ∃ k ∈ M, z ∈ Q k) :
+    ∃ (N : ℕ) (U : Fin N → Set ℂ), E ⊆ ⋃ j, U j ∧
+      (∀ j, ediam (U j) ≤ δ) ∧
+      ∑ j, extendedGauge h (ediam (U j)) ≤
+        C * ν A + ∑ k ∈ M, extendedGauge h (ediam (Q k)) := by
+  classical
+  obtain ⟨J, hJI, _, hselect, hcharge⟩ :=
+    select_disjoint_candidates_and_charge I x r R hr ν A hA
+      (fun i => extendedGauge h (ediam (closedBall (x i) (4 * r i)))) C hcost
+  let W : ({i // i ∈ J} ⊕ {k // k ∈ M}) → Set ℂ :=
+    Sum.elim (fun i => closedBall (x i) (4 * r i)) (fun k => Q k)
+  let e := Fintype.equivFin ({i // i ∈ J} ⊕ {k // k ∈ M})
+  refine ⟨_, W ∘ e.symm, ?_, ?_, ?_⟩
+  · intro z hz
+    rcases selected_and_missed_cover I J M
+        (fun i => closedBall (x i) (r i))
+        (fun i => closedBall (x i) (4 * r i)) Q E hselect hcell hz with hz | hz
+    · obtain ⟨i, hi, hzi⟩ := mem_iUnion₂.mp hz
+      exact mem_iUnion.mpr ⟨e (Sum.inl ⟨i, hi⟩), by simpa [W] using hzi⟩
+    · obtain ⟨k, hk, hzk⟩ := mem_iUnion₂.mp hz
+      exact mem_iUnion.mpr ⟨e (Sum.inr ⟨k, hk⟩), by simpa [W] using hzk⟩
+  · intro j
+    change ediam (W (e.symm j)) ≤ δ
+    rcases e.symm j with i | k
+    · exact hdisk i (hJI i.property)
+    · exact hmiss k k.property
+  · calc
+      _ = (∑ i ∈ J, extendedGauge h (ediam (closedBall (x i) (4 * r i)))) +
+          ∑ k ∈ M, extendedGauge h (ediam (Q k)) := by
+        rw [← Equiv.sum_comp e]
+        simp only [Function.comp_apply, Equiv.symm_apply_apply, Fintype.sum_sum_type,
+          W, Sum.elim_inl, Sum.elim_inr]
+        exact congrArg₂ (· + ·)
+          (Finset.sum_coe_sort J
+            (fun i => extendedGauge h (ediam (closedBall (x i) (4 * r i)))))
+          (Finset.sum_coe_sort M (fun k => extendedGauge h (ediam (Q k))))
+      _ ≤ _ := add_le_add hcharge le_rfl
+
+/-- The finite cover can be chosen independently at every sample and scale.
+Only its common sausage charge and missed-cell envelope need measurability. -/
+theorem expected_hausdorff_le_of_finite_sausage_covers
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (h : ℝ → ℝ) (E : Ω → Set ℂ) (δ : ℕ → ℝ≥0∞)
+    (hδ : Tendsto δ atTop (𝓝 0))
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (R : ℝ) (C : ℝ≥0)
+    (error : ℕ → Ω → ℝ≥0∞) (herr : ∀ n, AEMeasurable (error n) P)
+    (herrlim : Tendsto (fun n => ∫⁻ ω, error n ω ∂P) atTop (𝓝 0))
+    (hcovers : ∀ᵐ ω ∂P, ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
+      E ω ⊆ ⋃ j, U j ∧ (∀ j, ediam (U j) ≤ δ n) ∧
+        ∑ j, extendedGauge h (ediam (U j)) ≤
+          (C : ℝ≥0∞) * wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω) :
+    (∫⁻ ω, hausdorffGauge h (E ω) ∂P) ≤
+      (C : ℝ≥0∞) * wholeSausageChargeBound κ R := by
+  obtain ⟨_, hsausage⟩ := source_cover_charge P κ hκ hκ8 γ hSLE R
+  let Z (n : ℕ) (ω : Ω) := (C : ℝ≥0∞) *
+    wholeSausageCharge κ (γ ω) (v n) (closedBall 0 R) + error n ω
+  have hZ (n : ℕ) : AEMeasurable (Z n) P :=
+    ((hsausage (v n) (hv n)).1.const_mul _).add (herr n)
+  have hcost (n : ℕ) : (∫⁻ ω, Z n ω ∂P) ≤
+      (C : ℝ≥0∞) * wholeSausageChargeBound κ R + ∫⁻ ω, error n ω ∂P := by
+    dsimp [Z]
+    rw [lintegral_add_right' _ (herr n),
+      lintegral_const_mul' _ _ ENNReal.coe_ne_top]
+    exact add_le_add (mul_le_mul' le_rfl (hsausage (v n) (hv n)).2) le_rfl
+  apply expected_hausdorff_le_of_finite_cover_envelopes P h E δ hδ Z hZ hcovers
+  have ht : Tendsto (fun n => (C : ℝ≥0∞) * wholeSausageChargeBound κ R +
+      ∫⁻ ω, error n ω ∂P) atTop
+      (𝓝 ((C : ℝ≥0∞) * wholeSausageChargeBound κ R)) := by
+    simpa only [add_zero] using tendsto_const_nhds.add herrlim
+  exact (liminf_le_liminf (Eventually.of_forall hcost)).trans_eq ht.liminf_eq
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/FiniteMomentBounds.lean b/lean/OAI/Probability/SLE/Gauge/FiniteMomentBounds.lean
index 0ab2074d3f60532defdc45ca14e2bbaa43a88718..e15c88980c27156fc2295ce68e361b9a10165799 100644
--- a/lean/OAI/Probability/SLE/Gauge/FiniteMomentBounds.lean
+++ b/lean/OAI/Probability/SLE/Gauge/FiniteMomentBounds.lean
@@ -1,73 +1,73 @@
-import OAI.Probability.SLE.Gauge.FiniteMomentIdentification
-
-namespace OAI
-
-/-! Finite and uniformly bounded prescribed finite-level moments. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-include C in
-lemma levelCoefficient_aestronglyMeasurable
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
-    AEStronglyMeasurable (Target.levelCoefficient κ W ℓ m) (Target.angleLaw κ) := by
-  apply aestronglyMeasurable_of_tendsto_ae atTop
-    (fun n : ℕ => C.clippedCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw W hW
-      ℓ hℓ n (Nat.cast_nonneg n) m)
-  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-  exact C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
-
-include C in
-theorem uniform_finiteMoment_bound
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) (m : ℕ) :
-    ∃ M : ℝ, 0≤M ∧ ∀ ℓ : ℝ, 0≤ℓ → 0≤Target.finiteMoment κ W m ℓ ∧
-      Target.finiteMoment κ W m ℓ≤M := by
-  obtain ⟨M,hM,hbound⟩ := C.uniform_clippedCoefficient_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
-  let g := fun θ => M/(Real.sin θ)^greenExponent κ
-  have hg : Integrable g (Target.angleLaw κ) := by
-    simpa only [g,div_eq_mul_inv,mul_one,one_mul] using (Target.angle_inverse_sine_integrable κ).const_mul M
-  have hg0 : 0≤ᵐ[Target.angleLaw κ] g := by
-    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-    exact div_nonneg hM (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)
-  refine ⟨∫ θ, g θ ∂Target.angleLaw κ,integral_nonneg_of_ae hg0,?_⟩
-  intro ℓ hℓ
-  have hlim := C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW
-  have hb : ∀ᵐ θ ∂Target.angleLaw κ, 0≤Target.levelCoefficient κ W ℓ m θ ∧
-      Target.levelCoefficient κ W ℓ m θ≤g θ := by
-    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-    have ht := hlim θ hθ ℓ hℓ m
-    exact ⟨ge_of_tendsto ht (Eventually.of_forall (fun n =>
-      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).1)),
-      le_of_tendsto ht (Eventually.of_forall (fun n =>
-      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).2))⟩
-  have hi : Integrable (Target.levelCoefficient κ W ℓ m) (Target.angleLaw κ) := by
-    apply hg.mono' (C.levelCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m)
-    filter_upwards [hb] with θ hθ
-    rw [Real.norm_of_nonneg hθ.1]
-    exact hθ.2
-  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m]
-  exact ⟨integral_nonneg_of_ae (hb.mono (fun _ h => h.1)),
-    integral_mono_ae hi hg (hb.mono (fun _ h => h.2))⟩
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.FiniteMomentIdentification
+
+namespace OAI
+
+/-! Finite and uniformly bounded prescribed finite-level moments. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+include C in
+lemma levelCoefficient_aestronglyMeasurable
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
+    AEStronglyMeasurable (Target.levelCoefficient κ W ℓ m) (Target.angleLaw κ) := by
+  apply aestronglyMeasurable_of_tendsto_ae atTop
+    (fun n : ℕ => C.clippedCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw W hW
+      ℓ hℓ n (Nat.cast_nonneg n) m)
+  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+  exact C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
+
+include C in
+theorem uniform_finiteMoment_bound
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W) (m : ℕ) :
+    ∃ M : ℝ, 0≤M ∧ ∀ ℓ : ℝ, 0≤ℓ → 0≤Target.finiteMoment κ W m ℓ ∧
+      Target.finiteMoment κ W m ℓ≤M := by
+  obtain ⟨M,hM,hbound⟩ := C.uniform_clippedCoefficient_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
+  let g := fun θ => M/(Real.sin θ)^greenExponent κ
+  have hg : Integrable g (Target.angleLaw κ) := by
+    simpa only [g,div_eq_mul_inv,mul_one,one_mul] using (Target.angle_inverse_sine_integrable κ).const_mul M
+  have hg0 : 0≤ᵐ[Target.angleLaw κ] g := by
+    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+    exact div_nonneg hM (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2).le _)
+  refine ⟨∫ θ, g θ ∂Target.angleLaw κ,integral_nonneg_of_ae hg0,?_⟩
+  intro ℓ hℓ
+  have hlim := C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW
+  have hb : ∀ᵐ θ ∂Target.angleLaw κ, 0≤Target.levelCoefficient κ W ℓ m θ ∧
+      Target.levelCoefficient κ W ℓ m θ≤g θ := by
+    filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+    have ht := hlim θ hθ ℓ hℓ m
+    exact ⟨ge_of_tendsto ht (Eventually.of_forall (fun n =>
+      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).1)),
+      le_of_tendsto ht (Eventually.of_forall (fun n =>
+      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).2))⟩
+  have hi : Integrable (Target.levelCoefficient κ W ℓ m) (Target.angleLaw κ) := by
+    apply hg.mono' (C.levelCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m)
+    filter_upwards [hb] with θ hθ
+    rw [Real.norm_of_nonneg hθ.1]
+    exact hθ.2
+  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m]
+  exact ⟨integral_nonneg_of_ae (hb.mono (fun _ h => h.1)),
+    integral_mono_ae hi hg (hb.mono (fun _ h => h.2))⟩
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/FiniteMomentIdentification.lean b/lean/OAI/Probability/SLE/Gauge/FiniteMomentIdentification.lean
index e2e838d2c757924e0f9334b37d864cffb777b48f..84cdf051727e9d585c58c3e6a37376a302856f6f 100644
--- a/lean/OAI/Probability/SLE/Gauge/FiniteMomentIdentification.lean
+++ b/lean/OAI/Probability/SLE/Gauge/FiniteMomentIdentification.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Geometry.CoefficientAngularBound
-
-namespace OAI
-
-/-! The paper's iterated finite-level coefficient equals the invariant
-angle average of the actual stopped canonical disk moment. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Target
-
-def levelCoefficient (κ : ℝ) (W : WienerLaw) (ℓ : ℝ) (m : ℕ) (θ : ℝ) : ℝ :=
-  limUnder atTop (fun n : ℕ => clippedCoefficient κ W ℓ n m θ)
-
-end SLEExactGauge.Target
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-include C in
-lemma clippedCoefficient_aestronglyMeasurable
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (ℓ : ℝ) (hℓ : 0≤ℓ) (u : ℝ) (hu : 0≤u) (m : ℕ) :
-    AEStronglyMeasurable (Target.clippedCoefficient κ W ℓ u m) (Target.angleLaw κ) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  apply aestronglyMeasurable_of_tendsto_ae atTop
-    (fun n => (Target.aemeasurable_coefficient_integrand κ ℓ u W n m).aestronglyMeasurable.integral_prod_right')
-  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-  exact C.clippedCoefficient_tendsto hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ u hu m
-
-include C in
-lemma levelCoefficient_tendsto
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
-    Tendsto (fun n : ℕ => Target.clippedCoefficient κ W ℓ n m θ) atTop
-      (𝓝 (Target.levelCoefficient κ W ℓ m θ)) := by
-  have h := C.clippedCoefficient_unclip_limit hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
-  simpa only [Target.levelCoefficient,h.limUnder_eq] using h
-
-lemma levelCoefficient_eq_mass
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
-    Target.levelCoefficient κ W ℓ m θ =
-      ∫ ω, Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ *
-        (massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω)
-          (C.diskTerminalScalar θ hθ ω)
-          (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ))^m ∂P :=
-  (C.clippedCoefficient_unclip_limit hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m).limUnder_eq
-
-include C in
-theorem finiteMoment_eq_levelCoefficientIntegral
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
-    Target.finiteMoment κ W m ℓ =
-      ∫ θ, Target.levelCoefficient κ W ℓ m θ ∂Target.angleLaw κ := by
-  obtain ⟨M,hM,hbound⟩ := C.uniform_clippedCoefficient_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
-  have hi : Integrable (fun θ => M/(Real.sin θ)^greenExponent κ) (Target.angleLaw κ) := by
-    simpa only [div_eq_mul_inv,mul_one,one_mul] using (Target.angle_inverse_sine_integrable κ).const_mul M
-  have ht : Tendsto (fun n : ℕ => ∫ θ, Target.clippedCoefficient κ W ℓ n m θ ∂Target.angleLaw κ)
-      atTop (𝓝 (∫ θ, Target.levelCoefficient κ W ℓ m θ ∂Target.angleLaw κ)) := by
-    apply tendsto_integral_of_dominated_convergence (fun θ => M/(Real.sin θ)^greenExponent κ)
-    · intro n
-      exact C.clippedCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw W hW ℓ hℓ n (Nat.cast_nonneg n) m
-    · exact hi
-    · intro n
-      filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-      have hb := hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)
-      rw [Real.norm_of_nonneg hb.1]
-      exact hb.2
-    · filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-      exact C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
-  rw [C.finiteMoment_eq_outer_clipped_limit hB hκ hκ8 hc h0 F hw W hW ℓ hℓ m]
-  exact ht.limUnder_eq
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CoefficientAngularBound
+
+namespace OAI
+
+/-! The paper's iterated finite-level coefficient equals the invariant
+angle average of the actual stopped canonical disk moment. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Target
+
+def levelCoefficient (κ : ℝ) (W : WienerLaw) (ℓ : ℝ) (m : ℕ) (θ : ℝ) : ℝ :=
+  limUnder atTop (fun n : ℕ => clippedCoefficient κ W ℓ n m θ)
+
+end SLEExactGauge.Target
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+include C in
+lemma clippedCoefficient_aestronglyMeasurable
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (ℓ : ℝ) (hℓ : 0≤ℓ) (u : ℝ) (hu : 0≤u) (m : ℕ) :
+    AEStronglyMeasurable (Target.clippedCoefficient κ W ℓ u m) (Target.angleLaw κ) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  apply aestronglyMeasurable_of_tendsto_ae atTop
+    (fun n => (Target.aemeasurable_coefficient_integrand κ ℓ u W n m).aestronglyMeasurable.integral_prod_right')
+  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+  exact C.clippedCoefficient_tendsto hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ u hu m
+
+include C in
+lemma levelCoefficient_tendsto
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
+    Tendsto (fun n : ℕ => Target.clippedCoefficient κ W ℓ n m θ) atTop
+      (𝓝 (Target.levelCoefficient κ W ℓ m θ)) := by
+  have h := C.clippedCoefficient_unclip_limit hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
+  simpa only [Target.levelCoefficient,h.limUnder_eq] using h
+
+lemma levelCoefficient_eq_mass
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
+    Target.levelCoefficient κ W ℓ m θ =
+      ∫ ω, Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ *
+        (massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω)
+          (C.diskTerminalScalar θ hθ ω)
+          (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ))^m ∂P :=
+  (C.clippedCoefficient_unclip_limit hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m).limUnder_eq
+
+include C in
+theorem finiteMoment_eq_levelCoefficientIntegral
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
+    Target.finiteMoment κ W m ℓ =
+      ∫ θ, Target.levelCoefficient κ W ℓ m θ ∂Target.angleLaw κ := by
+  obtain ⟨M,hM,hbound⟩ := C.uniform_clippedCoefficient_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
+  have hi : Integrable (fun θ => M/(Real.sin θ)^greenExponent κ) (Target.angleLaw κ) := by
+    simpa only [div_eq_mul_inv,mul_one,one_mul] using (Target.angle_inverse_sine_integrable κ).const_mul M
+  have ht : Tendsto (fun n : ℕ => ∫ θ, Target.clippedCoefficient κ W ℓ n m θ ∂Target.angleLaw κ)
+      atTop (𝓝 (∫ θ, Target.levelCoefficient κ W ℓ m θ ∂Target.angleLaw κ)) := by
+    apply tendsto_integral_of_dominated_convergence (fun θ => M/(Real.sin θ)^greenExponent κ)
+    · intro n
+      exact C.clippedCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw W hW ℓ hℓ n (Nat.cast_nonneg n) m
+    · exact hi
+    · intro n
+      filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+      have hb := hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)
+      rw [Real.norm_of_nonneg hb.1]
+      exact hb.2
+    · filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+      exact C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
+  rw [C.finiteMoment_eq_outer_clipped_limit hB hκ hκ8 hc h0 F hw W hW ℓ hℓ m]
+  exact ht.limUnder_eq
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/FiniteMomentMonotonicity.lean b/lean/OAI/Probability/SLE/Gauge/FiniteMomentMonotonicity.lean
index cec130ea0cc54119a4d311c37c1028d98e2a143c..5a0ae689dcc8edf048c3a79cdfb9c3cc5073b43c 100644
--- a/lean/OAI/Probability/SLE/Gauge/FiniteMomentMonotonicity.lean
+++ b/lean/OAI/Probability/SLE/Gauge/FiniteMomentMonotonicity.lean
@@ -1,135 +1,135 @@
-import OAI.Probability.SLE.Geometry.CenteredDiskCoefficient
-
-namespace OAI
-
-/-! Monotonicity and convergence of the source's prescribed moments. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-include C in
-lemma levelCoefficient_nonneg_integrable
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
-    (∀ θ∈Ioo (0:ℝ) Real.pi, 0≤Target.levelCoefficient κ W ℓ m θ) ∧
-      Integrable (Target.levelCoefficient κ W ℓ m) (Target.angleLaw κ) := by
-  obtain ⟨M,hM,hbound⟩ := C.uniform_clippedCoefficient_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
-  have hb (θ) (hθ : θ∈Ioo (0:ℝ) Real.pi) : 0≤Target.levelCoefficient κ W ℓ m θ ∧
-      Target.levelCoefficient κ W ℓ m θ≤M/(Real.sin θ)^greenExponent κ := by
-    have ht := C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
-    exact ⟨ge_of_tendsto ht (Eventually.of_forall (fun n =>
-      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).1)),
-      le_of_tendsto ht (Eventually.of_forall (fun n =>
-      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).2))⟩
-  refine ⟨fun θ hθ => (hb θ hθ).1,?_⟩
-  have hi : Integrable (fun θ => M/(Real.sin θ)^greenExponent κ) (Target.angleLaw κ) := by
-    simpa only [div_eq_mul_inv,mul_one,one_mul] using (Target.angle_inverse_sine_integrable κ).const_mul M
-  apply hi.mono' (C.levelCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m)
-  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-  rw [Real.norm_of_nonneg (hb θ hθ).1]
-  exact (hb θ hθ).2
-
-include C in
-lemma levelCoefficient_mono
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (m : ℕ) :
-    MonotoneOn (fun ℓ => Target.levelCoefficient κ W ℓ m θ) (Ici 0) := by
-  intro ℓ hℓ L hL hℓL
-  let v := 4*ℓ/κ
-  let V := 4*L/κ
-  have hv : 0≤v := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
-  have hV : 0≤V := div_nonneg (mul_nonneg (by norm_num) hL) hκ.le
-  have hvV : v≤V := div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hℓL (by norm_num)) hκ.le
-  let u : ℕ → ℝ := fun n => if n=0 then v else V
-  have hu (n) : 0≤u n := by dsimp [u]; split_ifs <;> assumption
-  have hum : Monotone u := by
-    intro i j hij
-    dsimp only [u]
-    split_ifs with hi hj
-    · exact le_rfl
-    · exact hvV
-    · omega
-    · exact le_rfl
-  obtain ⟨τ,hτ,_,heτ⟩ := exists_monotone_radiusLevel_stops hB
-    (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)))
-    hBm hc h0 κ hκ hκ8 γ G F (Eventually.of_forall hw)
-    (Target.diskMapInv θ 0) (Target.diskMapInv_maps hθ (by simp)) u hu hum
-  have hs : ∀ᵐ ω ∂P, τ 0 ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) v := by
-    filter_upwards [heτ] with ω hω
-    simpa only [u,ite_true] using hω 0
-  have ht : ∀ᵐ ω ∂P, τ 1 ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) V := by
-    filter_upwards [heτ] with ω hω
-    simpa only [u,one_ne_zero,ite_false] using hω 1
-  have hmono := (C.compactWeightedMass_paths (Target.diskMassWeight κ θ hθ)
-    (Target.diskSourceHalfPlane θ) (Target.diskSourceHalfPlane_compact hθ)).mono
-      (fun _ h => h.1)
-  have h := finiteCenteredLaw_stoppedScalar_moment_mono hB hBm hc h0 κ hκ hκ8
-    γ G F (Eventually.of_forall hw) (Target.diskMapInv θ 0)
-    (Target.diskMapInv_maps hθ (by simp)) v V hv hvV
-    (C.diskScalar θ hθ) (C.diskScalar_progressive θ hθ) hmono (τ 0) (τ 1)
-    (hτ 0) (hτ 1) hs ht m
-  rw [← C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m (τ 0) (hτ 0) hs,
-    ← C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ L hL m (τ 1) (hτ 1) ht] at h
-  exact (ENNReal.ofReal_le_ofReal_iff
-    ((C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW L hL m).1 θ hθ)).mp h
-
-include C in
-theorem finiteMoment_mono
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (m : ℕ) : MonotoneOn (Target.finiteMoment κ W m) (Ici 0) := by
-  intro ℓ hℓ L hL hℓL
-  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m,
-    C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW L hL m]
-  apply integral_mono_ae
-    (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).2
-    (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW L hL m).2
-  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
-  exact C.levelCoefficient_mono hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ m hℓ hL hℓL
-
-include C in
-theorem finiteMoment_tendsto_moment
-    (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (m : ℕ) : Tendsto (fun ℓ : ℕ => Target.finiteMoment κ W m ℓ) atTop
-      (𝓝 (Target.moment κ W m)) := by
-  have hm : Monotone (fun ℓ : ℕ => Target.finiteMoment κ W m ℓ) := by
-    intro i j hij
-    exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW m
-      (by change (0:ℝ) ≤ (i:ℝ); exact Nat.cast_nonneg i)
-      (by change (0:ℝ) ≤ (j:ℝ); exact Nat.cast_nonneg j) (Nat.cast_le.mpr hij)
-  obtain ⟨M,_,hM⟩ := C.uniform_finiteMoment_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
-  have hb : BddAbove (range (fun ℓ : ℕ => Target.finiteMoment κ W m ℓ)) := by
-    refine ⟨M,?_⟩
-    rintro _ ⟨ℓ,rfl⟩
-    exact (hM ℓ (Nat.cast_nonneg ℓ)).2
-  have ht := tendsto_atTop_ciSup hm hb
-  simpa only [Target.moment,ht.limUnder_eq] using ht
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CenteredDiskCoefficient
+
+namespace OAI
+
+/-! Monotonicity and convergence of the source's prescribed moments. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+include C in
+lemma levelCoefficient_nonneg_integrable
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (ℓ : ℝ) (hℓ : 0≤ℓ) (m : ℕ) :
+    (∀ θ∈Ioo (0:ℝ) Real.pi, 0≤Target.levelCoefficient κ W ℓ m θ) ∧
+      Integrable (Target.levelCoefficient κ W ℓ m) (Target.angleLaw κ) := by
+  obtain ⟨M,hM,hbound⟩ := C.uniform_clippedCoefficient_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
+  have hb (θ) (hθ : θ∈Ioo (0:ℝ) Real.pi) : 0≤Target.levelCoefficient κ W ℓ m θ ∧
+      Target.levelCoefficient κ W ℓ m θ≤M/(Real.sin θ)^greenExponent κ := by
+    have ht := C.levelCoefficient_tendsto hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m
+    exact ⟨ge_of_tendsto ht (Eventually.of_forall (fun n =>
+      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).1)),
+      le_of_tendsto ht (Eventually.of_forall (fun n =>
+      (hbound θ hθ ℓ hℓ n (Nat.cast_nonneg n)).2))⟩
+  refine ⟨fun θ hθ => (hb θ hθ).1,?_⟩
+  have hi : Integrable (fun θ => M/(Real.sin θ)^greenExponent κ) (Target.angleLaw κ) := by
+    simpa only [div_eq_mul_inv,mul_one,one_mul] using (Target.angle_inverse_sine_integrable κ).const_mul M
+  apply hi.mono' (C.levelCoefficient_aestronglyMeasurable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m)
+  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+  rw [Real.norm_of_nonneg (hb θ hθ).1]
+  exact (hb θ hθ).2
+
+include C in
+lemma levelCoefficient_mono
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (m : ℕ) :
+    MonotoneOn (fun ℓ => Target.levelCoefficient κ W ℓ m θ) (Ici 0) := by
+  intro ℓ hℓ L hL hℓL
+  let v := 4*ℓ/κ
+  let V := 4*L/κ
+  have hv : 0≤v := div_nonneg (mul_nonneg (by norm_num) hℓ) hκ.le
+  have hV : 0≤V := div_nonneg (mul_nonneg (by norm_num) hL) hκ.le
+  have hvV : v≤V := div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hℓL (by norm_num)) hκ.le
+  let u : ℕ → ℝ := fun n => if n=0 then v else V
+  have hu (n) : 0≤u n := by dsimp [u]; split_ifs <;> assumption
+  have hum : Monotone u := by
+    intro i j hij
+    dsimp only [u]
+    split_ifs with hi hj
+    · exact le_rfl
+    · exact hvV
+    · omega
+    · exact le_rfl
+  obtain ⟨τ,hτ,_,heτ⟩ := exists_monotone_radiusLevel_stops hB
+    (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)))
+    hBm hc h0 κ hκ hκ8 γ G F (Eventually.of_forall hw)
+    (Target.diskMapInv θ 0) (Target.diskMapInv_maps hθ (by simp)) u hu hum
+  have hs : ∀ᵐ ω ∂P, τ 0 ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) v := by
+    filter_upwards [heτ] with ω hω
+    simpa only [u,ite_true] using hω 0
+  have ht : ∀ᵐ ω ∂P, τ 1 ω=radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) V := by
+    filter_upwards [heτ] with ω hω
+    simpa only [u,one_ne_zero,ite_false] using hω 1
+  have hmono := (C.compactWeightedMass_paths (Target.diskMassWeight κ θ hθ)
+    (Target.diskSourceHalfPlane θ) (Target.diskSourceHalfPlane_compact hθ)).mono
+      (fun _ h => h.1)
+  have h := finiteCenteredLaw_stoppedScalar_moment_mono hB hBm hc h0 κ hκ hκ8
+    γ G F (Eventually.of_forall hw) (Target.diskMapInv θ 0)
+    (Target.diskMapInv_maps hθ (by simp)) v V hv hvV
+    (C.diskScalar θ hθ) (C.diskScalar_progressive θ hθ) hmono (τ 0) (τ 1)
+    (hτ 0) (hτ 1) hs ht m
+  rw [← C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ ℓ hℓ m (τ 0) (hτ 0) hs,
+    ← C.levelCoefficient_eq_centered_moment hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ L hL m (τ 1) (hτ 1) ht] at h
+  exact (ENNReal.ofReal_le_ofReal_iff
+    ((C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW L hL m).1 θ hθ)).mp h
+
+include C in
+theorem finiteMoment_mono
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (m : ℕ) : MonotoneOn (Target.finiteMoment κ W m) (Ici 0) := by
+  intro ℓ hℓ L hL hℓL
+  rw [C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m,
+    C.finiteMoment_eq_levelCoefficientIntegral hB hc h0 hκ hκ8 F hw hgrowth W hW L hL m]
+  apply integral_mono_ae
+    (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW ℓ hℓ m).2
+    (C.levelCoefficient_nonneg_integrable hB hc h0 hκ hκ8 F hw hgrowth W hW L hL m).2
+  filter_upwards [Target.angleLaw_ae_mem κ] with θ hθ
+  exact C.levelCoefficient_mono hB hc h0 hκ hκ8 F hw hgrowth W hW θ hθ m hℓ hL hℓL
+
+include C in
+theorem finiteMoment_tendsto_moment
+    (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (m : ℕ) : Tendsto (fun ℓ : ℕ => Target.finiteMoment κ W m ℓ) atTop
+      (𝓝 (Target.moment κ W m)) := by
+  have hm : Monotone (fun ℓ : ℕ => Target.finiteMoment κ W m ℓ) := by
+    intro i j hij
+    exact C.finiteMoment_mono hB hc h0 hκ hκ8 F hw hgrowth W hW m
+      (by change (0:ℝ) ≤ (i:ℝ); exact Nat.cast_nonneg i)
+      (by change (0:ℝ) ≤ (j:ℝ); exact Nat.cast_nonneg j) (Nat.cast_le.mpr hij)
+  obtain ⟨M,_,hM⟩ := C.uniform_finiteMoment_bound hB hc h0 hκ hκ8 F hw hgrowth W hW m
+  have hb : BddAbove (range (fun ℓ : ℕ => Target.finiteMoment κ W m ℓ)) := by
+    refine ⟨M,?_⟩
+    rintro _ ⟨ℓ,rfl⟩
+    exact (hM ℓ (Nat.cast_nonneg ℓ)).2
+  have ht := tendsto_atTop_ciSup hm hb
+  simpa only [Target.moment,ht.limUnder_eq] using ht
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/FiniteStoppedThreshold.lean b/lean/OAI/Probability/SLE/Gauge/FiniteStoppedThreshold.lean
index 98ccd311a4f8e3bab9bbfd3e4afffaacd179ba89..462390acd7b9ac45930d38d5fbe73664b505a369 100644
--- a/lean/OAI/Probability/SLE/Gauge/FiniteStoppedThreshold.lean
+++ b/lean/OAI/Probability/SLE/Gauge/FiniteStoppedThreshold.lean
@@ -1,63 +1,63 @@
-import OAI.Probability.SLE.Stochastic.ProgressiveStoppedTime
-import OAI.Probability.SLE.Gauge.ExtendedExperimentStop
-
-namespace OAI
-
-/-! Threshold tests on the finite part of an extended stopping time belong
-  to its stopped past. No terminal mass is tested on the infinite event. -/
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge
-
-def finiteStoppedThreshold {Ω : Type*} (X : ℝ≥0 → Ω → ℝ≥0∞)
-    (S : Ω → WithTop ℝ≥0) (b : ℝ≥0∞) : Set Ω :=
-  {ω | S ω<⊤ ∧ b≤X ((S ω).untopD 0) ω}
-
-lemma finiteStoppedThreshold_measurable
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ}
-    (X : ℝ≥0 → Ω → ℝ≥0∞) (hX : IsProgressive ℱ X)
-    (S : Ω → WithTop ℝ≥0) (hS : IsStoppingTime ℱ S) (b : ℝ≥0∞) :
-    MeasurableSet[hS.measurableSpace] (finiteStoppedThreshold X S b) := by
-  have hE (H : ℝ≥0) : MeasurableSet[ℱ H]
-      ({ω | b≤X (cutoffStop S H ω) ω} ∩ {ω | S ω≤(H:WithTop ℝ≥0)}) := by
-    have hm := _root_.OAI.MeasureTheory.IsProgressive.bounded_stopped_measurable hX (cutoffStop S H)
-      (cutoffStop_stopping ℱ S hS H) H (fun ω => by
-        have h := min_le_right (S ω) (H:WithTop ℝ≥0)
-        rw [← cutoffStop_coe S H ω] at h
-        exact WithTop.coe_le_coe.mp h)
-    exact (measurableSet_le measurable_const hm).inter (hS H)
-  have he (H : ℝ≥0) : finiteStoppedThreshold X S b ∩ {ω | S ω≤(H:WithTop ℝ≥0)} =
-      {ω | b≤X (cutoffStop S H ω) ω} ∩ {ω | S ω≤(H:WithTop ℝ≥0)} := by
-    ext ω
-    simp only [finiteStoppedThreshold,mem_inter_iff,mem_ofPred_eq]
-    constructor
-    · rintro ⟨⟨_,hb⟩,hH⟩
-      exact ⟨by rwa [cutoffStop_eq_on_success S H ω hH],hH⟩
-    · rintro ⟨hb,hH⟩
-      exact ⟨⟨lt_of_le_of_lt hH (WithTop.coe_lt_top _),
-        by rwa [cutoffStop_eq_on_success S H ω hH] at hb⟩,hH⟩
-  constructor
-  · have hunion : finiteStoppedThreshold X S b = ⋃ n : ℕ,
-        finiteStoppedThreshold X S b ∩ {ω | S ω≤((n:ℝ≥0):WithTop ℝ≥0)} := by
-      ext ω
-      simp only [mem_iUnion,mem_inter_iff,mem_ofPred_eq]
-      constructor
-      · intro hω
-        lift S ω to ℝ≥0 using hω.1.ne with t ht
-        obtain ⟨n,hn⟩ := exists_nat_ge (t:ℝ)
-        exact ⟨n,hω,by exact_mod_cast hn⟩
-      · rintro ⟨_,hω,_⟩
-        exact hω
-    rw [hunion]
-    exact MeasurableSet.iUnion (fun n => le_iSup ℱ (n:ℝ≥0) _ (by rw [he]; exact hE _))
-  · intro H
-    rw [he]
-    exact hE H
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.ProgressiveStoppedTime
+import OAI.Probability.SLE.Gauge.ExtendedExperimentStop
+
+namespace OAI
+
+/-! Threshold tests on the finite part of an extended stopping time belong
+  to its stopped past. No terminal mass is tested on the infinite event. -/
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge
+
+def finiteStoppedThreshold {Ω : Type*} (X : ℝ≥0 → Ω → ℝ≥0∞)
+    (S : Ω → WithTop ℝ≥0) (b : ℝ≥0∞) : Set Ω :=
+  {ω | S ω<⊤ ∧ b≤X ((S ω).untopD 0) ω}
+
+lemma finiteStoppedThreshold_measurable
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ}
+    (X : ℝ≥0 → Ω → ℝ≥0∞) (hX : IsProgressive ℱ X)
+    (S : Ω → WithTop ℝ≥0) (hS : IsStoppingTime ℱ S) (b : ℝ≥0∞) :
+    MeasurableSet[hS.measurableSpace] (finiteStoppedThreshold X S b) := by
+  have hE (H : ℝ≥0) : MeasurableSet[ℱ H]
+      ({ω | b≤X (cutoffStop S H ω) ω} ∩ {ω | S ω≤(H:WithTop ℝ≥0)}) := by
+    have hm := _root_.OAI.MeasureTheory.IsProgressive.bounded_stopped_measurable hX (cutoffStop S H)
+      (cutoffStop_stopping ℱ S hS H) H (fun ω => by
+        have h := min_le_right (S ω) (H:WithTop ℝ≥0)
+        rw [← cutoffStop_coe S H ω] at h
+        exact WithTop.coe_le_coe.mp h)
+    exact (measurableSet_le measurable_const hm).inter (hS H)
+  have he (H : ℝ≥0) : finiteStoppedThreshold X S b ∩ {ω | S ω≤(H:WithTop ℝ≥0)} =
+      {ω | b≤X (cutoffStop S H ω) ω} ∩ {ω | S ω≤(H:WithTop ℝ≥0)} := by
+    ext ω
+    simp only [finiteStoppedThreshold,mem_inter_iff,mem_ofPred_eq]
+    constructor
+    · rintro ⟨⟨_,hb⟩,hH⟩
+      exact ⟨by rwa [cutoffStop_eq_on_success S H ω hH],hH⟩
+    · rintro ⟨hb,hH⟩
+      exact ⟨⟨lt_of_le_of_lt hH (WithTop.coe_lt_top _),
+        by rwa [cutoffStop_eq_on_success S H ω hH] at hb⟩,hH⟩
+  constructor
+  · have hunion : finiteStoppedThreshold X S b = ⋃ n : ℕ,
+        finiteStoppedThreshold X S b ∩ {ω | S ω≤((n:ℝ≥0):WithTop ℝ≥0)} := by
+      ext ω
+      simp only [mem_iUnion,mem_inter_iff,mem_ofPred_eq]
+      constructor
+      · intro hω
+        lift S ω to ℝ≥0 using hω.1.ne with t ht
+        obtain ⟨n,hn⟩ := exists_nat_ge (t:ℝ)
+        exact ⟨n,hω,by exact_mod_cast hn⟩
+      · rintro ⟨_,hω,_⟩
+        exact hω
+    rw [hunion]
+    exact MeasurableSet.iUnion (fun n => le_iSup ℱ (n:ℝ≥0) _ (by rw [he]; exact hE _))
+  · intro H
+    rw [he]
+    exact hE H
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/FirstFrontPassage.lean b/lean/OAI/Probability/SLE/Gauge/FirstFrontPassage.lean
index e2495e5744185d089a11d5e0264e26462e2adce4..71cbc69ef756031360f68bd8bb56abcc410c3602 100644
--- a/lean/OAI/Probability/SLE/Gauge/FirstFrontPassage.lean
+++ b/lean/OAI/Probability/SLE/Gauge/FirstFrontPassage.lean
@@ -1,239 +1,239 @@
-import OAI.Probability.SLE.Gauge.RetainedTubeSausage
-
-namespace OAI
-
-/-! The retained endpoint and pre-front margins give an actual first-front
-stopping time, with the retained middle interval strictly before it. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.FirstFrontPassage
-open PassageRetention RetainedTubeSausage
-
-/-- First time a scalar progress coordinate reaches the front. -/
-def firstFrontTime (x : ℝ≥0 → ℝ) (q : ℝ) : WithTop ℝ≥0 :=
-  hittingAfter (fun t (_ : Unit) => x t - q) {0} 0 ()
-
-/-- Finite-valued representative; it agrees with the actual first-front
-hitting time whenever a retained endpoint witnesses a crossing. -/
-def frontTime (x : ℝ≥0 → ℝ) (q : ℝ) : ℝ≥0 :=
-  (firstFrontTime x q).untopD 0
-
-theorem firstFrontTime_isStoppingTime
-    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
-    (x : ℝ≥0 → Ω → ℝ) (ha : Adapted ℱ x)
-    (hc : ∀ ω, Continuous (fun t => x t ω)) (q : ℝ) :
-    IsStoppingTime ℱ (fun ω => firstFrontTime (fun t => x t ω) q) := by
-  exact continuous_zero_hit_stopping ℱ (fun t ω => x t ω - q)
-    (fun t => (ha t).sub_const q) (fun ω => (hc ω).sub continuous_const) 0
-
-/-- A continuous crossing before a finite endpoint makes the infimum defining
-the stopping time an actual least crossing, even without global continuity. -/
-theorem exists_first_front
-    (x : ℝ≥0 → ℝ) (q : ℝ) (T : ℝ≥0)
-    (hc : ContinuousOn x (Icc 0 T)) (hstart : x 0 < q) (hend : q < x T) :
-    ∃ τ : ℝ≥0, firstFrontTime x q = (τ : WithTop ℝ≥0) ∧
-      0 < τ ∧ τ < T ∧ x τ = q ∧ ∀ t < τ, x t < q := by
-  let K := Icc (0 : ℝ≥0) T ∩ x ⁻¹' {q}
-  have hKc : IsClosed K := hc.preimage_isClosed_of_isClosed isClosed_Icc isClosed_singleton
-  have hK : IsCompact K := isCompact_Icc.of_isClosed_subset hKc inter_subset_left
-  have hKn : K.Nonempty := by
-    obtain ⟨t, ht, he⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ T from bot_le)
-      hc ⟨hstart.le, hend.le⟩
-    exact ⟨t, ht, he⟩
-  obtain ⟨τ, hτ, hmin⟩ := hK.exists_isLeast hKn
-  have hhit : x τ = q := hτ.2
-  have hpos : 0 < τ := lt_of_le_of_ne bot_le (by
-    intro he
-    exact (ne_of_lt hstart) (he ▸ hhit))
-  have hlt : τ < T := lt_of_le_of_ne hτ.1.2 (by
-    intro he
-    exact (ne_of_gt hend) (he ▸ hhit))
-  have hprior (t : ℝ≥0) (ht : t < τ) : x t < q := by
-    by_contra hn
-    obtain ⟨u, hu, hue⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ t from bot_le)
-      (hc.mono (Icc_subset_Icc_right (ht.le.trans hτ.1.2)))
-      ⟨hstart.le, le_of_not_gt hn⟩
-    exact (not_le_of_gt ht) ((hmin ⟨⟨bot_le, hu.2.trans (ht.le.trans hτ.1.2)⟩, hue⟩).trans hu.2)
-  have hleast : IsLeast {t : ℝ≥0 | 0 ≤ t ∧ x t - q ∈ ({0} : Set ℝ)} τ := by
-    refine ⟨⟨bot_le, by simp only [mem_singleton_iff, hhit, sub_self]⟩, ?_⟩
-    intro t ht
-    by_contra hn
-    exact (ne_of_lt (hprior t (lt_of_not_ge hn))) (sub_eq_zero.mp ht.2)
-  have htime : firstFrontTime x q = (τ : WithTop ℝ≥0) := by
-    unfold firstFrontTime hittingAfter
-    rw [ite_eq_left ⟨τ, hleast.1⟩, hleast.csInf_eq]
-  exact ⟨τ, htime, hpos, hlt, hhit, hprior⟩
-
-/-- Early containment and a retained pre-front prefix force the actual first
-front stop to fall strictly after the retained middle interval. -/
-theorem frontTime_between
-    (x : ℝ≥0 → ℝ) (q : ℝ) (σ b T : ℝ≥0) (_hσb : σ ≤ b) (_hbT : b < T)
-    (hc : ContinuousOn x (Icc 0 T))
-    (hearly : ∀ t ∈ Icc 0 σ, x t < q)
-    (hprefix : ∀ t ∈ Icc σ b, x t < q) (hend : q < x T) :
-    firstFrontTime x q = (frontTime x q : WithTop ℝ≥0) ∧
-      b < frontTime x q ∧ frontTime x q < T ∧ x (frontTime x q) = q := by
-  obtain ⟨τ, hτ, _, hτT, hhit, _⟩ := exists_first_front x q T hc
-    (hearly 0 ⟨le_rfl, bot_le⟩) hend
-  have hbτ : b < τ := by
-    by_contra hn
-    have hτb : τ ≤ b := le_of_not_gt hn
-    have hlt : x τ < q := by
-      by_cases hτσ : τ ≤ σ
-      · exact hearly τ ⟨bot_le, hτσ⟩
-      · exact hprefix τ ⟨le_of_not_ge hτσ, hτb⟩
-    exact (ne_of_lt hlt) hhit
-  have hfront : frontTime x q = τ := by simp only [frontTime, hτ, WithTop.untopD_coe]
-  exact ⟨hfront ▸ hτ, hfront ▸ hbτ, hfront ▸ hτT, hfront ▸ hhit⟩
-
-/-- Lower endpoint margins transfer by the same compact uniform-convergence
-argument as upper pre-front margins. -/
-theorem eventually_front_lt_on_compact_image
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hF : ContinuousOn F upperHalfPlane)
-    (hFconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
-    (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
-    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
-    (hKU : F '' K ⊆ U) (q : ℝ) (hbound : ∀ z ∈ K, q < p (F z)) :
-    ∀ᶠ n in atTop, ∀ z ∈ K, q < pₙ n (Fₙ n z) := by
-  have hnegconv : TendstoLocallyUniformlyOn (fun n z => -pₙ n z) (fun z => -p z) atTop U :=
-    hpconv.neg
-  have hh := eventually_front_lt_on_compact Fₙ F hF hFconv
-    (fun n z => -pₙ n z) (fun z => -p z) U hU hp.neg hnegconv K hK hKH hKU (-q)
-    (fun z hz => neg_lt_neg (hbound z hz))
-  filter_upwards [hh] with n hn z hz
-  exact neg_lt_neg_iff.mp (hn z hz)
-
-/-- Retained endpoint margins construct the actual finite first-front stop.
-The positive stopped sausage bound therefore needs no assumed front-stop fact. -/
-theorem retained_tube_firstFront_sausage_mass
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P)
-    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
-    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
-    (hp : ContinuousOn p U) (hpn : ∀ n, ContinuousOn (pₙ n) U)
-    (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
-    (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)})
-    (E : Set Ω) (hE : MeasurableSet E)
-    (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
-    (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3)
-    (R : Set ℂ) (hR : IsOpen R)
-    (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (Rₙ : ℕ → Set ℂ)
-    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
-    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
-    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
-    (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane →
-      η n ω t = Fₙ n (f ω t))
-    (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
-    (hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
-      MapsTo (η n ω) (Icc 0 (σ ω)) U ∧
-        ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1) :
-    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
-      ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤
-        segmentSausageCharge κ (η n ω) (σ ω) (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
-  let progress := fun z => p (F z)
-  have hprogress : ContinuousOn progress O :=
-    hp.comp (hF.continuousOn.mono hOH) hFOU
-  obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ :=
-    stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart
-  let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N
-    (beyondFrontRegion O progress) (enumerateRetainedPassage i)
-  let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1
-  let J : ℕ → ℝ≥0 × ℝ≥0 := fun i =>
-    (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1)
-  have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1
-  have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1
-  have hQR : ∀ i, F '' Q i ⊆ R := by
-    rintro i _ ⟨z, hz, rfl⟩
-    have hh := (hM _).2 hz
-    exact hmiddle z hh.1 hh.2.1 hh.2.2
-  have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _
-  have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _
-  have hnull : P (E \ ⋃ i, D i) = 0 := by
-    rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty]
-  have hretain : ∀ i ω, ω ∈ D i →
-      (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by
-    intro i ω hω
-    exact hω.1.2.2.2.1.image_subset
-  have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i →
-      σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
-        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by
-    intro i
-    let j := enumerateRetainedPassage i
-    have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1
-    have hLU : F '' L j.1.2.2.2 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hL _).2 hz).1
-    have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv
-      (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2)
-    have hKH : K j.1.2.1 ⊆ upperHalfPlane := fun z hz => hOH ((hK _).2 hz)
-    have hKU : F '' K j.1.2.1 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hK _).2 hz)
-    have himage := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
-      (K j.1.2.1) (hK _).1 hKH U hU hKU
-    have hNH : N j.2 ⊆ upperHalfPlane := fun z hz => hOH ((hN _).2 hz).1
-    have hNU : F '' N j.2 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hN _).2 hz).1
-    have hgt := eventually_front_lt_on_compact_image Fₙ F hF.continuousOn hconv pₙ p U hU
-      hp hpconv (N j.2) (hN _).1 hNH hNU 1 (fun z hz => ((hN _).2 hz).2)
-    filter_upwards [hlt, hgt, himage, hearly] with n hnL hnN hnK hearlyn
-    filter_upwards [hη n, hηc n, hearlyn] with ω hηω hηcω hearlyω
-    intro hω
-    have hDi : ω ∈ retainedPassageCertificate f σ K M L N
-        (beyondFrontRegion O progress) j := hω.2
-    obtain ⟨⟨hσs, _, hfull, hmid, hlow⟩, hend⟩ := hDi
-    obtain ⟨hearlyU, hearlylt⟩ := hearlyω hω.1
-    have hpathU : MapsTo (η n ω) (Icc 0 (passageEnd j.1)) U := by
-      intro t ht
-      by_cases hts : t ≤ σ ω
-      · exact hearlyU ⟨bot_le, hts⟩
-      · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
-        have hmem := hfull.2 ⟨hst, ht.2⟩
-        rw [hηω t hst (hKH hmem)]
-        exact hnK (mem_image_of_mem (Fₙ n) hmem)
-    have hc : ContinuousOn (fun t => pₙ n (η n ω t)) (Icc 0 (passageEnd j.1)) :=
-      (hpn n).comp hηcω.continuousOn hpathU
-    have hprefix : ∀ t ∈ Icc (σ ω) (intervalEnd j.1), pₙ n (η n ω t) < 1 := by
-      intro t ht
-      have hmem := hlow.2 ht
-      rw [hηω t ht.1 (hLH hmem)]
-      exact (hnL _ hmem).trans (by norm_num)
-    have hendgt : 1 < pₙ n (η n ω (passageEnd j.1)) := by
-      have hmem := hfull.2 ⟨hfull.1, le_rfl⟩
-      rw [hηω _ hfull.1 (hKH hmem)]
-      exact hnN _ hend
-    have hfront := frontTime_between (fun t => pₙ n (η n ω t)) 1
-      (σ ω) (intervalEnd j.1) (passageEnd j.1)
-      (hσs.le.trans (intervalStart_lt_end _).le) (intervalEnd_lt_passageEnd _)
-      hc hearlylt hprefix hendgt
-    refine ⟨hσs.le, hfront.2.1.le, ?_⟩
-    intro t ht
-    exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1)
-  exact countable_stopped_mass_ae_window_eventually P κ hκ hκ8 (fun ω => f ω) hSLE
-    Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D hE hD hnull hretain
-    v hv hlim η (fun _ => σ) (fun n ω => frontTime (fun t => pₙ n (η n ω t)) 1)
-    Rₙ hregion G q hq hG hwindow
-
-
-end SLEExactGauge.FirstFrontPassage
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RetainedTubeSausage
+
+namespace OAI
+
+/-! The retained endpoint and pre-front margins give an actual first-front
+stopping time, with the retained middle interval strictly before it. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.FirstFrontPassage
+open PassageRetention RetainedTubeSausage
+
+/-- First time a scalar progress coordinate reaches the front. -/
+def firstFrontTime (x : ℝ≥0 → ℝ) (q : ℝ) : WithTop ℝ≥0 :=
+  hittingAfter (fun t (_ : Unit) => x t - q) {0} 0 ()
+
+/-- Finite-valued representative; it agrees with the actual first-front
+hitting time whenever a retained endpoint witnesses a crossing. -/
+def frontTime (x : ℝ≥0 → ℝ) (q : ℝ) : ℝ≥0 :=
+  (firstFrontTime x q).untopD 0
+
+theorem firstFrontTime_isStoppingTime
+    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
+    (x : ℝ≥0 → Ω → ℝ) (ha : Adapted ℱ x)
+    (hc : ∀ ω, Continuous (fun t => x t ω)) (q : ℝ) :
+    IsStoppingTime ℱ (fun ω => firstFrontTime (fun t => x t ω) q) := by
+  exact continuous_zero_hit_stopping ℱ (fun t ω => x t ω - q)
+    (fun t => (ha t).sub_const q) (fun ω => (hc ω).sub continuous_const) 0
+
+/-- A continuous crossing before a finite endpoint makes the infimum defining
+the stopping time an actual least crossing, even without global continuity. -/
+theorem exists_first_front
+    (x : ℝ≥0 → ℝ) (q : ℝ) (T : ℝ≥0)
+    (hc : ContinuousOn x (Icc 0 T)) (hstart : x 0 < q) (hend : q < x T) :
+    ∃ τ : ℝ≥0, firstFrontTime x q = (τ : WithTop ℝ≥0) ∧
+      0 < τ ∧ τ < T ∧ x τ = q ∧ ∀ t < τ, x t < q := by
+  let K := Icc (0 : ℝ≥0) T ∩ x ⁻¹' {q}
+  have hKc : IsClosed K := hc.preimage_isClosed_of_isClosed isClosed_Icc isClosed_singleton
+  have hK : IsCompact K := isCompact_Icc.of_isClosed_subset hKc inter_subset_left
+  have hKn : K.Nonempty := by
+    obtain ⟨t, ht, he⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ T from bot_le)
+      hc ⟨hstart.le, hend.le⟩
+    exact ⟨t, ht, he⟩
+  obtain ⟨τ, hτ, hmin⟩ := hK.exists_isLeast hKn
+  have hhit : x τ = q := hτ.2
+  have hpos : 0 < τ := lt_of_le_of_ne bot_le (by
+    intro he
+    exact (ne_of_lt hstart) (he ▸ hhit))
+  have hlt : τ < T := lt_of_le_of_ne hτ.1.2 (by
+    intro he
+    exact (ne_of_gt hend) (he ▸ hhit))
+  have hprior (t : ℝ≥0) (ht : t < τ) : x t < q := by
+    by_contra hn
+    obtain ⟨u, hu, hue⟩ := intermediate_value_Icc (show (0 : ℝ≥0) ≤ t from bot_le)
+      (hc.mono (Icc_subset_Icc_right (ht.le.trans hτ.1.2)))
+      ⟨hstart.le, le_of_not_gt hn⟩
+    exact (not_le_of_gt ht) ((hmin ⟨⟨bot_le, hu.2.trans (ht.le.trans hτ.1.2)⟩, hue⟩).trans hu.2)
+  have hleast : IsLeast {t : ℝ≥0 | 0 ≤ t ∧ x t - q ∈ ({0} : Set ℝ)} τ := by
+    refine ⟨⟨bot_le, by simp only [mem_singleton_iff, hhit, sub_self]⟩, ?_⟩
+    intro t ht
+    by_contra hn
+    exact (ne_of_lt (hprior t (lt_of_not_ge hn))) (sub_eq_zero.mp ht.2)
+  have htime : firstFrontTime x q = (τ : WithTop ℝ≥0) := by
+    unfold firstFrontTime hittingAfter
+    rw [ite_eq_left ⟨τ, hleast.1⟩, hleast.csInf_eq]
+  exact ⟨τ, htime, hpos, hlt, hhit, hprior⟩
+
+/-- Early containment and a retained pre-front prefix force the actual first
+front stop to fall strictly after the retained middle interval. -/
+theorem frontTime_between
+    (x : ℝ≥0 → ℝ) (q : ℝ) (σ b T : ℝ≥0) (_hσb : σ ≤ b) (_hbT : b < T)
+    (hc : ContinuousOn x (Icc 0 T))
+    (hearly : ∀ t ∈ Icc 0 σ, x t < q)
+    (hprefix : ∀ t ∈ Icc σ b, x t < q) (hend : q < x T) :
+    firstFrontTime x q = (frontTime x q : WithTop ℝ≥0) ∧
+      b < frontTime x q ∧ frontTime x q < T ∧ x (frontTime x q) = q := by
+  obtain ⟨τ, hτ, _, hτT, hhit, _⟩ := exists_first_front x q T hc
+    (hearly 0 ⟨le_rfl, bot_le⟩) hend
+  have hbτ : b < τ := by
+    by_contra hn
+    have hτb : τ ≤ b := le_of_not_gt hn
+    have hlt : x τ < q := by
+      by_cases hτσ : τ ≤ σ
+      · exact hearly τ ⟨bot_le, hτσ⟩
+      · exact hprefix τ ⟨le_of_not_ge hτσ, hτb⟩
+    exact (ne_of_lt hlt) hhit
+  have hfront : frontTime x q = τ := by simp only [frontTime, hτ, WithTop.untopD_coe]
+  exact ⟨hfront ▸ hτ, hfront ▸ hbτ, hfront ▸ hτT, hfront ▸ hhit⟩
+
+/-- Lower endpoint margins transfer by the same compact uniform-convergence
+argument as upper pre-front margins. -/
+theorem eventually_front_lt_on_compact_image
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hF : ContinuousOn F upperHalfPlane)
+    (hFconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
+    (hp : ContinuousOn p U) (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
+    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
+    (hKU : F '' K ⊆ U) (q : ℝ) (hbound : ∀ z ∈ K, q < p (F z)) :
+    ∀ᶠ n in atTop, ∀ z ∈ K, q < pₙ n (Fₙ n z) := by
+  have hnegconv : TendstoLocallyUniformlyOn (fun n z => -pₙ n z) (fun z => -p z) atTop U :=
+    hpconv.neg
+  have hh := eventually_front_lt_on_compact Fₙ F hF hFconv
+    (fun n z => -pₙ n z) (fun z => -p z) U hU hp.neg hnegconv K hK hKH hKU (-q)
+    (fun z hz => neg_lt_neg (hbound z hz))
+  filter_upwards [hh] with n hn z hz
+  exact neg_lt_neg_iff.mp (hn z hz)
+
+/-- Retained endpoint margins construct the actual finite first-front stop.
+The positive stopped sausage bound therefore needs no assumed front-stop fact. -/
+theorem retained_tube_firstFront_sausage_mass
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P)
+    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
+    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
+    (hp : ContinuousOn p U) (hpn : ∀ n, ContinuousOn (pₙ n) U)
+    (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
+    (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)})
+    (E : Set Ω) (hE : MeasurableSet E)
+    (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
+    (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3)
+    (R : Set ℂ) (hR : IsOpen R)
+    (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (Rₙ : ℕ → Set ℂ)
+    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
+    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
+    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
+    (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane →
+      η n ω t = Fₙ n (f ω t))
+    (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
+    (hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
+      MapsTo (η n ω) (Icc 0 (σ ω)) U ∧
+        ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1) :
+    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
+      ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤
+        segmentSausageCharge κ (η n ω) (σ ω) (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
+  let progress := fun z => p (F z)
+  have hprogress : ContinuousOn progress O :=
+    hp.comp (hF.continuousOn.mono hOH) hFOU
+  obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ :=
+    stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart
+  let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N
+    (beyondFrontRegion O progress) (enumerateRetainedPassage i)
+  let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1
+  let J : ℕ → ℝ≥0 × ℝ≥0 := fun i =>
+    (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1)
+  have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1
+  have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1
+  have hQR : ∀ i, F '' Q i ⊆ R := by
+    rintro i _ ⟨z, hz, rfl⟩
+    have hh := (hM _).2 hz
+    exact hmiddle z hh.1 hh.2.1 hh.2.2
+  have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _
+  have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _
+  have hnull : P (E \ ⋃ i, D i) = 0 := by
+    rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty]
+  have hretain : ∀ i ω, ω ∈ D i →
+      (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by
+    intro i ω hω
+    exact hω.1.2.2.2.1.image_subset
+  have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i →
+      σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
+        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by
+    intro i
+    let j := enumerateRetainedPassage i
+    have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1
+    have hLU : F '' L j.1.2.2.2 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hL _).2 hz).1
+    have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv
+      (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2)
+    have hKH : K j.1.2.1 ⊆ upperHalfPlane := fun z hz => hOH ((hK _).2 hz)
+    have hKU : F '' K j.1.2.1 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hK _).2 hz)
+    have himage := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
+      (K j.1.2.1) (hK _).1 hKH U hU hKU
+    have hNH : N j.2 ⊆ upperHalfPlane := fun z hz => hOH ((hN _).2 hz).1
+    have hNU : F '' N j.2 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hN _).2 hz).1
+    have hgt := eventually_front_lt_on_compact_image Fₙ F hF.continuousOn hconv pₙ p U hU
+      hp hpconv (N j.2) (hN _).1 hNH hNU 1 (fun z hz => ((hN _).2 hz).2)
+    filter_upwards [hlt, hgt, himage, hearly] with n hnL hnN hnK hearlyn
+    filter_upwards [hη n, hηc n, hearlyn] with ω hηω hηcω hearlyω
+    intro hω
+    have hDi : ω ∈ retainedPassageCertificate f σ K M L N
+        (beyondFrontRegion O progress) j := hω.2
+    obtain ⟨⟨hσs, _, hfull, hmid, hlow⟩, hend⟩ := hDi
+    obtain ⟨hearlyU, hearlylt⟩ := hearlyω hω.1
+    have hpathU : MapsTo (η n ω) (Icc 0 (passageEnd j.1)) U := by
+      intro t ht
+      by_cases hts : t ≤ σ ω
+      · exact hearlyU ⟨bot_le, hts⟩
+      · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
+        have hmem := hfull.2 ⟨hst, ht.2⟩
+        rw [hηω t hst (hKH hmem)]
+        exact hnK (mem_image_of_mem (Fₙ n) hmem)
+    have hc : ContinuousOn (fun t => pₙ n (η n ω t)) (Icc 0 (passageEnd j.1)) :=
+      (hpn n).comp hηcω.continuousOn hpathU
+    have hprefix : ∀ t ∈ Icc (σ ω) (intervalEnd j.1), pₙ n (η n ω t) < 1 := by
+      intro t ht
+      have hmem := hlow.2 ht
+      rw [hηω t ht.1 (hLH hmem)]
+      exact (hnL _ hmem).trans (by norm_num)
+    have hendgt : 1 < pₙ n (η n ω (passageEnd j.1)) := by
+      have hmem := hfull.2 ⟨hfull.1, le_rfl⟩
+      rw [hηω _ hfull.1 (hKH hmem)]
+      exact hnN _ hend
+    have hfront := frontTime_between (fun t => pₙ n (η n ω t)) 1
+      (σ ω) (intervalEnd j.1) (passageEnd j.1)
+      (hσs.le.trans (intervalStart_lt_end _).le) (intervalEnd_lt_passageEnd _)
+      hc hearlylt hprefix hendgt
+    refine ⟨hσs.le, hfront.2.1.le, ?_⟩
+    intro t ht
+    exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1)
+  exact countable_stopped_mass_ae_window_eventually P κ hκ hκ8 (fun ω => f ω) hSLE
+    Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D hE hD hnull hretain
+    v hv hlim η (fun _ => σ) (fun n ω => frontTime (fun t => pₙ n (η n ω t)) 1)
+    Rₙ hregion G q hq hG hwindow
+
+
+end SLEExactGauge.FirstFrontPassage
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeBoundaryDischarge.lean b/lean/OAI/Probability/SLE/Gauge/GaugeBoundaryDischarge.lean
index 61e25287a6a813baea2cba3f5a29ec414924e4b3..2c57810bcd79a74a4afd124eadd61d1293ba355b 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeBoundaryDischarge.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeBoundaryDischarge.lean
@@ -1,28 +1,28 @@
-import OAI.Probability.SLE.Gauge.GaugeConformalReduction
-import OAI.Probability.SLE.Geometry.ActualFiniteBoundaryExtension
-
-namespace OAI
-
-/-! The exact explicit-gauge theorem using continuous finite boundary extensions. -/
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem ordinary_explicit_mainConclusion_of_support_crosscut_exit
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw) :
-    MainConclusion P (explicitGauge κ) γ :=
-  ordinary_explicit_mainConclusion_of_boundary_support_crosscut_exit P κ hκ hκ8 γ hSLE
-    actual_finiteLoewnerBoundaryExtension hsupport hstraight hexit
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.GaugeConformalReduction
+import OAI.Probability.SLE.Geometry.ActualFiniteBoundaryExtension
+
+namespace OAI
+
+/-! The exact explicit-gauge theorem using continuous finite boundary extensions. -/
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem ordinary_explicit_mainConclusion_of_support_crosscut_exit
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw) :
+    MainConclusion P (explicitGauge κ) γ :=
+  ordinary_explicit_mainConclusion_of_boundary_support_crosscut_exit P κ hκ hκ8 γ hSLE
+    actual_finiteLoewnerBoundaryExtension hsupport hstraight hexit
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeCompletion.lean b/lean/OAI/Probability/SLE/Gauge/GaugeCompletion.lean
index 1e0d3f38b774c97a8d0f6c952869466b12cce272..a41df62bf44e2553c63c5b576786d3404d7af373 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeCompletion.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeCompletion.lean
@@ -1,217 +1,217 @@
-import OAI.Probability.SLE.Gauge.ExactGaugeComparison
-
-namespace OAI
-
-/-! Continuous monotone completions of the explicit and exponent-one log-log gauges. -/
-
-noncomputable section
-
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-namespace GaugeCompletion
-
-def cutoff : ℝ := Real.exp 1
-
-def tailProfile (d q x : ℝ) : ℝ := Real.exp (-d * x) * (Real.log x) ^ q
-
-def completed (d q r : ℝ) : ℝ :=
-  if r ≤ 0 then 0 else tailProfile d q (max cutoff (Real.log (1 / r)))
-
-lemma cutoff_one_lt : 1 < cutoff := Real.one_lt_exp_iff.mpr zero_lt_one
-
-lemma cutoff_pos : 0 < cutoff := Real.exp_pos 1
-
-lemma log_one_le {x : ℝ} (hx : cutoff ≤ x) : 1 ≤ Real.log x := by
-  simpa only [cutoff, Real.log_exp] using Real.log_le_log cutoff_pos hx
-
-lemma tailProfile_pos {d q x : ℝ} (hx : cutoff ≤ x) : 0 < tailProfile d q x := by
-  exact mul_pos (Real.exp_pos _) (Real.rpow_pos_of_pos (zero_lt_one.trans_le (log_one_le hx)) _)
-
-lemma tailProfile_continuousAt {d q x : ℝ} (hx : cutoff ≤ x) :
-    ContinuousAt (tailProfile d q) x := by
-  have hxp : 0 < x := cutoff_pos.trans_le hx
-  have hlp : 0 < Real.log x := zero_lt_one.trans_le (log_one_le hx)
-  exact (Real.continuous_exp.continuousAt.comp (continuousAt_const.mul continuousAt_id)).mul
-    ((Real.continuousAt_log hxp.ne').rpow_const (Or.inl hlp.ne'))
-
-lemma tailProfile_hasDerivAt {d q x : ℝ} (hx : cutoff ≤ x) :
-    HasDerivAt (tailProfile d q)
-      (Real.exp (-d * x) * (Real.log x) ^ (q - 1) *
-        (q / x - d * Real.log x)) x := by
-  have hxp : 0 < x := cutoff_pos.trans_le hx
-  have hlp : 0 < Real.log x := zero_lt_one.trans_le (log_one_le hx)
-  have he := ((hasDerivAt_id x).const_mul (-d)).exp
-  have hl := (Real.hasDerivAt_log hxp.ne').rpow_const (p := q) (Or.inl hlp.ne')
-  convert he.mul hl using 1 <;> try rfl
-  simp only [id_eq]
-  rw [Real.rpow_sub_one hlp.ne']
-  field_simp
-  ring
-
-lemma tailProfile_antitoneOn {d q : ℝ} (hd : 1 ≤ d) (hq : q ≤ 1) :
-    AntitoneOn (tailProfile d q) (Ici cutoff) := by
-  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ici _)
-    (fun x hx => (tailProfile_continuousAt hx).continuousWithinAt)
-    (fun x hx => (tailProfile_hasDerivAt (interior_subset hx)).hasDerivWithinAt)
-  intro x hx
-  have hxc : cutoff ≤ x := interior_subset hx
-  have hxp : 0 < x := cutoff_pos.trans_le hxc
-  have hx1 : 1 ≤ x := cutoff_one_lt.le.trans hxc
-  have hl : 1 ≤ Real.log x := log_one_le hxc
-  have hdlog : 1 ≤ d * Real.log x := by
-    nlinarith [mul_nonneg (sub_nonneg.mpr hd) (sub_nonneg.mpr hl)]
-  have hdiv : q / x ≤ 1 := (div_le_one hxp).mpr (hq.trans hx1)
-  exact mul_nonpos_of_nonneg_of_nonpos (by positivity) (by linarith)
-
-lemma tailProfile_tendsto_zero {d q : ℝ} (hd : 0 < d) (hq : q ≤ 1) :
-    Tendsto (tailProfile d q) atTop (𝓝 0) := by
-  have hu : Tendsto (fun x : ℝ => x * Real.exp (-d * x)) atTop (𝓝 0) := by
-    simpa only [Real.rpow_one] using tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 d hd
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu
-  · filter_upwards [eventually_ge_atTop cutoff] with x hx
-    exact (tailProfile_pos hx).le
-  · filter_upwards [eventually_ge_atTop cutoff] with x hx
-    have hxp : 0 < x := cutoff_pos.trans_le hx
-    have hpow : (Real.log x) ^ q ≤ Real.log x := by
-      simpa only [Real.rpow_one] using
-        Real.rpow_le_rpow_of_exponent_le (log_one_le hx) hq
-    change Real.exp (-d * x) * (Real.log x) ^ q ≤ _
-    rw [mul_comm (Real.exp (-d * x))]
-    exact mul_le_mul_of_nonneg_right (hpow.trans (Real.log_le_self hxp.le))
-      (Real.exp_pos _).le
-
-lemma log_inv_tendsto_atTop :
-    Tendsto (fun r : ℝ => Real.log (1 / r)) (𝓝[>] 0) atTop := by
-  simpa only [one_div, Function.comp_def] using
-    Real.tendsto_log_atTop.comp tendsto_inv_nhdsGT_zero
-
-@[simp] lemma completed_zero (d q : ℝ) : completed d q 0 = 0 := by
-  simp [completed]
-
-lemma completed_eq_tail_eventually (d q : ℝ) :
-    completed d q =ᶠ[𝓝[>] 0] fun r => tailProfile d q (Real.log (1 / r)) := by
-  filter_upwards [self_mem_nhdsWithin,
-    log_inv_tendsto_atTop.eventually (eventually_ge_atTop cutoff)] with r hr hx
-  have hrp : 0 < r := hr
-  simp only [completed, not_le.mpr hrp, ↓reduceIte, max_eq_right hx]
-
-lemma completed_tendsto_zero {d q : ℝ} (hd : 0 < d) (hq : q ≤ 1) :
-    Tendsto (completed d q) (𝓝[>] 0) (𝓝 0) := by
-  exact ((tailProfile_tendsto_zero hd hq).comp log_inv_tendsto_atTop).congr'
-    (completed_eq_tail_eventually d q).symm
-
-lemma completed_continuousAt_pos {d q r : ℝ} (hr : 0 < r) :
-    ContinuousAt (completed d q) r := by
-  have hc : ContinuousAt
-      (fun r : ℝ => tailProfile d q (max cutoff (Real.log (1 / r)))) r := by
-    have hg : ContinuousAt (fun r : ℝ => max cutoff (Real.log (1 / r))) r :=
-      continuousAt_const.max
-      ((continuousAt_const.div continuousAt_id hr.ne').log (one_div_ne_zero hr.ne'))
-    exact ContinuousAt.comp (g := tailProfile d q)
-      (f := fun r : ℝ => max cutoff (Real.log (1 / r)))
-      (tailProfile_continuousAt (show cutoff ≤ max cutoff (Real.log (1 / r))
-        from le_max_left _ _)) hg
-  apply hc.congr_of_eventuallyEq
-  filter_upwards [Ioi_mem_nhds hr] with x hx
-  have hxp : 0 < x := hx
-  simp [completed, not_le.mpr hxp]
-
-lemma completed_monotoneOn {d q : ℝ} (hd : 1 ≤ d) (hq : q ≤ 1) :
-    MonotoneOn (completed d q) (Ici 0) := by
-  intro r hr s hs hrs
-  by_cases hrz : r = 0
-  · subst r
-    rw [completed_zero]
-    by_cases hsz : s = 0
-    · subst s
-      simp
-    · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hsz)
-      simp only [completed, not_le.mpr hsp, ↓reduceIte]
-      exact (tailProfile_pos (le_max_left _ _)).le
-  · have hrp : 0 < r := lt_of_le_of_ne hr (Ne.symm hrz)
-    have hsp : 0 < s := hrp.trans_le hrs
-    simp only [completed, not_le.mpr hrp, not_le.mpr hsp, ↓reduceIte]
-    apply (tailProfile_antitoneOn hd hq)
-      (show cutoff ≤ max cutoff (Real.log (1 / s)) from le_max_left _ _)
-      (show cutoff ≤ max cutoff (Real.log (1 / r)) from le_max_left _ _)
-    apply max_le_max_left
-    exact Real.log_le_log (one_div_pos.mpr hsp)
-      (one_div_le_one_div_of_le hrp hrs)
-
-theorem completed_isGauge {d q : ℝ} (hd : 1 ≤ d) (hq : q ≤ 1) :
-    IsGauge (completed d q) := by
-  refine ⟨?_, completed_monotoneOn hd hq, completed_zero d q, ?_⟩
-  · intro r hr
-    obtain rfl | hrp := (show 0 ≤ r from hr).eq_or_lt
-    · apply continuousWithinAt_Ioi_iff_Ici.mp
-      simpa only [ContinuousWithinAt, completed_zero] using
-        completed_tendsto_zero (zero_lt_one.trans_le hd) hq
-    · exact (completed_continuousAt_pos hrp).continuousWithinAt
-  · intro r hr
-    simp only [completed, not_le.mpr hr, ↓reduceIte]
-    exact tailProfile_pos (le_max_left _ _)
-
-lemma tailProfile_log_inv {d q r : ℝ} (hr : 0 < r) :
-    tailProfile d q (Real.log (1 / r)) =
-      r ^ d * (Real.log (Real.log (1 / r))) ^ q := by
-  unfold tailProfile
-  congr 1
-  rw [Real.rpow_def_of_pos hr]
-  congr 1
-  simp only [one_div, Real.log_inv]
-  ring
-
-lemma completed_small_formula (d q : ℝ) :
-    completed d q =ᶠ[𝓝[>] 0]
-      fun r => r ^ d * (Real.log (Real.log (1 / r))) ^ q := by
-  filter_upwards [completed_eq_tail_eventually d q, self_mem_nhdsWithin] with r he hr
-  exact he.trans (tailProfile_log_inv hr)
-
-end GaugeCompletion
-
-/-- A fixed, deterministic continuous nondecreasing completion of the explicit
-exact Hausdorff gauge. -/
-def explicitGauge (κ : ℝ) : ℝ → ℝ :=
-  GaugeCompletion.completed (dimension κ) (codimension κ / 2)
-
-/-- A deterministic completion of the exponent-one log-log gauge proposed by
-Schramm. -/
-def schrammGauge (κ : ℝ) : ℝ → ℝ :=
-  GaugeCompletion.completed (dimension κ) 1
-
-theorem explicitGauge_isGauge {κ : ℝ} (hκ : 0 < κ) : IsGauge (explicitGauge κ) := by
-  apply GaugeCompletion.completed_isGauge
-  · dsimp [dimension]
-    linarith
-  · dsimp [codimension, dimension]
-    linarith
-
-theorem explicitGauge_small_formula (κ : ℝ) :
-    HasSmallRadiusFormula (explicitGauge κ) (hFormula κ) :=
-  GaugeCompletion.completed_small_formula _ _
-
-theorem schrammGauge_isGauge {κ : ℝ} (hκ : 0 < κ) : IsGauge (schrammGauge κ) := by
-  apply GaugeCompletion.completed_isGauge
-  · dsimp [dimension]
-    linarith
-  · exact le_rfl
-
-theorem schrammGauge_small_formula (κ : ℝ) :
-    HasSmallRadiusFormula (schrammGauge κ) (gFormula κ) := by
-  change GaugeCompletion.completed (dimension κ) 1 =ᶠ[𝓝[>] 0]
-    fun r => r ^ dimension κ * Real.log (Real.log (1 / r))
-  simpa only [Real.rpow_one] using
-    GaugeCompletion.completed_small_formula (dimension κ) 1
-
-theorem exists_explicitGauge {κ : ℝ} (hκ : 0 < κ) :
-    ∃ h : ℝ → ℝ, IsGauge h ∧ HasSmallRadiusFormula h (hFormula κ) :=
-  ⟨explicitGauge κ, explicitGauge_isGauge hκ, explicitGauge_small_formula κ⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ExactGaugeComparison
+
+namespace OAI
+
+/-! Continuous monotone completions of the explicit and exponent-one log-log gauges. -/
+
+noncomputable section
+
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+namespace GaugeCompletion
+
+def cutoff : ℝ := Real.exp 1
+
+def tailProfile (d q x : ℝ) : ℝ := Real.exp (-d * x) * (Real.log x) ^ q
+
+def completed (d q r : ℝ) : ℝ :=
+  if r ≤ 0 then 0 else tailProfile d q (max cutoff (Real.log (1 / r)))
+
+lemma cutoff_one_lt : 1 < cutoff := Real.one_lt_exp_iff.mpr zero_lt_one
+
+lemma cutoff_pos : 0 < cutoff := Real.exp_pos 1
+
+lemma log_one_le {x : ℝ} (hx : cutoff ≤ x) : 1 ≤ Real.log x := by
+  simpa only [cutoff, Real.log_exp] using Real.log_le_log cutoff_pos hx
+
+lemma tailProfile_pos {d q x : ℝ} (hx : cutoff ≤ x) : 0 < tailProfile d q x := by
+  exact mul_pos (Real.exp_pos _) (Real.rpow_pos_of_pos (zero_lt_one.trans_le (log_one_le hx)) _)
+
+lemma tailProfile_continuousAt {d q x : ℝ} (hx : cutoff ≤ x) :
+    ContinuousAt (tailProfile d q) x := by
+  have hxp : 0 < x := cutoff_pos.trans_le hx
+  have hlp : 0 < Real.log x := zero_lt_one.trans_le (log_one_le hx)
+  exact (Real.continuous_exp.continuousAt.comp (continuousAt_const.mul continuousAt_id)).mul
+    ((Real.continuousAt_log hxp.ne').rpow_const (Or.inl hlp.ne'))
+
+lemma tailProfile_hasDerivAt {d q x : ℝ} (hx : cutoff ≤ x) :
+    HasDerivAt (tailProfile d q)
+      (Real.exp (-d * x) * (Real.log x) ^ (q - 1) *
+        (q / x - d * Real.log x)) x := by
+  have hxp : 0 < x := cutoff_pos.trans_le hx
+  have hlp : 0 < Real.log x := zero_lt_one.trans_le (log_one_le hx)
+  have he := ((hasDerivAt_id x).const_mul (-d)).exp
+  have hl := (Real.hasDerivAt_log hxp.ne').rpow_const (p := q) (Or.inl hlp.ne')
+  convert he.mul hl using 1 <;> try rfl
+  simp only [id_eq]
+  rw [Real.rpow_sub_one hlp.ne']
+  field_simp
+  ring
+
+lemma tailProfile_antitoneOn {d q : ℝ} (hd : 1 ≤ d) (hq : q ≤ 1) :
+    AntitoneOn (tailProfile d q) (Ici cutoff) := by
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ici _)
+    (fun x hx => (tailProfile_continuousAt hx).continuousWithinAt)
+    (fun x hx => (tailProfile_hasDerivAt (interior_subset hx)).hasDerivWithinAt)
+  intro x hx
+  have hxc : cutoff ≤ x := interior_subset hx
+  have hxp : 0 < x := cutoff_pos.trans_le hxc
+  have hx1 : 1 ≤ x := cutoff_one_lt.le.trans hxc
+  have hl : 1 ≤ Real.log x := log_one_le hxc
+  have hdlog : 1 ≤ d * Real.log x := by
+    nlinarith [mul_nonneg (sub_nonneg.mpr hd) (sub_nonneg.mpr hl)]
+  have hdiv : q / x ≤ 1 := (div_le_one hxp).mpr (hq.trans hx1)
+  exact mul_nonpos_of_nonneg_of_nonpos (by positivity) (by linarith)
+
+lemma tailProfile_tendsto_zero {d q : ℝ} (hd : 0 < d) (hq : q ≤ 1) :
+    Tendsto (tailProfile d q) atTop (𝓝 0) := by
+  have hu : Tendsto (fun x : ℝ => x * Real.exp (-d * x)) atTop (𝓝 0) := by
+    simpa only [Real.rpow_one] using tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 d hd
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu
+  · filter_upwards [eventually_ge_atTop cutoff] with x hx
+    exact (tailProfile_pos hx).le
+  · filter_upwards [eventually_ge_atTop cutoff] with x hx
+    have hxp : 0 < x := cutoff_pos.trans_le hx
+    have hpow : (Real.log x) ^ q ≤ Real.log x := by
+      simpa only [Real.rpow_one] using
+        Real.rpow_le_rpow_of_exponent_le (log_one_le hx) hq
+    change Real.exp (-d * x) * (Real.log x) ^ q ≤ _
+    rw [mul_comm (Real.exp (-d * x))]
+    exact mul_le_mul_of_nonneg_right (hpow.trans (Real.log_le_self hxp.le))
+      (Real.exp_pos _).le
+
+lemma log_inv_tendsto_atTop :
+    Tendsto (fun r : ℝ => Real.log (1 / r)) (𝓝[>] 0) atTop := by
+  simpa only [one_div, Function.comp_def] using
+    Real.tendsto_log_atTop.comp tendsto_inv_nhdsGT_zero
+
+@[simp] lemma completed_zero (d q : ℝ) : completed d q 0 = 0 := by
+  simp [completed]
+
+lemma completed_eq_tail_eventually (d q : ℝ) :
+    completed d q =ᶠ[𝓝[>] 0] fun r => tailProfile d q (Real.log (1 / r)) := by
+  filter_upwards [self_mem_nhdsWithin,
+    log_inv_tendsto_atTop.eventually (eventually_ge_atTop cutoff)] with r hr hx
+  have hrp : 0 < r := hr
+  simp only [completed, not_le.mpr hrp, ↓reduceIte, max_eq_right hx]
+
+lemma completed_tendsto_zero {d q : ℝ} (hd : 0 < d) (hq : q ≤ 1) :
+    Tendsto (completed d q) (𝓝[>] 0) (𝓝 0) := by
+  exact ((tailProfile_tendsto_zero hd hq).comp log_inv_tendsto_atTop).congr'
+    (completed_eq_tail_eventually d q).symm
+
+lemma completed_continuousAt_pos {d q r : ℝ} (hr : 0 < r) :
+    ContinuousAt (completed d q) r := by
+  have hc : ContinuousAt
+      (fun r : ℝ => tailProfile d q (max cutoff (Real.log (1 / r)))) r := by
+    have hg : ContinuousAt (fun r : ℝ => max cutoff (Real.log (1 / r))) r :=
+      continuousAt_const.max
+      ((continuousAt_const.div continuousAt_id hr.ne').log (one_div_ne_zero hr.ne'))
+    exact ContinuousAt.comp (g := tailProfile d q)
+      (f := fun r : ℝ => max cutoff (Real.log (1 / r)))
+      (tailProfile_continuousAt (show cutoff ≤ max cutoff (Real.log (1 / r))
+        from le_max_left _ _)) hg
+  apply hc.congr_of_eventuallyEq
+  filter_upwards [Ioi_mem_nhds hr] with x hx
+  have hxp : 0 < x := hx
+  simp [completed, not_le.mpr hxp]
+
+lemma completed_monotoneOn {d q : ℝ} (hd : 1 ≤ d) (hq : q ≤ 1) :
+    MonotoneOn (completed d q) (Ici 0) := by
+  intro r hr s hs hrs
+  by_cases hrz : r = 0
+  · subst r
+    rw [completed_zero]
+    by_cases hsz : s = 0
+    · subst s
+      simp
+    · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hsz)
+      simp only [completed, not_le.mpr hsp, ↓reduceIte]
+      exact (tailProfile_pos (le_max_left _ _)).le
+  · have hrp : 0 < r := lt_of_le_of_ne hr (Ne.symm hrz)
+    have hsp : 0 < s := hrp.trans_le hrs
+    simp only [completed, not_le.mpr hrp, not_le.mpr hsp, ↓reduceIte]
+    apply (tailProfile_antitoneOn hd hq)
+      (show cutoff ≤ max cutoff (Real.log (1 / s)) from le_max_left _ _)
+      (show cutoff ≤ max cutoff (Real.log (1 / r)) from le_max_left _ _)
+    apply max_le_max_left
+    exact Real.log_le_log (one_div_pos.mpr hsp)
+      (one_div_le_one_div_of_le hrp hrs)
+
+theorem completed_isGauge {d q : ℝ} (hd : 1 ≤ d) (hq : q ≤ 1) :
+    IsGauge (completed d q) := by
+  refine ⟨?_, completed_monotoneOn hd hq, completed_zero d q, ?_⟩
+  · intro r hr
+    obtain rfl | hrp := (show 0 ≤ r from hr).eq_or_lt
+    · apply continuousWithinAt_Ioi_iff_Ici.mp
+      simpa only [ContinuousWithinAt, completed_zero] using
+        completed_tendsto_zero (zero_lt_one.trans_le hd) hq
+    · exact (completed_continuousAt_pos hrp).continuousWithinAt
+  · intro r hr
+    simp only [completed, not_le.mpr hr, ↓reduceIte]
+    exact tailProfile_pos (le_max_left _ _)
+
+lemma tailProfile_log_inv {d q r : ℝ} (hr : 0 < r) :
+    tailProfile d q (Real.log (1 / r)) =
+      r ^ d * (Real.log (Real.log (1 / r))) ^ q := by
+  unfold tailProfile
+  congr 1
+  rw [Real.rpow_def_of_pos hr]
+  congr 1
+  simp only [one_div, Real.log_inv]
+  ring
+
+lemma completed_small_formula (d q : ℝ) :
+    completed d q =ᶠ[𝓝[>] 0]
+      fun r => r ^ d * (Real.log (Real.log (1 / r))) ^ q := by
+  filter_upwards [completed_eq_tail_eventually d q, self_mem_nhdsWithin] with r he hr
+  exact he.trans (tailProfile_log_inv hr)
+
+end GaugeCompletion
+
+/-- A fixed, deterministic continuous nondecreasing completion of the explicit
+exact Hausdorff gauge. -/
+def explicitGauge (κ : ℝ) : ℝ → ℝ :=
+  GaugeCompletion.completed (dimension κ) (codimension κ / 2)
+
+/-- A deterministic completion of the exponent-one log-log gauge proposed by
+Schramm. -/
+def schrammGauge (κ : ℝ) : ℝ → ℝ :=
+  GaugeCompletion.completed (dimension κ) 1
+
+theorem explicitGauge_isGauge {κ : ℝ} (hκ : 0 < κ) : IsGauge (explicitGauge κ) := by
+  apply GaugeCompletion.completed_isGauge
+  · dsimp [dimension]
+    linarith
+  · dsimp [codimension, dimension]
+    linarith
+
+theorem explicitGauge_small_formula (κ : ℝ) :
+    HasSmallRadiusFormula (explicitGauge κ) (hFormula κ) :=
+  GaugeCompletion.completed_small_formula _ _
+
+theorem schrammGauge_isGauge {κ : ℝ} (hκ : 0 < κ) : IsGauge (schrammGauge κ) := by
+  apply GaugeCompletion.completed_isGauge
+  · dsimp [dimension]
+    linarith
+  · exact le_rfl
+
+theorem schrammGauge_small_formula (κ : ℝ) :
+    HasSmallRadiusFormula (schrammGauge κ) (gFormula κ) := by
+  change GaugeCompletion.completed (dimension κ) 1 =ᶠ[𝓝[>] 0]
+    fun r => r ^ dimension κ * Real.log (Real.log (1 / r))
+  simpa only [Real.rpow_one] using
+    GaugeCompletion.completed_small_formula (dimension κ) 1
+
+theorem exists_explicitGauge {κ : ℝ} (hκ : 0 < κ) :
+    ∃ h : ℝ → ℝ, IsGauge h ∧ HasSmallRadiusFormula h (hFormula κ) :=
+  ⟨explicitGauge κ, explicitGauge_isGauge hκ, explicitGauge_small_formula κ⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeConformalReduction.lean b/lean/OAI/Probability/SLE/Gauge/GaugeConformalReduction.lean
index 4f103dd311b45a31009f1d32c5f1ff7841665839..97e8125f0e3fbbf760720facd1675e06c3b25467 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeConformalReduction.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeConformalReduction.lean
@@ -1,59 +1,59 @@
-import OAI.Probability.SLE.Gauge.GaugeTraceDischarge
-import OAI.Probability.SLE.Geometry.HalfPlaneCompactnessFromGrowth
-import OAI.Probability.SLE.Geometry.HalfPlaneCompactnessBloch
-import OAI.Probability.SLE.Stochastic.PlanarDiskOpenness
-
-namespace OAI
-
-/-! The explicit gauge needs no independent normal-family input once
-the disk growth estimate is available. -/
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem ordinary_explicit_mainConclusion_of_boundary_support_crosscut_exit
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw) :
-    MainConclusion P (explicitGauge κ) γ :=
-  ordinary_explicit_mainConclusion_of_conformal_inputs P κ hκ hκ8 γ hSLE
-    actual_halfPlaneUnivalentCompactness hboundary hsupport actual_planarDiskOpenness hstraight hexit
-
-theorem ordinary_explicit_mainConclusion_of_geometric_inputs
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw) :
-    MainConclusion P (explicitGauge κ) γ :=
-  ordinary_explicit_mainConclusion_of_conformal_inputs P κ hκ hκ8 γ hSLE
-    actual_halfPlaneUnivalentCompactness hboundary hsupport hplanar hstraight hexit
-
-theorem ordinary_explicit_mainConclusion_of_growth_and_geometric_inputs
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw) :
-    MainConclusion P (explicitGauge κ) γ :=
-  ordinary_explicit_mainConclusion_of_conformal_inputs P κ hκ hκ8 γ hSLE
-    (halfPlaneUnivalentCompactness_of_diskGrowth hgrowth) hboundary hsupport hplanar hstraight hexit
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.GaugeTraceDischarge
+import OAI.Probability.SLE.Geometry.HalfPlaneCompactnessFromGrowth
+import OAI.Probability.SLE.Geometry.HalfPlaneCompactnessBloch
+import OAI.Probability.SLE.Stochastic.PlanarDiskOpenness
+
+namespace OAI
+
+/-! The explicit gauge needs no independent normal-family input once
+the disk growth estimate is available. -/
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem ordinary_explicit_mainConclusion_of_boundary_support_crosscut_exit
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw) :
+    MainConclusion P (explicitGauge κ) γ :=
+  ordinary_explicit_mainConclusion_of_conformal_inputs P κ hκ hκ8 γ hSLE
+    actual_halfPlaneUnivalentCompactness hboundary hsupport actual_planarDiskOpenness hstraight hexit
+
+theorem ordinary_explicit_mainConclusion_of_geometric_inputs
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw) :
+    MainConclusion P (explicitGauge κ) γ :=
+  ordinary_explicit_mainConclusion_of_conformal_inputs P κ hκ hκ8 γ hSLE
+    actual_halfPlaneUnivalentCompactness hboundary hsupport hplanar hstraight hexit
+
+theorem ordinary_explicit_mainConclusion_of_growth_and_geometric_inputs
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw) :
+    MainConclusion P (explicitGauge κ) γ :=
+  ordinary_explicit_mainConclusion_of_conformal_inputs P κ hκ hκ8 γ hSLE
+    (halfPlaneUnivalentCompactness_of_diskGrowth hgrowth) hboundary hsupport hplanar hstraight hexit
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeCrosscutDischarge.lean b/lean/OAI/Probability/SLE/Gauge/GaugeCrosscutDischarge.lean
index b45358fc73555d1ddcc10c68c9fec38bd93fa378..b48efe83e76b491adb52da593dafacf140cbfbe3 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeCrosscutDischarge.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeCrosscutDischarge.lean
@@ -1,26 +1,26 @@
-import OAI.Probability.SLE.Gauge.GaugeExitDischarge
-import OAI.Probability.SLE.Geometry.ActualCrosscutStraightening
-
-namespace OAI
-
-/-! The exact explicit-gauge theorem using relative crosscut straightening. -/
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem ordinary_explicit_mainConclusion_of_support
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hsupport : PublishedGeometricTubeSupport P γ) :
-    MainConclusion P (explicitGauge κ) γ :=
-  ordinary_explicit_mainConclusion_of_support_crosscut P κ hκ hκ8 γ hSLE
-    hsupport actual_halfPlaneCrosscutStraightening
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.GaugeExitDischarge
+import OAI.Probability.SLE.Geometry.ActualCrosscutStraightening
+
+namespace OAI
+
+/-! The exact explicit-gauge theorem using relative crosscut straightening. -/
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem ordinary_explicit_mainConclusion_of_support
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hsupport : PublishedGeometricTubeSupport P γ) :
+    MainConclusion P (explicitGauge κ) γ :=
+  ordinary_explicit_mainConclusion_of_support_crosscut P κ hκ hκ8 γ hSLE
+    hsupport actual_halfPlaneCrosscutStraightening
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeEntrySmallness.lean b/lean/OAI/Probability/SLE/Gauge/GaugeEntrySmallness.lean
index 396978b01c618f0da4c7b7096840ead76ff1c373..1f3adb30c424d4a6c5fd2e892c6d792c4c4cff5b 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeEntrySmallness.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeEntrySmallness.lean
@@ -1,55 +1,55 @@
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLE.LoewnerCharts
-
-namespace OAI
-
-/-! All radii in a late prescribed batch are uniformly small. Hence every
-marked point above a fixed height has a uniformly long initial radius clock. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology
-
-namespace SLEExactGauge.Target
-
-lemma entryRadius_le_batch_exp {κ : ℝ} (hκ : 0<κ) (W : WienerLaw) (e : Entry) :
-    entryRadius κ W e ≤ Real.exp (-(e.1:ℝ)^2) := by
-  unfold entryRadius
-  apply Real.exp_le_exp.mpr
-  have ha := internal_a_pos hκ
-  have hnonneg : 0≤2*a κ*(entryPosition κ W e:ℝ)*((batchLevel κ W e.1:ℝ)+(e.1:ℝ)) := by
-    positivity
-  linarith
-
-lemma radii_uniformly_small {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
-    (δ : ℝ) (hδ : 0<δ) :
-    ∃ N : ℕ, ∀ e : GaugeIndex κ W, N≤e.val.1 → radii κ W e<δ := by
-  obtain ⟨N,hN⟩ := exists_nat_gt (max 1 (-Real.log δ))
-  refine ⟨N,?_⟩
-  intro e he
-  have hNe : (N:ℝ)≤(e.val.1:ℝ) := Nat.cast_le.mpr he
-  have hn1 : (1:ℝ)≤(e.val.1:ℝ) := (le_max_left _ _).trans (hN.le.trans hNe)
-  have hlog : -Real.log δ < (e.val.1:ℝ) := (le_max_right _ _).trans_lt (hN.trans_le hNe)
-  apply (entryRadius_le_batch_exp hκ W e.val).trans_lt
-  rw [← Real.exp_log hδ]
-  apply Real.exp_lt_exp.mpr
-  nlinarith
-
-lemma uniform_marked_radius_gap {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
-    (η A L : ℝ) (hη : 0<η) (hA : 0<A) :
-    ∃ N : ℕ, ∀ e : GaugeIndex κ W, N≤e.val.1 → ∀ z : ℂ, η≤2*z.im →
-      L < Real.log ((2*z.im)/(A*radii κ W e)) := by
-  obtain ⟨N,hN⟩ := radii_uniformly_small hκ W (η/(A*Real.exp L)) (by positivity)
-  refine ⟨N,?_⟩
-  intro e he z hz
-  have hr := radii_pos κ W e
-  have hsmall := (lt_div_iff₀ (mul_pos hA (Real.exp_pos L))).mp (hN e he)
-  apply (Real.lt_log_iff_exp_lt (div_pos (hη.trans_le hz) (mul_pos hA hr))).mpr
-  apply (lt_div_iff₀ (mul_pos hA hr)).mpr
-  nlinarith
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLE.LoewnerCharts
+
+namespace OAI
+
+/-! All radii in a late prescribed batch are uniformly small. Hence every
+marked point above a fixed height has a uniformly long initial radius clock. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology
+
+namespace SLEExactGauge.Target
+
+lemma entryRadius_le_batch_exp {κ : ℝ} (hκ : 0<κ) (W : WienerLaw) (e : Entry) :
+    entryRadius κ W e ≤ Real.exp (-(e.1:ℝ)^2) := by
+  unfold entryRadius
+  apply Real.exp_le_exp.mpr
+  have ha := internal_a_pos hκ
+  have hnonneg : 0≤2*a κ*(entryPosition κ W e:ℝ)*((batchLevel κ W e.1:ℝ)+(e.1:ℝ)) := by
+    positivity
+  linarith
+
+lemma radii_uniformly_small {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
+    (δ : ℝ) (hδ : 0<δ) :
+    ∃ N : ℕ, ∀ e : GaugeIndex κ W, N≤e.val.1 → radii κ W e<δ := by
+  obtain ⟨N,hN⟩ := exists_nat_gt (max 1 (-Real.log δ))
+  refine ⟨N,?_⟩
+  intro e he
+  have hNe : (N:ℝ)≤(e.val.1:ℝ) := Nat.cast_le.mpr he
+  have hn1 : (1:ℝ)≤(e.val.1:ℝ) := (le_max_left _ _).trans (hN.le.trans hNe)
+  have hlog : -Real.log δ < (e.val.1:ℝ) := (le_max_right _ _).trans_lt (hN.trans_le hNe)
+  apply (entryRadius_le_batch_exp hκ W e.val).trans_lt
+  rw [← Real.exp_log hδ]
+  apply Real.exp_lt_exp.mpr
+  nlinarith
+
+lemma uniform_marked_radius_gap {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
+    (η A L : ℝ) (hη : 0<η) (hA : 0<A) :
+    ∃ N : ℕ, ∀ e : GaugeIndex κ W, N≤e.val.1 → ∀ z : ℂ, η≤2*z.im →
+      L < Real.log ((2*z.im)/(A*radii κ W e)) := by
+  obtain ⟨N,hN⟩ := radii_uniformly_small hκ W (η/(A*Real.exp L)) (by positivity)
+  refine ⟨N,?_⟩
+  intro e he z hz
+  have hr := radii_pos κ W e
+  have hsmall := (lt_div_iff₀ (mul_pos hA (Real.exp_pos L))).mp (hN e he)
+  apply (Real.lt_log_iff_exp_lt (div_pos (hη.trans_le hz) (mul_pos hA hr))).mpr
+  apply (lt_div_iff₀ (mul_pos hA hr)).mpr
+  nlinarith
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeExitDischarge.lean b/lean/OAI/Probability/SLE/Gauge/GaugeExitDischarge.lean
index 12b97eb835706bd1b64d48e663d3d30c5d62efa4..e8eded1b9b0f31b1e3adef7cb8602c82be115731 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeExitDischarge.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeExitDischarge.lean
@@ -1,27 +1,27 @@
-import OAI.Probability.SLE.Gauge.GaugeBoundaryDischarge
-import OAI.Probability.SLE.Geometry.ActualConformalExitLaw
-
-namespace OAI
-
-/-! The exact explicit-gauge theorem using conformal Brownian exit laws. -/
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem ordinary_explicit_mainConclusion_of_support_crosscut
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening) :
-    MainConclusion P (explicitGauge κ) γ :=
-  ordinary_explicit_mainConclusion_of_support_crosscut_exit P κ hκ hκ8 γ hSLE
-    hsupport hstraight actual_halfPlaneConformalExitLaw
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.GaugeBoundaryDischarge
+import OAI.Probability.SLE.Geometry.ActualConformalExitLaw
+
+namespace OAI
+
+/-! The exact explicit-gauge theorem using conformal Brownian exit laws. -/
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem ordinary_explicit_mainConclusion_of_support_crosscut
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening) :
+    MainConclusion P (explicitGauge κ) γ :=
+  ordinary_explicit_mainConclusion_of_support_crosscut_exit P κ hκ hκ8 γ hSLE
+    hsupport hstraight actual_halfPlaneConformalExitLaw
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeGrowthDischarge.lean b/lean/OAI/Probability/SLE/Gauge/GaugeGrowthDischarge.lean
index 4153c10931a1e362eced511c574eded4de66a960..588c7b266b66964c1a8c2c7831ea48e420291899 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeGrowthDischarge.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeGrowthDischarge.lean
@@ -1,55 +1,55 @@
-import OAI.Probability.SLE.Loewner.ActualDiskGrowth
-import OAI.Probability.SLE.Gauge.GaugeTraceDischarge
-import OAI.Probability.SLE.Multipoint.MultipointOrdinaryProbability
-import OAI.Probability.SLE.Mass.CanonicalMassConstruction
-
-namespace OAI
-
-/-! The prescribed gauge, multipoint estimate, and canonical mass
-construction for ordinary chordal SLE. -/
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem ordinary_prescribed_mainConclusion
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (W : Target.WienerLaw) (hW : Target.IsWiener W) :
-    MainConclusion P (Target.gauge κ W) γ :=
-  ordinary_prescribed_mainConclusion_of_univalent_growth P κ hκ hκ8 γ hSLE
-    actual_diskUnivalentGrowth W hW
-
-theorem RegularSLEPresentation.physical_multipoint
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (m : ℕ) (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane) (z : Fin m → ℂ)
-    (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
-    (b e : Fin m → ℝ) (hb : ∀ i, 0 < b i) (he : ∀ i, 0 < e i)
-    (hbR : ∀ i, b i ≤ 2*(z i).im*‖deriv J (z i)‖)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
-    (P.comap (Subtype.val : a.samples → Ω))
-      {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-        physicalConformalRadius J (a.G ω) t (z i) ≤ e i} ≤
-      ENNReal.ofReal (multipointConstant κ m * ∏ i, min 1 ((e i/b i)^(1-κ/8))) :=
-  a.physical_multipoint_main hκ hκ8 actual_diskUnivalentGrowth m J hJ hJinj z hz hne
-    b e hb he hbR hsep
-
-theorem RegularSLEPresentation.canonical_mass
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) :
-    Nonempty (Multipoint.CanonicalMass (P.comap (Subtype.val : a.samples → Ω)) κ a.B
-      (measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B)
-      (fun ω => γ ω) a.G) :=
-  a.canonical_mass_main hκ hκ8 actual_diskUnivalentGrowth
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.ActualDiskGrowth
+import OAI.Probability.SLE.Gauge.GaugeTraceDischarge
+import OAI.Probability.SLE.Multipoint.MultipointOrdinaryProbability
+import OAI.Probability.SLE.Mass.CanonicalMassConstruction
+
+namespace OAI
+
+/-! The prescribed gauge, multipoint estimate, and canonical mass
+construction for ordinary chordal SLE. -/
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem ordinary_prescribed_mainConclusion
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (W : Target.WienerLaw) (hW : Target.IsWiener W) :
+    MainConclusion P (Target.gauge κ W) γ :=
+  ordinary_prescribed_mainConclusion_of_univalent_growth P κ hκ hκ8 γ hSLE
+    actual_diskUnivalentGrowth W hW
+
+theorem RegularSLEPresentation.physical_multipoint
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (m : ℕ) (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane) (z : Fin m → ℂ)
+    (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
+    (b e : Fin m → ℝ) (hb : ∀ i, 0 < b i) (he : ∀ i, 0 < e i)
+    (hbR : ∀ i, b i ≤ 2*(z i).im*‖deriv J (z i)‖)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
+    (P.comap (Subtype.val : a.samples → Ω))
+      {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+        physicalConformalRadius J (a.G ω) t (z i) ≤ e i} ≤
+      ENNReal.ofReal (multipointConstant κ m * ∏ i, min 1 ((e i/b i)^(1-κ/8))) :=
+  a.physical_multipoint_main hκ hκ8 actual_diskUnivalentGrowth m J hJ hJinj z hz hne
+    b e hb he hbR hsep
+
+theorem RegularSLEPresentation.canonical_mass
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) :
+    Nonempty (Multipoint.CanonicalMass (P.comap (Subtype.val : a.samples → Ω)) κ a.B
+      (measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B)
+      (fun ω => γ ω) a.G) :=
+  a.canonical_mass_main hκ hκ8 actual_diskUnivalentGrowth
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeTailSum.lean b/lean/OAI/Probability/SLE/Gauge/GaugeTailSum.lean
index 8940f2c066e126a69505ceeff9eb88a312480e0b..8a78450d575ef637fe530c3efacbb4027298744f 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeTailSum.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeTailSum.lean
@@ -1,62 +1,62 @@
-import OAI.Probability.SLE.Geometry.FiniteLawTailENN
-
-namespace OAI
-
-/-! The prescribed gauge entries retain their multiplicities when tail
-probabilities are summed. Cleanup entries occur exactly once per batch. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Target
-
-abbrev TailCode (κ : ℝ) (W : WienerLaw) :=
-  ℕ ⊕ (Σ n : ℕ, Σ k : ℤ, Fin (multiplicity κ W (n+1) k))
-
-def gaugeTailEquiv (κ : ℝ) (W : WienerLaw) : GaugeIndex κ W ≃ TailCode κ W where
-  toFun e := match h : e.val.2 with
-    | none => Sum.inl (e.val.1-1)
-    | some (k,j) => Sum.inr ⟨e.val.1-1,k,j,by
-        have hj := e.property.2
-        rw [h] at hj
-        simpa only [Nat.sub_add_cancel e.property.1] using hj⟩
-  invFun c := match c with
-    | Sum.inl n => ⟨(n+1,none),by simp [ValidEntry]⟩
-    | Sum.inr ⟨n,k,j⟩ => ⟨(n+1,some (k,j.val)),by
-        exact ⟨by omega,j.isLt⟩⟩
-  left_inv e := by
-    apply Subtype.ext
-    rcases e with ⟨⟨n,o⟩,hn,ho⟩
-    cases o with
-    | none => simp only [Nat.sub_add_cancel hn]
-    | some p => rcases p with ⟨k,j⟩; simp only [Nat.sub_add_cancel hn]
-  right_inv c := by
-    cases c with
-    | inl n => simp
-    | inr p => rcases p with ⟨n,k,j⟩; simp
-
-lemma gauge_tail_sum_eq (κ : ℝ) (W : WienerLaw) (p : ℝ → ℝ≥0∞) :
-    (∑' e : GaugeIndex κ W, p (16*weights κ W e)) =
-      (∑' n : ℕ, p (16*((n:ℝ)+1))) +
-      ∑' n : ℕ, ∑' k : ℤ, (multiplicity κ W (n+1) k : ℝ≥0∞)*
-        p (16*MomentBatches.dyadic k) := by
-  rw [← (gaugeTailEquiv κ W).symm.tsum_eq (fun e => p (16*weights κ W e))]
-  rw [ENNReal.summable.tsum_sum ENNReal.summable]
-  congr 1
-  · apply tsum_congr
-    intro n
-    simp [gaugeTailEquiv,weights,entryValue]
-  · rw [ENNReal.tsum_sigma']
-    apply tsum_congr
-    intro n
-    rw [ENNReal.tsum_sigma']
-    apply tsum_congr
-    intro k
-    simp [gaugeTailEquiv,weights,entryValue,MomentBatches.dyadic,tsum_fintype]
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.FiniteLawTailENN
+
+namespace OAI
+
+/-! The prescribed gauge entries retain their multiplicities when tail
+probabilities are summed. Cleanup entries occur exactly once per batch. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Target
+
+abbrev TailCode (κ : ℝ) (W : WienerLaw) :=
+  ℕ ⊕ (Σ n : ℕ, Σ k : ℤ, Fin (multiplicity κ W (n+1) k))
+
+def gaugeTailEquiv (κ : ℝ) (W : WienerLaw) : GaugeIndex κ W ≃ TailCode κ W where
+  toFun e := match h : e.val.2 with
+    | none => Sum.inl (e.val.1-1)
+    | some (k,j) => Sum.inr ⟨e.val.1-1,k,j,by
+        have hj := e.property.2
+        rw [h] at hj
+        simpa only [Nat.sub_add_cancel e.property.1] using hj⟩
+  invFun c := match c with
+    | Sum.inl n => ⟨(n+1,none),by simp [ValidEntry]⟩
+    | Sum.inr ⟨n,k,j⟩ => ⟨(n+1,some (k,j.val)),by
+        exact ⟨by omega,j.isLt⟩⟩
+  left_inv e := by
+    apply Subtype.ext
+    rcases e with ⟨⟨n,o⟩,hn,ho⟩
+    cases o with
+    | none => simp only [Nat.sub_add_cancel hn]
+    | some p => rcases p with ⟨k,j⟩; simp only [Nat.sub_add_cancel hn]
+  right_inv c := by
+    cases c with
+    | inl n => simp
+    | inr p => rcases p with ⟨n,k,j⟩; simp
+
+lemma gauge_tail_sum_eq (κ : ℝ) (W : WienerLaw) (p : ℝ → ℝ≥0∞) :
+    (∑' e : GaugeIndex κ W, p (16*weights κ W e)) =
+      (∑' n : ℕ, p (16*((n:ℝ)+1))) +
+      ∑' n : ℕ, ∑' k : ℤ, (multiplicity κ W (n+1) k : ℝ≥0∞)*
+        p (16*MomentBatches.dyadic k) := by
+  rw [← (gaugeTailEquiv κ W).symm.tsum_eq (fun e => p (16*weights κ W e))]
+  rw [ENNReal.summable.tsum_sum ENNReal.summable]
+  congr 1
+  · apply tsum_congr
+    intro n
+    simp [gaugeTailEquiv,weights,entryValue]
+  · rw [ENNReal.tsum_sigma']
+    apply tsum_congr
+    intro n
+    rw [ENNReal.tsum_sigma']
+    apply tsum_congr
+    intro k
+    simp [gaugeTailEquiv,weights,entryValue,MomentBatches.dyadic,tsum_fintype]
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GaugeTraceDischarge.lean b/lean/OAI/Probability/SLE/Gauge/GaugeTraceDischarge.lean
index 6e033354fc31e208a8811ed86d105996571df356..d95558f54abf523d97af2556bb7b8cde7cf54c5f 100644
--- a/lean/OAI/Probability/SLE/Gauge/GaugeTraceDischarge.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GaugeTraceDischarge.lean
@@ -1,57 +1,57 @@
-import OAI.Probability.SLE.Loewner.CanonicalTraceDischarge
-import OAI.Probability.SLE.Gauge.PrescribedPublishedMain
-import OAI.Probability.SLE.Geometry.ExplicitInputDischarge
-
-namespace OAI
-
-/-! Exact gauge theorems using canonical trace generation and explicit
-conformal hypotheses. -/
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance gaugeTraceRealMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance gaugeTraceRealBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-theorem RegularSLEPresentation.canonical_driver_trace_generation
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) :
-    PublishedTraceGeneration
-      ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B))
-      κ (fun t p => p t) :=
-  canonical_driver_trace_generation_of_witness κ hκ hκ8 a.B a.brownian
-    a.measurable_B a.continuous_B a.zero_B (fun ω => γ ω) a.G a.F a.witness
-
-theorem ordinary_prescribed_mainConclusion_of_univalent_growth
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (W : Target.WienerLaw) (hW : Target.IsWiener W) :
-    MainConclusion P (Target.gauge κ W) γ :=
-  ordinary_prescribed_mainConclusion_of_published_inputs P κ hκ hκ8 γ hSLE hgrowth
-    ((ordinaryRegularPresentation hSLE).canonical_driver_trace_generation hκ hκ8) W hW
-
-theorem ordinary_explicit_mainConclusion_of_conformal_inputs
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hcompact : PublishedHalfPlaneUnivalentCompactness)
-    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (hstraight : PublishedHalfPlaneCrosscutStraightening)
-    (hexit : PublishedHalfPlaneConformalExitLaw) :
-    MainConclusion P (explicitGauge κ) γ :=
-  ordinary_explicit_mainConclusion_of_remaining_inputs P κ hκ hκ8 γ hSLE
-    ((ordinaryRegularPresentation hSLE).canonical_driver_trace_generation hκ hκ8)
-    hcompact hboundary hsupport hplanar hstraight hexit
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.CanonicalTraceDischarge
+import OAI.Probability.SLE.Gauge.PrescribedPublishedMain
+import OAI.Probability.SLE.Geometry.ExplicitInputDischarge
+
+namespace OAI
+
+/-! Exact gauge theorems using canonical trace generation and explicit
+conformal hypotheses. -/
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance gaugeTraceRealMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance gaugeTraceRealBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+theorem RegularSLEPresentation.canonical_driver_trace_generation
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) :
+    PublishedTraceGeneration
+      ((P.comap (Subtype.val : a.samples → Ω)).map (sourceDriverPath a.B a.continuous_B))
+      κ (fun t p => p t) :=
+  canonical_driver_trace_generation_of_witness κ hκ hκ8 a.B a.brownian
+    a.measurable_B a.continuous_B a.zero_B (fun ω => γ ω) a.G a.F a.witness
+
+theorem ordinary_prescribed_mainConclusion_of_univalent_growth
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (W : Target.WienerLaw) (hW : Target.IsWiener W) :
+    MainConclusion P (Target.gauge κ W) γ :=
+  ordinary_prescribed_mainConclusion_of_published_inputs P κ hκ hκ8 γ hSLE hgrowth
+    ((ordinaryRegularPresentation hSLE).canonical_driver_trace_generation hκ hκ8) W hW
+
+theorem ordinary_explicit_mainConclusion_of_conformal_inputs
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hcompact : PublishedHalfPlaneUnivalentCompactness)
+    (hboundary : PublishedFiniteLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (hstraight : PublishedHalfPlaneCrosscutStraightening)
+    (hexit : PublishedHalfPlaneConformalExitLaw) :
+    MainConclusion P (explicitGauge κ) γ :=
+  ordinary_explicit_mainConclusion_of_remaining_inputs P κ hκ hκ8 γ hSLE
+    ((ordinaryRegularPresentation hSLE).canonical_driver_trace_generation hκ hκ8)
+    hcompact hboundary hsupport hplanar hstraight hexit
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/GoodGridRestriction.lean b/lean/OAI/Probability/SLE/Gauge/GoodGridRestriction.lean
index dd7698835793e87007a64ca156b9885dabfe87ed..02bd6e36f2c7f991b4d012bf8cc897ce87691530 100644
--- a/lean/OAI/Probability/SLE/Gauge/GoodGridRestriction.lean
+++ b/lean/OAI/Probability/SLE/Gauge/GoodGridRestriction.lean
@@ -1,133 +1,133 @@
-import OAI.Probability.SLE.Gauge.PrescribedGridLower
-
-namespace OAI
-
-/-! Summable bad-cell mass leaves positive restrictions satisfying every
-sufficiently late prescribed-grid bound. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def badGridCells (μ : Measure ℂ) (r H : ℝ) : Set ℂ :=
-  ⋃ i : ℤ×ℤ, if ENNReal.ofReal H < μ (closedBall (coverGridCenter r i) (2*r)) then
-    closedBall (coverGridCenter r i) (2*r) else ∅
-
-lemma measurableSet_badGridCells (μ : Measure ℂ) (r H : ℝ) :
-    MeasurableSet (badGridCells μ r H) := by
-  classical
-  apply MeasurableSet.iUnion
-  intro i
-  split_ifs <;> simp only [measurableSet_closedBall,MeasurableSet.empty]
-
-def prescribedBadGrid (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
-    (e : Target.GaugeIndex κ W) : Set ℂ :=
-  badGridCells μ (Target.radii κ W e)
-    (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ)
-
-def prescribedGoodGrid (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
-    (N : ℕ) : Set ℂ :=
-  ⋂ e : Target.GaugeIndex κ W,
-    if N ≤ e.val.1 then (prescribedBadGrid μ κ W D e)ᶜ else univ
-
-lemma measurableSet_prescribedGoodGrid (μ : Measure ℂ) (κ : ℝ)
-    (W : Target.WienerLaw) (D : ℝ) (N : ℕ) :
-    MeasurableSet (prescribedGoodGrid μ κ W D N) := by
-  classical
-  apply MeasurableSet.iInter
-  intro e
-  split_ifs
-  · exact (measurableSet_badGridCells _ _ _).compl
-  · exact MeasurableSet.univ
-
-lemma ae_mem_prescribedGoodGrid_of_summable
-    (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
-    (hsum : (∑' e : Target.GaugeIndex κ W, μ (prescribedBadGrid μ κ W D e)) ≠ ∞) :
-    ∀ᵐ x ∂μ, ∃ N : ℕ, x ∈ prescribedGoodGrid μ κ W D N := by
-  classical
-  filter_upwards [ae_finite_setOfPred_mem hsum] with x hx
-  obtain ⟨N,hN⟩ := (hx.image (fun e => e.val.1)).bddAbove
-  refine ⟨N+1,mem_iInter.mpr ?_⟩
-  intro e
-  by_cases he : N+1 ≤ e.val.1
-  · rw [ite_eq_left he]
-    intro hbad
-    have hh := hN (mem_image_of_mem (fun e : Target.GaugeIndex κ W => e.val.1) hbad)
-    omega
-  · rw [ite_eq_right he]
-    trivial
-
-lemma exists_positive_prescribedGoodGrid
-    (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
-    (hsum : (∑' e : Target.GaugeIndex κ W, μ (prescribedBadGrid μ κ W D e)) ≠ ∞)
-    (E : Set ℂ) (hE : 0 < μ E) :
-    ∃ N : ℕ, 0 < μ (E ∩ prescribedGoodGrid μ κ W D N) := by
-  by_contra! hn
-  have hz : μ (⋃ N : ℕ, E ∩ prescribedGoodGrid μ κ W D N)=0 :=
-    measure_iUnion_null (fun N => le_zero_iff.mp (hn N))
-  have he : E =ᵐ[μ] ⋃ N : ℕ, E ∩ prescribedGoodGrid μ κ W D N := by
-    filter_upwards [ae_mem_prescribedGoodGrid_of_summable μ κ W D hsum] with x hx
-    obtain ⟨N,hN⟩ := hx
-    apply propext
-    constructor
-    · intro hEx
-      exact mem_iUnion.mpr ⟨N,hEx,hN⟩
-    · intro hEx
-      obtain ⟨_,hEx,_⟩ := mem_iUnion.mp hEx
-      exact hEx
-  exact hE.ne' ((measure_congr he).trans hz)
-
-lemma prescribedGoodGrid_cell_bound
-    (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ) (N : ℕ)
-    (e : Target.Entry) (he : Target.ValidEntry κ W e) (hN : N ≤ e.1) (i : ℤ×ℤ) :
-    (μ.restrict (prescribedGoodGrid μ κ W D N))
-      (closedBall (coverGridCenter (Target.entryRadius κ W e) i) (2*Target.entryRadius κ W e)) ≤
-      ENNReal.ofReal (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ) := by
-  classical
-  let Q := closedBall (coverGridCenter (Target.entryRadius κ W e) i) (2*Target.entryRadius κ W e)
-  let H := ENNReal.ofReal (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ)
-  change (μ.restrict (prescribedGoodGrid μ κ W D N)) Q ≤ H
-  by_cases hbad : H < μ Q
-  · have hd : Q ∩ prescribedGoodGrid μ κ W D N=∅ := by
-      apply eq_empty_iff_forall_notMem.mpr
-      rintro x ⟨hxQ,hxG⟩
-      have hn := mem_iInter.mp hxG (⟨e,he⟩ : Target.GaugeIndex κ W)
-      rw [ite_eq_left hN] at hn
-      apply hn
-      apply mem_iUnion.mpr
-      refine ⟨i,?_⟩
-      change x ∈ if H < μ Q then Q else ∅
-      rw [ite_eq_left hbad]
-      exact hxQ
-    rw [Measure.restrict_apply measurableSet_closedBall,hd,measure_empty]
-    exact bot_le
-  · exact (Measure.restrict_le_self Q).trans (le_of_not_gt hbad)
-
-lemma positive_prescribed_hausdorff_of_summable_bad_mass
-    (μ : Measure ℂ) [IsFiniteMeasure μ] (κ : ℝ) (W : Target.WienerLaw)
-    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
-    (hm1 : 0 < Target.moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
-    (D : ℝ) (hD : 0 ≤ D)
-    (hsum : (∑' e : Target.GaugeIndex κ W, μ (prescribedBadGrid μ κ W D e)) ≠ ∞)
-    (E : Set ℂ) (hE : MeasurableSet E) (hpos : 0 < μ E) :
-    0 < Target.hausdorff (Target.gauge κ W) E := by
-  obtain ⟨N,hN⟩ := exists_positive_prescribedGoodGrid μ κ W D hsum E hpos
-  let ν := μ.restrict (prescribedGoodGrid μ κ W D N)
-  obtain ⟨C,hC,hbound⟩ := prescribed_grid_mass_distribution ν κ W hg hm1 hJ N D hD
-    (prescribedGoodGrid_cell_bound μ κ W D N)
-  have hp : 0 < ν E := by
-    simpa only [ν,Measure.restrict_apply hE] using hN
-  have hb := hbound E
-  change ν E ≤ ENNReal.ofReal C*Target.hausdorff (Target.gauge κ W) E at hb
-  by_contra! hz
-  rw [le_zero_iff.mp hz,mul_zero] at hb
-  exact hp.not_ge hb
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedGridLower
+
+namespace OAI
+
+/-! Summable bad-cell mass leaves positive restrictions satisfying every
+sufficiently late prescribed-grid bound. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def badGridCells (μ : Measure ℂ) (r H : ℝ) : Set ℂ :=
+  ⋃ i : ℤ×ℤ, if ENNReal.ofReal H < μ (closedBall (coverGridCenter r i) (2*r)) then
+    closedBall (coverGridCenter r i) (2*r) else ∅
+
+lemma measurableSet_badGridCells (μ : Measure ℂ) (r H : ℝ) :
+    MeasurableSet (badGridCells μ r H) := by
+  classical
+  apply MeasurableSet.iUnion
+  intro i
+  split_ifs <;> simp only [measurableSet_closedBall,MeasurableSet.empty]
+
+def prescribedBadGrid (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
+    (e : Target.GaugeIndex κ W) : Set ℂ :=
+  badGridCells μ (Target.radii κ W e)
+    (D*Target.weights κ W e*(Target.radii κ W e)^Target.d κ)
+
+def prescribedGoodGrid (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
+    (N : ℕ) : Set ℂ :=
+  ⋂ e : Target.GaugeIndex κ W,
+    if N ≤ e.val.1 then (prescribedBadGrid μ κ W D e)ᶜ else univ
+
+lemma measurableSet_prescribedGoodGrid (μ : Measure ℂ) (κ : ℝ)
+    (W : Target.WienerLaw) (D : ℝ) (N : ℕ) :
+    MeasurableSet (prescribedGoodGrid μ κ W D N) := by
+  classical
+  apply MeasurableSet.iInter
+  intro e
+  split_ifs
+  · exact (measurableSet_badGridCells _ _ _).compl
+  · exact MeasurableSet.univ
+
+lemma ae_mem_prescribedGoodGrid_of_summable
+    (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
+    (hsum : (∑' e : Target.GaugeIndex κ W, μ (prescribedBadGrid μ κ W D e)) ≠ ∞) :
+    ∀ᵐ x ∂μ, ∃ N : ℕ, x ∈ prescribedGoodGrid μ κ W D N := by
+  classical
+  filter_upwards [ae_finite_setOfPred_mem hsum] with x hx
+  obtain ⟨N,hN⟩ := (hx.image (fun e => e.val.1)).bddAbove
+  refine ⟨N+1,mem_iInter.mpr ?_⟩
+  intro e
+  by_cases he : N+1 ≤ e.val.1
+  · rw [ite_eq_left he]
+    intro hbad
+    have hh := hN (mem_image_of_mem (fun e : Target.GaugeIndex κ W => e.val.1) hbad)
+    omega
+  · rw [ite_eq_right he]
+    trivial
+
+lemma exists_positive_prescribedGoodGrid
+    (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ)
+    (hsum : (∑' e : Target.GaugeIndex κ W, μ (prescribedBadGrid μ κ W D e)) ≠ ∞)
+    (E : Set ℂ) (hE : 0 < μ E) :
+    ∃ N : ℕ, 0 < μ (E ∩ prescribedGoodGrid μ κ W D N) := by
+  by_contra! hn
+  have hz : μ (⋃ N : ℕ, E ∩ prescribedGoodGrid μ κ W D N)=0 :=
+    measure_iUnion_null (fun N => le_zero_iff.mp (hn N))
+  have he : E =ᵐ[μ] ⋃ N : ℕ, E ∩ prescribedGoodGrid μ κ W D N := by
+    filter_upwards [ae_mem_prescribedGoodGrid_of_summable μ κ W D hsum] with x hx
+    obtain ⟨N,hN⟩ := hx
+    apply propext
+    constructor
+    · intro hEx
+      exact mem_iUnion.mpr ⟨N,hEx,hN⟩
+    · intro hEx
+      obtain ⟨_,hEx,_⟩ := mem_iUnion.mp hEx
+      exact hEx
+  exact hE.ne' ((measure_congr he).trans hz)
+
+lemma prescribedGoodGrid_cell_bound
+    (μ : Measure ℂ) (κ : ℝ) (W : Target.WienerLaw) (D : ℝ) (N : ℕ)
+    (e : Target.Entry) (he : Target.ValidEntry κ W e) (hN : N ≤ e.1) (i : ℤ×ℤ) :
+    (μ.restrict (prescribedGoodGrid μ κ W D N))
+      (closedBall (coverGridCenter (Target.entryRadius κ W e) i) (2*Target.entryRadius κ W e)) ≤
+      ENNReal.ofReal (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ) := by
+  classical
+  let Q := closedBall (coverGridCenter (Target.entryRadius κ W e) i) (2*Target.entryRadius κ W e)
+  let H := ENNReal.ofReal (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ)
+  change (μ.restrict (prescribedGoodGrid μ κ W D N)) Q ≤ H
+  by_cases hbad : H < μ Q
+  · have hd : Q ∩ prescribedGoodGrid μ κ W D N=∅ := by
+      apply eq_empty_iff_forall_notMem.mpr
+      rintro x ⟨hxQ,hxG⟩
+      have hn := mem_iInter.mp hxG (⟨e,he⟩ : Target.GaugeIndex κ W)
+      rw [ite_eq_left hN] at hn
+      apply hn
+      apply mem_iUnion.mpr
+      refine ⟨i,?_⟩
+      change x ∈ if H < μ Q then Q else ∅
+      rw [ite_eq_left hbad]
+      exact hxQ
+    rw [Measure.restrict_apply measurableSet_closedBall,hd,measure_empty]
+    exact bot_le
+  · exact (Measure.restrict_le_self Q).trans (le_of_not_gt hbad)
+
+lemma positive_prescribed_hausdorff_of_summable_bad_mass
+    (μ : Measure ℂ) [IsFiniteMeasure μ] (κ : ℝ) (W : Target.WienerLaw)
+    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
+    (hm1 : 0 < Target.moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
+    (D : ℝ) (hD : 0 ≤ D)
+    (hsum : (∑' e : Target.GaugeIndex κ W, μ (prescribedBadGrid μ κ W D e)) ≠ ∞)
+    (E : Set ℂ) (hE : MeasurableSet E) (hpos : 0 < μ E) :
+    0 < Target.hausdorff (Target.gauge κ W) E := by
+  obtain ⟨N,hN⟩ := exists_positive_prescribedGoodGrid μ κ W D hsum E hpos
+  let ν := μ.restrict (prescribedGoodGrid μ κ W D N)
+  obtain ⟨C,hC,hbound⟩ := prescribed_grid_mass_distribution ν κ W hg hm1 hJ N D hD
+    (prescribedGoodGrid_cell_bound μ κ W D N)
+  have hp : 0 < ν E := by
+    simpa only [ν,Measure.restrict_apply hE] using hN
+  have hb := hbound E
+  change ν E ≤ ENNReal.ofReal C*Target.hausdorff (Target.gauge κ W) E at hb
+  by_contra! hz
+  rw [le_zero_iff.mp hz,mul_zero] at hb
+  exact hp.not_ge hb
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/LargerGaugeMain.lean b/lean/OAI/Probability/SLE/Gauge/LargerGaugeMain.lean
index 365e7487d4fafa37388704232cf1fdd382a4c50a..734a29d2bcb4ad7c64c074d199e4b60eb0a9af1a 100644
--- a/lean/OAI/Probability/SLE/Gauge/LargerGaugeMain.lean
+++ b/lean/OAI/Probability/SLE/Gauge/LargerGaugeMain.lean
@@ -1,78 +1,78 @@
-import OAI.Probability.SLE.LowerPositivity
-import OAI.Probability.SLE.Gauge.ExactGaugeComparison
-import OAI.Probability.SLE.Gauge.GaugeCompletion
-
-namespace OAI
-
-/-! The larger-gauge corollary in the introduction of the explicit-gauge paper. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-/-- On one event, every positive-time compact SLE segment has non-sigma-finite
-Hausdorff measure for Schramm's proposed exponent-one log-log gauge, by
-comparison with the explicit gauge's positive Hausdorff measure. -/
-theorem larger_gauge_not_sigmaFinite_on_all_segments
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (h g : ℝ → ℝ) (hh : IsGauge h) (hg : IsGauge g)
-    (hf : HasSmallRadiusFormula h (hFormula κ))
-    (gf : HasSmallRadiusFormula g (gFormula κ)) :
-    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
-      ¬ SigmaFinite ((hausdorffGauge g).restrict (segment (γ ω) s t)) := by
-  have hlo := sourceLowerMain_proved Ω inferInstance P inferInstance
-    κ hκ hκ8 γ hSLE h hh hf
-  filter_upwards [hlo] with ω hω s t hs hst
-  exact larger_logLog_gauge_not_sigmaFinite hκ hh hg hf gf (hω s t hs hst)
-
-/-- The same simultaneous event gives infinite total larger-gauge measure. -/
-theorem larger_gauge_infinite_on_all_segments
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (h g : ℝ → ℝ) (hh : IsGauge h) (hg : IsGauge g)
-    (hf : HasSmallRadiusFormula h (hFormula κ))
-    (gf : HasSmallRadiusFormula g (gFormula κ)) :
-    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
-      hausdorffGauge g (segment (γ ω) s t) = ∞ := by
-  have hlo := sourceLowerMain_proved Ω inferInstance P inferInstance
-    κ hκ hκ8 γ hSLE h hh hf
-  filter_upwards [hlo] with ω hω s t hs hst
-  exact hausdorffGauge_larger_eq_top_of_positive hh hg
-    (gauge_ratio_tendsto_zero hκ hf gf) (hω s t hs hst)
-
-/-- The paper's larger-gauge corollary with deterministic gauges constructed
-in Lean, so no existence or completion hypothesis remains. -/
-theorem schramm_gauge_not_sigmaFinite_on_all_segments
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) :
-    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
-      ¬ SigmaFinite ((hausdorffGauge (schrammGauge κ)).restrict
-        (segment (γ ω) s t)) :=
-  larger_gauge_not_sigmaFinite_on_all_segments P κ hκ hκ8 γ hSLE
-    (explicitGauge κ) (schrammGauge κ)
-    (explicitGauge_isGauge hκ) (schrammGauge_isGauge hκ)
-    (explicitGauge_small_formula κ) (schrammGauge_small_formula κ)
-
-/-- Infinite total measure for the concrete exponent-one log-log gauge. -/
-theorem schramm_gauge_infinite_on_all_segments
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) :
-    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
-      hausdorffGauge (schrammGauge κ) (segment (γ ω) s t) = ∞ :=
-  larger_gauge_infinite_on_all_segments P κ hκ hκ8 γ hSLE
-    (explicitGauge κ) (schrammGauge κ)
-    (explicitGauge_isGauge hκ) (schrammGauge_isGauge hκ)
-    (explicitGauge_small_formula κ) (schrammGauge_small_formula κ)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.LowerPositivity
+import OAI.Probability.SLE.Gauge.ExactGaugeComparison
+import OAI.Probability.SLE.Gauge.GaugeCompletion
+
+namespace OAI
+
+/-! The larger-gauge corollary in the introduction of the explicit-gauge paper. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+/-- On one event, every positive-time compact SLE segment has non-sigma-finite
+Hausdorff measure for Schramm's proposed exponent-one log-log gauge, by
+comparison with the explicit gauge's positive Hausdorff measure. -/
+theorem larger_gauge_not_sigmaFinite_on_all_segments
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (h g : ℝ → ℝ) (hh : IsGauge h) (hg : IsGauge g)
+    (hf : HasSmallRadiusFormula h (hFormula κ))
+    (gf : HasSmallRadiusFormula g (gFormula κ)) :
+    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
+      ¬ SigmaFinite ((hausdorffGauge g).restrict (segment (γ ω) s t)) := by
+  have hlo := sourceLowerMain_proved Ω inferInstance P inferInstance
+    κ hκ hκ8 γ hSLE h hh hf
+  filter_upwards [hlo] with ω hω s t hs hst
+  exact larger_logLog_gauge_not_sigmaFinite hκ hh hg hf gf (hω s t hs hst)
+
+/-- The same simultaneous event gives infinite total larger-gauge measure. -/
+theorem larger_gauge_infinite_on_all_segments
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (h g : ℝ → ℝ) (hh : IsGauge h) (hg : IsGauge g)
+    (hf : HasSmallRadiusFormula h (hFormula κ))
+    (gf : HasSmallRadiusFormula g (gFormula κ)) :
+    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
+      hausdorffGauge g (segment (γ ω) s t) = ∞ := by
+  have hlo := sourceLowerMain_proved Ω inferInstance P inferInstance
+    κ hκ hκ8 γ hSLE h hh hf
+  filter_upwards [hlo] with ω hω s t hs hst
+  exact hausdorffGauge_larger_eq_top_of_positive hh hg
+    (gauge_ratio_tendsto_zero hκ hf gf) (hω s t hs hst)
+
+/-- The paper's larger-gauge corollary with deterministic gauges constructed
+in Lean, so no existence or completion hypothesis remains. -/
+theorem schramm_gauge_not_sigmaFinite_on_all_segments
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) :
+    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
+      ¬ SigmaFinite ((hausdorffGauge (schrammGauge κ)).restrict
+        (segment (γ ω) s t)) :=
+  larger_gauge_not_sigmaFinite_on_all_segments P κ hκ hκ8 γ hSLE
+    (explicitGauge κ) (schrammGauge κ)
+    (explicitGauge_isGauge hκ) (schrammGauge_isGauge hκ)
+    (explicitGauge_small_formula κ) (schrammGauge_small_formula κ)
+
+/-- Infinite total measure for the concrete exponent-one log-log gauge. -/
+theorem schramm_gauge_infinite_on_all_segments
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) :
+    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
+      hausdorffGauge (schrammGauge κ) (segment (γ ω) s t) = ∞ :=
+  larger_gauge_infinite_on_all_segments P κ hκ hκ8 γ hSLE
+    (explicitGauge κ) (schrammGauge κ)
+    (explicitGauge_isGauge hκ) (schrammGauge_isGauge hκ)
+    (explicitGauge_small_formula κ) (schrammGauge_small_formula κ)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/LocalAffinePassage.lean b/lean/OAI/Probability/SLE/Gauge/LocalAffinePassage.lean
index 3d567906505e6a0d365b6fbae3d1e14e82662c50..c44372a40f855e8e27f2fd647298d860a4bd09f2 100644
--- a/lean/OAI/Probability/SLE/Gauge/LocalAffinePassage.lean
+++ b/lean/OAI/Probability/SLE/Gauge/LocalAffinePassage.lean
@@ -1,96 +1,96 @@
-import OAI.Probability.SLE.Gauge.RibbonPassageScaling
-import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit
-
-namespace OAI
-
-/-! Affine transport uses the progress coordinate only on the successful
-physical envelope. This avoids imposing any condition on the arbitrary
-off-image values of `coordinateInverse`. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open FirstFrontPassage PassageSuccess FrontSuccessMeasurability RibbonCoordinates
-open RibbonPassageScaling
-
-theorem frontSuccess_mass_eq_of_eqOn
-    {Ω : Type*} (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ)
-    (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (a : ℕ → ℂ) (ε : ℝ) (n : ℕ)
-    (hpeq : EqOn (p n) (p' n) (V n)) (κ v c : ℝ) (R : Set ℂ) :
-    {ω | ω ∈ FrontSuccess η p σ V a ε n ∧ ENNReal.ofReal c ≤
-      segmentSausageCharge κ (η n ω) (σ ω)
-        (frontTime (fun t => p n (η n ω t)) 1) v R} =
-    {ω | ω ∈ FrontSuccess η p' σ V a ε n ∧ ENNReal.ofReal c ≤
-      segmentSausageCharge κ (η n ω) (σ ω)
-        (frontTime (fun t => p' n (η n ω t)) 1) v R} := by
-  have heq := frontSuccess_eq_of_eqOn η p p' σ V a ε n hpeq
-  ext ω
-  by_cases hω : ω ∈ FrontSuccess η p σ V a ε n
-  · have hω' := heq ▸ hω
-    have hs := hω
-    obtain ⟨hfirst, hhit, _, hpath, _⟩ := hω
-    have hfirst' := firstFrontTime_eq_of_prefix _ _ 1 _ hfirst hhit
-      (fun t ht => hpeq (hpath ht))
-    have ht : frontTime (fun t => p' n (η n ω t)) 1 =
-        frontTime (fun t => p n (η n ω t)) 1 := by
-      simp only [frontTime, hfirst', WithTop.untopD_coe]
-    simp only [mem_ofPred_eq, hs, hω', true_and, ht]
-  · have hω' : ω ∉ FrontSuccess η p' σ V a ε n := by
-      simpa only [← heq] using hω
-    simp only [mem_ofPred_eq, hω, hω', false_and]
-
-def affineInverse (a : ℂ) (ell : ℝ) (z : ℂ) : ℂ := (z - a) / (ell : ℂ)
-
-theorem affineInverse_left (a : ℂ) {ell : ℝ} (hell : 0 < ell) (z : ℂ) :
-    affineInverse a ell (affine a ell z) = z := by
-  simp only [affineInverse, affine, add_sub_cancel_left]
-  exact mul_div_cancel_left₀ z (Complex.ofReal_ne_zero.mpr hell.ne')
-
-theorem frontSuccess_mass_probability_affine_local
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ)
-    (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (b : ℕ → ℂ) (ε : ℝ)
-    (a : ℂ) {ell : ℝ} (hell : 0 < ell) (n : ℕ)
-    (hp : ∀ z ∈ V n, p' n (affine a ell z) = p n z)
-    (κ : ℝ) (R : Set ℂ) {v : ℝ} (hv : 0 < v) (c : ℝ) :
-    P {ω | ω ∈ FrontSuccess (fun j ω t => affine a ell (η j ω t)) p' σ
-          (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n ∧
-        ENNReal.ofReal (c * ell ^ dimension κ) ≤
-          segmentSausageCharge κ (fun t => affine a ell (η n ω t)) (σ ω)
-            (frontTime (fun t => p' n (affine a ell (η n ω t))) 1)
-            (ell * v) (affine a ell '' R)} =
-      P {ω | ω ∈ FrontSuccess η p σ V b ε n ∧ ENNReal.ofReal c ≤
-        segmentSausageCharge κ (η n ω) (σ ω)
-          (frontTime (fun t => p n (η n ω t)) 1) v R} := by
-  let pp := fun j z => p j (affineInverse a ell z)
-  have heq : EqOn (pp n) (p' n) (affine a ell '' V n) := by
-    rintro z ⟨w, hw, rfl⟩
-    dsimp only [pp]
-    rw [affineInverse_left a hell]
-    exact (hp w hw).symm
-  rw [← frontSuccess_mass_eq_of_eqOn
-    (fun j ω t => affine a ell (η j ω t)) pp p' σ
-    (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n
-    heq κ (ell * v) (c * ell ^ dimension κ) (affine a ell '' R)]
-  exact frontSuccess_mass_probability_affine P η p pp σ V b ε a hell n
-    (fun z => by dsimp only [pp]; rw [affineInverse_left a hell]) κ R hv c
-
-theorem longitudinalCoordinate_affine_on_image
-    (P : ℂ → ℂ) (K : Set ℂ) (hiP : InjOn P K) (a : ℂ)
-    {ell : ℝ} (hell : 0 < ell) (k : ℝ) :
-    ∀ z ∈ P '' K,
-      longitudinalCoordinate (fun w => affine a ell (P w)) K k (affine a ell z) =
-        longitudinalCoordinate P K k z := by
-  have hi : InjOn (fun w => affine a ell (P w)) K :=
-    (affine_bijective a hell).1.injOn.comp hiP (mapsTo_univ _ _)
-  rintro z ⟨w, hw, rfl⟩
-  simp only [longitudinalCoordinate, coordinateInverse_left hi hw,
-    coordinateInverse_left hiP hw]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RibbonPassageScaling
+import OAI.Probability.SLE.Gauge.RibbonCoordinateLimit
+
+namespace OAI
+
+/-! Affine transport uses the progress coordinate only on the successful
+physical envelope. This avoids imposing any condition on the arbitrary
+off-image values of `coordinateInverse`. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open FirstFrontPassage PassageSuccess FrontSuccessMeasurability RibbonCoordinates
+open RibbonPassageScaling
+
+theorem frontSuccess_mass_eq_of_eqOn
+    {Ω : Type*} (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ)
+    (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (a : ℕ → ℂ) (ε : ℝ) (n : ℕ)
+    (hpeq : EqOn (p n) (p' n) (V n)) (κ v c : ℝ) (R : Set ℂ) :
+    {ω | ω ∈ FrontSuccess η p σ V a ε n ∧ ENNReal.ofReal c ≤
+      segmentSausageCharge κ (η n ω) (σ ω)
+        (frontTime (fun t => p n (η n ω t)) 1) v R} =
+    {ω | ω ∈ FrontSuccess η p' σ V a ε n ∧ ENNReal.ofReal c ≤
+      segmentSausageCharge κ (η n ω) (σ ω)
+        (frontTime (fun t => p' n (η n ω t)) 1) v R} := by
+  have heq := frontSuccess_eq_of_eqOn η p p' σ V a ε n hpeq
+  ext ω
+  by_cases hω : ω ∈ FrontSuccess η p σ V a ε n
+  · have hω' := heq ▸ hω
+    have hs := hω
+    obtain ⟨hfirst, hhit, _, hpath, _⟩ := hω
+    have hfirst' := firstFrontTime_eq_of_prefix _ _ 1 _ hfirst hhit
+      (fun t ht => hpeq (hpath ht))
+    have ht : frontTime (fun t => p' n (η n ω t)) 1 =
+        frontTime (fun t => p n (η n ω t)) 1 := by
+      simp only [frontTime, hfirst', WithTop.untopD_coe]
+    simp only [mem_ofPred_eq, hs, hω', true_and, ht]
+  · have hω' : ω ∉ FrontSuccess η p' σ V a ε n := by
+      simpa only [← heq] using hω
+    simp only [mem_ofPred_eq, hω, hω', false_and]
+
+def affineInverse (a : ℂ) (ell : ℝ) (z : ℂ) : ℂ := (z - a) / (ell : ℂ)
+
+theorem affineInverse_left (a : ℂ) {ell : ℝ} (hell : 0 < ell) (z : ℂ) :
+    affineInverse a ell (affine a ell z) = z := by
+  simp only [affineInverse, affine, add_sub_cancel_left]
+  exact mul_div_cancel_left₀ z (Complex.ofReal_ne_zero.mpr hell.ne')
+
+theorem frontSuccess_mass_probability_affine_local
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (p p' : ℕ → ℂ → ℝ)
+    (σ : Ω → ℝ≥0) (V : ℕ → Set ℂ) (b : ℕ → ℂ) (ε : ℝ)
+    (a : ℂ) {ell : ℝ} (hell : 0 < ell) (n : ℕ)
+    (hp : ∀ z ∈ V n, p' n (affine a ell z) = p n z)
+    (κ : ℝ) (R : Set ℂ) {v : ℝ} (hv : 0 < v) (c : ℝ) :
+    P {ω | ω ∈ FrontSuccess (fun j ω t => affine a ell (η j ω t)) p' σ
+          (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n ∧
+        ENNReal.ofReal (c * ell ^ dimension κ) ≤
+          segmentSausageCharge κ (fun t => affine a ell (η n ω t)) (σ ω)
+            (frontTime (fun t => p' n (affine a ell (η n ω t))) 1)
+            (ell * v) (affine a ell '' R)} =
+      P {ω | ω ∈ FrontSuccess η p σ V b ε n ∧ ENNReal.ofReal c ≤
+        segmentSausageCharge κ (η n ω) (σ ω)
+          (frontTime (fun t => p n (η n ω t)) 1) v R} := by
+  let pp := fun j z => p j (affineInverse a ell z)
+  have heq : EqOn (pp n) (p' n) (affine a ell '' V n) := by
+    rintro z ⟨w, hw, rfl⟩
+    dsimp only [pp]
+    rw [affineInverse_left a hell]
+    exact (hp w hw).symm
+  rw [← frontSuccess_mass_eq_of_eqOn
+    (fun j ω t => affine a ell (η j ω t)) pp p' σ
+    (fun j => affine a ell '' V j) (fun j => affine a ell (b j)) (ell * ε) n
+    heq κ (ell * v) (c * ell ^ dimension κ) (affine a ell '' R)]
+  exact frontSuccess_mass_probability_affine P η p pp σ V b ε a hell n
+    (fun z => by dsimp only [pp]; rw [affineInverse_left a hell]) κ R hv c
+
+theorem longitudinalCoordinate_affine_on_image
+    (P : ℂ → ℂ) (K : Set ℂ) (hiP : InjOn P K) (a : ℂ)
+    {ell : ℝ} (hell : 0 < ell) (k : ℝ) :
+    ∀ z ∈ P '' K,
+      longitudinalCoordinate (fun w => affine a ell (P w)) K k (affine a ell z) =
+        longitudinalCoordinate P K k z := by
+  have hi : InjOn (fun w => affine a ell (P w)) K :=
+    (affine_bijective a hell).1.injOn.comp hiP (mapsTo_univ _ _)
+  rintro z ⟨w, hw, rfl⟩
+  simp only [longitudinalCoordinate, coordinateInverse_left hi hw,
+    coordinateInverse_left hiP hw]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/MissedCellCleanup.lean b/lean/OAI/Probability/SLE/Gauge/MissedCellCleanup.lean
index 0025dd5d267a683a26088a8f09bbe372541d2f23..d96b9cf894f975060f5eb6021689ec2324315f28 100644
--- a/lean/OAI/Probability/SLE/Gauge/MissedCellCleanup.lean
+++ b/lean/OAI/Probability/SLE/Gauge/MissedCellCleanup.lean
@@ -1,79 +1,79 @@
-import OAI.Probability.SLE.Gauge.CoverNumerics
-
-namespace OAI
-
-/-! The planar grid count cancels the gauge and hitting exponents in the
-missed-cell contribution to the dense cover. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-theorem missed_cell_sum_bound {ι : Type*} (I : Finset ι)
-    (cost prob : ι → ℝ≥0∞) (κ r n c A B D : ℝ)
-    (hr : 0 < r) (hn : 0 ≤ Real.log n) (_hA : 0 ≤ A) (hB : 0 ≤ B) (hD : 0 ≤ D)
-    (hcard : (I.card : ℝ) ≤ A * r ^ (-2 : ℝ))
-    (hcost : ∀ i ∈ I, cost i ≤ ENNReal.ofReal
-      (B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)))
-    (hprob : ∀ i ∈ I, prob i ≤ ENNReal.ofReal
-      (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n))) :
-    ∑ i ∈ I, cost i * prob i ≤ ENNReal.ofReal
-      (A * B * D * (Real.log n) ^ (codimension κ / 2) *
-        Real.exp (-c * Real.sqrt n)) := by
-  have hb0 : 0 ≤ B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2) :=
-    mul_nonneg (mul_nonneg hB (Real.rpow_nonneg hr.le _)) (Real.rpow_nonneg hn _)
-  have hd0 : 0 ≤ D * r ^ codimension κ * Real.exp (-c * Real.sqrt n) :=
-    mul_nonneg (mul_nonneg hD (Real.rpow_nonneg hr.le _)) (Real.exp_pos _).le
-  calc
-    _ ≤ ∑ _i ∈ I, ENNReal.ofReal
-        (B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)) *
-        ENNReal.ofReal (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n)) :=
-      Finset.sum_le_sum (fun i hi => mul_le_mul' (hcost i hi) (hprob i hi))
-    _ = ENNReal.ofReal ((I.card : ℝ) *
-        ((B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)) *
-          (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n)))) := by
-      rw [Finset.sum_const, nsmul_eq_mul, ← ENNReal.ofReal_mul hb0,
-        ← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _)]
-    _ ≤ ENNReal.ofReal ((A * r ^ (-2 : ℝ)) *
-        ((B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)) *
-          (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n)))) :=
-      ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hcard (mul_nonneg hb0 hd0))
-    _ = _ := by
-      congr 1
-      calc
-        _ = (r ^ (-2 : ℝ) * r ^ dimension κ * r ^ codimension κ) *
-            (A * B * D * (Real.log n) ^ (codimension κ / 2) *
-              Real.exp (-c * Real.sqrt n)) := by ring
-        _ = _ := by rw [dense_cover_power_cancellation κ hr, one_mul]
-
-/-- The missed-cell cost vanishes even when its finite prefactor depends on
-the compact interior cutoff. The leading sausage charge does not depend on it. -/
-theorem missed_cell_sum_tendsto_zero {ι : Type*}
-    (I : ℕ → Finset ι) (cost prob : ℕ → ι → ℝ≥0∞)
-    (κ : ℝ) (hκ : 0 < κ) (r : ℕ → ℝ) (hr : ∀ n, 0 < r n)
-    (c : ℝ) (hc : 0 < c) (A B D : ℝ)
-    (hA : 0 ≤ A) (hB : 0 ≤ B) (hD : 0 ≤ D)
-    (hcard : ∀ n, ((I n).card : ℝ) ≤ A * r n ^ (-2 : ℝ))
-    (hcost : ∀ n i, i ∈ I n → cost n i ≤ ENNReal.ofReal
-      (B * r n ^ dimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2)))
-    (hprob : ∀ n i, i ∈ I n → prob n i ≤ ENNReal.ofReal
-      (D * r n ^ codimension κ * Real.exp (-c * Real.sqrt (n : ℝ)))) :
-    Tendsto (fun n => ∑ i ∈ I n, cost n i * prob n i) atTop (𝓝 0) := by
-  have ht : Tendsto (fun n : ℕ => ENNReal.ofReal
-      (A * B * D * (Real.log (n : ℝ)) ^ (codimension κ / 2) *
-        Real.exp (-c * Real.sqrt (n : ℝ)))) atTop (𝓝 0) := by
-    have ht := (explicit_cleanup_tendsto_zero hκ hc).const_mul (A * B * D)
-    simpa only [Function.comp_def, mul_assoc, mul_zero, ENNReal.ofReal_zero] using
-      ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht
-  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht
-    (fun _ => bot_le)
-    (fun n => missed_cell_sum_bound (I n) (cost n) (prob n) κ (r n) n c A B D
-      (hr n) (Real.log_natCast_nonneg n) hA hB hD (hcard n) (hcost n) (hprob n))
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverNumerics
+
+namespace OAI
+
+/-! The planar grid count cancels the gauge and hitting exponents in the
+missed-cell contribution to the dense cover. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+theorem missed_cell_sum_bound {ι : Type*} (I : Finset ι)
+    (cost prob : ι → ℝ≥0∞) (κ r n c A B D : ℝ)
+    (hr : 0 < r) (hn : 0 ≤ Real.log n) (_hA : 0 ≤ A) (hB : 0 ≤ B) (hD : 0 ≤ D)
+    (hcard : (I.card : ℝ) ≤ A * r ^ (-2 : ℝ))
+    (hcost : ∀ i ∈ I, cost i ≤ ENNReal.ofReal
+      (B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)))
+    (hprob : ∀ i ∈ I, prob i ≤ ENNReal.ofReal
+      (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n))) :
+    ∑ i ∈ I, cost i * prob i ≤ ENNReal.ofReal
+      (A * B * D * (Real.log n) ^ (codimension κ / 2) *
+        Real.exp (-c * Real.sqrt n)) := by
+  have hb0 : 0 ≤ B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2) :=
+    mul_nonneg (mul_nonneg hB (Real.rpow_nonneg hr.le _)) (Real.rpow_nonneg hn _)
+  have hd0 : 0 ≤ D * r ^ codimension κ * Real.exp (-c * Real.sqrt n) :=
+    mul_nonneg (mul_nonneg hD (Real.rpow_nonneg hr.le _)) (Real.exp_pos _).le
+  calc
+    _ ≤ ∑ _i ∈ I, ENNReal.ofReal
+        (B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)) *
+        ENNReal.ofReal (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n)) :=
+      Finset.sum_le_sum (fun i hi => mul_le_mul' (hcost i hi) (hprob i hi))
+    _ = ENNReal.ofReal ((I.card : ℝ) *
+        ((B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)) *
+          (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n)))) := by
+      rw [Finset.sum_const, nsmul_eq_mul, ← ENNReal.ofReal_mul hb0,
+        ← ENNReal.ofReal_natCast, ← ENNReal.ofReal_mul (Nat.cast_nonneg _)]
+    _ ≤ ENNReal.ofReal ((A * r ^ (-2 : ℝ)) *
+        ((B * r ^ dimension κ * (Real.log n) ^ (codimension κ / 2)) *
+          (D * r ^ codimension κ * Real.exp (-c * Real.sqrt n)))) :=
+      ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hcard (mul_nonneg hb0 hd0))
+    _ = _ := by
+      congr 1
+      calc
+        _ = (r ^ (-2 : ℝ) * r ^ dimension κ * r ^ codimension κ) *
+            (A * B * D * (Real.log n) ^ (codimension κ / 2) *
+              Real.exp (-c * Real.sqrt n)) := by ring
+        _ = _ := by rw [dense_cover_power_cancellation κ hr, one_mul]
+
+/-- The missed-cell cost vanishes even when its finite prefactor depends on
+the compact interior cutoff. The leading sausage charge does not depend on it. -/
+theorem missed_cell_sum_tendsto_zero {ι : Type*}
+    (I : ℕ → Finset ι) (cost prob : ℕ → ι → ℝ≥0∞)
+    (κ : ℝ) (hκ : 0 < κ) (r : ℕ → ℝ) (hr : ∀ n, 0 < r n)
+    (c : ℝ) (hc : 0 < c) (A B D : ℝ)
+    (hA : 0 ≤ A) (hB : 0 ≤ B) (hD : 0 ≤ D)
+    (hcard : ∀ n, ((I n).card : ℝ) ≤ A * r n ^ (-2 : ℝ))
+    (hcost : ∀ n i, i ∈ I n → cost n i ≤ ENNReal.ofReal
+      (B * r n ^ dimension κ * (Real.log (n : ℝ)) ^ (codimension κ / 2)))
+    (hprob : ∀ n i, i ∈ I n → prob n i ≤ ENNReal.ofReal
+      (D * r n ^ codimension κ * Real.exp (-c * Real.sqrt (n : ℝ)))) :
+    Tendsto (fun n => ∑ i ∈ I n, cost n i * prob n i) atTop (𝓝 0) := by
+  have ht : Tendsto (fun n : ℕ => ENNReal.ofReal
+      (A * B * D * (Real.log (n : ℝ)) ^ (codimension κ / 2) *
+        Real.exp (-c * Real.sqrt (n : ℝ)))) atTop (𝓝 0) := by
+    have ht := (explicit_cleanup_tendsto_zero hκ hc).const_mul (A * B * D)
+    simpa only [Function.comp_def, mul_assoc, mul_zero, ENNReal.ofReal_zero] using
+      ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht
+    (fun _ => bot_le)
+    (fun n => missed_cell_sum_bound (I n) (cost n) (prob n) κ (r n) n c A B D
+      (hr n) (Real.log_natCast_nonneg n) hA hB hD (hcard n) (hcost n) (hprob n))
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/MomentBatches.lean b/lean/OAI/Probability/SLE/Gauge/MomentBatches.lean
index 5181b6749e2c44eab39079b525ca6f792528d4f4..9d627ff393d33134483b4b24e30337a430818729 100644
--- a/lean/OAI/Probability/SLE/Gauge/MomentBatches.lean
+++ b/lean/OAI/Probability/SLE/Gauge/MomentBatches.lean
@@ -1,472 +1,472 @@
-import OAI.Probability.SLE.MomentGauge
-
-namespace OAI
-
-/-! The dyadic moment batches in Lemma `gauge:multiplicity` of the first paper. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge.MomentBatches
-
-def dyadic (k : ℤ) : ℝ := (2 : ℝ) ^ k
-
-def weight (n : ℕ) (m : ℝ) (k : ℤ) : ℝ := (4 : ℝ) ^ n * dyadic k ^ n / m
-
-def copies (n : ℕ) (m T : ℝ) (k : ℤ) : ℕ :=
-  if dyadic k ≤ T then ⌊weight n m k⌋₊ else 0
-
-lemma dyadic_pos (k : ℤ) : 0 < dyadic k := zpow_pos (by norm_num) k
-
-lemma dyadic_step (k : ℤ) : dyadic (k - 1) = dyadic k / 2 := by
-  simp [dyadic, zpow_sub₀]
-
-lemma denominator_pos {n : ℕ} (hn : 1 ≤ n) : 0 < 1 - (1 / 2 : ℝ) ^ n := by
-  exact sub_pos.mpr (pow_lt_one₀ (by norm_num) (by norm_num) (by omega))
-
-/-- A finite selection of dyadic levels below `x` is bounded by the complete
-geometric tail. Missing levels only improve the bound. -/
-lemma dyadic_sum_le {n : ℕ} (hn : 1 ≤ n) (S : Finset ℤ)
-    {x : ℝ} (hx : 0 ≤ x) (hS : ∀ k ∈ S, dyadic k ≤ x) :
-    ∑ k ∈ S, dyadic k ^ n ≤ x ^ n / (1 - (1 / 2 : ℝ) ^ n) := by
-  classical
-  induction S using Finset.induction_on_max generalizing x with
-  | empty =>
-      rw [Finset.sum_empty]
-      exact div_nonneg (pow_nonneg hx _) (denominator_pos hn).le
-  | insert a S hmax ih =>
-      have ha : a ∉ S := fun ha => (hmax a ha).false
-      have hprev : ∀ k ∈ S, dyadic k ≤ dyadic (a - 1) := by
-        intro k hk
-        exact zpow_le_zpow_right₀ (by norm_num) (show k ≤ a - 1 by have := hmax k hk; omega)
-      have hi := ih (dyadic_pos (a - 1)).le hprev
-      have hax := hS a (Finset.mem_insert_self _ _)
-      rw [Finset.sum_insert ha]
-      calc
-        _ ≤ dyadic a ^ n + dyadic (a - 1) ^ n / (1 - (1 / 2 : ℝ) ^ n) :=
-          add_le_add le_rfl hi
-        _ = dyadic a ^ n / (1 - (1 / 2 : ℝ) ^ n) := by
-          have hp : dyadic (a - 1) ^ n = dyadic a ^ n * (1 / 2 : ℝ) ^ n := by
-            rw [dyadic_step, div_pow, one_div_pow]
-            ring
-          apply (eq_div_iff (denominator_pos hn).ne').mpr
-          rw [add_mul, div_mul_cancel₀ _ (denominator_pos hn).ne', hp]
-          ring
-        _ ≤ x ^ n / (1 - (1 / 2 : ℝ) ^ n) :=
-          div_le_div_of_nonneg_right (pow_le_pow_left₀ (dyadic_pos a).le hax n)
-            (denominator_pos hn).le
-
-lemma weight_nonneg {n : ℕ} {m : ℝ} (hm : 0 < m) (k : ℤ) : 0 ≤ weight n m k := by
-  unfold weight
-  exact div_nonneg (mul_nonneg (pow_nonneg (by norm_num) _) (pow_nonneg (dyadic_pos k).le _)) hm.le
-
-lemma copies_le_weight {n : ℕ} {m T : ℝ} (hm : 0 < m) (k : ℤ) :
-    (copies n m T k : ℝ) ≤ weight n m k := by
-  unfold copies
-  split_ifs
-  · exact Nat.floor_le (weight_nonneg hm k)
-  · simpa only [Nat.cast_zero] using weight_nonneg (n := n) hm k
-
-lemma hit_count_upper {n : ℕ} (hn : 1 ≤ n) {m T x : ℝ}
-    (hm : 0 < m) (hx : 0 ≤ x) (S : Finset ℤ) :
-    ∑ k ∈ S, (if 16 * dyadic k ≤ x then (copies n m T k : ℝ) else 0) ≤
-      ((4 : ℝ) ^ n / (m * 16 ^ n * (1 - (1 / 2 : ℝ) ^ n))) * x ^ n := by
-  classical
-  rw [← Finset.sum_filter]
-  let A := S.filter (fun k => 16 * dyadic k ≤ x)
-  have hA : ∀ k ∈ A, dyadic k ≤ x / 16 := by
-    intro k hk
-    exact (le_div_iff₀ (by norm_num : (0 : ℝ) < 16)).mpr
-      (by simpa [mul_comm] using (Finset.mem_filter.mp hk).2)
-  calc
-    _ ≤ ∑ k ∈ A, weight n m k := Finset.sum_le_sum (fun k _ => copies_le_weight hm k)
-    _ = ((4 : ℝ) ^ n / m) * ∑ k ∈ A, dyadic k ^ n := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro k hk
-      dsimp [weight]
-      ring
-    _ ≤ ((4 : ℝ) ^ n / m) * ((x / 16) ^ n / (1 - (1 / 2 : ℝ) ^ n)) :=
-      mul_le_mul_of_nonneg_left (dyadic_sum_le hn A (by positivity) hA) (by positivity)
-    _ = _ := by rw [div_pow]; field_simp
-
-lemma integral_hit_count {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
-    (n : ℕ) (m T c : ℝ) (S : Finset ℤ) :
-    (∫ ω, ∑ k ∈ S, if c * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0 ∂P) =
-      ∑ k ∈ S, (copies n m T k : ℝ) * P.real {ω | c * dyadic k ≤ X ω} := by
-  classical
-  have he (k : ℤ) : MeasurableSet {ω | c * dyadic k ≤ X ω} :=
-    measurableSet_le measurable_const hX
-  have hi (k : ℤ) : Integrable
-      (fun ω => if c * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0) P := by
-    exact (integrable_const _).indicator (he k)
-  rw [integral_finsetSum S (fun k _ => hi k)]
-  apply Finset.sum_congr rfl
-  intro k hk
-  simpa only [Set.indicator, Set.mem_ofPred_eq, smul_eq_mul, mul_comm] using
-    integral_indicator_const (μ := P) (copies n m T k : ℝ) (he k)
-
-/-- The upper batch probability estimate with its precise geometric constant.
-Only the positive finite `n`th moment of the nonnegative random mass is used. -/
-theorem upper_batch_probability {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
-    (hX0 : ∀ ω, 0 ≤ X ω) {n : ℕ} (hn : 1 ≤ n)
-    {m T : ℝ} (hm : 0 < m) (hInt : Integrable (fun ω => X ω ^ n) P)
-    (hMoment : ∫ ω, X ω ^ n ∂P = m) :
-    ∑' k : ℤ, (copies n m T k : ℝ) * P.real {ω | 16 * dyadic k ≤ X ω} ≤
-      (4 : ℝ) ^ n / 16 ^ n / (1 - (1 / 2 : ℝ) ^ n) := by
-  classical
-  apply Real.tsum_le_of_sum_le (fun k => mul_nonneg (Nat.cast_nonneg _) (measureReal_nonneg))
-  intro S
-  let C := (4 : ℝ) ^ n / (m * 16 ^ n * (1 - (1 / 2 : ℝ) ^ n))
-  have hi (k : ℤ) : Integrable
-      (fun ω => if 16 * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0) P :=
-    (integrable_const _).indicator (measurableSet_le measurable_const hX)
-  calc
-    _ = ∫ ω, ∑ k ∈ S, if 16 * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0 ∂P :=
-      (integral_hit_count P X hX n m T 16 S).symm
-    _ ≤ ∫ ω, C * X ω ^ n ∂P := integral_mono
-      (integrable_finsetSum S (fun k _ => hi k)) (hInt.const_mul C)
-      (fun ω => hit_count_upper hn hm (hX0 ω) S)
-    _ = C * m := by rw [integral_const_mul, hMoment]
-    _ = _ := by dsimp [C]; field_simp
-
-lemma floor_half_sub_half {w : ℝ} (_hw : 0 ≤ w) : w / 2 - 1 / 2 ≤ (⌊w⌋₊ : ℝ) := by
-  by_cases h1 : 1 ≤ w
-  · exact (by linarith : w / 2 - 1 / 2 ≤ w / 2).trans (Nat.div_two_lt_floor h1).le
-  · exact (by linarith : w / 2 - 1 / 2 ≤ 0).trans (Nat.cast_nonneg _)
-
-lemma dyadic_level {x : ℝ} (hx : 0 < x) :
-    ∃ k : ℤ, x / 2 < dyadic k ∧ dyadic k ≤ x := by
-  obtain ⟨k, hlo, hhi⟩ := exists_mem_Ico_zpow hx (by norm_num : (1 : ℝ) < 2)
-  refine ⟨k, ?_, hlo⟩
-  have hs : (2 : ℝ) ^ (k + 1) = 2 * dyadic k := by
-    simp [dyadic, zpow_add₀, mul_comm]
-  rw [hs] at hhi
-  linarith
-
-/-- One well-chosen dyadic level already gives the required lower count.
-This avoids charging a floor error at every dyadic level. -/
-lemma hit_count_lower {n : ℕ} (hn : 1 ≤ n) {m T x : ℝ} (hm : 0 < m)
-    (hx : 0 ≤ x) (hxT : x ≤ T) (S : Finset ℤ)
-    (hS : ∀ k, copies n m T k ≠ 0 → k ∈ S) :
-    (2 : ℝ) ^ n / (2 * m) * x ^ n - 1 / 2 ≤
-      ∑ k ∈ S, if dyadic k ≤ x then (copies n m T k : ℝ) else 0 := by
-  classical
-  have hnonneg : 0 ≤ ∑ k ∈ S,
-      if dyadic k ≤ x then (copies n m T k : ℝ) else 0 :=
-    Finset.sum_nonneg (fun k _ => by split_ifs <;> positivity)
-  obtain rfl | hxp := hx.eq_or_lt
-  · simpa only [zero_pow (by omega : n ≠ 0), mul_zero, zero_sub] using
-      (show -(1 / 2 : ℝ) ≤ _ from (by norm_num : -(1 / 2 : ℝ) ≤ 0).trans hnonneg)
-  obtain ⟨k, hklo, hkhi⟩ := dyadic_level hxp
-  have hkT : dyadic k ≤ T := hkhi.trans hxT
-  have hw : (2 : ℝ) ^ n * x ^ n / m ≤ weight n m k := by
-    have hp := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ x / 2) hklo.le n
-    have hp' := mul_le_mul_of_nonneg_left hp (show 0 ≤ (4 : ℝ) ^ n / m by positivity)
-    unfold weight
-    calc
-      _ = ((4 : ℝ) ^ n / m) * (x / 2) ^ n := by
-        rw [div_pow, show (4 : ℝ) ^ n = 2 ^ n * 2 ^ n by rw [← mul_pow]; norm_num]
-        field_simp
-      _ ≤ ((4 : ℝ) ^ n / m) * dyadic k ^ n := hp'
-      _ = _ := by ring
-  have hfloor : weight n m k / 2 - 1 / 2 ≤ (copies n m T k : ℝ) := by
-    simpa only [copies, ite_eq_left hkT] using floor_half_sub_half (weight_nonneg hm k)
-  have hsingle : (copies n m T k : ℝ) ≤
-      ∑ j ∈ S, if dyadic j ≤ x then (copies n m T j : ℝ) else 0 := by
-    by_cases hz : copies n m T k = 0
-    · simpa only [hz, Nat.cast_zero] using hnonneg
-    · have hs := Finset.single_le_sum
-        (fun j (_ : j ∈ S) => show 0 ≤ (if dyadic j ≤ x then (copies n m T j : ℝ) else 0)
-          from by split_ifs <;> positivity) (hS k hz)
-      simpa only [ite_eq_left hkhi] using hs
-  have hcoeff : (2 : ℝ) ^ n / (2 * m) * x ^ n = ((2 : ℝ) ^ n * x ^ n / m) / 2 := by
-    ring
-  rw [hcoeff]
-  linarith
-
-/-- The `(n+1)`st moment bounds the loss above the paper's cutoff
-`T = 4 M / m`; a retained `3m/4` moment leaves at least `m/2`. -/
-theorem retained_truncated_moment {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (Y : Ω → ℝ) (hY : Measurable Y) (hY0 : ∀ ω, 0 ≤ Y ω)
-    (n : ℕ) {m M : ℝ} (hm : 0 < m) (hM : 0 < M)
-    (hi : Integrable (fun ω => Y ω ^ n) P)
-    (hi' : Integrable (fun ω => Y ω ^ (n + 1)) P)
-    (hretained : 3 * m / 4 ≤ ∫ ω, Y ω ^ n ∂P)
-    (hnext : (∫ ω, Y ω ^ (n + 1) ∂P) ≤ M) :
-    m / 2 ≤ ∫ ω in {ω | Y ω ≤ 4 * M / m}, Y ω ^ n ∂P := by
-  let T := 4 * M / m
-  have hT : 0 < T := by dsimp [T]; positivity
-  have hset : MeasurableSet {ω | Y ω ≤ T} := measurableSet_le hY measurable_const
-  have htail : (∫ ω in {ω | Y ω ≤ T}ᶜ, Y ω ^ n ∂P) ≤ M / T := by
-    calc
-      _ = ∫ ω, {ω | Y ω ≤ T}ᶜ.indicator (fun ω => Y ω ^ n) ω ∂P :=
-        (integral_indicator hset.compl).symm
-      _ ≤ ∫ ω, Y ω ^ (n + 1) / T ∂P := by
-        apply integral_mono (hi.indicator hset.compl) (hi'.div_const T)
-        intro ω
-        by_cases hω : Y ω ≤ T
-        · simp only [Set.indicator, Set.mem_compl_iff, Set.mem_ofPred_eq, hω, not_true_eq_false,
-            ↓reduceIte]
-          exact div_nonneg (pow_nonneg (hY0 ω) _) hT.le
-        · simp only [Set.indicator, Set.mem_compl_iff, Set.mem_ofPred_eq, hω, not_false_eq_true,
-            ↓reduceIte]
-          apply (le_div_iff₀ hT).mpr
-          calc
-            Y ω ^ n * T ≤ Y ω ^ n * Y ω :=
-              mul_le_mul_of_nonneg_left (le_of_not_ge hω) (pow_nonneg (hY0 ω) _)
-            _ = Y ω ^ (n + 1) := (pow_succ _ _).symm
-      _ ≤ M / T := by rw [integral_div]; exact div_le_div_of_nonneg_right hnext hT.le
-  have hdecomp := integral_add_compl hset hi
-  have hMT : M / T = m / 4 := by dsimp [T]; field_simp
-  rw [hMT] at htail
-  change m / 2 ≤ ∫ ω in {ω | Y ω ≤ T}, Y ω ^ n ∂P
-  linarith
-
-/-- The lower batch probability estimate. The `1/2` loss is slightly stronger
-than the paper's `1 / (2 * (1 - 2⁻ⁿ))` loss. -/
-theorem lower_batch_probability_finset {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsProbabilityMeasure P] (Y : Ω → ℝ) (hY : Measurable Y)
-    (hY0 : ∀ ω, 0 ≤ Y ω) {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m)
-    (hi : Integrable (fun ω => Y ω ^ n) P)
-    (htrunc : m / 2 ≤ ∫ ω in {ω | Y ω ≤ T}, Y ω ^ n ∂P)
-    (S : Finset ℤ) (hS : ∀ k, copies n m T k ≠ 0 → k ∈ S) :
-    (2 : ℝ) ^ n / 4 - 1 / 2 ≤
-      ∑ k ∈ S, (copies n m T k : ℝ) * P.real {ω | dyadic k ≤ Y ω} := by
-  classical
-  let C := (2 : ℝ) ^ n / (2 * m)
-  let F := {ω | Y ω ≤ T}.indicator (fun ω => Y ω ^ n)
-  have hC : 0 ≤ C := by dsimp [C]; positivity
-  have hset : MeasurableSet {ω | Y ω ≤ T} := measurableSet_le hY measurable_const
-  have hFi : Integrable F P := hi.indicator hset
-  have hcount (k : ℤ) : Integrable
-      (fun ω => if dyadic k ≤ Y ω then (copies n m T k : ℝ) else 0) P :=
-    (integrable_const _).indicator (measurableSet_le measurable_const hY)
-  have hbound (ω : Ω) : C * F ω - 1 / 2 ≤
-      ∑ k ∈ S, if dyadic k ≤ Y ω then (copies n m T k : ℝ) else 0 := by
-    by_cases hω : Y ω ≤ T
-    · simpa only [F, Set.indicator, Set.mem_ofPred_eq, ite_eq_left hω, C] using
-        hit_count_lower hn hm (hY0 ω) hω S hS
-    · simp only [F, Set.indicator, Set.mem_ofPred_eq, ite_eq_right hω, mul_zero, zero_sub]
-      exact (by norm_num : -(1 / 2 : ℝ) ≤ 0).trans
-        (Finset.sum_nonneg (fun k _ => by split_ifs <;> positivity))
-  have hint := integral_mono ((hFi.const_mul C).sub (integrable_const (1 / 2 : ℝ)))
-    (integrable_finsetSum S (fun k _ => hcount k)) hbound
-  change (∫ ω, C * F ω - 1 / 2 ∂P) ≤
-    ∫ ω, ∑ k ∈ S, if dyadic k ≤ Y ω then (copies n m T k : ℝ) else 0 ∂P at hint
-  rw [integral_sub (hFi.const_mul C) (integrable_const (1 / 2 : ℝ)), integral_const_mul,
-    integral_const] at hint
-  simp only [measureReal_def, measure_univ, ENNReal.toReal_one, one_smul] at hint
-  have hF : ∫ ω, F ω ∂P = ∫ ω in {ω | Y ω ≤ T}, Y ω ^ n ∂P := integral_indicator hset
-  rw [hF] at hint
-  have hl := mul_le_mul_of_nonneg_left htrunc hC
-  have he : C * (m / 2) = (2 : ℝ) ^ n / 4 := by dsimp [C]; field_simp; ring
-  rw [he] at hl
-  have hprob := integral_hit_count P Y hY n m T 1 S
-  simp only [one_mul] at hprob
-  rw [hprob] at hint
-  linarith
-
-lemma lower_constant {n : ℕ} (hn : 3 ≤ n) :
-    (2 : ℝ) ^ n / 8 ≤ (2 : ℝ) ^ n / 4 - 1 / 2 := by
-  have hp : (4 : ℝ) ≤ 2 ^ n := by
-    have h := pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) (show 2 ≤ n by omega)
-    norm_num at h ⊢
-    exact h
-  linarith
-
-lemma copies_positive_bounds {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m)
-    {k : ℤ} (hk : copies n m T k ≠ 0) :
-    min 1 (m / (4 : ℝ) ^ n) ≤ dyadic k ∧ dyadic k ≤ T := by
-  have hcut : dyadic k ≤ T := by
-    by_contra hc
-    simp only [copies, ite_eq_right hc] at hk
-    exact hk rfl
-  have hfloor : 0 < ⌊weight n m k⌋₊ := by
-    simpa only [copies, ite_eq_left hcut, Nat.pos_iff_ne_zero] using hk
-  have hw : 1 ≤ weight n m k := Nat.floor_pos.mp hfloor
-  have hnum : m / (4 : ℝ) ^ n ≤ dyadic k ^ n := by
-    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 ^ n)).mpr
-    have hm' := (le_div_iff₀ hm).mp hw
-    simpa only [weight, one_mul, mul_one, mul_comm] using hm'
-  refine ⟨?_, hcut⟩
-  rcases le_total 1 (dyadic k) with hlarge | hsmall
-  · exact (min_le_left _ _).trans hlarge
-  · have hp : dyadic k ^ n ≤ dyadic k := by
-      simpa only [Real.rpow_natCast] using Real.rpow_le_self_of_le_one
-        (dyadic_pos k).le hsmall (show (1 : ℝ) ≤ n by exact_mod_cast hn)
-    exact (min_le_right _ _).trans (hnum.trans hp)
-
-lemma copies_support_finite {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m) :
-    Set.Finite {k : ℤ | copies n m T k ≠ 0} := by
-  let b := min 1 (m / (4 : ℝ) ^ n)
-  have hb : 0 < b := lt_min zero_lt_one (by positivity)
-  have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  apply (Set.finite_Icc (⌈Real.log b / Real.log 2⌉ : ℤ)
-    (⌊Real.log T / Real.log 2⌋ : ℤ)).subset
-  intro k hk
-  obtain ⟨hlo, hhi⟩ := copies_positive_bounds hn hm hk
-  constructor
-  · apply Int.ceil_le.mpr
-    apply (div_le_iff₀ hlog).mpr
-    exact (Real.le_zpow_iff_log_le hb (by norm_num : (0 : ℝ) < 2)).mp hlo
-  · apply Int.le_floor.mpr
-    apply (le_div_iff₀ hlog).mpr
-    exact Real.le_log_of_zpow_le (by norm_num : (0 : ℝ) < 2) hhi
-
-/-- The lower batch inequality for the whole integer sum, with no auxiliary
-finite support assumption. -/
-theorem lower_batch_probability {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsProbabilityMeasure P] (Y : Ω → ℝ) (hY : Measurable Y)
-    (hY0 : ∀ ω, 0 ≤ Y ω) {n : ℕ} (hn : 3 ≤ n) {m M : ℝ}
-    (hm : 0 < m) (hM : 0 < M)
-    (hi : Integrable (fun ω => Y ω ^ n) P)
-    (hi' : Integrable (fun ω => Y ω ^ (n + 1)) P)
-    (hretained : 3 * m / 4 ≤ ∫ ω, Y ω ^ n ∂P)
-    (hnext : (∫ ω, Y ω ^ (n + 1) ∂P) ≤ M) :
-    (2 : ℝ) ^ n / 8 ≤ ∑' k : ℤ,
-      (copies n m (4 * M / m) k : ℝ) * P.real {ω | dyadic k ≤ Y ω} := by
-  classical
-  have hn1 : 1 ≤ n := by omega
-  let S := (copies_support_finite hn1 (T := 4 * M / m) hm).toFinset
-  have hs : ∀ k, copies n m (4 * M / m) k ≠ 0 → k ∈ S := by
-    intro k hk
-    exact (copies_support_finite hn1 (T := 4 * M / m) hm).mem_toFinset.mpr hk
-  rw [tsum_eq_sum (s := S) (fun k hk => by
-    have hz : copies n m (4 * M / m) k = 0 := by
-      by_contra hc
-      exact hk (hs k hc)
-    simp only [hz, Nat.cast_zero, zero_mul])]
-  apply (lower_constant hn).trans
-  exact lower_batch_probability_finset P Y hY hY0 hn1 hm hi
-    (retained_truncated_moment P Y hY hY0 n hm hM hi hi' hretained hnext) S hs
-
-lemma copies_eq_target (κ : ℝ) (W : SLEExactGauge.Target.WienerLaw) (n : ℕ) (k : ℤ) :
-    copies n (SLEExactGauge.Target.moment κ W n)
-      (SLEExactGauge.Target.threshold κ W n) k = SLEExactGauge.Target.multiplicity κ W n k := rfl
-
-/-- The upper inequality in the exact notation of the moment-gauge construction. -/
-theorem target_upper_batch_probability {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
-    (hX0 : ∀ ω, 0 ≤ X ω) (κ : ℝ) (W : SLEExactGauge.Target.WienerLaw)
-    {n : ℕ} (hn : 1 ≤ n) (hm : 0 < SLEExactGauge.Target.moment κ W n)
-    (hi : Integrable (fun ω => X ω ^ n) P)
-    (he : ∫ ω, X ω ^ n ∂P = SLEExactGauge.Target.moment κ W n) :
-    ∑' k : ℤ, (SLEExactGauge.Target.multiplicity κ W n k : ℝ) *
-      P.real {ω | 16 * (2 : ℝ) ^ k ≤ X ω} ≤
-        (4 : ℝ) ^ n / 16 ^ n / (1 - (1 / 2 : ℝ) ^ n) :=
-  upper_batch_probability P X hX hX0 hn hm hi he
-
-/-- The success inequality in the exact notation of the moment-gauge construction. -/
-theorem target_lower_batch_probability {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsProbabilityMeasure P] (Y : Ω → ℝ) (hY : Measurable Y)
-    (hY0 : ∀ ω, 0 ≤ Y ω) (κ : ℝ) (W : SLEExactGauge.Target.WienerLaw)
-    {n : ℕ} (hn : 3 ≤ n)
-    (hm : 0 < SLEExactGauge.Target.moment κ W n)
-    (hM : 0 < SLEExactGauge.Target.moment κ W (n + 1))
-    (hi : Integrable (fun ω => Y ω ^ n) P)
-    (hi' : Integrable (fun ω => Y ω ^ (n + 1)) P)
-    (hretained : 3 * SLEExactGauge.Target.moment κ W n / 4 ≤ ∫ ω, Y ω ^ n ∂P)
-    (hnext : (∫ ω, Y ω ^ (n + 1) ∂P) ≤ SLEExactGauge.Target.moment κ W (n + 1)) :
-    (2 : ℝ) ^ n / 8 ≤ ∑' k : ℤ, (SLEExactGauge.Target.multiplicity κ W n k : ℝ) *
-      P.real {ω | (2 : ℝ) ^ k ≤ Y ω} :=
-  lower_batch_probability P Y hY hY0 hn hm hM hi hi' hretained hnext
-
-lemma cleanup_count_le {x : ℝ} (hx : 0 ≤ x) (S : Finset ℕ) :
-    ∑ n ∈ S, (if 16 * ((n : ℝ) + 1) ≤ x then (1 : ℝ) else 0) ≤ x / 16 := by
-  classical
-  rw [← Finset.sum_filter]
-  let A : Finset ℕ := S.filter (fun n : ℕ => 16 * ((n : ℝ) + 1) ≤ x)
-  have hsub : A ⊆ Finset.range ⌊x / 16⌋₊ := by
-    intro n hn
-    have hreal : (n : ℝ) + 1 ≤ x / 16 := by
-      apply (le_div_iff₀ (by norm_num : (0 : ℝ) < 16)).mpr
-      simpa only [mul_comm] using (Finset.mem_filter.mp hn).2
-    have hnat : n + 1 ≤ ⌊x / 16⌋₊ := Nat.le_floor (by simpa using hreal)
-    exact Finset.mem_range.mpr (by omega)
-  change (∑ n ∈ A, (1 : ℝ)) ≤ x / 16
-  rw [Finset.sum_const, nsmul_eq_mul, mul_one]
-  calc
-    (A.card : ℝ) ≤ (⌊x / 16⌋₊ : ℝ) := by
-      exact_mod_cast (Finset.card_le_card hsub).trans_eq (Finset.card_range _)
-    _ ≤ x / 16 := Nat.floor_le (by positivity)
-
-/-- The cleanup entries have summable tail probabilities, bounded by the first
-moment divided by 16. Indices `n + 1` are the paper's positive integers. -/
-theorem cleanup_tail_bound {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
-    (hX0 : ∀ ω, 0 ≤ X ω) (hi : Integrable X P) :
-    Summable (fun n : ℕ => P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω}) ∧
-      (∑' n : ℕ, P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω}) ≤ (∫ ω, X ω ∂P) / 16 := by
-  classical
-  have hsum (S : Finset ℕ) :
-      ∑ n ∈ S, P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω} ≤ (∫ ω, X ω ∂P) / 16 := by
-    have hset (n : ℕ) : MeasurableSet {ω | 16 * ((n : ℝ) + 1) ≤ X ω} :=
-      measurableSet_le measurable_const hX
-    have hcount (n : ℕ) : Integrable
-        (fun ω => if 16 * ((n : ℝ) + 1) ≤ X ω then (1 : ℝ) else 0) P :=
-      (integrable_const _).indicator (hset n)
-    calc
-      _ = ∫ ω, ∑ n ∈ S, if 16 * ((n : ℝ) + 1) ≤ X ω then (1 : ℝ) else 0 ∂P := by
-        rw [integral_finsetSum S (fun n _ => hcount n)]
-        apply Finset.sum_congr rfl
-        intro n hn
-        simpa only [Set.indicator, Set.mem_ofPred_eq, smul_eq_mul, mul_one] using
-          (integral_indicator_const (μ := P) (1 : ℝ) (hset n)).symm
-      _ ≤ ∫ ω, X ω / 16 ∂P := integral_mono
-        (integrable_finsetSum S (fun n _ => hcount n)) (hi.div_const 16)
-        (fun ω => cleanup_count_le (hX0 ω) S)
-      _ = _ := integral_div _ _
-  exact ⟨summable_of_sum_le (fun _ => measureReal_nonneg) hsum,
-    Real.tsum_le_of_sum_le (fun _ => measureReal_nonneg) hsum⟩
-
-lemma upper_constant_le_geometric {n : ℕ} (hn : 1 ≤ n) :
-    (4 : ℝ) ^ n / 16 ^ n / (1 - (1 / 2 : ℝ) ^ n) ≤ 2 * (1 / 4 : ℝ) ^ n := by
-  have hpow : (1 / 2 : ℝ) ^ n ≤ 1 / 2 := by
-    obtain ⟨j, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n ≠ 0)
-    rw [pow_succ]
-    exact (mul_le_mul_of_nonneg_right (pow_le_one₀ (by norm_num) (by norm_num))
-      (by norm_num : (0 : ℝ) ≤ 1 / 2)).trans_eq (one_mul _)
-  have he : (4 : ℝ) ^ n / 16 ^ n = (1 / 4 : ℝ) ^ n := by
-    rw [← div_pow]
-    norm_num
-  rw [he]
-  apply (div_le_iff₀ (denominator_pos hn)).mpr
-  have hnonneg : 0 ≤ (1 / 4 : ℝ) ^ n := by positivity
-  nlinarith
-
-/-- Both types of entries in all positive-order batches have summable exceedance
-probabilities. The hypotheses are precisely the positive finite moments of one
-nonnegative mass, not any SLE conclusion. -/
-theorem all_batch_tails_summable {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
-    (hX0 : ∀ ω, 0 ≤ X ω) (m T : ℕ → ℝ)
-    (hm : ∀ n, 1 ≤ n → 0 < m n)
-    (hi : ∀ n, 1 ≤ n → Integrable (fun ω => X ω ^ n) P)
-    (hMoment : ∀ n, 1 ≤ n → (∫ ω, X ω ^ n ∂P) = m n) :
-    Summable (fun n : ℕ =>
-      (∑' k : ℤ, (copies (n + 1) (m (n + 1)) (T (n + 1)) k : ℝ) *
-        P.real {ω | 16 * dyadic k ≤ X ω}) +
-      P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω}) := by
-  have hgeom : Summable (fun n : ℕ => 2 * (1 / 4 : ℝ) ^ (n + 1)) := by
-    simpa only [pow_succ, mul_assoc, mul_comm, mul_left_comm] using
-      (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 4)
-        (by norm_num : (1 / 4 : ℝ) < 1)).mul_left (2 * (1 / 4 : ℝ))
-  have hbatch : Summable (fun n : ℕ =>
-      ∑' k : ℤ, (copies (n + 1) (m (n + 1)) (T (n + 1)) k : ℝ) *
-        P.real {ω | 16 * dyadic k ≤ X ω}) := by
-    apply Summable.of_nonneg_of_le (fun n => tsum_nonneg (fun _ => by positivity)) _ hgeom
-    intro n
-    exact (upper_batch_probability P X hX hX0 (by omega) (hm _ (by omega))
-      (hi _ (by omega)) (hMoment _ (by omega))).trans (upper_constant_le_geometric (by omega))
-  have hfirst : Integrable X P := by simpa only [pow_one] using hi 1 le_rfl
-  exact hbatch.add (cleanup_tail_bound P X hX hX0 hfirst).1
-
-end SLEExactGauge.MomentBatches
-
-end
-
-end OAI
+import OAI.Probability.SLE.MomentGauge
+
+namespace OAI
+
+/-! The dyadic moment batches in Lemma `gauge:multiplicity` of the first paper. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge.MomentBatches
+
+def dyadic (k : ℤ) : ℝ := (2 : ℝ) ^ k
+
+def weight (n : ℕ) (m : ℝ) (k : ℤ) : ℝ := (4 : ℝ) ^ n * dyadic k ^ n / m
+
+def copies (n : ℕ) (m T : ℝ) (k : ℤ) : ℕ :=
+  if dyadic k ≤ T then ⌊weight n m k⌋₊ else 0
+
+lemma dyadic_pos (k : ℤ) : 0 < dyadic k := zpow_pos (by norm_num) k
+
+lemma dyadic_step (k : ℤ) : dyadic (k - 1) = dyadic k / 2 := by
+  simp [dyadic, zpow_sub₀]
+
+lemma denominator_pos {n : ℕ} (hn : 1 ≤ n) : 0 < 1 - (1 / 2 : ℝ) ^ n := by
+  exact sub_pos.mpr (pow_lt_one₀ (by norm_num) (by norm_num) (by omega))
+
+/-- A finite selection of dyadic levels below `x` is bounded by the complete
+geometric tail. Missing levels only improve the bound. -/
+lemma dyadic_sum_le {n : ℕ} (hn : 1 ≤ n) (S : Finset ℤ)
+    {x : ℝ} (hx : 0 ≤ x) (hS : ∀ k ∈ S, dyadic k ≤ x) :
+    ∑ k ∈ S, dyadic k ^ n ≤ x ^ n / (1 - (1 / 2 : ℝ) ^ n) := by
+  classical
+  induction S using Finset.induction_on_max generalizing x with
+  | empty =>
+      rw [Finset.sum_empty]
+      exact div_nonneg (pow_nonneg hx _) (denominator_pos hn).le
+  | insert a S hmax ih =>
+      have ha : a ∉ S := fun ha => (hmax a ha).false
+      have hprev : ∀ k ∈ S, dyadic k ≤ dyadic (a - 1) := by
+        intro k hk
+        exact zpow_le_zpow_right₀ (by norm_num) (show k ≤ a - 1 by have := hmax k hk; omega)
+      have hi := ih (dyadic_pos (a - 1)).le hprev
+      have hax := hS a (Finset.mem_insert_self _ _)
+      rw [Finset.sum_insert ha]
+      calc
+        _ ≤ dyadic a ^ n + dyadic (a - 1) ^ n / (1 - (1 / 2 : ℝ) ^ n) :=
+          add_le_add le_rfl hi
+        _ = dyadic a ^ n / (1 - (1 / 2 : ℝ) ^ n) := by
+          have hp : dyadic (a - 1) ^ n = dyadic a ^ n * (1 / 2 : ℝ) ^ n := by
+            rw [dyadic_step, div_pow, one_div_pow]
+            ring
+          apply (eq_div_iff (denominator_pos hn).ne').mpr
+          rw [add_mul, div_mul_cancel₀ _ (denominator_pos hn).ne', hp]
+          ring
+        _ ≤ x ^ n / (1 - (1 / 2 : ℝ) ^ n) :=
+          div_le_div_of_nonneg_right (pow_le_pow_left₀ (dyadic_pos a).le hax n)
+            (denominator_pos hn).le
+
+lemma weight_nonneg {n : ℕ} {m : ℝ} (hm : 0 < m) (k : ℤ) : 0 ≤ weight n m k := by
+  unfold weight
+  exact div_nonneg (mul_nonneg (pow_nonneg (by norm_num) _) (pow_nonneg (dyadic_pos k).le _)) hm.le
+
+lemma copies_le_weight {n : ℕ} {m T : ℝ} (hm : 0 < m) (k : ℤ) :
+    (copies n m T k : ℝ) ≤ weight n m k := by
+  unfold copies
+  split_ifs
+  · exact Nat.floor_le (weight_nonneg hm k)
+  · simpa only [Nat.cast_zero] using weight_nonneg (n := n) hm k
+
+lemma hit_count_upper {n : ℕ} (hn : 1 ≤ n) {m T x : ℝ}
+    (hm : 0 < m) (hx : 0 ≤ x) (S : Finset ℤ) :
+    ∑ k ∈ S, (if 16 * dyadic k ≤ x then (copies n m T k : ℝ) else 0) ≤
+      ((4 : ℝ) ^ n / (m * 16 ^ n * (1 - (1 / 2 : ℝ) ^ n))) * x ^ n := by
+  classical
+  rw [← Finset.sum_filter]
+  let A := S.filter (fun k => 16 * dyadic k ≤ x)
+  have hA : ∀ k ∈ A, dyadic k ≤ x / 16 := by
+    intro k hk
+    exact (le_div_iff₀ (by norm_num : (0 : ℝ) < 16)).mpr
+      (by simpa [mul_comm] using (Finset.mem_filter.mp hk).2)
+  calc
+    _ ≤ ∑ k ∈ A, weight n m k := Finset.sum_le_sum (fun k _ => copies_le_weight hm k)
+    _ = ((4 : ℝ) ^ n / m) * ∑ k ∈ A, dyadic k ^ n := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro k hk
+      dsimp [weight]
+      ring
+    _ ≤ ((4 : ℝ) ^ n / m) * ((x / 16) ^ n / (1 - (1 / 2 : ℝ) ^ n)) :=
+      mul_le_mul_of_nonneg_left (dyadic_sum_le hn A (by positivity) hA) (by positivity)
+    _ = _ := by rw [div_pow]; field_simp
+
+lemma integral_hit_count {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
+    (n : ℕ) (m T c : ℝ) (S : Finset ℤ) :
+    (∫ ω, ∑ k ∈ S, if c * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0 ∂P) =
+      ∑ k ∈ S, (copies n m T k : ℝ) * P.real {ω | c * dyadic k ≤ X ω} := by
+  classical
+  have he (k : ℤ) : MeasurableSet {ω | c * dyadic k ≤ X ω} :=
+    measurableSet_le measurable_const hX
+  have hi (k : ℤ) : Integrable
+      (fun ω => if c * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0) P := by
+    exact (integrable_const _).indicator (he k)
+  rw [integral_finsetSum S (fun k _ => hi k)]
+  apply Finset.sum_congr rfl
+  intro k hk
+  simpa only [Set.indicator, Set.mem_ofPred_eq, smul_eq_mul, mul_comm] using
+    integral_indicator_const (μ := P) (copies n m T k : ℝ) (he k)
+
+/-- The upper batch probability estimate with its precise geometric constant.
+Only the positive finite `n`th moment of the nonnegative random mass is used. -/
+theorem upper_batch_probability {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
+    (hX0 : ∀ ω, 0 ≤ X ω) {n : ℕ} (hn : 1 ≤ n)
+    {m T : ℝ} (hm : 0 < m) (hInt : Integrable (fun ω => X ω ^ n) P)
+    (hMoment : ∫ ω, X ω ^ n ∂P = m) :
+    ∑' k : ℤ, (copies n m T k : ℝ) * P.real {ω | 16 * dyadic k ≤ X ω} ≤
+      (4 : ℝ) ^ n / 16 ^ n / (1 - (1 / 2 : ℝ) ^ n) := by
+  classical
+  apply Real.tsum_le_of_sum_le (fun k => mul_nonneg (Nat.cast_nonneg _) (measureReal_nonneg))
+  intro S
+  let C := (4 : ℝ) ^ n / (m * 16 ^ n * (1 - (1 / 2 : ℝ) ^ n))
+  have hi (k : ℤ) : Integrable
+      (fun ω => if 16 * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0) P :=
+    (integrable_const _).indicator (measurableSet_le measurable_const hX)
+  calc
+    _ = ∫ ω, ∑ k ∈ S, if 16 * dyadic k ≤ X ω then (copies n m T k : ℝ) else 0 ∂P :=
+      (integral_hit_count P X hX n m T 16 S).symm
+    _ ≤ ∫ ω, C * X ω ^ n ∂P := integral_mono
+      (integrable_finsetSum S (fun k _ => hi k)) (hInt.const_mul C)
+      (fun ω => hit_count_upper hn hm (hX0 ω) S)
+    _ = C * m := by rw [integral_const_mul, hMoment]
+    _ = _ := by dsimp [C]; field_simp
+
+lemma floor_half_sub_half {w : ℝ} (_hw : 0 ≤ w) : w / 2 - 1 / 2 ≤ (⌊w⌋₊ : ℝ) := by
+  by_cases h1 : 1 ≤ w
+  · exact (by linarith : w / 2 - 1 / 2 ≤ w / 2).trans (Nat.div_two_lt_floor h1).le
+  · exact (by linarith : w / 2 - 1 / 2 ≤ 0).trans (Nat.cast_nonneg _)
+
+lemma dyadic_level {x : ℝ} (hx : 0 < x) :
+    ∃ k : ℤ, x / 2 < dyadic k ∧ dyadic k ≤ x := by
+  obtain ⟨k, hlo, hhi⟩ := exists_mem_Ico_zpow hx (by norm_num : (1 : ℝ) < 2)
+  refine ⟨k, ?_, hlo⟩
+  have hs : (2 : ℝ) ^ (k + 1) = 2 * dyadic k := by
+    simp [dyadic, zpow_add₀, mul_comm]
+  rw [hs] at hhi
+  linarith
+
+/-- One well-chosen dyadic level already gives the required lower count.
+This avoids charging a floor error at every dyadic level. -/
+lemma hit_count_lower {n : ℕ} (hn : 1 ≤ n) {m T x : ℝ} (hm : 0 < m)
+    (hx : 0 ≤ x) (hxT : x ≤ T) (S : Finset ℤ)
+    (hS : ∀ k, copies n m T k ≠ 0 → k ∈ S) :
+    (2 : ℝ) ^ n / (2 * m) * x ^ n - 1 / 2 ≤
+      ∑ k ∈ S, if dyadic k ≤ x then (copies n m T k : ℝ) else 0 := by
+  classical
+  have hnonneg : 0 ≤ ∑ k ∈ S,
+      if dyadic k ≤ x then (copies n m T k : ℝ) else 0 :=
+    Finset.sum_nonneg (fun k _ => by split_ifs <;> positivity)
+  obtain rfl | hxp := hx.eq_or_lt
+  · simpa only [zero_pow (by omega : n ≠ 0), mul_zero, zero_sub] using
+      (show -(1 / 2 : ℝ) ≤ _ from (by norm_num : -(1 / 2 : ℝ) ≤ 0).trans hnonneg)
+  obtain ⟨k, hklo, hkhi⟩ := dyadic_level hxp
+  have hkT : dyadic k ≤ T := hkhi.trans hxT
+  have hw : (2 : ℝ) ^ n * x ^ n / m ≤ weight n m k := by
+    have hp := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ x / 2) hklo.le n
+    have hp' := mul_le_mul_of_nonneg_left hp (show 0 ≤ (4 : ℝ) ^ n / m by positivity)
+    unfold weight
+    calc
+      _ = ((4 : ℝ) ^ n / m) * (x / 2) ^ n := by
+        rw [div_pow, show (4 : ℝ) ^ n = 2 ^ n * 2 ^ n by rw [← mul_pow]; norm_num]
+        field_simp
+      _ ≤ ((4 : ℝ) ^ n / m) * dyadic k ^ n := hp'
+      _ = _ := by ring
+  have hfloor : weight n m k / 2 - 1 / 2 ≤ (copies n m T k : ℝ) := by
+    simpa only [copies, ite_eq_left hkT] using floor_half_sub_half (weight_nonneg hm k)
+  have hsingle : (copies n m T k : ℝ) ≤
+      ∑ j ∈ S, if dyadic j ≤ x then (copies n m T j : ℝ) else 0 := by
+    by_cases hz : copies n m T k = 0
+    · simpa only [hz, Nat.cast_zero] using hnonneg
+    · have hs := Finset.single_le_sum
+        (fun j (_ : j ∈ S) => show 0 ≤ (if dyadic j ≤ x then (copies n m T j : ℝ) else 0)
+          from by split_ifs <;> positivity) (hS k hz)
+      simpa only [ite_eq_left hkhi] using hs
+  have hcoeff : (2 : ℝ) ^ n / (2 * m) * x ^ n = ((2 : ℝ) ^ n * x ^ n / m) / 2 := by
+    ring
+  rw [hcoeff]
+  linarith
+
+/-- The `(n+1)`st moment bounds the loss above the paper's cutoff
+`T = 4 M / m`; a retained `3m/4` moment leaves at least `m/2`. -/
+theorem retained_truncated_moment {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (Y : Ω → ℝ) (hY : Measurable Y) (hY0 : ∀ ω, 0 ≤ Y ω)
+    (n : ℕ) {m M : ℝ} (hm : 0 < m) (hM : 0 < M)
+    (hi : Integrable (fun ω => Y ω ^ n) P)
+    (hi' : Integrable (fun ω => Y ω ^ (n + 1)) P)
+    (hretained : 3 * m / 4 ≤ ∫ ω, Y ω ^ n ∂P)
+    (hnext : (∫ ω, Y ω ^ (n + 1) ∂P) ≤ M) :
+    m / 2 ≤ ∫ ω in {ω | Y ω ≤ 4 * M / m}, Y ω ^ n ∂P := by
+  let T := 4 * M / m
+  have hT : 0 < T := by dsimp [T]; positivity
+  have hset : MeasurableSet {ω | Y ω ≤ T} := measurableSet_le hY measurable_const
+  have htail : (∫ ω in {ω | Y ω ≤ T}ᶜ, Y ω ^ n ∂P) ≤ M / T := by
+    calc
+      _ = ∫ ω, {ω | Y ω ≤ T}ᶜ.indicator (fun ω => Y ω ^ n) ω ∂P :=
+        (integral_indicator hset.compl).symm
+      _ ≤ ∫ ω, Y ω ^ (n + 1) / T ∂P := by
+        apply integral_mono (hi.indicator hset.compl) (hi'.div_const T)
+        intro ω
+        by_cases hω : Y ω ≤ T
+        · simp only [Set.indicator, Set.mem_compl_iff, Set.mem_ofPred_eq, hω, not_true_eq_false,
+            ↓reduceIte]
+          exact div_nonneg (pow_nonneg (hY0 ω) _) hT.le
+        · simp only [Set.indicator, Set.mem_compl_iff, Set.mem_ofPred_eq, hω, not_false_eq_true,
+            ↓reduceIte]
+          apply (le_div_iff₀ hT).mpr
+          calc
+            Y ω ^ n * T ≤ Y ω ^ n * Y ω :=
+              mul_le_mul_of_nonneg_left (le_of_not_ge hω) (pow_nonneg (hY0 ω) _)
+            _ = Y ω ^ (n + 1) := (pow_succ _ _).symm
+      _ ≤ M / T := by rw [integral_div]; exact div_le_div_of_nonneg_right hnext hT.le
+  have hdecomp := integral_add_compl hset hi
+  have hMT : M / T = m / 4 := by dsimp [T]; field_simp
+  rw [hMT] at htail
+  change m / 2 ≤ ∫ ω in {ω | Y ω ≤ T}, Y ω ^ n ∂P
+  linarith
+
+/-- The lower batch probability estimate. The `1/2` loss is slightly stronger
+than the paper's `1 / (2 * (1 - 2⁻ⁿ))` loss. -/
+theorem lower_batch_probability_finset {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsProbabilityMeasure P] (Y : Ω → ℝ) (hY : Measurable Y)
+    (hY0 : ∀ ω, 0 ≤ Y ω) {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m)
+    (hi : Integrable (fun ω => Y ω ^ n) P)
+    (htrunc : m / 2 ≤ ∫ ω in {ω | Y ω ≤ T}, Y ω ^ n ∂P)
+    (S : Finset ℤ) (hS : ∀ k, copies n m T k ≠ 0 → k ∈ S) :
+    (2 : ℝ) ^ n / 4 - 1 / 2 ≤
+      ∑ k ∈ S, (copies n m T k : ℝ) * P.real {ω | dyadic k ≤ Y ω} := by
+  classical
+  let C := (2 : ℝ) ^ n / (2 * m)
+  let F := {ω | Y ω ≤ T}.indicator (fun ω => Y ω ^ n)
+  have hC : 0 ≤ C := by dsimp [C]; positivity
+  have hset : MeasurableSet {ω | Y ω ≤ T} := measurableSet_le hY measurable_const
+  have hFi : Integrable F P := hi.indicator hset
+  have hcount (k : ℤ) : Integrable
+      (fun ω => if dyadic k ≤ Y ω then (copies n m T k : ℝ) else 0) P :=
+    (integrable_const _).indicator (measurableSet_le measurable_const hY)
+  have hbound (ω : Ω) : C * F ω - 1 / 2 ≤
+      ∑ k ∈ S, if dyadic k ≤ Y ω then (copies n m T k : ℝ) else 0 := by
+    by_cases hω : Y ω ≤ T
+    · simpa only [F, Set.indicator, Set.mem_ofPred_eq, ite_eq_left hω, C] using
+        hit_count_lower hn hm (hY0 ω) hω S hS
+    · simp only [F, Set.indicator, Set.mem_ofPred_eq, ite_eq_right hω, mul_zero, zero_sub]
+      exact (by norm_num : -(1 / 2 : ℝ) ≤ 0).trans
+        (Finset.sum_nonneg (fun k _ => by split_ifs <;> positivity))
+  have hint := integral_mono ((hFi.const_mul C).sub (integrable_const (1 / 2 : ℝ)))
+    (integrable_finsetSum S (fun k _ => hcount k)) hbound
+  change (∫ ω, C * F ω - 1 / 2 ∂P) ≤
+    ∫ ω, ∑ k ∈ S, if dyadic k ≤ Y ω then (copies n m T k : ℝ) else 0 ∂P at hint
+  rw [integral_sub (hFi.const_mul C) (integrable_const (1 / 2 : ℝ)), integral_const_mul,
+    integral_const] at hint
+  simp only [measureReal_def, measure_univ, ENNReal.toReal_one, one_smul] at hint
+  have hF : ∫ ω, F ω ∂P = ∫ ω in {ω | Y ω ≤ T}, Y ω ^ n ∂P := integral_indicator hset
+  rw [hF] at hint
+  have hl := mul_le_mul_of_nonneg_left htrunc hC
+  have he : C * (m / 2) = (2 : ℝ) ^ n / 4 := by dsimp [C]; field_simp; ring
+  rw [he] at hl
+  have hprob := integral_hit_count P Y hY n m T 1 S
+  simp only [one_mul] at hprob
+  rw [hprob] at hint
+  linarith
+
+lemma lower_constant {n : ℕ} (hn : 3 ≤ n) :
+    (2 : ℝ) ^ n / 8 ≤ (2 : ℝ) ^ n / 4 - 1 / 2 := by
+  have hp : (4 : ℝ) ≤ 2 ^ n := by
+    have h := pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) (show 2 ≤ n by omega)
+    norm_num at h ⊢
+    exact h
+  linarith
+
+lemma copies_positive_bounds {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m)
+    {k : ℤ} (hk : copies n m T k ≠ 0) :
+    min 1 (m / (4 : ℝ) ^ n) ≤ dyadic k ∧ dyadic k ≤ T := by
+  have hcut : dyadic k ≤ T := by
+    by_contra hc
+    simp only [copies, ite_eq_right hc] at hk
+    exact hk rfl
+  have hfloor : 0 < ⌊weight n m k⌋₊ := by
+    simpa only [copies, ite_eq_left hcut, Nat.pos_iff_ne_zero] using hk
+  have hw : 1 ≤ weight n m k := Nat.floor_pos.mp hfloor
+  have hnum : m / (4 : ℝ) ^ n ≤ dyadic k ^ n := by
+    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 ^ n)).mpr
+    have hm' := (le_div_iff₀ hm).mp hw
+    simpa only [weight, one_mul, mul_one, mul_comm] using hm'
+  refine ⟨?_, hcut⟩
+  rcases le_total 1 (dyadic k) with hlarge | hsmall
+  · exact (min_le_left _ _).trans hlarge
+  · have hp : dyadic k ^ n ≤ dyadic k := by
+      simpa only [Real.rpow_natCast] using Real.rpow_le_self_of_le_one
+        (dyadic_pos k).le hsmall (show (1 : ℝ) ≤ n by exact_mod_cast hn)
+    exact (min_le_right _ _).trans (hnum.trans hp)
+
+lemma copies_support_finite {n : ℕ} (hn : 1 ≤ n) {m T : ℝ} (hm : 0 < m) :
+    Set.Finite {k : ℤ | copies n m T k ≠ 0} := by
+  let b := min 1 (m / (4 : ℝ) ^ n)
+  have hb : 0 < b := lt_min zero_lt_one (by positivity)
+  have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  apply (Set.finite_Icc (⌈Real.log b / Real.log 2⌉ : ℤ)
+    (⌊Real.log T / Real.log 2⌋ : ℤ)).subset
+  intro k hk
+  obtain ⟨hlo, hhi⟩ := copies_positive_bounds hn hm hk
+  constructor
+  · apply Int.ceil_le.mpr
+    apply (div_le_iff₀ hlog).mpr
+    exact (Real.le_zpow_iff_log_le hb (by norm_num : (0 : ℝ) < 2)).mp hlo
+  · apply Int.le_floor.mpr
+    apply (le_div_iff₀ hlog).mpr
+    exact Real.le_log_of_zpow_le (by norm_num : (0 : ℝ) < 2) hhi
+
+/-- The lower batch inequality for the whole integer sum, with no auxiliary
+finite support assumption. -/
+theorem lower_batch_probability {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsProbabilityMeasure P] (Y : Ω → ℝ) (hY : Measurable Y)
+    (hY0 : ∀ ω, 0 ≤ Y ω) {n : ℕ} (hn : 3 ≤ n) {m M : ℝ}
+    (hm : 0 < m) (hM : 0 < M)
+    (hi : Integrable (fun ω => Y ω ^ n) P)
+    (hi' : Integrable (fun ω => Y ω ^ (n + 1)) P)
+    (hretained : 3 * m / 4 ≤ ∫ ω, Y ω ^ n ∂P)
+    (hnext : (∫ ω, Y ω ^ (n + 1) ∂P) ≤ M) :
+    (2 : ℝ) ^ n / 8 ≤ ∑' k : ℤ,
+      (copies n m (4 * M / m) k : ℝ) * P.real {ω | dyadic k ≤ Y ω} := by
+  classical
+  have hn1 : 1 ≤ n := by omega
+  let S := (copies_support_finite hn1 (T := 4 * M / m) hm).toFinset
+  have hs : ∀ k, copies n m (4 * M / m) k ≠ 0 → k ∈ S := by
+    intro k hk
+    exact (copies_support_finite hn1 (T := 4 * M / m) hm).mem_toFinset.mpr hk
+  rw [tsum_eq_sum (s := S) (fun k hk => by
+    have hz : copies n m (4 * M / m) k = 0 := by
+      by_contra hc
+      exact hk (hs k hc)
+    simp only [hz, Nat.cast_zero, zero_mul])]
+  apply (lower_constant hn).trans
+  exact lower_batch_probability_finset P Y hY hY0 hn1 hm hi
+    (retained_truncated_moment P Y hY hY0 n hm hM hi hi' hretained hnext) S hs
+
+lemma copies_eq_target (κ : ℝ) (W : SLEExactGauge.Target.WienerLaw) (n : ℕ) (k : ℤ) :
+    copies n (SLEExactGauge.Target.moment κ W n)
+      (SLEExactGauge.Target.threshold κ W n) k = SLEExactGauge.Target.multiplicity κ W n k := rfl
+
+/-- The upper inequality in the exact notation of the moment-gauge construction. -/
+theorem target_upper_batch_probability {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
+    (hX0 : ∀ ω, 0 ≤ X ω) (κ : ℝ) (W : SLEExactGauge.Target.WienerLaw)
+    {n : ℕ} (hn : 1 ≤ n) (hm : 0 < SLEExactGauge.Target.moment κ W n)
+    (hi : Integrable (fun ω => X ω ^ n) P)
+    (he : ∫ ω, X ω ^ n ∂P = SLEExactGauge.Target.moment κ W n) :
+    ∑' k : ℤ, (SLEExactGauge.Target.multiplicity κ W n k : ℝ) *
+      P.real {ω | 16 * (2 : ℝ) ^ k ≤ X ω} ≤
+        (4 : ℝ) ^ n / 16 ^ n / (1 - (1 / 2 : ℝ) ^ n) :=
+  upper_batch_probability P X hX hX0 hn hm hi he
+
+/-- The success inequality in the exact notation of the moment-gauge construction. -/
+theorem target_lower_batch_probability {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsProbabilityMeasure P] (Y : Ω → ℝ) (hY : Measurable Y)
+    (hY0 : ∀ ω, 0 ≤ Y ω) (κ : ℝ) (W : SLEExactGauge.Target.WienerLaw)
+    {n : ℕ} (hn : 3 ≤ n)
+    (hm : 0 < SLEExactGauge.Target.moment κ W n)
+    (hM : 0 < SLEExactGauge.Target.moment κ W (n + 1))
+    (hi : Integrable (fun ω => Y ω ^ n) P)
+    (hi' : Integrable (fun ω => Y ω ^ (n + 1)) P)
+    (hretained : 3 * SLEExactGauge.Target.moment κ W n / 4 ≤ ∫ ω, Y ω ^ n ∂P)
+    (hnext : (∫ ω, Y ω ^ (n + 1) ∂P) ≤ SLEExactGauge.Target.moment κ W (n + 1)) :
+    (2 : ℝ) ^ n / 8 ≤ ∑' k : ℤ, (SLEExactGauge.Target.multiplicity κ W n k : ℝ) *
+      P.real {ω | (2 : ℝ) ^ k ≤ Y ω} :=
+  lower_batch_probability P Y hY hY0 hn hm hM hi hi' hretained hnext
+
+lemma cleanup_count_le {x : ℝ} (hx : 0 ≤ x) (S : Finset ℕ) :
+    ∑ n ∈ S, (if 16 * ((n : ℝ) + 1) ≤ x then (1 : ℝ) else 0) ≤ x / 16 := by
+  classical
+  rw [← Finset.sum_filter]
+  let A : Finset ℕ := S.filter (fun n : ℕ => 16 * ((n : ℝ) + 1) ≤ x)
+  have hsub : A ⊆ Finset.range ⌊x / 16⌋₊ := by
+    intro n hn
+    have hreal : (n : ℝ) + 1 ≤ x / 16 := by
+      apply (le_div_iff₀ (by norm_num : (0 : ℝ) < 16)).mpr
+      simpa only [mul_comm] using (Finset.mem_filter.mp hn).2
+    have hnat : n + 1 ≤ ⌊x / 16⌋₊ := Nat.le_floor (by simpa using hreal)
+    exact Finset.mem_range.mpr (by omega)
+  change (∑ n ∈ A, (1 : ℝ)) ≤ x / 16
+  rw [Finset.sum_const, nsmul_eq_mul, mul_one]
+  calc
+    (A.card : ℝ) ≤ (⌊x / 16⌋₊ : ℝ) := by
+      exact_mod_cast (Finset.card_le_card hsub).trans_eq (Finset.card_range _)
+    _ ≤ x / 16 := Nat.floor_le (by positivity)
+
+/-- The cleanup entries have summable tail probabilities, bounded by the first
+moment divided by 16. Indices `n + 1` are the paper's positive integers. -/
+theorem cleanup_tail_bound {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
+    (hX0 : ∀ ω, 0 ≤ X ω) (hi : Integrable X P) :
+    Summable (fun n : ℕ => P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω}) ∧
+      (∑' n : ℕ, P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω}) ≤ (∫ ω, X ω ∂P) / 16 := by
+  classical
+  have hsum (S : Finset ℕ) :
+      ∑ n ∈ S, P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω} ≤ (∫ ω, X ω ∂P) / 16 := by
+    have hset (n : ℕ) : MeasurableSet {ω | 16 * ((n : ℝ) + 1) ≤ X ω} :=
+      measurableSet_le measurable_const hX
+    have hcount (n : ℕ) : Integrable
+        (fun ω => if 16 * ((n : ℝ) + 1) ≤ X ω then (1 : ℝ) else 0) P :=
+      (integrable_const _).indicator (hset n)
+    calc
+      _ = ∫ ω, ∑ n ∈ S, if 16 * ((n : ℝ) + 1) ≤ X ω then (1 : ℝ) else 0 ∂P := by
+        rw [integral_finsetSum S (fun n _ => hcount n)]
+        apply Finset.sum_congr rfl
+        intro n hn
+        simpa only [Set.indicator, Set.mem_ofPred_eq, smul_eq_mul, mul_one] using
+          (integral_indicator_const (μ := P) (1 : ℝ) (hset n)).symm
+      _ ≤ ∫ ω, X ω / 16 ∂P := integral_mono
+        (integrable_finsetSum S (fun n _ => hcount n)) (hi.div_const 16)
+        (fun ω => cleanup_count_le (hX0 ω) S)
+      _ = _ := integral_div _ _
+  exact ⟨summable_of_sum_le (fun _ => measureReal_nonneg) hsum,
+    Real.tsum_le_of_sum_le (fun _ => measureReal_nonneg) hsum⟩
+
+lemma upper_constant_le_geometric {n : ℕ} (hn : 1 ≤ n) :
+    (4 : ℝ) ^ n / 16 ^ n / (1 - (1 / 2 : ℝ) ^ n) ≤ 2 * (1 / 4 : ℝ) ^ n := by
+  have hpow : (1 / 2 : ℝ) ^ n ≤ 1 / 2 := by
+    obtain ⟨j, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n ≠ 0)
+    rw [pow_succ]
+    exact (mul_le_mul_of_nonneg_right (pow_le_one₀ (by norm_num) (by norm_num))
+      (by norm_num : (0 : ℝ) ≤ 1 / 2)).trans_eq (one_mul _)
+  have he : (4 : ℝ) ^ n / 16 ^ n = (1 / 4 : ℝ) ^ n := by
+    rw [← div_pow]
+    norm_num
+  rw [he]
+  apply (div_le_iff₀ (denominator_pos hn)).mpr
+  have hnonneg : 0 ≤ (1 / 4 : ℝ) ^ n := by positivity
+  nlinarith
+
+/-- Both types of entries in all positive-order batches have summable exceedance
+probabilities. The hypotheses are precisely the positive finite moments of one
+nonnegative mass, not any SLE conclusion. -/
+theorem all_batch_tails_summable {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsFiniteMeasure P] (X : Ω → ℝ) (hX : Measurable X)
+    (hX0 : ∀ ω, 0 ≤ X ω) (m T : ℕ → ℝ)
+    (hm : ∀ n, 1 ≤ n → 0 < m n)
+    (hi : ∀ n, 1 ≤ n → Integrable (fun ω => X ω ^ n) P)
+    (hMoment : ∀ n, 1 ≤ n → (∫ ω, X ω ^ n ∂P) = m n) :
+    Summable (fun n : ℕ =>
+      (∑' k : ℤ, (copies (n + 1) (m (n + 1)) (T (n + 1)) k : ℝ) *
+        P.real {ω | 16 * dyadic k ≤ X ω}) +
+      P.real {ω | 16 * ((n : ℝ) + 1) ≤ X ω}) := by
+  have hgeom : Summable (fun n : ℕ => 2 * (1 / 4 : ℝ) ^ (n + 1)) := by
+    simpa only [pow_succ, mul_assoc, mul_comm, mul_left_comm] using
+      (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 4)
+        (by norm_num : (1 / 4 : ℝ) < 1)).mul_left (2 * (1 / 4 : ℝ))
+  have hbatch : Summable (fun n : ℕ =>
+      ∑' k : ℤ, (copies (n + 1) (m (n + 1)) (T (n + 1)) k : ℝ) *
+        P.real {ω | 16 * dyadic k ≤ X ω}) := by
+    apply Summable.of_nonneg_of_le (fun n => tsum_nonneg (fun _ => by positivity)) _ hgeom
+    intro n
+    exact (upper_batch_probability P X hX hX0 (by omega) (hm _ (by omega))
+      (hi _ (by omega)) (hMoment _ (by omega))).trans (upper_constant_le_geometric (by omega))
+  have hfirst : Integrable X P := by simpa only [pow_one] using hi 1 le_rfl
+  exact hbatch.add (cleanup_tail_bound P X hX hX0 hfirst).1
+
+end SLEExactGauge.MomentBatches
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/MomentCoefficientConvergence.lean b/lean/OAI/Probability/SLE/Gauge/MomentCoefficientConvergence.lean
index 9df660c5f0656e30c8ee470a49e0eeb30c800049..7a1654bccb5c9ab0e2819af99c32ecf43688a888 100644
--- a/lean/OAI/Probability/SLE/Gauge/MomentCoefficientConvergence.lean
+++ b/lean/OAI/Probability/SLE/Gauge/MomentCoefficientConvergence.lean
@@ -1,53 +1,53 @@
-import OAI.Probability.SLE.Mass.MassCompensatorSelection
-
-namespace OAI
-
-/-! Bounded truncation of the prescribed moment coefficients. An
-integrable finite-level density is enough to pass from convergence in
-probability of the discrete mass to its clipped weighted moments. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma continuous_clip (u : ℝ) : Continuous (Target.clip u) := by
-  exact (continuous_id.max continuous_const).min continuous_const
-
-lemma clip_nonneg {u : ℝ} (hu : 0 ≤ u) (x : ℝ) : 0 ≤ Target.clip u x :=
-  le_min (le_max_right x 0) hu
-
-lemma clip_le (u x : ℝ) : Target.clip u x ≤ u := min_le_right _ _
-
-/-- This is the inner limit in the paper's explicit coefficient formula;
-the density need only be integrable, not uniformly bounded. -/
-theorem weighted_clip_pow_integral_tendsto
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (X : ℕ → Ω → ℝ) (Y w : Ω → ℝ)
-    (hX : ∀ n, AEStronglyMeasurable (X n) P) (hw : Integrable w P)
-    (hXY : TendstoInMeasure P X atTop Y) (u : ℝ) (hu : 0 ≤ u) (k : ℕ) :
-    Tendsto (fun n => ∫ ω, w ω*(Target.clip u (X n ω))^k ∂P) atTop
-      (𝓝 (∫ ω, w ω*(Target.clip u (Y ω))^k ∂P)) := by
-  apply tendsto_of_subseq_tendsto
-  intro ns hns
-  obtain ⟨ms,_,hms⟩ := (hXY.comp hns).exists_seq_tendsto_ae
-  refine ⟨ms,?_⟩
-  apply tendsto_integral_of_dominated_convergence (fun ω => ‖w ω‖*u^k)
-  · intro n
-    exact hw.aestronglyMeasurable.mul
-      (((continuous_clip u).comp_aestronglyMeasurable (hX (ns (ms n)))).pow k)
-  · exact hw.norm.mul_const (u^k)
-  · intro n
-    filter_upwards [] with ω
-    rw [norm_mul,norm_pow,Real.norm_of_nonneg (clip_nonneg hu _)]
-    exact mul_le_mul_of_nonneg_left
-      (pow_le_pow_left₀ (clip_nonneg hu _) (clip_le _ _) k) (norm_nonneg _)
-  · filter_upwards [hms] with ω hω
-    exact tendsto_const_nhds.mul (((continuous_clip u).continuousAt.tendsto.comp hω).pow k)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.MassCompensatorSelection
+
+namespace OAI
+
+/-! Bounded truncation of the prescribed moment coefficients. An
+integrable finite-level density is enough to pass from convergence in
+probability of the discrete mass to its clipped weighted moments. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma continuous_clip (u : ℝ) : Continuous (Target.clip u) := by
+  exact (continuous_id.max continuous_const).min continuous_const
+
+lemma clip_nonneg {u : ℝ} (hu : 0 ≤ u) (x : ℝ) : 0 ≤ Target.clip u x :=
+  le_min (le_max_right x 0) hu
+
+lemma clip_le (u x : ℝ) : Target.clip u x ≤ u := min_le_right _ _
+
+/-- This is the inner limit in the paper's explicit coefficient formula;
+the density need only be integrable, not uniformly bounded. -/
+theorem weighted_clip_pow_integral_tendsto
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (X : ℕ → Ω → ℝ) (Y w : Ω → ℝ)
+    (hX : ∀ n, AEStronglyMeasurable (X n) P) (hw : Integrable w P)
+    (hXY : TendstoInMeasure P X atTop Y) (u : ℝ) (hu : 0 ≤ u) (k : ℕ) :
+    Tendsto (fun n => ∫ ω, w ω*(Target.clip u (X n ω))^k ∂P) atTop
+      (𝓝 (∫ ω, w ω*(Target.clip u (Y ω))^k ∂P)) := by
+  apply tendsto_of_subseq_tendsto
+  intro ns hns
+  obtain ⟨ms,_,hms⟩ := (hXY.comp hns).exists_seq_tendsto_ae
+  refine ⟨ms,?_⟩
+  apply tendsto_integral_of_dominated_convergence (fun ω => ‖w ω‖*u^k)
+  · intro n
+    exact hw.aestronglyMeasurable.mul
+      (((continuous_clip u).comp_aestronglyMeasurable (hX (ns (ms n)))).pow k)
+  · exact hw.norm.mul_const (u^k)
+  · intro n
+    filter_upwards [] with ω
+    rw [norm_mul,norm_pow,Real.norm_of_nonneg (clip_nonneg hu _)]
+    exact mul_le_mul_of_nonneg_left
+      (pow_le_pow_left₀ (clip_nonneg hu _) (clip_le _ _) k) (norm_nonneg _)
+  · filter_upwards [hms] with ω hω
+    exact tendsto_const_nhds.mul (((continuous_clip u).continuousAt.tendsto.comp hω).pow k)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/MomentUnclipping.lean b/lean/OAI/Probability/SLE/Gauge/MomentUnclipping.lean
index 44d7f5aa644abb31916111019114b3c653fb957f..859ff3e6bfd9ff02493f730734fa97787aed0da8 100644
--- a/lean/OAI/Probability/SLE/Gauge/MomentUnclipping.lean
+++ b/lean/OAI/Probability/SLE/Gauge/MomentUnclipping.lean
@@ -1,71 +1,71 @@
-import OAI.Probability.SLE.Gauge.MomentCoefficientConvergence
-
-namespace OAI
-
-/-! Removing the prescribed integer clipping levels under the proved
-integrable terminal-moment bound. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma clip_nat_eventually_eq {x : ℝ} (hx : 0≤x) :
-    ∀ᶠ n : ℕ in atTop, Target.clip n x=x := by
-  obtain ⟨N,hN⟩ := exists_nat_ge x
-  filter_upwards [eventually_ge_atTop N] with n hn
-  unfold Target.clip
-  rw [max_eq_left hx,min_eq_left (hN.trans (Nat.cast_le.mpr hn))]
-
-lemma weighted_clip_integral_bound
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (Y Z w : Ω → ℝ) (hY : AEStronglyMeasurable Y P) (hw : AEStronglyMeasurable w P)
-    (hdom : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤Z ω)
-    (m : ℕ) (hi : Integrable (fun ω => w ω*(Z ω)^m) P) (u : ℝ) (hu : 0≤u) :
-    0≤(∫ ω, w ω*(Target.clip u (Y ω))^m ∂P) ∧
-      (∫ ω, w ω*(Target.clip u (Y ω))^m ∂P)≤∫ ω, w ω*(Z ω)^m ∂P := by
-  have hbound : ∀ᵐ ω ∂P, w ω*(Target.clip u (Y ω))^m≤w ω*(Z ω)^m := by
-    filter_upwards [hdom] with ω hω
-    apply mul_le_mul_of_nonneg_left _ hω.1
-    apply pow_le_pow_left₀ (clip_nonneg hu _)
-    exact (min_le_left _ _).trans (by simpa only [max_eq_left hω.2.1] using hω.2.2)
-  have hn : 0≤ᵐ[P] (fun ω => w ω*(Target.clip u (Y ω))^m) := by
-    filter_upwards [hdom] with ω hω
-    exact mul_nonneg hω.1 (pow_nonneg (clip_nonneg hu _) m)
-  have hiC : Integrable (fun ω => w ω*(Target.clip u (Y ω))^m) P := by
-    apply hi.mono' (hw.mul (((continuous_clip u).comp_aestronglyMeasurable hY).pow m))
-    filter_upwards [hn,hbound] with ω hω hb
-    change ‖w ω*(Target.clip u (Y ω))^m‖≤w ω*(Z ω)^m
-    rwa [Real.norm_of_nonneg hω]
-  exact ⟨integral_nonneg_of_ae hn,integral_mono_ae hiC hi hbound⟩
-
-theorem weighted_unclip_pow_tendsto
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (Y Z w : Ω → ℝ) (hY : AEStronglyMeasurable Y P) (hw : AEStronglyMeasurable w P)
-    (hdom : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤Z ω)
-    (m : ℕ) (hi : Integrable (fun ω => w ω*(Z ω)^m) P) :
-    Tendsto (fun n : ℕ => ∫ ω, w ω*(Target.clip n (Y ω))^m ∂P) atTop
-      (𝓝 (∫ ω, w ω*(Y ω)^m ∂P)) := by
-  apply tendsto_integral_of_dominated_convergence (fun ω => ‖w ω*(Z ω)^m‖)
-  · intro n
-    exact hw.mul (((continuous_clip n).comp_aestronglyMeasurable hY).pow m)
-  · exact hi.norm
-  · intro n
-    filter_upwards [hdom] with ω hω
-    have hZ : 0≤Z ω := hω.2.1.trans hω.2.2
-    rw [Real.norm_of_nonneg (mul_nonneg hω.1 (pow_nonneg (clip_nonneg (Nat.cast_nonneg n) _) m)),
-      Real.norm_of_nonneg (mul_nonneg hω.1 (pow_nonneg hZ m))]
-    apply mul_le_mul_of_nonneg_left _ hω.1
-    apply pow_le_pow_left₀ (clip_nonneg (Nat.cast_nonneg n) _)
-    exact (min_le_left _ _).trans (by simpa only [max_eq_left hω.2.1] using hω.2.2)
-  · filter_upwards [hdom] with ω hω
-    apply tendsto_const_nhds.congr'
-    filter_upwards [clip_nat_eventually_eq hω.2.1] with n hn
-    rw [hn]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.MomentCoefficientConvergence
+
+namespace OAI
+
+/-! Removing the prescribed integer clipping levels under the proved
+integrable terminal-moment bound. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma clip_nat_eventually_eq {x : ℝ} (hx : 0≤x) :
+    ∀ᶠ n : ℕ in atTop, Target.clip n x=x := by
+  obtain ⟨N,hN⟩ := exists_nat_ge x
+  filter_upwards [eventually_ge_atTop N] with n hn
+  unfold Target.clip
+  rw [max_eq_left hx,min_eq_left (hN.trans (Nat.cast_le.mpr hn))]
+
+lemma weighted_clip_integral_bound
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (Y Z w : Ω → ℝ) (hY : AEStronglyMeasurable Y P) (hw : AEStronglyMeasurable w P)
+    (hdom : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤Z ω)
+    (m : ℕ) (hi : Integrable (fun ω => w ω*(Z ω)^m) P) (u : ℝ) (hu : 0≤u) :
+    0≤(∫ ω, w ω*(Target.clip u (Y ω))^m ∂P) ∧
+      (∫ ω, w ω*(Target.clip u (Y ω))^m ∂P)≤∫ ω, w ω*(Z ω)^m ∂P := by
+  have hbound : ∀ᵐ ω ∂P, w ω*(Target.clip u (Y ω))^m≤w ω*(Z ω)^m := by
+    filter_upwards [hdom] with ω hω
+    apply mul_le_mul_of_nonneg_left _ hω.1
+    apply pow_le_pow_left₀ (clip_nonneg hu _)
+    exact (min_le_left _ _).trans (by simpa only [max_eq_left hω.2.1] using hω.2.2)
+  have hn : 0≤ᵐ[P] (fun ω => w ω*(Target.clip u (Y ω))^m) := by
+    filter_upwards [hdom] with ω hω
+    exact mul_nonneg hω.1 (pow_nonneg (clip_nonneg hu _) m)
+  have hiC : Integrable (fun ω => w ω*(Target.clip u (Y ω))^m) P := by
+    apply hi.mono' (hw.mul (((continuous_clip u).comp_aestronglyMeasurable hY).pow m))
+    filter_upwards [hn,hbound] with ω hω hb
+    change ‖w ω*(Target.clip u (Y ω))^m‖≤w ω*(Z ω)^m
+    rwa [Real.norm_of_nonneg hω]
+  exact ⟨integral_nonneg_of_ae hn,integral_mono_ae hiC hi hbound⟩
+
+theorem weighted_unclip_pow_tendsto
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (Y Z w : Ω → ℝ) (hY : AEStronglyMeasurable Y P) (hw : AEStronglyMeasurable w P)
+    (hdom : ∀ᵐ ω ∂P, 0≤w ω ∧ 0≤Y ω ∧ Y ω≤Z ω)
+    (m : ℕ) (hi : Integrable (fun ω => w ω*(Z ω)^m) P) :
+    Tendsto (fun n : ℕ => ∫ ω, w ω*(Target.clip n (Y ω))^m ∂P) atTop
+      (𝓝 (∫ ω, w ω*(Y ω)^m ∂P)) := by
+  apply tendsto_integral_of_dominated_convergence (fun ω => ‖w ω*(Z ω)^m‖)
+  · intro n
+    exact hw.mul (((continuous_clip n).comp_aestronglyMeasurable hY).pow m)
+  · exact hi.norm
+  · intro n
+    filter_upwards [hdom] with ω hω
+    have hZ : 0≤Z ω := hω.2.1.trans hω.2.2
+    rw [Real.norm_of_nonneg (mul_nonneg hω.1 (pow_nonneg (clip_nonneg (Nat.cast_nonneg n) _) m)),
+      Real.norm_of_nonneg (mul_nonneg hω.1 (pow_nonneg hZ m))]
+    apply mul_le_mul_of_nonneg_left _ hω.1
+    apply pow_le_pow_left₀ (clip_nonneg (Nat.cast_nonneg n) _)
+    exact (min_le_left _ _).trans (by simpa only [max_eq_left hω.2.1] using hω.2.2)
+  · filter_upwards [hdom] with ω hω
+    apply tendsto_const_nhds.congr'
+    filter_upwards [clip_nat_eventually_eq hω.2.1] with n hn
+    rw [hn]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/NormalRibbonInjectivity.lean b/lean/OAI/Probability/SLE/Gauge/NormalRibbonInjectivity.lean
index f4727c258c8b082382f3b3c15c13bfd96d88dc4a..d6fd3b391af8fcd842fd10be0bc6e541df5389de 100644
--- a/lean/OAI/Probability/SLE/Gauge/NormalRibbonInjectivity.lean
+++ b/lean/OAI/Probability/SLE/Gauge/NormalRibbonInjectivity.lean
@@ -1,52 +1,52 @@
-import OAI.Probability.SLE.Gauge.NormalRibbonPatch
-
-namespace OAI
-
-/-! A global chord bound and the local ribbon estimate rule out all
-self-contacts of a width-one normal strip. -/
-
-noncomputable section
-open Set Metric
-open scoped NNReal
-
-namespace SLEExactGauge.Serpentine
-
-theorem normalRibbon_injective_of_chord_bound (σ tangent : ℝ → ℂ)
-    (hd : ∀ t, HasDerivAt σ (tangent t) t)
-    (hunit : ∀ t, ‖tangent t‖ = 1)
-    (hL : LipschitzWith (1 / 50) tangent)
-    (hchord : ∀ s t : ℝ, 0 ≤ s → 0 ≤ t → |s - t| ≤ 2 * ‖σ s - σ t‖) :
-    InjOn (normalRibbon σ tangent) {z : ℂ | 0 ≤ z.re ∧ |z.im| ≤ 1} := by
-  intro z hz w hw he
-  have hnz : ‖(z.im : ℂ) * Complex.I * tangent z.re‖ ≤ 1 := by
-    rw [norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I,
-      hunit, mul_one, mul_one]
-    exact hz.2
-  have hnw : ‖(w.im : ℂ) * Complex.I * tangent w.re‖ ≤ 1 := by
-    rw [norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I,
-      hunit, mul_one, mul_one]
-    exact hw.2
-  have heq : σ z.re - σ w.re =
-      (w.im : ℂ) * Complex.I * tangent w.re - (z.im : ℂ) * Complex.I * tangent z.re := by
-    unfold normalRibbon at he
-    exact sub_eq_sub_iff_add_eq_add.mpr (by simpa only [add_comm] using he)
-  have hn : ‖σ z.re - σ w.re‖ ≤ 2 := by
-    rw [heq]
-    exact (norm_sub_le _ _).trans (by linarith)
-  have hc : |z.re - w.re| ≤ 4 := (hchord _ _ hz.1 hw.1).trans (by linarith)
-  have hab : max z.re w.re - min z.re w.re ≤ 4 := by
-    rw [max_sub_min_eq_abs]
-    simpa only [abs_sub_comm] using hc
-  have hz' : z.re ∈ Icc (min z.re w.re) (max z.re w.re) := ⟨min_le_left _ _, le_max_left _ _⟩
-  have hw' : w.re ∈ Icc (min z.re w.re) (max z.re w.re) := ⟨min_le_right _ _, le_max_right _ _⟩
-  have hlower := (normalRibbon_patch_bounds σ tangent _ _ hab
-    (fun t _ => (hd t).hasDerivWithinAt) (fun t _ => hunit t) hL.lipschitzOnWith
-    z w ⟨hz', hz.2⟩ ⟨hw', hw.2⟩).2
-  rw [he, dist_self, mul_zero] at hlower
-  exact dist_eq_zero.mp (le_antisymm hlower dist_nonneg)
-
-end SLEExactGauge.Serpentine
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.NormalRibbonPatch
+
+namespace OAI
+
+/-! A global chord bound and the local ribbon estimate rule out all
+self-contacts of a width-one normal strip. -/
+
+noncomputable section
+open Set Metric
+open scoped NNReal
+
+namespace SLEExactGauge.Serpentine
+
+theorem normalRibbon_injective_of_chord_bound (σ tangent : ℝ → ℂ)
+    (hd : ∀ t, HasDerivAt σ (tangent t) t)
+    (hunit : ∀ t, ‖tangent t‖ = 1)
+    (hL : LipschitzWith (1 / 50) tangent)
+    (hchord : ∀ s t : ℝ, 0 ≤ s → 0 ≤ t → |s - t| ≤ 2 * ‖σ s - σ t‖) :
+    InjOn (normalRibbon σ tangent) {z : ℂ | 0 ≤ z.re ∧ |z.im| ≤ 1} := by
+  intro z hz w hw he
+  have hnz : ‖(z.im : ℂ) * Complex.I * tangent z.re‖ ≤ 1 := by
+    rw [norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I,
+      hunit, mul_one, mul_one]
+    exact hz.2
+  have hnw : ‖(w.im : ℂ) * Complex.I * tangent w.re‖ ≤ 1 := by
+    rw [norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I,
+      hunit, mul_one, mul_one]
+    exact hw.2
+  have heq : σ z.re - σ w.re =
+      (w.im : ℂ) * Complex.I * tangent w.re - (z.im : ℂ) * Complex.I * tangent z.re := by
+    unfold normalRibbon at he
+    exact sub_eq_sub_iff_add_eq_add.mpr (by simpa only [add_comm] using he)
+  have hn : ‖σ z.re - σ w.re‖ ≤ 2 := by
+    rw [heq]
+    exact (norm_sub_le _ _).trans (by linarith)
+  have hc : |z.re - w.re| ≤ 4 := (hchord _ _ hz.1 hw.1).trans (by linarith)
+  have hab : max z.re w.re - min z.re w.re ≤ 4 := by
+    rw [max_sub_min_eq_abs]
+    simpa only [abs_sub_comm] using hc
+  have hz' : z.re ∈ Icc (min z.re w.re) (max z.re w.re) := ⟨min_le_left _ _, le_max_left _ _⟩
+  have hw' : w.re ∈ Icc (min z.re w.re) (max z.re w.re) := ⟨min_le_right _ _, le_max_right _ _⟩
+  have hlower := (normalRibbon_patch_bounds σ tangent _ _ hab
+    (fun t _ => (hd t).hasDerivWithinAt) (fun t _ => hunit t) hL.lipschitzOnWith
+    z w ⟨hz', hz.2⟩ ⟨hw', hw.2⟩).2
+  rw [he, dist_self, mul_zero] at hlower
+  exact dist_eq_zero.mp (le_antisymm hlower dist_nonneg)
+
+end SLEExactGauge.Serpentine
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/NormalRibbonPatch.lean b/lean/OAI/Probability/SLE/Gauge/NormalRibbonPatch.lean
index cc17a5f689f427295c55a7347a718382e7ecd05d..69ca98619894e826bbb5b974d67382e5e8f03bb2 100644
--- a/lean/OAI/Probability/SLE/Gauge/NormalRibbonPatch.lean
+++ b/lean/OAI/Probability/SLE/Gauge/NormalRibbonPatch.lean
@@ -1,103 +1,103 @@
-import OAI.Probability.SLE.Gauge.RibbonStepPatch
-
-namespace OAI
-
-/-! Fixed bi-Lipschitz constants for the normal ribbon of a unit-speed
-curve whose tangent changes at rate at most 1/50. This is the local
-analytic estimate for the explicit serpentine's straight and circular pieces. -/
-
-noncomputable section
-open Set Metric
-open scoped NNReal
-
-namespace SLEExactGauge.Serpentine
-
-def normalRibbon (σ tangent : ℝ → ℂ) (z : ℂ) : ℂ :=
-  σ z.re + (z.im : ℂ) * Complex.I * tangent z.re
-
-lemma curve_linear_error
-    (σ tangent : ℝ → ℂ) (a b : ℝ) (hab : b - a ≤ 4)
-    (hd : ∀ t ∈ Icc a b, HasDerivWithinAt σ (tangent t) (Icc a b) t)
-    (hL : LipschitzOnWith (1 / 50) tangent (Icc a b))
-    {s t : ℝ} (hs : s ∈ Icc a b) (ht : t ∈ Icc a b) :
-    ‖σ s - σ t - ((s - t : ℝ) : ℂ) * tangent t‖ ≤ (4 / 50 : ℝ) * |s - t| := by
-  let f := fun u : ℝ => σ u - (u : ℂ) * tangent t
-  have hf (u : ℝ) (hu : u ∈ Icc a b) :
-      HasDerivWithinAt f (tangent u - tangent t) (Icc a b) u := by
-    convert (hd u hu).sub
-      ((Complex.ofRealCLM.hasDerivAt (x := u)).mul_const (tangent t)).hasDerivWithinAt using 1 <;>
-      simp only [f, Complex.ofRealCLM_apply, Complex.ofReal_one, one_mul]
-    rfl
-  have hbound (u : ℝ) (hu : u ∈ Icc a b) : ‖tangent u - tangent t‖ ≤ (4 / 50 : ℝ) := by
-    have hdut : dist u t ≤ 4 := by
-      rw [Real.dist_eq, abs_le]
-      constructor <;> linarith [hu.1, hu.2, ht.1, ht.2]
-    have hh := hL.dist_le_mul u hu t ht
-    rw [dist_eq_norm] at hh
-    exact hh.trans (by norm_num; linarith)
-  have hh := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hf hbound
-    (convex_Icc a b) ht hs
-  have he : f s - f t = σ s - σ t - ((s - t : ℝ) : ℂ) * tangent t := by
-    dsimp only [f]
-    push_cast
-    ring
-  rw [he, Real.norm_eq_abs] at hh
-  exact hh
-
-/-- A length-four patch has fixed constants 2 in both directions. -/
-theorem normalRibbon_patch_bounds
-    (σ tangent : ℝ → ℂ) (a b : ℝ) (hab : b - a ≤ 4)
-    (hd : ∀ t ∈ Icc a b, HasDerivWithinAt σ (tangent t) (Icc a b) t)
-    (hunit : ∀ t ∈ Icc a b, ‖tangent t‖ = 1)
-    (hL : LipschitzOnWith (1 / 50) tangent (Icc a b))
-    (z w : ℂ) (hz : z.re ∈ Icc a b ∧ |z.im| ≤ 1)
-    (hw : w.re ∈ Icc a b ∧ |w.im| ≤ 1) :
-    dist (normalRibbon σ tangent z) (normalRibbon σ tangent w) ≤ 2 * dist z w ∧
-      dist z w ≤ 2 * dist (normalRibbon σ tangent z) (normalRibbon σ tangent w) := by
-  let E := σ z.re - σ w.re - ((z.re - w.re : ℝ) : ℂ) * tangent w.re
-  have hE : ‖E‖ ≤ (4 / 50 : ℝ) * |z.re - w.re| :=
-    curve_linear_error σ tangent a b hab hd hL hz.1 hw.1
-  have hT := hL.dist_le_mul z.re hz.1 w.re hw.1
-  rw [dist_eq_norm, Real.dist_eq] at hT
-  have hnorm : ‖(z.im : ℂ) * Complex.I * (tangent z.re - tangent w.re)‖ ≤
-      (1 / 50 : ℝ) * |z.re - w.re| := by
-    rw [norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, mul_one]
-    calc
-      _ ≤ 1 * ‖tangent z.re - tangent w.re‖ := mul_le_mul_of_nonneg_right hz.2 (norm_nonneg _)
-      _ ≤ _ := by simpa using hT
-  let D := normalRibbon σ tangent z - normalRibbon σ tangent w
-  let A := (z - w) * tangent w.re
-  have he : D - A = E + (z.im : ℂ) * Complex.I * (tangent z.re - tangent w.re) := by
-    have hzw : z - w = ((z.re - w.re : ℝ) : ℂ) +
-        ((z.im - w.im : ℝ) : ℂ) * Complex.I := by
-      apply Complex.ext <;> simp
-    dsimp only [D, A, E, normalRibbon]
-    rw [hzw]
-    push_cast
-    ring
-  have her : ‖D - A‖ ≤ (1 / 10 : ℝ) * ‖z - w‖ := by
-    rw [he]
-    have hh := (norm_add_le E _).trans (add_le_add hE hnorm)
-    have hre : |z.re - w.re| ≤ ‖z - w‖ := by
-      simpa only [Complex.sub_re] using Complex.abs_re_le_norm (z - w)
-    nlinarith
-  have hA : ‖A‖ = ‖z - w‖ := by
-    change ‖(z - w) * tangent w.re‖ = _
-    rw [norm_mul, hunit _ hw.1, mul_one]
-  have hupper : ‖D‖ ≤ ‖z - w‖ + ‖D - A‖ := by
-    have hh := norm_add_le A (D - A)
-    rw [add_sub_cancel, hA] at hh
-    exact hh
-  have hlower : ‖z - w‖ ≤ ‖D‖ + ‖D - A‖ := by
-    have hh := norm_sub_le D (D - A)
-    rw [sub_sub_cancel, hA] at hh
-    exact hh
-  simp only [dist_eq_norm]
-  change ‖D‖ ≤ 2 * ‖z - w‖ ∧ ‖z - w‖ ≤ 2 * ‖D‖
-  constructor <;> nlinarith [norm_nonneg D, norm_nonneg (z - w)]
-
-end SLEExactGauge.Serpentine
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RibbonStepPatch
+
+namespace OAI
+
+/-! Fixed bi-Lipschitz constants for the normal ribbon of a unit-speed
+curve whose tangent changes at rate at most 1/50. This is the local
+analytic estimate for the explicit serpentine's straight and circular pieces. -/
+
+noncomputable section
+open Set Metric
+open scoped NNReal
+
+namespace SLEExactGauge.Serpentine
+
+def normalRibbon (σ tangent : ℝ → ℂ) (z : ℂ) : ℂ :=
+  σ z.re + (z.im : ℂ) * Complex.I * tangent z.re
+
+lemma curve_linear_error
+    (σ tangent : ℝ → ℂ) (a b : ℝ) (hab : b - a ≤ 4)
+    (hd : ∀ t ∈ Icc a b, HasDerivWithinAt σ (tangent t) (Icc a b) t)
+    (hL : LipschitzOnWith (1 / 50) tangent (Icc a b))
+    {s t : ℝ} (hs : s ∈ Icc a b) (ht : t ∈ Icc a b) :
+    ‖σ s - σ t - ((s - t : ℝ) : ℂ) * tangent t‖ ≤ (4 / 50 : ℝ) * |s - t| := by
+  let f := fun u : ℝ => σ u - (u : ℂ) * tangent t
+  have hf (u : ℝ) (hu : u ∈ Icc a b) :
+      HasDerivWithinAt f (tangent u - tangent t) (Icc a b) u := by
+    convert (hd u hu).sub
+      ((Complex.ofRealCLM.hasDerivAt (x := u)).mul_const (tangent t)).hasDerivWithinAt using 1 <;>
+      simp only [f, Complex.ofRealCLM_apply, Complex.ofReal_one, one_mul]
+    rfl
+  have hbound (u : ℝ) (hu : u ∈ Icc a b) : ‖tangent u - tangent t‖ ≤ (4 / 50 : ℝ) := by
+    have hdut : dist u t ≤ 4 := by
+      rw [Real.dist_eq, abs_le]
+      constructor <;> linarith [hu.1, hu.2, ht.1, ht.2]
+    have hh := hL.dist_le_mul u hu t ht
+    rw [dist_eq_norm] at hh
+    exact hh.trans (by norm_num; linarith)
+  have hh := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hf hbound
+    (convex_Icc a b) ht hs
+  have he : f s - f t = σ s - σ t - ((s - t : ℝ) : ℂ) * tangent t := by
+    dsimp only [f]
+    push_cast
+    ring
+  rw [he, Real.norm_eq_abs] at hh
+  exact hh
+
+/-- A length-four patch has fixed constants 2 in both directions. -/
+theorem normalRibbon_patch_bounds
+    (σ tangent : ℝ → ℂ) (a b : ℝ) (hab : b - a ≤ 4)
+    (hd : ∀ t ∈ Icc a b, HasDerivWithinAt σ (tangent t) (Icc a b) t)
+    (hunit : ∀ t ∈ Icc a b, ‖tangent t‖ = 1)
+    (hL : LipschitzOnWith (1 / 50) tangent (Icc a b))
+    (z w : ℂ) (hz : z.re ∈ Icc a b ∧ |z.im| ≤ 1)
+    (hw : w.re ∈ Icc a b ∧ |w.im| ≤ 1) :
+    dist (normalRibbon σ tangent z) (normalRibbon σ tangent w) ≤ 2 * dist z w ∧
+      dist z w ≤ 2 * dist (normalRibbon σ tangent z) (normalRibbon σ tangent w) := by
+  let E := σ z.re - σ w.re - ((z.re - w.re : ℝ) : ℂ) * tangent w.re
+  have hE : ‖E‖ ≤ (4 / 50 : ℝ) * |z.re - w.re| :=
+    curve_linear_error σ tangent a b hab hd hL hz.1 hw.1
+  have hT := hL.dist_le_mul z.re hz.1 w.re hw.1
+  rw [dist_eq_norm, Real.dist_eq] at hT
+  have hnorm : ‖(z.im : ℂ) * Complex.I * (tangent z.re - tangent w.re)‖ ≤
+      (1 / 50 : ℝ) * |z.re - w.re| := by
+    rw [norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, mul_one]
+    calc
+      _ ≤ 1 * ‖tangent z.re - tangent w.re‖ := mul_le_mul_of_nonneg_right hz.2 (norm_nonneg _)
+      _ ≤ _ := by simpa using hT
+  let D := normalRibbon σ tangent z - normalRibbon σ tangent w
+  let A := (z - w) * tangent w.re
+  have he : D - A = E + (z.im : ℂ) * Complex.I * (tangent z.re - tangent w.re) := by
+    have hzw : z - w = ((z.re - w.re : ℝ) : ℂ) +
+        ((z.im - w.im : ℝ) : ℂ) * Complex.I := by
+      apply Complex.ext <;> simp
+    dsimp only [D, A, E, normalRibbon]
+    rw [hzw]
+    push_cast
+    ring
+  have her : ‖D - A‖ ≤ (1 / 10 : ℝ) * ‖z - w‖ := by
+    rw [he]
+    have hh := (norm_add_le E _).trans (add_le_add hE hnorm)
+    have hre : |z.re - w.re| ≤ ‖z - w‖ := by
+      simpa only [Complex.sub_re] using Complex.abs_re_le_norm (z - w)
+    nlinarith
+  have hA : ‖A‖ = ‖z - w‖ := by
+    change ‖(z - w) * tangent w.re‖ = _
+    rw [norm_mul, hunit _ hw.1, mul_one]
+  have hupper : ‖D‖ ≤ ‖z - w‖ + ‖D - A‖ := by
+    have hh := norm_add_le A (D - A)
+    rw [add_sub_cancel, hA] at hh
+    exact hh
+  have hlower : ‖z - w‖ ≤ ‖D‖ + ‖D - A‖ := by
+    have hh := norm_sub_le D (D - A)
+    rw [sub_sub_cancel, hA] at hh
+    exact hh
+  simp only [dist_eq_norm]
+  change ‖D‖ ≤ 2 * ‖z - w‖ ∧ ‖z - w‖ ≤ 2 * ‖D‖
+  constructor <;> nlinarith [norm_nonneg D, norm_nonneg (z - w)]
+
+end SLEExactGauge.Serpentine
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PassageRetention.lean b/lean/OAI/Probability/SLE/Gauge/PassageRetention.lean
index 0ea1f38f06c0018631d0576497c086b03033e04c..e16ab306b36a9b409ae07bfca1a4fca387a2972c 100644
--- a/lean/OAI/Probability/SLE/Gauge/PassageRetention.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PassageRetention.lean
@@ -1,427 +1,427 @@
-import OAI.Probability.SLE.Geometry.CountableInteriorIntervals
-import OAI.Probability.SLE.Geometry.ActualStoppedTubeSupport
-import OAI.Probability.SLE.Geometry.FirstHeightBounded
-
-namespace OAI
-
-/-! Countable, measurable retention of actual middle visits and continuation
-prefixes. The middle interval is produced from a continuous progress crossing;
-the compact certificates are then extracted from fixed countable exhaustions. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal
-
-namespace SLEExactGauge.PassageRetention
-
-/-- First crossing of a scalar progress level on a compact time interval. -/
-theorem first_progress_crossing
-    (x : ℝ → ℝ) {a b q : ℝ} (hab : a < b)
-    (hx : ContinuousOn x (Icc a b)) (ha : x a < q) (hb : q < x b) :
-    ∃ θ ∈ Ioo a b, x θ = q ∧ ∀ u ∈ Ico a θ, x u < q := by
-  let K := Icc a b ∩ x ⁻¹' {q}
-  have hKc : IsClosed K := hx.preimage_isClosed_of_isClosed isClosed_Icc isClosed_singleton
-  have hK : IsCompact K := isCompact_Icc.of_isClosed_subset hKc inter_subset_left
-  have hKn : K.Nonempty := by
-    obtain ⟨t, ht, he⟩ := intermediate_value_Icc hab.le hx ⟨ha.le, hb.le⟩
-    exact ⟨t, ht, he⟩
-  obtain ⟨θ, hθ, hmin⟩ := hK.exists_isLeast hKn
-  have he : x θ = q := hθ.2
-  have haθ : a < θ := lt_of_le_of_ne hθ.1.1 (by
-    intro h
-    exact (ne_of_lt ha) (h ▸ he))
-  have hθb : θ < b := lt_of_le_of_ne hθ.1.2 (by
-    intro h
-    exact (ne_of_gt hb) (h ▸ he))
-  refine ⟨θ, ⟨haθ, hθb⟩, he, ?_⟩
-  intro u hu
-  by_contra hn
-  obtain ⟨v, hv, hvx⟩ := intermediate_value_Icc hu.1
-    (hx.mono (Icc_subset_Icc_right (hu.2.le.trans hθ.1.2))) ⟨ha.le, le_of_not_gt hn⟩
-  have hvK : v ∈ K := ⟨⟨hv.1, hv.2.trans (hu.2.le.trans hθ.1.2)⟩, hvx⟩
-  exact (not_le_of_gt hu.2) ((hmin hvK).trans hv.2)
-
-def middleRegion (O : Set ℂ) (progress : ℂ → ℝ) : Set ℂ :=
-  O ∩ progress ⁻¹' Ioo (1 / 3 : ℝ) (2 / 3)
-
-def beforeFrontRegion (O : Set ℂ) (progress : ℂ → ℝ) : Set ℂ :=
-  O ∩ progress ⁻¹' Iio (2 / 3 : ℝ)
-
-def beyondFrontRegion (O : Set ℂ) (progress : ℂ → ℝ) : Set ℂ :=
-  O ∩ progress ⁻¹' Ioi (1 : ℝ)
-
-/-- Progress from below one third to beyond the front at one forces a rational
-middle interval before the first front crossing and a later rational endpoint. -/
-theorem rational_middle_before_front
-    (f : ℝ → ℂ) (hf : Continuous f) (O : Set ℂ) (hO : IsOpen O)
-    (progress : ℂ → ℝ) (hp : ContinuousOn progress O)
-    {a b : ℝ} (hab : a < b) (hfO : MapsTo f (Icc a b) O)
-    (hstart : progress (f a) < 1 / 3) (hend : 1 < progress (f b)) :
-    ∃ s t u : ℚ, a < (s : ℝ) ∧ (s : ℝ) < t ∧ (t : ℝ) < u ∧ (u : ℝ) < b ∧
-      MapsTo f (Icc (s : ℝ) t) (middleRegion O progress) ∧
-      MapsTo f (Icc a t) (beforeFrontRegion O progress) ∧
-      f u ∈ beyondFrontRegion O progress := by
-  let x := progress ∘ f
-  have hx : ContinuousOn x (Icc a b) := hp.comp hf.continuousOn hfO
-  obtain ⟨θ, hθ, hxθ, hprior⟩ := first_progress_crossing x hab hx
-    (show x a < 2 / 3 by dsimp [x]; linarith) (by dsimp [x]; linarith)
-  obtain ⟨v, hv, hxv⟩ := intermediate_value_Icc hθ.1.le
-    (hx.mono (Icc_subset_Icc_right hθ.2.le))
-    (show (1 / 2 : ℝ) ∈ Icc (x a) (x θ) by dsimp [x] at *; rw [hxθ]; constructor <;> linarith)
-  have hav : a < v := lt_of_le_of_ne hv.1 (by
-    intro he
-    subst v
-    dsimp [x] at hxv
-    linarith)
-  have hvθ : v < θ := lt_of_le_of_ne hv.2 (by
-    intro he
-    subst v
-    rw [hxθ] at hxv
-    norm_num at hxv)
-  have hM : IsOpen (middleRegion O progress) := hp.isOpen_inter_preimage hO isOpen_Ioo
-  have hvM : f v ∈ middleRegion O progress :=
-    ⟨hfO ⟨hav.le, hvθ.le.trans hθ.2.le⟩, by change (1 / 3 : ℝ) < x v ∧ x v < 2 / 3; rw [hxv]; norm_num⟩
-  obtain ⟨s, t, has, hst, htθ, hmid⟩ :=
-    exists_rational_interval_in_open_visit f hf _ hM a θ ⟨v, ⟨hav, hvθ⟩, hvM⟩
-  have hθb : θ < b := hθ.2
-  let q := ((1 : ℝ) + x b) / 2
-  have hq : 1 < q ∧ q < x b := by dsimp [q, x]; constructor <;> linarith
-  obtain ⟨w, hw, hxw⟩ := intermediate_value_Icc hθb.le
-    (hx.mono (Icc_subset_Icc_left hθ.1.le))
-    (show q ∈ Icc (x θ) (x b) by rw [hxθ]; exact ⟨by linarith [hq.1], hq.2.le⟩)
-  have hθw : θ < w := lt_of_le_of_ne hw.1 (by intro he; subst w; rw [hxθ] at hxw; linarith [hq.1])
-  have hwb : w < b := lt_of_le_of_ne hw.2 (by intro he; subst w; linarith [hq.2])
-  have hW : IsOpen (beyondFrontRegion O progress) := hp.isOpen_inter_preimage hO isOpen_Ioi
-  have hwW : f w ∈ beyondFrontRegion O progress :=
-    ⟨hfO ⟨hθ.1.le.trans hw.1, hw.2⟩, by change 1 < x w; rw [hxw]; exact hq.1⟩
-  obtain ⟨u, u', hθu, huu', hu'b, hhigh⟩ :=
-    exists_rational_interval_in_open_visit f hf _ hW θ b ⟨w, ⟨hθw, hwb⟩, hwW⟩
-  refine ⟨s, t, u, has, hst, htθ.trans hθu, huu'.trans hu'b, hmid, ?_, ?_⟩
-  · intro v hv
-    exact ⟨hfO ⟨hv.1, hv.2.trans (htθ.le.trans hθ.2.le)⟩,
-      hprior v ⟨hv.1, hv.2.trans_lt htθ⟩⟩
-  · exact hhigh ⟨le_rfl, huu'.le⟩
-
-/-- Fixed compact exhaustions provide countably many certificates for both
-the whole continuation and the middle subinterval. -/
-theorem exists_compact_passage_certificates
-    (O : Set ℂ) (hO : IsOpen O) (progress : ℂ → ℝ)
-    (hp : ContinuousOn progress O) :
-    ∃ K M L : ℕ → Set ℂ,
-      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
-      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
-      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
-      ∀ (f : ℝ → ℂ), Continuous f → ∀ a b : ℝ, a < b →
-        MapsTo f (Icc a b) O → progress (f a) < 1 / 3 → 1 < progress (f b) →
-        ∃ (s t u : ℚ) (i j k : ℕ),
-          a < (s : ℝ) ∧ (s : ℝ) < t ∧ (t : ℝ) < u ∧ (u : ℝ) < b ∧
-          f '' Icc a u ⊆ K i ∧ f '' Icc (s : ℝ) t ⊆ M j ∧
-          f '' Icc a t ⊆ L k ∧ f u ∈ beyondFrontRegion O progress := by
-  obtain ⟨K, hK, hKO, hKcap⟩ := exists_countable_compact_exhaustion_open O hO
-  obtain ⟨M, hM, hMO, hMcap⟩ := exists_countable_compact_exhaustion_open
-    (middleRegion O progress) (hp.isOpen_inter_preimage hO isOpen_Ioo)
-  obtain ⟨L, hL, hLO, hLcap⟩ := exists_countable_compact_exhaustion_open
-    (beforeFrontRegion O progress) (hp.isOpen_inter_preimage hO isOpen_Iio)
-  refine ⟨K, M, L, fun n => ⟨hK n, hKO n⟩, fun n => ⟨hM n, hMO n⟩,
-    fun n => ⟨hL n, hLO n⟩, ?_⟩
-  intro f hf a b hab hfO hstart hend
-  obtain ⟨s, t, u, has, hst, htu, hub, hmid, hlow, hhigh⟩ :=
-    rational_middle_before_front f hf O hO progress hp hab hfO hstart hend
-  obtain ⟨i, hi⟩ := hKcap (f '' Icc a u) (isCompact_Icc.image hf)
-    ((hfO.mono (Icc_subset_Icc_right hub.le) Subset.rfl).image_subset)
-  obtain ⟨j, hj⟩ := hMcap (f '' Icc (s : ℝ) t) (isCompact_Icc.image hf) hmid.image_subset
-  obtain ⟨k, hk⟩ := hLcap (f '' Icc a t) (isCompact_Icc.image hf) hlow.image_subset
-  exact ⟨s, t, u, i, j, k, has, hst, htu, hub, hi, hj, hk, hhigh⟩
-
-local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-/-- Compact containment from a variable starting time is a closed path test. -/
-theorem isClosed_variable_start_compact (u : ℝ≥0) {K : Set ℂ} (hK : IsClosed K) :
-    IsClosed {x : ℝ≥0 × C(ℝ≥0, ℂ) | x.1 ≤ u ∧ MapsTo x.2 (Icc x.1 u) K} := by
-  have he : {x : ℝ≥0 × C(ℝ≥0, ℂ) | x.1 ≤ u ∧ MapsTo x.2 (Icc x.1 u) K} =
-      {x | x.1 ≤ u} ∩ ⋂ t ∈ Icc (0 : ℝ≥0) u, {x | x.2 (max x.1 t) ∈ K} := by
-    ext x
-    simp only [mem_ofPred_eq, mem_inter_iff, mem_iInter]
-    constructor
-    · rintro ⟨hxu, hm⟩
-      exact ⟨hxu, fun t ht => hm ⟨le_max_left _ _, max_le hxu ht.2⟩⟩
-    · rintro ⟨hxu, hm⟩
-      refine ⟨hxu, fun t ht => ?_⟩
-      simpa only [max_eq_right ht.1] using hm t ⟨bot_le, ht.2⟩
-  rw [he]
-  exact (isClosed_le continuous_fst continuous_const).inter
-    (isClosed_iInter fun t => isClosed_iInter fun _ =>
-      hK.preimage (continuous_eval.comp
-        (continuous_snd.prodMk (continuous_fst.max continuous_const))))
-
-theorem measurableSet_variable_start_compact
-    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (σ : Ω → ℝ≥0) (hσ : Measurable σ) (u : ℝ≥0)
-    {K : Set ℂ} (hK : IsCompact K) :
-    MeasurableSet {ω | σ ω ≤ u ∧ MapsTo (f ω) (Icc (σ ω) u) K} :=
-  (isClosed_variable_start_compact u hK.isClosed).measurableSet.preimage (hσ.prodMk hf)
-
-def ratTime (q : ℚ) : ℝ≥0 := Real.toNNReal (q : ℝ)
-
-@[simp] theorem ratTime_coe {q : ℚ} (hq : 0 ≤ q) : (ratTime q : ℝ) = (q : ℝ) := by
-  exact Real.coe_toNNReal _ (by exact_mod_cast hq)
-
-theorem ratTime_lt {p q : ℚ} (hp : 0 ≤ p) (hpq : p < q) : ratTime p < ratTime q := by
-  apply NNReal.coe_lt_coe.mp
-  rw [ratTime_coe hp, ratTime_coe (hp.trans hpq.le)]
-  exact_mod_cast hpq
-
-abbrev RationalPassageTimes := {q : ℚ × ℚ × ℚ // 0 < q.1 ∧ q.1 < q.2.1 ∧ q.2.1 < q.2.2}
-abbrev PassageIndex := RationalPassageTimes × (ℕ × ℕ × ℕ)
-
-def intervalStart (i : PassageIndex) : ℝ≥0 := ratTime i.1.1.1
-def intervalEnd (i : PassageIndex) : ℝ≥0 := ratTime i.1.1.2.1
-def passageEnd (i : PassageIndex) : ℝ≥0 := ratTime i.1.1.2.2
-
-theorem intervalStart_lt_end (i : PassageIndex) : intervalStart i < intervalEnd i :=
-  ratTime_lt i.1.2.1.le i.1.2.2.1
-
-theorem intervalEnd_lt_passageEnd (i : PassageIndex) : intervalEnd i < passageEnd i :=
-  ratTime_lt (i.1.2.1.le.trans i.1.2.2.1.le) i.1.2.2.2
-
-/-- The three compact tests retain the full prefix, a middle interval, and
-a prefix ending before any two-thirds front crossing. -/
-def compactPassageCertificate {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
-    (K M L : ℕ → Set ℂ) (W : Set ℂ) (i : PassageIndex) : Set Ω :=
-  {ω | σ ω < intervalStart i ∧ f ω (passageEnd i) ∈ W ∧
-    (σ ω ≤ passageEnd i ∧ MapsTo (f ω) (Icc (σ ω) (passageEnd i)) (K i.2.1)) ∧
-    MapsTo (f ω) (Icc (intervalStart i) (intervalEnd i)) (M i.2.2.1) ∧
-    (σ ω ≤ intervalEnd i ∧ MapsTo (f ω) (Icc (σ ω) (intervalEnd i)) (L i.2.2.2))}
-
-theorem compactPassageCertificate_measurable
-    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
-    (K M L : ℕ → Set ℂ) (hK : ∀ n, IsCompact (K n))
-    (hM : ∀ n, IsCompact (M n)) (hL : ∀ n, IsCompact (L n))
-    (W : Set ℂ) (hW : MeasurableSet W) (i : PassageIndex) :
-    MeasurableSet (compactPassageCertificate f σ K M L W i) :=
-  (measurableSet_lt hσ measurable_const).inter
-    ((hW.preimage ((continuous_eval_const (passageEnd i)).measurable.comp hf)).inter
-      ((measurableSet_variable_start_compact f hf σ hσ _ (hK _)).inter
-        ((measurableSet_interval_compact_certificate f hf _ _ (hM _)).inter
-          (measurableSet_variable_start_compact f hf σ hσ _ (hL _)))))
-
-/-- Fixed compact containment supplies an actual positive open margin. -/
-theorem retained_compact_open_margin {K O : Set ℂ} (hK : IsCompact K)
-    (hO : IsOpen O) (hKO : K ⊆ O) :
-    ∃ δ : ℝ, 0 < δ ∧ cthickening δ K ⊆ O :=
-  hK.exists_cthickening_subset_open hO hKO
-
-/-- A continuous nonnegative-time path with an actual progress crossing
-belongs to one of the fixed countable compact certificates. -/
-theorem countable_measurable_passage_cover
-    {Ω : Type*} [MeasurableSpace Ω]
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (σ : Ω → ℝ≥0) (hσ : Measurable σ)
-    (O : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
-    (progress : ℂ → ℝ) (hp : ContinuousOn progress O) (E : Set Ω)
-    (hpass : ∀ ω ∈ E, ∃ b : ℝ≥0, σ ω < b ∧ MapsTo (f ω) (Icc (σ ω) b) O ∧
-      progress (f ω (σ ω)) < 1 / 3 ∧ 1 < progress (f ω b)) :
-    ∃ K M L : ℕ → Set ℂ,
-      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
-      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
-      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
-      (∀ n, K n ⊆ upperHalfPlane) ∧ (∀ n, M n ⊆ upperHalfPlane) ∧
-      (∀ i, MeasurableSet (compactPassageCertificate f σ K M L
-        (beyondFrontRegion O progress) i)) ∧
-      E ⊆ ⋃ i : PassageIndex, compactPassageCertificate f σ K M L
-        (beyondFrontRegion O progress) i := by
-  obtain ⟨K, M, L, hK, hM, hL, hcert⟩ := exists_compact_passage_certificates O hO progress hp
-  refine ⟨K, M, L, hK, hM, hL, fun n => (hK n).2.trans hOH,
-    fun n z hz => hOH ((hM n).2 hz).1, ?_, ?_⟩
-  · exact fun i => compactPassageCertificate_measurable f hf σ hσ K M L
-      (fun n => (hK n).1) (fun n => (hM n).1) (fun n => (hL n).1)
-      _ (hp.isOpen_inter_preimage hO isOpen_Ioi).measurableSet i
-  · intro ω hω
-    obtain ⟨b, hσb, hfO, hstart, hend⟩ := hpass ω hω
-    let fR : ℝ → ℂ := fun t => f ω (Real.toNNReal t)
-    have hfR : Continuous fR := (f ω).continuous.comp continuous_real_toNNReal
-    have hfRO : MapsTo fR (Icc (σ ω : ℝ) (b : ℝ)) O := by
-      intro t ht
-      have ht0 : 0 ≤ t := (σ ω).coe_nonneg.trans ht.1
-      apply hfO
-      constructor
-      · apply NNReal.coe_le_coe.mp
-        simpa only [Real.coe_toNNReal _ ht0] using ht.1
-      · apply NNReal.coe_le_coe.mp
-        simpa only [Real.coe_toNNReal _ ht0] using ht.2
-    obtain ⟨s, t, u, i, j, k, has, hst, htu, hub, hpre, hmid, hlow, hhigh⟩ :=
-      hcert fR hfR (σ ω) b hσb hfRO
-        (by simpa only [fR, Real.toNNReal_coe] using hstart)
-        (by simpa only [fR, Real.toNNReal_coe] using hend)
-    have hs : 0 < s := by exact_mod_cast ((σ ω).coe_nonneg.trans_lt has)
-    have hst' : s < t := by exact_mod_cast hst
-    have htu' : t < u := by exact_mod_cast htu
-    have ht : 0 ≤ t := hs.le.trans hst'.le
-    have hu : 0 ≤ u := ht.trans htu'.le
-    let idx : PassageIndex := ⟨⟨(s, t, u), hs, hst', htu'⟩, (i, j, k)⟩
-    apply mem_iUnion.mpr
-    refine ⟨idx, ?_⟩
-    change σ ω < ratTime s ∧ f ω (ratTime u) ∈ beyondFrontRegion O progress ∧
-      (σ ω ≤ ratTime u ∧ MapsTo (f ω) (Icc (σ ω) (ratTime u)) (K i)) ∧
-      MapsTo (f ω) (Icc (ratTime s) (ratTime t)) (M j) ∧
-      (σ ω ≤ ratTime t ∧ MapsTo (f ω) (Icc (σ ω) (ratTime t)) (L k))
-    have has' : σ ω < ratTime s := by
-      apply NNReal.coe_lt_coe.mp
-      simpa only [ratTime_coe hs.le] using has
-    have lift (a : ℝ≥0) (q : ℚ) (hq : 0 ≤ q) (A : Set ℂ)
-        (hA : fR '' Icc (a : ℝ) (q : ℝ) ⊆ A) : MapsTo (f ω) (Icc a (ratTime q)) A := by
-      intro v hv
-      apply hA
-      refine ⟨(v : ℝ), ?_, ?_⟩
-      · constructor
-        · exact_mod_cast hv.1
-        · have hh : (v : ℝ) ≤ (ratTime q : ℝ) := hv.2
-          simpa only [ratTime_coe hq] using hh
-      · simp only [fR, Real.toNNReal_coe]
-    refine ⟨has', hhigh, ⟨has'.le.trans ((ratTime_lt hs.le hst').le.trans
-      (ratTime_lt ht htu').le), lift _ _ hu _ hpre⟩, ?_,
-      ⟨has'.le.trans (ratTime_lt hs.le hst').le, lift _ _ ht _ hlow⟩⟩
-    apply lift _ _ ht
-    simpa only [ratTime_coe hs.le] using hmid
-
-/-- The actual future tube event supplies the progress-crossing hypothesis;
-no compact or interval certificate is assumed. -/
-theorem stopped_tube_has_countable_passage_cover
-    {Ω : Type*} [MeasurableSpace Ω]
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (σ : Ω → ℝ≥0) (hσ : Measurable σ)
-    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
-    (progress : ℂ → ℝ) (hp : ContinuousOn progress O)
-    (hW : W ⊆ {z | 1 < progress z}) (E : Set Ω)
-    (hE : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
-    (hstart : ∀ ω ∈ E, progress (f ω (σ ω)) < 1 / 3) :
-    ∃ K M L : ℕ → Set ℂ,
-      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
-      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
-      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
-      (∀ n, K n ⊆ upperHalfPlane) ∧ (∀ n, M n ⊆ upperHalfPlane) ∧
-      (∀ i, MeasurableSet (compactPassageCertificate f σ K M L
-        (beyondFrontRegion O progress) i)) ∧
-      E ⊆ ⋃ i : PassageIndex, compactPassageCertificate f σ K M L
-        (beyondFrontRegion O progress) i := by
-  apply countable_measurable_passage_cover f hf σ hσ O hO hOH progress hp E
-  intro ω hω
-  obtain ⟨T, hT, hpath, hend⟩ := hE hω
-  refine ⟨σ ω + T, lt_add_of_pos_right _ hT, ?_, hstart ω hω, hW hend⟩
-  intro t ht
-  have hdiff : t - σ ω ∈ Icc (0 : ℝ≥0) T := by
-    exact ⟨bot_le, tsub_le_iff_right.mpr (by simpa only [add_comm] using ht.2)⟩
-  simpa only [add_tsub_cancel_of_le ht.1] using hpath hdiff
-
-abbrev RetainedPassageIndex := PassageIndex × ℕ
-
-def retainedPassageCertificate {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
-    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (i : RetainedPassageIndex) : Set Ω :=
-  compactPassageCertificate f σ K M L W i.1 ∩
-    {ω | f ω (passageEnd i.1) ∈ N i.2}
-
-/-- Actual tube passages admit a countable measurable cover with compact
-endpoint margins as well as compact prefix and middle margins. -/
-theorem stopped_tube_retained_cover
-    {Ω : Type*} [MeasurableSpace Ω]
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (σ : Ω → ℝ≥0) (hσ : Measurable σ)
-    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
-    (progress : ℂ → ℝ) (hp : ContinuousOn progress O)
-    (hW : W ⊆ {z | 1 < progress z}) (E : Set Ω)
-    (hE : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
-    (hstart : ∀ ω ∈ E, progress (f ω (σ ω)) < 1 / 3) :
-    ∃ K M L N : ℕ → Set ℂ,
-      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
-      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
-      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
-      (∀ n, IsCompact (N n) ∧ N n ⊆ beyondFrontRegion O progress) ∧
-      (∀ i, MeasurableSet (retainedPassageCertificate f σ K M L N
-        (beyondFrontRegion O progress) i)) ∧
-      E ⊆ ⋃ i : RetainedPassageIndex, retainedPassageCertificate f σ K M L N
-        (beyondFrontRegion O progress) i := by
-  obtain ⟨K, M, L, hK, hM, hL, _, _, hDm, hcover⟩ :=
-    stopped_tube_has_countable_passage_cover f hf σ hσ O W hO hOH progress hp hW E hE hstart
-  obtain ⟨N, hN, hNO, hNcap⟩ := exists_countable_compact_exhaustion_open
-    (beyondFrontRegion O progress) (hp.isOpen_inter_preimage hO isOpen_Ioi)
-  refine ⟨K, M, L, N, hK, hM, hL, fun n => ⟨hN n, hNO n⟩, ?_, ?_⟩
-  · intro i
-    exact (hDm i.1).inter ((hN i.2).measurableSet.preimage
-      ((continuous_eval_const (passageEnd i.1)).measurable.comp hf))
-  · intro ω hω
-    obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hω)
-    have hend : f ω (passageEnd i) ∈ beyondFrontRegion O progress := hi.2.1
-    obtain ⟨j, hj⟩ := hNcap {f ω (passageEnd i)} isCompact_singleton
-      (singleton_subset_iff.mpr hend)
-    exact mem_iUnion.mpr ⟨(i, j), hi, hj (mem_singleton _)⟩
-
-/-- On each retained certificate all strict passage conditions transfer
-uniformly to sufficiently late locally convergent maps. -/
-theorem retained_certificate_transfers
-    {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
-    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (i : RetainedPassageIndex)
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (hF : ContinuousOn F upperHalfPlane)
-    (hK : IsCompact (K i.1.2.1)) (hKH : K i.1.2.1 ⊆ upperHalfPlane)
-    (hM : IsCompact (M i.1.2.2.1)) (hMH : M i.1.2.2.1 ⊆ upperHalfPlane)
-    (hL : IsCompact (L i.1.2.2.2)) (hLH : L i.1.2.2.2 ⊆ upperHalfPlane)
-    (hN : IsCompact (N i.2)) (hNH : N i.2 ⊆ upperHalfPlane)
-    (R Rmid Rlow Rhigh : Set ℂ)
-    (hR : IsOpen R) (hRmid : IsOpen Rmid) (hRlow : IsOpen Rlow) (hRhigh : IsOpen Rhigh)
-    (hKR : F '' K i.1.2.1 ⊆ R) (hMR : F '' M i.1.2.2.1 ⊆ Rmid)
-    (hLR : F '' L i.1.2.2.2 ⊆ Rlow) (hNR : F '' N i.2 ⊆ Rhigh) :
-    ∀ᶠ n in atTop, ∀ ω ∈ retainedPassageCertificate f σ K M L N W i,
-      MapsTo (fun t => Fₙ n (f ω t)) (Icc (σ ω) (passageEnd i.1)) R ∧
-      MapsTo (fun t => Fₙ n (f ω t)) (Icc (intervalStart i.1) (intervalEnd i.1)) Rmid ∧
-      MapsTo (fun t => Fₙ n (f ω t)) (Icc (σ ω) (intervalEnd i.1)) Rlow ∧
-      Fₙ n (f ω (passageEnd i.1)) ∈ Rhigh := by
-  filter_upwards [eventually_image_compact_subset_open Fₙ F hconv hF _ hK hKH R hR hKR,
-    eventually_image_compact_subset_open Fₙ F hconv hF _ hM hMH Rmid hRmid hMR,
-    eventually_image_compact_subset_open Fₙ F hconv hF _ hL hLH Rlow hRlow hLR,
-    eventually_image_compact_subset_open Fₙ F hconv hF _ hN hNH Rhigh hRhigh hNR]
-    with n hnK hnM hnL hnN
-  rintro ω ⟨⟨_, _, hpre, hmid, hlow⟩, hend⟩
-  exact ⟨fun t ht => hnK (mem_image_of_mem _ (hpre.2 ht)),
-    fun t ht => hnM (mem_image_of_mem _ (hmid ht)),
-    fun t ht => hnL (mem_image_of_mem _ (hlow.2 ht)), hnN (mem_image_of_mem _ hend)⟩
-
-/-- The middle interval lies before every later time which attains the
-front level; in particular it precedes the first such passage stop. -/
-theorem retained_interval_before_front
-    {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
-    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (i : RetainedPassageIndex)
-    (progress : ℂ → ℝ) (hL : ∀ z ∈ L i.1.2.2.2, progress z < 2 / 3)
-    (ω : Ω) (hω : ω ∈ retainedPassageCertificate f σ K M L N W i)
-    (τ : ℝ≥0) (hστ : σ ω ≤ τ) (hτ : 2 / 3 ≤ progress (f ω τ)) :
-    intervalEnd i.1 < τ := by
-  by_contra hn
-  have hlow := hω.1.2.2.2.2.2
-  have hh := hL _ (hlow ⟨hστ, le_of_not_gt hn⟩)
-  exact (not_le_of_gt hh) hτ
-
-instance rationalPassageTimes_nonempty : Nonempty RationalPassageTimes :=
-  ⟨⟨(1, 2, 3), by norm_num⟩⟩
-
-/-- Enumerating the concrete rational/compact indices gives the natural
-indexing expected by finite-retention and unconditional sausage lemmas. -/
-def enumerateRetainedPassage : ℕ → RetainedPassageIndex :=
-  (exists_surjective_nat RetainedPassageIndex).choose
-
-theorem enumerateRetainedPassage_surjective : Function.Surjective enumerateRetainedPassage :=
-  (exists_surjective_nat RetainedPassageIndex).choose_spec
-
-theorem nat_retained_cover
-    {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
-    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (E : Set Ω)
-    (hcover : E ⊆ ⋃ i : RetainedPassageIndex, retainedPassageCertificate f σ K M L N W i) :
-    E ⊆ ⋃ n : ℕ, retainedPassageCertificate f σ K M L N W (enumerateRetainedPassage n) := by
-  intro ω hω
-  obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hω)
-  obtain ⟨n, rfl⟩ := enumerateRetainedPassage_surjective i
-  exact mem_iUnion.mpr ⟨n, hi⟩
-
-end SLEExactGauge.PassageRetention
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CountableInteriorIntervals
+import OAI.Probability.SLE.Geometry.ActualStoppedTubeSupport
+import OAI.Probability.SLE.Geometry.FirstHeightBounded
+
+namespace OAI
+
+/-! Countable, measurable retention of actual middle visits and continuation
+prefixes. The middle interval is produced from a continuous progress crossing;
+the compact certificates are then extracted from fixed countable exhaustions. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal
+
+namespace SLEExactGauge.PassageRetention
+
+/-- First crossing of a scalar progress level on a compact time interval. -/
+theorem first_progress_crossing
+    (x : ℝ → ℝ) {a b q : ℝ} (hab : a < b)
+    (hx : ContinuousOn x (Icc a b)) (ha : x a < q) (hb : q < x b) :
+    ∃ θ ∈ Ioo a b, x θ = q ∧ ∀ u ∈ Ico a θ, x u < q := by
+  let K := Icc a b ∩ x ⁻¹' {q}
+  have hKc : IsClosed K := hx.preimage_isClosed_of_isClosed isClosed_Icc isClosed_singleton
+  have hK : IsCompact K := isCompact_Icc.of_isClosed_subset hKc inter_subset_left
+  have hKn : K.Nonempty := by
+    obtain ⟨t, ht, he⟩ := intermediate_value_Icc hab.le hx ⟨ha.le, hb.le⟩
+    exact ⟨t, ht, he⟩
+  obtain ⟨θ, hθ, hmin⟩ := hK.exists_isLeast hKn
+  have he : x θ = q := hθ.2
+  have haθ : a < θ := lt_of_le_of_ne hθ.1.1 (by
+    intro h
+    exact (ne_of_lt ha) (h ▸ he))
+  have hθb : θ < b := lt_of_le_of_ne hθ.1.2 (by
+    intro h
+    exact (ne_of_gt hb) (h ▸ he))
+  refine ⟨θ, ⟨haθ, hθb⟩, he, ?_⟩
+  intro u hu
+  by_contra hn
+  obtain ⟨v, hv, hvx⟩ := intermediate_value_Icc hu.1
+    (hx.mono (Icc_subset_Icc_right (hu.2.le.trans hθ.1.2))) ⟨ha.le, le_of_not_gt hn⟩
+  have hvK : v ∈ K := ⟨⟨hv.1, hv.2.trans (hu.2.le.trans hθ.1.2)⟩, hvx⟩
+  exact (not_le_of_gt hu.2) ((hmin hvK).trans hv.2)
+
+def middleRegion (O : Set ℂ) (progress : ℂ → ℝ) : Set ℂ :=
+  O ∩ progress ⁻¹' Ioo (1 / 3 : ℝ) (2 / 3)
+
+def beforeFrontRegion (O : Set ℂ) (progress : ℂ → ℝ) : Set ℂ :=
+  O ∩ progress ⁻¹' Iio (2 / 3 : ℝ)
+
+def beyondFrontRegion (O : Set ℂ) (progress : ℂ → ℝ) : Set ℂ :=
+  O ∩ progress ⁻¹' Ioi (1 : ℝ)
+
+/-- Progress from below one third to beyond the front at one forces a rational
+middle interval before the first front crossing and a later rational endpoint. -/
+theorem rational_middle_before_front
+    (f : ℝ → ℂ) (hf : Continuous f) (O : Set ℂ) (hO : IsOpen O)
+    (progress : ℂ → ℝ) (hp : ContinuousOn progress O)
+    {a b : ℝ} (hab : a < b) (hfO : MapsTo f (Icc a b) O)
+    (hstart : progress (f a) < 1 / 3) (hend : 1 < progress (f b)) :
+    ∃ s t u : ℚ, a < (s : ℝ) ∧ (s : ℝ) < t ∧ (t : ℝ) < u ∧ (u : ℝ) < b ∧
+      MapsTo f (Icc (s : ℝ) t) (middleRegion O progress) ∧
+      MapsTo f (Icc a t) (beforeFrontRegion O progress) ∧
+      f u ∈ beyondFrontRegion O progress := by
+  let x := progress ∘ f
+  have hx : ContinuousOn x (Icc a b) := hp.comp hf.continuousOn hfO
+  obtain ⟨θ, hθ, hxθ, hprior⟩ := first_progress_crossing x hab hx
+    (show x a < 2 / 3 by dsimp [x]; linarith) (by dsimp [x]; linarith)
+  obtain ⟨v, hv, hxv⟩ := intermediate_value_Icc hθ.1.le
+    (hx.mono (Icc_subset_Icc_right hθ.2.le))
+    (show (1 / 2 : ℝ) ∈ Icc (x a) (x θ) by dsimp [x] at *; rw [hxθ]; constructor <;> linarith)
+  have hav : a < v := lt_of_le_of_ne hv.1 (by
+    intro he
+    subst v
+    dsimp [x] at hxv
+    linarith)
+  have hvθ : v < θ := lt_of_le_of_ne hv.2 (by
+    intro he
+    subst v
+    rw [hxθ] at hxv
+    norm_num at hxv)
+  have hM : IsOpen (middleRegion O progress) := hp.isOpen_inter_preimage hO isOpen_Ioo
+  have hvM : f v ∈ middleRegion O progress :=
+    ⟨hfO ⟨hav.le, hvθ.le.trans hθ.2.le⟩, by change (1 / 3 : ℝ) < x v ∧ x v < 2 / 3; rw [hxv]; norm_num⟩
+  obtain ⟨s, t, has, hst, htθ, hmid⟩ :=
+    exists_rational_interval_in_open_visit f hf _ hM a θ ⟨v, ⟨hav, hvθ⟩, hvM⟩
+  have hθb : θ < b := hθ.2
+  let q := ((1 : ℝ) + x b) / 2
+  have hq : 1 < q ∧ q < x b := by dsimp [q, x]; constructor <;> linarith
+  obtain ⟨w, hw, hxw⟩ := intermediate_value_Icc hθb.le
+    (hx.mono (Icc_subset_Icc_left hθ.1.le))
+    (show q ∈ Icc (x θ) (x b) by rw [hxθ]; exact ⟨by linarith [hq.1], hq.2.le⟩)
+  have hθw : θ < w := lt_of_le_of_ne hw.1 (by intro he; subst w; rw [hxθ] at hxw; linarith [hq.1])
+  have hwb : w < b := lt_of_le_of_ne hw.2 (by intro he; subst w; linarith [hq.2])
+  have hW : IsOpen (beyondFrontRegion O progress) := hp.isOpen_inter_preimage hO isOpen_Ioi
+  have hwW : f w ∈ beyondFrontRegion O progress :=
+    ⟨hfO ⟨hθ.1.le.trans hw.1, hw.2⟩, by change 1 < x w; rw [hxw]; exact hq.1⟩
+  obtain ⟨u, u', hθu, huu', hu'b, hhigh⟩ :=
+    exists_rational_interval_in_open_visit f hf _ hW θ b ⟨w, ⟨hθw, hwb⟩, hwW⟩
+  refine ⟨s, t, u, has, hst, htθ.trans hθu, huu'.trans hu'b, hmid, ?_, ?_⟩
+  · intro v hv
+    exact ⟨hfO ⟨hv.1, hv.2.trans (htθ.le.trans hθ.2.le)⟩,
+      hprior v ⟨hv.1, hv.2.trans_lt htθ⟩⟩
+  · exact hhigh ⟨le_rfl, huu'.le⟩
+
+/-- Fixed compact exhaustions provide countably many certificates for both
+the whole continuation and the middle subinterval. -/
+theorem exists_compact_passage_certificates
+    (O : Set ℂ) (hO : IsOpen O) (progress : ℂ → ℝ)
+    (hp : ContinuousOn progress O) :
+    ∃ K M L : ℕ → Set ℂ,
+      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
+      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
+      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
+      ∀ (f : ℝ → ℂ), Continuous f → ∀ a b : ℝ, a < b →
+        MapsTo f (Icc a b) O → progress (f a) < 1 / 3 → 1 < progress (f b) →
+        ∃ (s t u : ℚ) (i j k : ℕ),
+          a < (s : ℝ) ∧ (s : ℝ) < t ∧ (t : ℝ) < u ∧ (u : ℝ) < b ∧
+          f '' Icc a u ⊆ K i ∧ f '' Icc (s : ℝ) t ⊆ M j ∧
+          f '' Icc a t ⊆ L k ∧ f u ∈ beyondFrontRegion O progress := by
+  obtain ⟨K, hK, hKO, hKcap⟩ := exists_countable_compact_exhaustion_open O hO
+  obtain ⟨M, hM, hMO, hMcap⟩ := exists_countable_compact_exhaustion_open
+    (middleRegion O progress) (hp.isOpen_inter_preimage hO isOpen_Ioo)
+  obtain ⟨L, hL, hLO, hLcap⟩ := exists_countable_compact_exhaustion_open
+    (beforeFrontRegion O progress) (hp.isOpen_inter_preimage hO isOpen_Iio)
+  refine ⟨K, M, L, fun n => ⟨hK n, hKO n⟩, fun n => ⟨hM n, hMO n⟩,
+    fun n => ⟨hL n, hLO n⟩, ?_⟩
+  intro f hf a b hab hfO hstart hend
+  obtain ⟨s, t, u, has, hst, htu, hub, hmid, hlow, hhigh⟩ :=
+    rational_middle_before_front f hf O hO progress hp hab hfO hstart hend
+  obtain ⟨i, hi⟩ := hKcap (f '' Icc a u) (isCompact_Icc.image hf)
+    ((hfO.mono (Icc_subset_Icc_right hub.le) Subset.rfl).image_subset)
+  obtain ⟨j, hj⟩ := hMcap (f '' Icc (s : ℝ) t) (isCompact_Icc.image hf) hmid.image_subset
+  obtain ⟨k, hk⟩ := hLcap (f '' Icc a t) (isCompact_Icc.image hf) hlow.image_subset
+  exact ⟨s, t, u, i, j, k, has, hst, htu, hub, hi, hj, hk, hhigh⟩
+
+local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+/-- Compact containment from a variable starting time is a closed path test. -/
+theorem isClosed_variable_start_compact (u : ℝ≥0) {K : Set ℂ} (hK : IsClosed K) :
+    IsClosed {x : ℝ≥0 × C(ℝ≥0, ℂ) | x.1 ≤ u ∧ MapsTo x.2 (Icc x.1 u) K} := by
+  have he : {x : ℝ≥0 × C(ℝ≥0, ℂ) | x.1 ≤ u ∧ MapsTo x.2 (Icc x.1 u) K} =
+      {x | x.1 ≤ u} ∩ ⋂ t ∈ Icc (0 : ℝ≥0) u, {x | x.2 (max x.1 t) ∈ K} := by
+    ext x
+    simp only [mem_ofPred_eq, mem_inter_iff, mem_iInter]
+    constructor
+    · rintro ⟨hxu, hm⟩
+      exact ⟨hxu, fun t ht => hm ⟨le_max_left _ _, max_le hxu ht.2⟩⟩
+    · rintro ⟨hxu, hm⟩
+      refine ⟨hxu, fun t ht => ?_⟩
+      simpa only [max_eq_right ht.1] using hm t ⟨bot_le, ht.2⟩
+  rw [he]
+  exact (isClosed_le continuous_fst continuous_const).inter
+    (isClosed_iInter fun t => isClosed_iInter fun _ =>
+      hK.preimage (continuous_eval.comp
+        (continuous_snd.prodMk (continuous_fst.max continuous_const))))
+
+theorem measurableSet_variable_start_compact
+    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (σ : Ω → ℝ≥0) (hσ : Measurable σ) (u : ℝ≥0)
+    {K : Set ℂ} (hK : IsCompact K) :
+    MeasurableSet {ω | σ ω ≤ u ∧ MapsTo (f ω) (Icc (σ ω) u) K} :=
+  (isClosed_variable_start_compact u hK.isClosed).measurableSet.preimage (hσ.prodMk hf)
+
+def ratTime (q : ℚ) : ℝ≥0 := Real.toNNReal (q : ℝ)
+
+@[simp] theorem ratTime_coe {q : ℚ} (hq : 0 ≤ q) : (ratTime q : ℝ) = (q : ℝ) := by
+  exact Real.coe_toNNReal _ (by exact_mod_cast hq)
+
+theorem ratTime_lt {p q : ℚ} (hp : 0 ≤ p) (hpq : p < q) : ratTime p < ratTime q := by
+  apply NNReal.coe_lt_coe.mp
+  rw [ratTime_coe hp, ratTime_coe (hp.trans hpq.le)]
+  exact_mod_cast hpq
+
+abbrev RationalPassageTimes := {q : ℚ × ℚ × ℚ // 0 < q.1 ∧ q.1 < q.2.1 ∧ q.2.1 < q.2.2}
+abbrev PassageIndex := RationalPassageTimes × (ℕ × ℕ × ℕ)
+
+def intervalStart (i : PassageIndex) : ℝ≥0 := ratTime i.1.1.1
+def intervalEnd (i : PassageIndex) : ℝ≥0 := ratTime i.1.1.2.1
+def passageEnd (i : PassageIndex) : ℝ≥0 := ratTime i.1.1.2.2
+
+theorem intervalStart_lt_end (i : PassageIndex) : intervalStart i < intervalEnd i :=
+  ratTime_lt i.1.2.1.le i.1.2.2.1
+
+theorem intervalEnd_lt_passageEnd (i : PassageIndex) : intervalEnd i < passageEnd i :=
+  ratTime_lt (i.1.2.1.le.trans i.1.2.2.1.le) i.1.2.2.2
+
+/-- The three compact tests retain the full prefix, a middle interval, and
+a prefix ending before any two-thirds front crossing. -/
+def compactPassageCertificate {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
+    (K M L : ℕ → Set ℂ) (W : Set ℂ) (i : PassageIndex) : Set Ω :=
+  {ω | σ ω < intervalStart i ∧ f ω (passageEnd i) ∈ W ∧
+    (σ ω ≤ passageEnd i ∧ MapsTo (f ω) (Icc (σ ω) (passageEnd i)) (K i.2.1)) ∧
+    MapsTo (f ω) (Icc (intervalStart i) (intervalEnd i)) (M i.2.2.1) ∧
+    (σ ω ≤ intervalEnd i ∧ MapsTo (f ω) (Icc (σ ω) (intervalEnd i)) (L i.2.2.2))}
+
+theorem compactPassageCertificate_measurable
+    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
+    (K M L : ℕ → Set ℂ) (hK : ∀ n, IsCompact (K n))
+    (hM : ∀ n, IsCompact (M n)) (hL : ∀ n, IsCompact (L n))
+    (W : Set ℂ) (hW : MeasurableSet W) (i : PassageIndex) :
+    MeasurableSet (compactPassageCertificate f σ K M L W i) :=
+  (measurableSet_lt hσ measurable_const).inter
+    ((hW.preimage ((continuous_eval_const (passageEnd i)).measurable.comp hf)).inter
+      ((measurableSet_variable_start_compact f hf σ hσ _ (hK _)).inter
+        ((measurableSet_interval_compact_certificate f hf _ _ (hM _)).inter
+          (measurableSet_variable_start_compact f hf σ hσ _ (hL _)))))
+
+/-- Fixed compact containment supplies an actual positive open margin. -/
+theorem retained_compact_open_margin {K O : Set ℂ} (hK : IsCompact K)
+    (hO : IsOpen O) (hKO : K ⊆ O) :
+    ∃ δ : ℝ, 0 < δ ∧ cthickening δ K ⊆ O :=
+  hK.exists_cthickening_subset_open hO hKO
+
+/-- A continuous nonnegative-time path with an actual progress crossing
+belongs to one of the fixed countable compact certificates. -/
+theorem countable_measurable_passage_cover
+    {Ω : Type*} [MeasurableSpace Ω]
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (σ : Ω → ℝ≥0) (hσ : Measurable σ)
+    (O : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
+    (progress : ℂ → ℝ) (hp : ContinuousOn progress O) (E : Set Ω)
+    (hpass : ∀ ω ∈ E, ∃ b : ℝ≥0, σ ω < b ∧ MapsTo (f ω) (Icc (σ ω) b) O ∧
+      progress (f ω (σ ω)) < 1 / 3 ∧ 1 < progress (f ω b)) :
+    ∃ K M L : ℕ → Set ℂ,
+      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
+      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
+      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
+      (∀ n, K n ⊆ upperHalfPlane) ∧ (∀ n, M n ⊆ upperHalfPlane) ∧
+      (∀ i, MeasurableSet (compactPassageCertificate f σ K M L
+        (beyondFrontRegion O progress) i)) ∧
+      E ⊆ ⋃ i : PassageIndex, compactPassageCertificate f σ K M L
+        (beyondFrontRegion O progress) i := by
+  obtain ⟨K, M, L, hK, hM, hL, hcert⟩ := exists_compact_passage_certificates O hO progress hp
+  refine ⟨K, M, L, hK, hM, hL, fun n => (hK n).2.trans hOH,
+    fun n z hz => hOH ((hM n).2 hz).1, ?_, ?_⟩
+  · exact fun i => compactPassageCertificate_measurable f hf σ hσ K M L
+      (fun n => (hK n).1) (fun n => (hM n).1) (fun n => (hL n).1)
+      _ (hp.isOpen_inter_preimage hO isOpen_Ioi).measurableSet i
+  · intro ω hω
+    obtain ⟨b, hσb, hfO, hstart, hend⟩ := hpass ω hω
+    let fR : ℝ → ℂ := fun t => f ω (Real.toNNReal t)
+    have hfR : Continuous fR := (f ω).continuous.comp continuous_real_toNNReal
+    have hfRO : MapsTo fR (Icc (σ ω : ℝ) (b : ℝ)) O := by
+      intro t ht
+      have ht0 : 0 ≤ t := (σ ω).coe_nonneg.trans ht.1
+      apply hfO
+      constructor
+      · apply NNReal.coe_le_coe.mp
+        simpa only [Real.coe_toNNReal _ ht0] using ht.1
+      · apply NNReal.coe_le_coe.mp
+        simpa only [Real.coe_toNNReal _ ht0] using ht.2
+    obtain ⟨s, t, u, i, j, k, has, hst, htu, hub, hpre, hmid, hlow, hhigh⟩ :=
+      hcert fR hfR (σ ω) b hσb hfRO
+        (by simpa only [fR, Real.toNNReal_coe] using hstart)
+        (by simpa only [fR, Real.toNNReal_coe] using hend)
+    have hs : 0 < s := by exact_mod_cast ((σ ω).coe_nonneg.trans_lt has)
+    have hst' : s < t := by exact_mod_cast hst
+    have htu' : t < u := by exact_mod_cast htu
+    have ht : 0 ≤ t := hs.le.trans hst'.le
+    have hu : 0 ≤ u := ht.trans htu'.le
+    let idx : PassageIndex := ⟨⟨(s, t, u), hs, hst', htu'⟩, (i, j, k)⟩
+    apply mem_iUnion.mpr
+    refine ⟨idx, ?_⟩
+    change σ ω < ratTime s ∧ f ω (ratTime u) ∈ beyondFrontRegion O progress ∧
+      (σ ω ≤ ratTime u ∧ MapsTo (f ω) (Icc (σ ω) (ratTime u)) (K i)) ∧
+      MapsTo (f ω) (Icc (ratTime s) (ratTime t)) (M j) ∧
+      (σ ω ≤ ratTime t ∧ MapsTo (f ω) (Icc (σ ω) (ratTime t)) (L k))
+    have has' : σ ω < ratTime s := by
+      apply NNReal.coe_lt_coe.mp
+      simpa only [ratTime_coe hs.le] using has
+    have lift (a : ℝ≥0) (q : ℚ) (hq : 0 ≤ q) (A : Set ℂ)
+        (hA : fR '' Icc (a : ℝ) (q : ℝ) ⊆ A) : MapsTo (f ω) (Icc a (ratTime q)) A := by
+      intro v hv
+      apply hA
+      refine ⟨(v : ℝ), ?_, ?_⟩
+      · constructor
+        · exact_mod_cast hv.1
+        · have hh : (v : ℝ) ≤ (ratTime q : ℝ) := hv.2
+          simpa only [ratTime_coe hq] using hh
+      · simp only [fR, Real.toNNReal_coe]
+    refine ⟨has', hhigh, ⟨has'.le.trans ((ratTime_lt hs.le hst').le.trans
+      (ratTime_lt ht htu').le), lift _ _ hu _ hpre⟩, ?_,
+      ⟨has'.le.trans (ratTime_lt hs.le hst').le, lift _ _ ht _ hlow⟩⟩
+    apply lift _ _ ht
+    simpa only [ratTime_coe hs.le] using hmid
+
+/-- The actual future tube event supplies the progress-crossing hypothesis;
+no compact or interval certificate is assumed. -/
+theorem stopped_tube_has_countable_passage_cover
+    {Ω : Type*} [MeasurableSpace Ω]
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (σ : Ω → ℝ≥0) (hσ : Measurable σ)
+    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
+    (progress : ℂ → ℝ) (hp : ContinuousOn progress O)
+    (hW : W ⊆ {z | 1 < progress z}) (E : Set Ω)
+    (hE : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
+    (hstart : ∀ ω ∈ E, progress (f ω (σ ω)) < 1 / 3) :
+    ∃ K M L : ℕ → Set ℂ,
+      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
+      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
+      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
+      (∀ n, K n ⊆ upperHalfPlane) ∧ (∀ n, M n ⊆ upperHalfPlane) ∧
+      (∀ i, MeasurableSet (compactPassageCertificate f σ K M L
+        (beyondFrontRegion O progress) i)) ∧
+      E ⊆ ⋃ i : PassageIndex, compactPassageCertificate f σ K M L
+        (beyondFrontRegion O progress) i := by
+  apply countable_measurable_passage_cover f hf σ hσ O hO hOH progress hp E
+  intro ω hω
+  obtain ⟨T, hT, hpath, hend⟩ := hE hω
+  refine ⟨σ ω + T, lt_add_of_pos_right _ hT, ?_, hstart ω hω, hW hend⟩
+  intro t ht
+  have hdiff : t - σ ω ∈ Icc (0 : ℝ≥0) T := by
+    exact ⟨bot_le, tsub_le_iff_right.mpr (by simpa only [add_comm] using ht.2)⟩
+  simpa only [add_tsub_cancel_of_le ht.1] using hpath hdiff
+
+abbrev RetainedPassageIndex := PassageIndex × ℕ
+
+def retainedPassageCertificate {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
+    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (i : RetainedPassageIndex) : Set Ω :=
+  compactPassageCertificate f σ K M L W i.1 ∩
+    {ω | f ω (passageEnd i.1) ∈ N i.2}
+
+/-- Actual tube passages admit a countable measurable cover with compact
+endpoint margins as well as compact prefix and middle margins. -/
+theorem stopped_tube_retained_cover
+    {Ω : Type*} [MeasurableSpace Ω]
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (σ : Ω → ℝ≥0) (hσ : Measurable σ)
+    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
+    (progress : ℂ → ℝ) (hp : ContinuousOn progress O)
+    (hW : W ⊆ {z | 1 < progress z}) (E : Set Ω)
+    (hE : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
+    (hstart : ∀ ω ∈ E, progress (f ω (σ ω)) < 1 / 3) :
+    ∃ K M L N : ℕ → Set ℂ,
+      (∀ n, IsCompact (K n) ∧ K n ⊆ O) ∧
+      (∀ n, IsCompact (M n) ∧ M n ⊆ middleRegion O progress) ∧
+      (∀ n, IsCompact (L n) ∧ L n ⊆ beforeFrontRegion O progress) ∧
+      (∀ n, IsCompact (N n) ∧ N n ⊆ beyondFrontRegion O progress) ∧
+      (∀ i, MeasurableSet (retainedPassageCertificate f σ K M L N
+        (beyondFrontRegion O progress) i)) ∧
+      E ⊆ ⋃ i : RetainedPassageIndex, retainedPassageCertificate f σ K M L N
+        (beyondFrontRegion O progress) i := by
+  obtain ⟨K, M, L, hK, hM, hL, _, _, hDm, hcover⟩ :=
+    stopped_tube_has_countable_passage_cover f hf σ hσ O W hO hOH progress hp hW E hE hstart
+  obtain ⟨N, hN, hNO, hNcap⟩ := exists_countable_compact_exhaustion_open
+    (beyondFrontRegion O progress) (hp.isOpen_inter_preimage hO isOpen_Ioi)
+  refine ⟨K, M, L, N, hK, hM, hL, fun n => ⟨hN n, hNO n⟩, ?_, ?_⟩
+  · intro i
+    exact (hDm i.1).inter ((hN i.2).measurableSet.preimage
+      ((continuous_eval_const (passageEnd i.1)).measurable.comp hf))
+  · intro ω hω
+    obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hω)
+    have hend : f ω (passageEnd i) ∈ beyondFrontRegion O progress := hi.2.1
+    obtain ⟨j, hj⟩ := hNcap {f ω (passageEnd i)} isCompact_singleton
+      (singleton_subset_iff.mpr hend)
+    exact mem_iUnion.mpr ⟨(i, j), hi, hj (mem_singleton _)⟩
+
+/-- On each retained certificate all strict passage conditions transfer
+uniformly to sufficiently late locally convergent maps. -/
+theorem retained_certificate_transfers
+    {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
+    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (i : RetainedPassageIndex)
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (hF : ContinuousOn F upperHalfPlane)
+    (hK : IsCompact (K i.1.2.1)) (hKH : K i.1.2.1 ⊆ upperHalfPlane)
+    (hM : IsCompact (M i.1.2.2.1)) (hMH : M i.1.2.2.1 ⊆ upperHalfPlane)
+    (hL : IsCompact (L i.1.2.2.2)) (hLH : L i.1.2.2.2 ⊆ upperHalfPlane)
+    (hN : IsCompact (N i.2)) (hNH : N i.2 ⊆ upperHalfPlane)
+    (R Rmid Rlow Rhigh : Set ℂ)
+    (hR : IsOpen R) (hRmid : IsOpen Rmid) (hRlow : IsOpen Rlow) (hRhigh : IsOpen Rhigh)
+    (hKR : F '' K i.1.2.1 ⊆ R) (hMR : F '' M i.1.2.2.1 ⊆ Rmid)
+    (hLR : F '' L i.1.2.2.2 ⊆ Rlow) (hNR : F '' N i.2 ⊆ Rhigh) :
+    ∀ᶠ n in atTop, ∀ ω ∈ retainedPassageCertificate f σ K M L N W i,
+      MapsTo (fun t => Fₙ n (f ω t)) (Icc (σ ω) (passageEnd i.1)) R ∧
+      MapsTo (fun t => Fₙ n (f ω t)) (Icc (intervalStart i.1) (intervalEnd i.1)) Rmid ∧
+      MapsTo (fun t => Fₙ n (f ω t)) (Icc (σ ω) (intervalEnd i.1)) Rlow ∧
+      Fₙ n (f ω (passageEnd i.1)) ∈ Rhigh := by
+  filter_upwards [eventually_image_compact_subset_open Fₙ F hconv hF _ hK hKH R hR hKR,
+    eventually_image_compact_subset_open Fₙ F hconv hF _ hM hMH Rmid hRmid hMR,
+    eventually_image_compact_subset_open Fₙ F hconv hF _ hL hLH Rlow hRlow hLR,
+    eventually_image_compact_subset_open Fₙ F hconv hF _ hN hNH Rhigh hRhigh hNR]
+    with n hnK hnM hnL hnN
+  rintro ω ⟨⟨_, _, hpre, hmid, hlow⟩, hend⟩
+  exact ⟨fun t ht => hnK (mem_image_of_mem _ (hpre.2 ht)),
+    fun t ht => hnM (mem_image_of_mem _ (hmid ht)),
+    fun t ht => hnL (mem_image_of_mem _ (hlow.2 ht)), hnN (mem_image_of_mem _ hend)⟩
+
+/-- The middle interval lies before every later time which attains the
+front level; in particular it precedes the first such passage stop. -/
+theorem retained_interval_before_front
+    {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
+    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (i : RetainedPassageIndex)
+    (progress : ℂ → ℝ) (hL : ∀ z ∈ L i.1.2.2.2, progress z < 2 / 3)
+    (ω : Ω) (hω : ω ∈ retainedPassageCertificate f σ K M L N W i)
+    (τ : ℝ≥0) (hστ : σ ω ≤ τ) (hτ : 2 / 3 ≤ progress (f ω τ)) :
+    intervalEnd i.1 < τ := by
+  by_contra hn
+  have hlow := hω.1.2.2.2.2.2
+  have hh := hL _ (hlow ⟨hστ, le_of_not_gt hn⟩)
+  exact (not_le_of_gt hh) hτ
+
+instance rationalPassageTimes_nonempty : Nonempty RationalPassageTimes :=
+  ⟨⟨(1, 2, 3), by norm_num⟩⟩
+
+/-- Enumerating the concrete rational/compact indices gives the natural
+indexing expected by finite-retention and unconditional sausage lemmas. -/
+def enumerateRetainedPassage : ℕ → RetainedPassageIndex :=
+  (exists_surjective_nat RetainedPassageIndex).choose
+
+theorem enumerateRetainedPassage_surjective : Function.Surjective enumerateRetainedPassage :=
+  (exists_surjective_nat RetainedPassageIndex).choose_spec
+
+theorem nat_retained_cover
+    {Ω : Type*} (f : Ω → C(ℝ≥0, ℂ)) (σ : Ω → ℝ≥0)
+    (K M L N : ℕ → Set ℂ) (W : Set ℂ) (E : Set Ω)
+    (hcover : E ⊆ ⋃ i : RetainedPassageIndex, retainedPassageCertificate f σ K M L N W i) :
+    E ⊆ ⋃ n : ℕ, retainedPassageCertificate f σ K M L N W (enumerateRetainedPassage n) := by
+  intro ω hω
+  obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hω)
+  obtain ⟨n, rfl⟩ := enumerateRetainedPassage_surjective i
+  exact mem_iUnion.mpr ⟨n, hi⟩
+
+end SLEExactGauge.PassageRetention
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PassageSuccess.lean b/lean/OAI/Probability/SLE/Gauge/PassageSuccess.lean
index aa6f11414bf3d06b24edfdbdff147405985547a1..b49d07f91057e3bd44c41c6389f746bf96d7b5b1 100644
--- a/lean/OAI/Probability/SLE/Gauge/PassageSuccess.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PassageSuccess.lean
@@ -1,233 +1,233 @@
-import OAI.Probability.SLE.Gauge.FirstFrontPassage
-
-namespace OAI
-
-/-! The same retained continuation certifies the whole passage, its terminal
-position, and its sausage mass. No success or stopping-window event is assumed. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.PassageSuccess
-open PassageRetention RetainedTubeSausage FirstFrontPassage
-
-/-- A single finite retention preserves a positive fraction of the varying
-probability and certifies every eventual success property of the retained sets. -/
-theorem eventual_retained_success_probability
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (E : Set Ω) (D : ℕ → Set Ω) (G S : ℕ → Set Ω)
-    (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i))
-    (hcover : P (E \ ⋃ i, D i) = 0) (q : ℝ) (hq : 0 < q)
-    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
-    (hS : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → ω ∈ S n) :
-    ∀ᶠ n in atTop, ENNReal.ofReal (q / 2) ≤ P ((G n ∩ S n) ∩ E) := by
-  obtain ⟨N, hsmall⟩ := finite_retention_of_countable_cover P E D hE hD hcover
-    (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity))
-  have hs : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P,
-      ω ∈ G n ∩ E ∩ D i → ω ∈ S n :=
-    (Finset.eventually_all _).mpr (fun i _ => hS i)
-  filter_upwards [hG, hs] with n hgn hsn
-  let A := ⋃ i ∈ Finset.range N, D i
-  have hsub : G n ∩ E ⊆ (G n ∩ E ∩ A) ∪ (E \ A) := by
-    intro ω hω
-    by_cases hωA : ω ∈ A
-    · exact Or.inl ⟨hω, hωA⟩
-    · exact Or.inr ⟨hω.2, hωA⟩
-  have hb := hgn.trans ((measure_mono hsub).trans (measure_union_le _ _))
-  have hret : ENNReal.ofReal (q / 2) ≤ P (G n ∩ E ∩ A) := by
-    by_contra h
-    have he : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = ENNReal.ofReal q := by
-      rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
-      congr 1
-      ring
-    exact (not_lt_of_ge hb) (he ▸ ENNReal.add_lt_add (lt_of_not_ge h) hsmall)
-  apply hret.trans
-  apply measure_mono_ae
-  filter_upwards [(Finset.eventually_all _).mpr hsn] with ω hω
-  intro hm
-  obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hm.2
-  exact ⟨⟨hm.1.1, hω i hi ⟨hm.1, hDi⟩⟩, hm.1.2⟩
-
-/-- Concrete success of the first front stop, including the full preceding
-path and the required terminal offset. -/
-def FrontSuccess {Ω : Type*}
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (pₙ : ℕ → ℂ → ℝ) (σ : Ω → ℝ≥0)
-    (Vₙ : ℕ → Set ℂ) (aₙ : ℕ → ℂ) (ε : ℝ) (n : ℕ) : Set Ω :=
-  {ω | let τ := frontTime (fun t => pₙ n (η n ω t)) 1
-    firstFrontTime (fun t => pₙ n (η n ω t)) 1 = (τ : WithTop ℝ≥0) ∧
-    pₙ n (η n ω τ) = 1 ∧ σ ω < τ ∧ MapsTo (η n ω) (Icc 0 τ) (Vₙ n) ∧ ‖η n ω τ - aₙ n‖ < ε / 4}
-
-/-- Actual successful passage and sausage mass from the same retained tube. -/
-theorem retained_tube_firstFront_success_mass
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P)
-    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
-    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
-    (hp : ContinuousOn p U) (hpn : ∀ n, ContinuousOn (pₙ n) U)
-    (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
-    (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)})
-    (E : Set Ω) (hE : MeasurableSet E)
-    (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
-    (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3)
-    (V : Set ℂ) (hV : IsOpen V) (hFV : MapsTo F O V)
-    (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n)
-    (aₙ : ℕ → ℂ) (ε : ℝ)
-    (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - aₙ n‖ < ε / 4)
-    (R : Set ℂ) (hR : IsOpen R)
-    (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (Rₙ : ℕ → Set ℂ)
-    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
-    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
-    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
-    (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane →
-      η n ω t = Fₙ n (f ω t))
-    (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
-    (hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
-      MapsTo (η n ω) (Icc 0 (σ ω)) U ∧
-        ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1)
-    (hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
-      MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n)) :
-    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
-      ENNReal.ofReal (q / 8) ≤ P {ω |
-        ω ∈ FrontSuccess η pₙ σ Vₙ aₙ ε n ∧ ENNReal.ofReal a ≤
-        segmentSausageCharge κ (η n ω) (σ ω) (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
-  let progress := fun z => p (F z)
-  have hprogress : ContinuousOn progress O :=
-    hp.comp (hF.continuousOn.mono hOH) hFOU
-  obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ :=
-    stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart
-  let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N
-    (beyondFrontRegion O progress) (enumerateRetainedPassage i)
-  let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1
-  let J : ℕ → ℝ≥0 × ℝ≥0 := fun i =>
-    (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1)
-  have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1
-  have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1
-  have hQR : ∀ i, F '' Q i ⊆ R := by
-    rintro i _ ⟨z, hz, rfl⟩
-    have hh := (hM _).2 hz
-    exact hmiddle z hh.1 hh.2.1 hh.2.2
-  have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _
-  have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _
-  have hnull : P (E \ ⋃ i, D i) = 0 := by
-    rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty]
-  have hretain : ∀ i ω, ω ∈ D i →
-      (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by
-    intro i ω hω
-    exact hω.1.2.2.2.1.image_subset
-  let S := FrontSuccess η pₙ σ Vₙ aₙ ε
-  have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i →
-      (σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
-        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2)) ∧ ω ∈ S n := by
-    intro i
-    let j := enumerateRetainedPassage i
-    have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1
-    have hLU : F '' L j.1.2.2.2 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hL _).2 hz).1
-    have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv
-      (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2)
-    have hKH : K j.1.2.1 ⊆ upperHalfPlane := fun z hz => hOH ((hK _).2 hz)
-    have hKU : F '' K j.1.2.1 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hK _).2 hz)
-    have himage := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
-      (K j.1.2.1) (hK _).1 hKH U hU hKU
-    have hKV : F '' K j.1.2.1 ⊆ V := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFV ((hK _).2 hz)
-    have himageV := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
-      (K j.1.2.1) (hK _).1 hKH V hV hKV
-    have hNH : N j.2 ⊆ upperHalfPlane := fun z hz => hOH ((hN _).2 hz).1
-    have hNU : F '' N j.2 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hN _).2 hz).1
-    have hgt := eventually_front_lt_on_compact_image Fₙ F hF.continuousOn hconv pₙ p U hU
-      hp hpconv (N j.2) (hN _).1 hNH hNU 1 (fun z hz => ((hN _).2 hz).2)
-    filter_upwards [hlt, hgt, himage, himageV, hVVₙ, hearly, hearlyV] with
-      n hnL hnN hnK hnKV hnV hearlyn hearlyVn
-    filter_upwards [hη n, hηc n, hearlyn, hearlyVn] with ω hηω hηcω hearlyω hearlyVω
-    intro hω
-    have hDi : ω ∈ retainedPassageCertificate f σ K M L N
-        (beyondFrontRegion O progress) j := hω.2
-    obtain ⟨⟨hσs, _, hfull, hmid, hlow⟩, hend⟩ := hDi
-    obtain ⟨hearlyU, hearlylt⟩ := hearlyω hω.1
-    have hpathU : MapsTo (η n ω) (Icc 0 (passageEnd j.1)) U := by
-      intro t ht
-      by_cases hts : t ≤ σ ω
-      · exact hearlyU ⟨bot_le, hts⟩
-      · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
-        have hmem := hfull.2 ⟨hst, ht.2⟩
-        rw [hηω t hst (hKH hmem)]
-        exact hnK (mem_image_of_mem (Fₙ n) hmem)
-    have hc : ContinuousOn (fun t => pₙ n (η n ω t)) (Icc 0 (passageEnd j.1)) :=
-      (hpn n).comp hηcω.continuousOn hpathU
-    have hprefix : ∀ t ∈ Icc (σ ω) (intervalEnd j.1), pₙ n (η n ω t) < 1 := by
-      intro t ht
-      have hmem := hlow.2 ht
-      rw [hηω t ht.1 (hLH hmem)]
-      exact (hnL _ hmem).trans (by norm_num)
-    have hendgt : 1 < pₙ n (η n ω (passageEnd j.1)) := by
-      have hmem := hfull.2 ⟨hfull.1, le_rfl⟩
-      rw [hηω _ hfull.1 (hKH hmem)]
-      exact hnN _ hend
-    have hfront := frontTime_between (fun t => pₙ n (η n ω t)) 1
-      (σ ω) (intervalEnd j.1) (passageEnd j.1)
-      (hσs.le.trans (intervalStart_lt_end _).le) (intervalEnd_lt_passageEnd _)
-      hc hearlylt hprefix hendgt
-    refine ⟨⟨hσs.le, hfront.2.1.le, ?_⟩, ?_⟩
-    · intro t ht
-      exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1)
-    · have hpathV : MapsTo (η n ω)
-          (Icc 0 (frontTime (fun t => pₙ n (η n ω t)) 1)) (Vₙ n) := by
-        intro t ht
-        by_cases hts : t ≤ σ ω
-        · exact hearlyVω hω.1 ⟨bot_le, hts⟩
-        · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
-          have hmem := hfull.2 ⟨hst, ht.2.trans hfront.2.2.1.le⟩
-          rw [hηω t hst (hKH hmem)]
-          exact hnV (hnKV (mem_image_of_mem (Fₙ n) hmem))
-      exact ⟨hfront.1, hfront.2.2.2,
-        hσs.le.trans_lt ((intervalStart_lt_end _).trans hfront.2.1), hpathV, hterminal n _ (hpathV ⟨bot_le, le_rfl⟩) hfront.2.2.2⟩
-  have hsuccess : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P,
-      ω ∈ G n ∩ E ∩ D i → ω ∈ S n := by
-    intro i
-    filter_upwards [hwindow i] with n hn
-    filter_upwards [hn] with ω hω
-    exact fun hw => (hω hw).2
-  have hprob := eventual_retained_success_probability P E D G S hE hD hnull q hq hG hsuccess
-  have hwin : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ (G n ∩ S n) ∩ E ∩ D i →
-      σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
-        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by
-    intro i
-    filter_upwards [hwindow i] with n hn
-    filter_upwards [hn] with ω hω
-    exact fun hw => (hω ⟨⟨hw.1.1.1, hw.1.2⟩, hw.2⟩).1
-  obtain ⟨a, ha, hmass⟩ := countable_stopped_mass_ae_window_eventually P κ hκ hκ8
-    (fun ω => f ω) hSLE Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D
-    hE hD hnull hretain v hv hlim η (fun _ => σ)
-    (fun n ω => frontTime (fun t => pₙ n (η n ω t)) 1)
-    Rₙ hregion (fun n => G n ∩ S n) (q / 2) (by positivity) hprob hwin
-  refine ⟨a, ha, ?_⟩
-  filter_upwards [hmass] with n hn
-  have heq : q / 2 / 4 = q / 8 := by ring
-  rw [heq] at hn
-  exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1.2, hω.2⟩))
-
-
-end SLEExactGauge.PassageSuccess
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.FirstFrontPassage
+
+namespace OAI
+
+/-! The same retained continuation certifies the whole passage, its terminal
+position, and its sausage mass. No success or stopping-window event is assumed. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.PassageSuccess
+open PassageRetention RetainedTubeSausage FirstFrontPassage
+
+/-- A single finite retention preserves a positive fraction of the varying
+probability and certifies every eventual success property of the retained sets. -/
+theorem eventual_retained_success_probability
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (E : Set Ω) (D : ℕ → Set Ω) (G S : ℕ → Set Ω)
+    (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i))
+    (hcover : P (E \ ⋃ i, D i) = 0) (q : ℝ) (hq : 0 < q)
+    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
+    (hS : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i → ω ∈ S n) :
+    ∀ᶠ n in atTop, ENNReal.ofReal (q / 2) ≤ P ((G n ∩ S n) ∩ E) := by
+  obtain ⟨N, hsmall⟩ := finite_retention_of_countable_cover P E D hE hD hcover
+    (ENNReal.ofReal (q / 2)) (ENNReal.ofReal_pos.mpr (by positivity))
+  have hs : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ᵐ ω ∂P,
+      ω ∈ G n ∩ E ∩ D i → ω ∈ S n :=
+    (Finset.eventually_all _).mpr (fun i _ => hS i)
+  filter_upwards [hG, hs] with n hgn hsn
+  let A := ⋃ i ∈ Finset.range N, D i
+  have hsub : G n ∩ E ⊆ (G n ∩ E ∩ A) ∪ (E \ A) := by
+    intro ω hω
+    by_cases hωA : ω ∈ A
+    · exact Or.inl ⟨hω, hωA⟩
+    · exact Or.inr ⟨hω.2, hωA⟩
+  have hb := hgn.trans ((measure_mono hsub).trans (measure_union_le _ _))
+  have hret : ENNReal.ofReal (q / 2) ≤ P (G n ∩ E ∩ A) := by
+    by_contra h
+    have he : ENNReal.ofReal (q / 2) + ENNReal.ofReal (q / 2) = ENNReal.ofReal q := by
+      rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
+      congr 1
+      ring
+    exact (not_lt_of_ge hb) (he ▸ ENNReal.add_lt_add (lt_of_not_ge h) hsmall)
+  apply hret.trans
+  apply measure_mono_ae
+  filter_upwards [(Finset.eventually_all _).mpr hsn] with ω hω
+  intro hm
+  obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hm.2
+  exact ⟨⟨hm.1.1, hω i hi ⟨hm.1, hDi⟩⟩, hm.1.2⟩
+
+/-- Concrete success of the first front stop, including the full preceding
+path and the required terminal offset. -/
+def FrontSuccess {Ω : Type*}
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (pₙ : ℕ → ℂ → ℝ) (σ : Ω → ℝ≥0)
+    (Vₙ : ℕ → Set ℂ) (aₙ : ℕ → ℂ) (ε : ℝ) (n : ℕ) : Set Ω :=
+  {ω | let τ := frontTime (fun t => pₙ n (η n ω t)) 1
+    firstFrontTime (fun t => pₙ n (η n ω t)) 1 = (τ : WithTop ℝ≥0) ∧
+    pₙ n (η n ω τ) = 1 ∧ σ ω < τ ∧ MapsTo (η n ω) (Icc 0 τ) (Vₙ n) ∧ ‖η n ω τ - aₙ n‖ < ε / 4}
+
+/-- Actual successful passage and sausage mass from the same retained tube. -/
+theorem retained_tube_firstFront_success_mass
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P)
+    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
+    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
+    (hp : ContinuousOn p U) (hpn : ∀ n, ContinuousOn (pₙ n) U)
+    (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
+    (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)})
+    (E : Set Ω) (hE : MeasurableSet E)
+    (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
+    (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3)
+    (V : Set ℂ) (hV : IsOpen V) (hFV : MapsTo F O V)
+    (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n)
+    (aₙ : ℕ → ℂ) (ε : ℝ)
+    (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - aₙ n‖ < ε / 4)
+    (R : Set ℂ) (hR : IsOpen R)
+    (hmiddle : ∀ z ∈ O, 1 / 3 < p (F z) → p (F z) < 2 / 3 → F z ∈ R)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (Rₙ : ℕ → Set ℂ)
+    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
+    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
+    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
+    (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane →
+      η n ω t = Fₙ n (f ω t))
+    (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
+    (hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
+      MapsTo (η n ω) (Icc 0 (σ ω)) U ∧
+        ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1)
+    (hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
+      MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n)) :
+    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
+      ENNReal.ofReal (q / 8) ≤ P {ω |
+        ω ∈ FrontSuccess η pₙ σ Vₙ aₙ ε n ∧ ENNReal.ofReal a ≤
+        segmentSausageCharge κ (η n ω) (σ ω) (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
+  let progress := fun z => p (F z)
+  have hprogress : ContinuousOn progress O :=
+    hp.comp (hF.continuousOn.mono hOH) hFOU
+  obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ :=
+    stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart
+  let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N
+    (beyondFrontRegion O progress) (enumerateRetainedPassage i)
+  let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1
+  let J : ℕ → ℝ≥0 × ℝ≥0 := fun i =>
+    (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1)
+  have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1
+  have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1
+  have hQR : ∀ i, F '' Q i ⊆ R := by
+    rintro i _ ⟨z, hz, rfl⟩
+    have hh := (hM _).2 hz
+    exact hmiddle z hh.1 hh.2.1 hh.2.2
+  have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _
+  have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _
+  have hnull : P (E \ ⋃ i, D i) = 0 := by
+    rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty]
+  have hretain : ∀ i ω, ω ∈ D i →
+      (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by
+    intro i ω hω
+    exact hω.1.2.2.2.1.image_subset
+  let S := FrontSuccess η pₙ σ Vₙ aₙ ε
+  have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i →
+      (σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
+        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2)) ∧ ω ∈ S n := by
+    intro i
+    let j := enumerateRetainedPassage i
+    have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1
+    have hLU : F '' L j.1.2.2.2 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hL _).2 hz).1
+    have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv
+      (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2)
+    have hKH : K j.1.2.1 ⊆ upperHalfPlane := fun z hz => hOH ((hK _).2 hz)
+    have hKU : F '' K j.1.2.1 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hK _).2 hz)
+    have himage := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
+      (K j.1.2.1) (hK _).1 hKH U hU hKU
+    have hKV : F '' K j.1.2.1 ⊆ V := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFV ((hK _).2 hz)
+    have himageV := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
+      (K j.1.2.1) (hK _).1 hKH V hV hKV
+    have hNH : N j.2 ⊆ upperHalfPlane := fun z hz => hOH ((hN _).2 hz).1
+    have hNU : F '' N j.2 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hN _).2 hz).1
+    have hgt := eventually_front_lt_on_compact_image Fₙ F hF.continuousOn hconv pₙ p U hU
+      hp hpconv (N j.2) (hN _).1 hNH hNU 1 (fun z hz => ((hN _).2 hz).2)
+    filter_upwards [hlt, hgt, himage, himageV, hVVₙ, hearly, hearlyV] with
+      n hnL hnN hnK hnKV hnV hearlyn hearlyVn
+    filter_upwards [hη n, hηc n, hearlyn, hearlyVn] with ω hηω hηcω hearlyω hearlyVω
+    intro hω
+    have hDi : ω ∈ retainedPassageCertificate f σ K M L N
+        (beyondFrontRegion O progress) j := hω.2
+    obtain ⟨⟨hσs, _, hfull, hmid, hlow⟩, hend⟩ := hDi
+    obtain ⟨hearlyU, hearlylt⟩ := hearlyω hω.1
+    have hpathU : MapsTo (η n ω) (Icc 0 (passageEnd j.1)) U := by
+      intro t ht
+      by_cases hts : t ≤ σ ω
+      · exact hearlyU ⟨bot_le, hts⟩
+      · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
+        have hmem := hfull.2 ⟨hst, ht.2⟩
+        rw [hηω t hst (hKH hmem)]
+        exact hnK (mem_image_of_mem (Fₙ n) hmem)
+    have hc : ContinuousOn (fun t => pₙ n (η n ω t)) (Icc 0 (passageEnd j.1)) :=
+      (hpn n).comp hηcω.continuousOn hpathU
+    have hprefix : ∀ t ∈ Icc (σ ω) (intervalEnd j.1), pₙ n (η n ω t) < 1 := by
+      intro t ht
+      have hmem := hlow.2 ht
+      rw [hηω t ht.1 (hLH hmem)]
+      exact (hnL _ hmem).trans (by norm_num)
+    have hendgt : 1 < pₙ n (η n ω (passageEnd j.1)) := by
+      have hmem := hfull.2 ⟨hfull.1, le_rfl⟩
+      rw [hηω _ hfull.1 (hKH hmem)]
+      exact hnN _ hend
+    have hfront := frontTime_between (fun t => pₙ n (η n ω t)) 1
+      (σ ω) (intervalEnd j.1) (passageEnd j.1)
+      (hσs.le.trans (intervalStart_lt_end _).le) (intervalEnd_lt_passageEnd _)
+      hc hearlylt hprefix hendgt
+    refine ⟨⟨hσs.le, hfront.2.1.le, ?_⟩, ?_⟩
+    · intro t ht
+      exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1)
+    · have hpathV : MapsTo (η n ω)
+          (Icc 0 (frontTime (fun t => pₙ n (η n ω t)) 1)) (Vₙ n) := by
+        intro t ht
+        by_cases hts : t ≤ σ ω
+        · exact hearlyVω hω.1 ⟨bot_le, hts⟩
+        · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
+          have hmem := hfull.2 ⟨hst, ht.2.trans hfront.2.2.1.le⟩
+          rw [hηω t hst (hKH hmem)]
+          exact hnV (hnKV (mem_image_of_mem (Fₙ n) hmem))
+      exact ⟨hfront.1, hfront.2.2.2,
+        hσs.le.trans_lt ((intervalStart_lt_end _).trans hfront.2.1), hpathV, hterminal n _ (hpathV ⟨bot_le, le_rfl⟩) hfront.2.2.2⟩
+  have hsuccess : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P,
+      ω ∈ G n ∩ E ∩ D i → ω ∈ S n := by
+    intro i
+    filter_upwards [hwindow i] with n hn
+    filter_upwards [hn] with ω hω
+    exact fun hw => (hω hw).2
+  have hprob := eventual_retained_success_probability P E D G S hE hD hnull q hq hG hsuccess
+  have hwin : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ (G n ∩ S n) ∩ E ∩ D i →
+      σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
+        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by
+    intro i
+    filter_upwards [hwindow i] with n hn
+    filter_upwards [hn] with ω hω
+    exact fun hw => (hω ⟨⟨hw.1.1.1, hw.1.2⟩, hw.2⟩).1
+  obtain ⟨a, ha, hmass⟩ := countable_stopped_mass_ae_window_eventually P κ hκ hκ8
+    (fun ω => f ω) hSLE Fₙ F hFn hinj hF hne hconv Q hQ hQH R hR hQR J hJ E D
+    hE hD hnull hretain v hv hlim η (fun _ => σ)
+    (fun n ω => frontTime (fun t => pₙ n (η n ω t)) 1)
+    Rₙ hregion (fun n => G n ∩ S n) (q / 2) (by positivity) hprob hwin
+  refine ⟨a, ha, ?_⟩
+  filter_upwards [hmass] with n hn
+  have heq : q / 2 / 4 = q / 8 := by ring
+  rw [heq] at hn
+  exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1.2, hω.2⟩))
+
+
+end SLEExactGauge.PassageSuccess
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PassageSuccessRegions.lean b/lean/OAI/Probability/SLE/Gauge/PassageSuccessRegions.lean
index 469999441ae5a3db12f88518726a92dc035ba76e..5a9276c63d12df85fca159d5d133db9c78dd79fe 100644
--- a/lean/OAI/Probability/SLE/Gauge/PassageSuccessRegions.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PassageSuccessRegions.lean
@@ -1,189 +1,189 @@
-import OAI.Probability.SLE.Gauge.PassageSuccess
-import OAI.Probability.SLE.Gauge.RetainedRegionMargins
-
-namespace OAI
-
-/-! The actual front-success theorem with moving middle regions. The
-physical neighborhoods are selected only after finitely many compact
-middle certificates have been retained. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.PassageSuccess
-open PassageRetention RetainedTubeSausage FirstFrontPassage
-
-/-- Full front success and mass in the moving middle strip. Local uniform
-coordinate convergence supplies its margin on every retained compact;
-there is no inclusion assumption on the whole limiting open strip. -/
-theorem retained_tube_firstFront_success_mass_middle_regions
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
-    (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P)
-    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
-    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
-    (hp : ContinuousOn p U) (hpn : ∀ᶠ n in atTop, ContinuousOn (pₙ n) U)
-    (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
-    (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)})
-    (E : Set Ω) (hE : MeasurableSet E)
-    (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
-    (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3)
-    (V : Set ℂ) (hV : IsOpen V) (hFV : MapsTo F O V)
-    (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n)
-    (aₙ : ℕ → ℂ) (ε : ℝ)
-    (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - aₙ n‖ < ε / 4)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (Rₙ : ℕ → Set ℂ)
-    (hregion : ∀ᶠ n in atTop, U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} ⊆ Rₙ n)
-    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
-    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
-    (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane →
-      η n ω t = Fₙ n (f ω t))
-    (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
-    (hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
-      MapsTo (η n ω) (Icc 0 (σ ω)) U ∧
-        ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1)
-    (hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
-      MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n)) :
-    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
-      ENNReal.ofReal (q / 8) ≤ P {ω |
-        ω ∈ FrontSuccess η pₙ σ Vₙ aₙ ε n ∧ ENNReal.ofReal a ≤
-        segmentSausageCharge κ (η n ω) (σ ω) (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
-  let progress := fun z => p (F z)
-  have hprogress : ContinuousOn progress O :=
-    hp.comp (hF.continuousOn.mono hOH) hFOU
-  obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ :=
-    stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart
-  let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N
-    (beyondFrontRegion O progress) (enumerateRetainedPassage i)
-  let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1
-  let J : ℕ → ℝ≥0 × ℝ≥0 := fun i =>
-    (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1)
-  have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1
-  have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1
-  have hQR : ∀ i, F '' Q i ⊆ U ∩ {z | (1 / 3 : ℝ) < p z ∧ p z < 2 / 3} := by
-    rintro i _ ⟨z, hz, rfl⟩
-    have hh := (hM _).2 hz
-    exact ⟨hFOU hh.1, hh.2⟩
-  have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _
-  have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _
-  have hnull : P (E \ ⋃ i, D i) = 0 := by
-    rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty]
-  have hretain : ∀ i ω, ω ∈ D i →
-      (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by
-    intro i ω hω
-    exact hω.1.2.2.2.1.image_subset
-  let S := FrontSuccess η pₙ σ Vₙ aₙ ε
-  have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i →
-      (σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
-        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2)) ∧ ω ∈ S n := by
-    intro i
-    let j := enumerateRetainedPassage i
-    have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1
-    have hLU : F '' L j.1.2.2.2 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hL _).2 hz).1
-    have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv
-      (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2)
-    have hKH : K j.1.2.1 ⊆ upperHalfPlane := fun z hz => hOH ((hK _).2 hz)
-    have hKU : F '' K j.1.2.1 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hK _).2 hz)
-    have himage := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
-      (K j.1.2.1) (hK _).1 hKH U hU hKU
-    have hKV : F '' K j.1.2.1 ⊆ V := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFV ((hK _).2 hz)
-    have himageV := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
-      (K j.1.2.1) (hK _).1 hKH V hV hKV
-    have hNH : N j.2 ⊆ upperHalfPlane := fun z hz => hOH ((hN _).2 hz).1
-    have hNU : F '' N j.2 ⊆ U := by
-      rintro _ ⟨z, hz, rfl⟩
-      exact hFOU ((hN _).2 hz).1
-    have hgt := eventually_front_lt_on_compact_image Fₙ F hF.continuousOn hconv pₙ p U hU
-      hp hpconv (N j.2) (hN _).1 hNH hNU 1 (fun z hz => ((hN _).2 hz).2)
-    filter_upwards [hlt, hgt, himage, himageV, hVVₙ, hearly, hearlyV, hpn] with
-      n hnL hnN hnK hnKV hnV hearlyn hearlyVn hpnn
-    filter_upwards [hη n, hηc n, hearlyn, hearlyVn] with ω hηω hηcω hearlyω hearlyVω
-    intro hω
-    have hDi : ω ∈ retainedPassageCertificate f σ K M L N
-        (beyondFrontRegion O progress) j := hω.2
-    obtain ⟨⟨hσs, _, hfull, hmid, hlow⟩, hend⟩ := hDi
-    obtain ⟨hearlyU, hearlylt⟩ := hearlyω hω.1
-    have hpathU : MapsTo (η n ω) (Icc 0 (passageEnd j.1)) U := by
-      intro t ht
-      by_cases hts : t ≤ σ ω
-      · exact hearlyU ⟨bot_le, hts⟩
-      · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
-        have hmem := hfull.2 ⟨hst, ht.2⟩
-        rw [hηω t hst (hKH hmem)]
-        exact hnK (mem_image_of_mem (Fₙ n) hmem)
-    have hc : ContinuousOn (fun t => pₙ n (η n ω t)) (Icc 0 (passageEnd j.1)) :=
-      hpnn.comp hηcω.continuousOn hpathU
-    have hprefix : ∀ t ∈ Icc (σ ω) (intervalEnd j.1), pₙ n (η n ω t) < 1 := by
-      intro t ht
-      have hmem := hlow.2 ht
-      rw [hηω t ht.1 (hLH hmem)]
-      exact (hnL _ hmem).trans (by norm_num)
-    have hendgt : 1 < pₙ n (η n ω (passageEnd j.1)) := by
-      have hmem := hfull.2 ⟨hfull.1, le_rfl⟩
-      rw [hηω _ hfull.1 (hKH hmem)]
-      exact hnN _ hend
-    have hfront := frontTime_between (fun t => pₙ n (η n ω t)) 1
-      (σ ω) (intervalEnd j.1) (passageEnd j.1)
-      (hσs.le.trans (intervalStart_lt_end _).le) (intervalEnd_lt_passageEnd _)
-      hc hearlylt hprefix hendgt
-    refine ⟨⟨hσs.le, hfront.2.1.le, ?_⟩, ?_⟩
-    · intro t ht
-      exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1)
-    · have hpathV : MapsTo (η n ω)
-          (Icc 0 (frontTime (fun t => pₙ n (η n ω t)) 1)) (Vₙ n) := by
-        intro t ht
-        by_cases hts : t ≤ σ ω
-        · exact hearlyVω hω.1 ⟨bot_le, hts⟩
-        · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
-          have hmem := hfull.2 ⟨hst, ht.2.trans hfront.2.2.1.le⟩
-          rw [hηω t hst (hKH hmem)]
-          exact hnV (hnKV (mem_image_of_mem (Fₙ n) hmem))
-      exact ⟨hfront.1, hfront.2.2.2,
-        hσs.le.trans_lt ((intervalStart_lt_end _).trans hfront.2.1), hpathV, hterminal n _ (hpathV ⟨bot_le, le_rfl⟩) hfront.2.2.2⟩
-  have hsuccess : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P,
-      ω ∈ G n ∩ E ∩ D i → ω ∈ S n := by
-    intro i
-    filter_upwards [hwindow i] with n hn
-    filter_upwards [hn] with ω hω
-    exact fun hw => (hω hw).2
-  have hprob := eventual_retained_success_probability P E D G S hE hD hnull q hq hG hsuccess
-  have hwin : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ (G n ∩ S n) ∩ E ∩ D i →
-      σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
-        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by
-    intro i
-    filter_upwards [hwindow i] with n hn
-    filter_upwards [hn] with ω hω
-    exact fun hw => (hω ⟨⟨hw.1.1.1, hw.1.2⟩, hw.2⟩).1
-  obtain ⟨a, ha, hmass⟩ := countable_stopped_mass_ae_window_middle_regions P κ hκ hκ8
-    (fun ω => f ω) hSLE Fₙ F hFn hinj hF hne hconv Q hQ hQH pₙ p U hU hp hpconv hQR J hJ E D
-    hE hD hnull hretain v hv hlim η (fun _ => σ)
-    (fun n ω => frontTime (fun t => pₙ n (η n ω t)) 1)
-    Rₙ hregion (fun n => G n ∩ S n) (q / 2) (by positivity) hprob hwin
-  refine ⟨a, ha, ?_⟩
-  filter_upwards [hmass] with n hn
-  have heq : q / 2 / 4 = q / 8 := by ring
-  rw [heq] at hn
-  exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1.2, hω.2⟩))
-
-
-end SLEExactGauge.PassageSuccess
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PassageSuccess
+import OAI.Probability.SLE.Gauge.RetainedRegionMargins
+
+namespace OAI
+
+/-! The actual front-success theorem with moving middle regions. The
+physical neighborhoods are selected only after finitely many compact
+middle certificates have been retained. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.PassageSuccess
+open PassageRetention RetainedTubeSausage FirstFrontPassage
+
+/-- Full front success and mass in the moving middle strip. Local uniform
+coordinate convergence supplies its margin on every retained compact;
+there is no inclusion assumption on the whole limiting open strip. -/
+theorem retained_tube_firstFront_success_mass_middle_regions
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f)
+    (hSLE : IsOrdinaryChordalSLE κ (fun ω => f ω) P)
+    (σ : Ω → ℝ≥0) (hσ : Measurable σ)
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (O W : Set ℂ) (hO : IsOpen O) (hOH : O ⊆ upperHalfPlane)
+    (pₙ : ℕ → ℂ → ℝ) (p : ℂ → ℝ) (U : Set ℂ) (hU : IsOpen U)
+    (hp : ContinuousOn p U) (hpn : ∀ᶠ n in atTop, ContinuousOn (pₙ n) U)
+    (hpconv : TendstoLocallyUniformlyOn pₙ p atTop U)
+    (hFOU : MapsTo F O U) (hW : W ⊆ {z | 1 < p (F z)})
+    (E : Set Ω) (hE : MeasurableSet E)
+    (htube : E ⊆ stoppedPhysicalTube (fun ω => f ω) σ O W)
+    (hstart : ∀ ω ∈ E, p (F (f ω (σ ω))) < 1 / 3)
+    (V : Set ℂ) (hV : IsOpen V) (hFV : MapsTo F O V)
+    (Vₙ : ℕ → Set ℂ) (hVVₙ : ∀ᶠ n in atTop, V ⊆ Vₙ n)
+    (aₙ : ℕ → ℂ) (ε : ℝ)
+    (hterminal : ∀ n z, z ∈ Vₙ n → pₙ n z = 1 → ‖z - aₙ n‖ < ε / 4)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (Rₙ : ℕ → Set ℂ)
+    (hregion : ∀ᶠ n in atTop, U ∩ {z | (1 / 3 : ℝ) < pₙ n z ∧ pₙ n z < 2 / 3} ⊆ Rₙ n)
+    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
+    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n ∩ E))
+    (hη : ∀ n, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, σ ω ≤ t → f ω t ∈ upperHalfPlane →
+      η n ω t = Fₙ n (f ω t))
+    (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
+    (hearly : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
+      MapsTo (η n ω) (Icc 0 (σ ω)) U ∧
+        ∀ t ∈ Icc 0 (σ ω), pₙ n (η n ω t) < 1)
+    (hearlyV : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E →
+      MapsTo (η n ω) (Icc 0 (σ ω)) (Vₙ n)) :
+    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
+      ENNReal.ofReal (q / 8) ≤ P {ω |
+        ω ∈ FrontSuccess η pₙ σ Vₙ aₙ ε n ∧ ENNReal.ofReal a ≤
+        segmentSausageCharge κ (η n ω) (σ ω) (frontTime (fun t => pₙ n (η n ω t)) 1) (v n) (Rₙ n)} := by
+  let progress := fun z => p (F z)
+  have hprogress : ContinuousOn progress O :=
+    hp.comp (hF.continuousOn.mono hOH) hFOU
+  obtain ⟨K, M, L, N, hK, hM, hL, hN, hDm, hcover⟩ :=
+    stopped_tube_retained_cover f hf σ hσ O W hO hOH progress hprogress hW E htube hstart
+  let D : ℕ → Set Ω := fun i => retainedPassageCertificate f σ K M L N
+    (beyondFrontRegion O progress) (enumerateRetainedPassage i)
+  let Q : ℕ → Set ℂ := fun i => M (enumerateRetainedPassage i).1.2.2.1
+  let J : ℕ → ℝ≥0 × ℝ≥0 := fun i =>
+    (intervalStart (enumerateRetainedPassage i).1, intervalEnd (enumerateRetainedPassage i).1)
+  have hQ : ∀ i, IsCompact (Q i) := fun i => (hM _).1
+  have hQH : ∀ i, Q i ⊆ upperHalfPlane := fun i z hz => hOH ((hM _).2 hz).1
+  have hQR : ∀ i, F '' Q i ⊆ U ∩ {z | (1 / 3 : ℝ) < p z ∧ p z < 2 / 3} := by
+    rintro i _ ⟨z, hz, rfl⟩
+    have hh := (hM _).2 hz
+    exact ⟨hFOU hh.1, hh.2⟩
+  have hJ : ∀ i, (J i).1 < (J i).2 := fun i => intervalStart_lt_end _
+  have hD : ∀ i, MeasurableSet (D i) := fun i => hDm _
+  have hnull : P (E \ ⋃ i, D i) = 0 := by
+    rw [sdiff_eq_empty.mpr (nat_retained_cover f σ K M L N _ E hcover), measure_empty]
+  have hretain : ∀ i ω, ω ∈ D i →
+      (fun t => f ω t) '' Icc (J i).1 (J i).2 ⊆ Q i := by
+    intro i ω hω
+    exact hω.1.2.2.2.1.image_subset
+  let S := FrontSuccess η pₙ σ Vₙ aₙ ε
+  have hwindow : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n ∩ E ∩ D i →
+      (σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
+        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2)) ∧ ω ∈ S n := by
+    intro i
+    let j := enumerateRetainedPassage i
+    have hLH : L j.1.2.2.2 ⊆ upperHalfPlane := fun z hz => hOH ((hL _).2 hz).1
+    have hLU : F '' L j.1.2.2.2 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hL _).2 hz).1
+    have hlt := eventually_front_lt_on_compact Fₙ F hF.continuousOn hconv pₙ p U hU hp hpconv
+      (L j.1.2.2.2) (hL _).1 hLH hLU (2 / 3) (fun z hz => ((hL _).2 hz).2)
+    have hKH : K j.1.2.1 ⊆ upperHalfPlane := fun z hz => hOH ((hK _).2 hz)
+    have hKU : F '' K j.1.2.1 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hK _).2 hz)
+    have himage := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
+      (K j.1.2.1) (hK _).1 hKH U hU hKU
+    have hKV : F '' K j.1.2.1 ⊆ V := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFV ((hK _).2 hz)
+    have himageV := eventually_image_compact_subset_open Fₙ F hconv hF.continuousOn
+      (K j.1.2.1) (hK _).1 hKH V hV hKV
+    have hNH : N j.2 ⊆ upperHalfPlane := fun z hz => hOH ((hN _).2 hz).1
+    have hNU : F '' N j.2 ⊆ U := by
+      rintro _ ⟨z, hz, rfl⟩
+      exact hFOU ((hN _).2 hz).1
+    have hgt := eventually_front_lt_on_compact_image Fₙ F hF.continuousOn hconv pₙ p U hU
+      hp hpconv (N j.2) (hN _).1 hNH hNU 1 (fun z hz => ((hN _).2 hz).2)
+    filter_upwards [hlt, hgt, himage, himageV, hVVₙ, hearly, hearlyV, hpn] with
+      n hnL hnN hnK hnKV hnV hearlyn hearlyVn hpnn
+    filter_upwards [hη n, hηc n, hearlyn, hearlyVn] with ω hηω hηcω hearlyω hearlyVω
+    intro hω
+    have hDi : ω ∈ retainedPassageCertificate f σ K M L N
+        (beyondFrontRegion O progress) j := hω.2
+    obtain ⟨⟨hσs, _, hfull, hmid, hlow⟩, hend⟩ := hDi
+    obtain ⟨hearlyU, hearlylt⟩ := hearlyω hω.1
+    have hpathU : MapsTo (η n ω) (Icc 0 (passageEnd j.1)) U := by
+      intro t ht
+      by_cases hts : t ≤ σ ω
+      · exact hearlyU ⟨bot_le, hts⟩
+      · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
+        have hmem := hfull.2 ⟨hst, ht.2⟩
+        rw [hηω t hst (hKH hmem)]
+        exact hnK (mem_image_of_mem (Fₙ n) hmem)
+    have hc : ContinuousOn (fun t => pₙ n (η n ω t)) (Icc 0 (passageEnd j.1)) :=
+      hpnn.comp hηcω.continuousOn hpathU
+    have hprefix : ∀ t ∈ Icc (σ ω) (intervalEnd j.1), pₙ n (η n ω t) < 1 := by
+      intro t ht
+      have hmem := hlow.2 ht
+      rw [hηω t ht.1 (hLH hmem)]
+      exact (hnL _ hmem).trans (by norm_num)
+    have hendgt : 1 < pₙ n (η n ω (passageEnd j.1)) := by
+      have hmem := hfull.2 ⟨hfull.1, le_rfl⟩
+      rw [hηω _ hfull.1 (hKH hmem)]
+      exact hnN _ hend
+    have hfront := frontTime_between (fun t => pₙ n (η n ω t)) 1
+      (σ ω) (intervalEnd j.1) (passageEnd j.1)
+      (hσs.le.trans (intervalStart_lt_end _).le) (intervalEnd_lt_passageEnd _)
+      hc hearlylt hprefix hendgt
+    refine ⟨⟨hσs.le, hfront.2.1.le, ?_⟩, ?_⟩
+    · intro t ht
+      exact hηω t (hσs.le.trans ht.1) (hOH ((hM _).2 (hmid ht)).1)
+    · have hpathV : MapsTo (η n ω)
+          (Icc 0 (frontTime (fun t => pₙ n (η n ω t)) 1)) (Vₙ n) := by
+        intro t ht
+        by_cases hts : t ≤ σ ω
+        · exact hearlyVω hω.1 ⟨bot_le, hts⟩
+        · have hst : σ ω ≤ t := (lt_of_not_ge hts).le
+          have hmem := hfull.2 ⟨hst, ht.2.trans hfront.2.2.1.le⟩
+          rw [hηω t hst (hKH hmem)]
+          exact hnV (hnKV (mem_image_of_mem (Fₙ n) hmem))
+      exact ⟨hfront.1, hfront.2.2.2,
+        hσs.le.trans_lt ((intervalStart_lt_end _).trans hfront.2.1), hpathV, hterminal n _ (hpathV ⟨bot_le, le_rfl⟩) hfront.2.2.2⟩
+  have hsuccess : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P,
+      ω ∈ G n ∩ E ∩ D i → ω ∈ S n := by
+    intro i
+    filter_upwards [hwindow i] with n hn
+    filter_upwards [hn] with ω hω
+    exact fun hw => (hω hw).2
+  have hprob := eventual_retained_success_probability P E D G S hE hD hnull q hq hG hsuccess
+  have hwin : ∀ i, ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ (G n ∩ S n) ∩ E ∩ D i →
+      σ ω ≤ (J i).1 ∧ (J i).2 ≤ frontTime (fun t => pₙ n (η n ω t)) 1 ∧
+        EqOn (η n ω) (fun u => Fₙ n (f ω u)) (Icc (J i).1 (J i).2) := by
+    intro i
+    filter_upwards [hwindow i] with n hn
+    filter_upwards [hn] with ω hω
+    exact fun hw => (hω ⟨⟨hw.1.1.1, hw.1.2⟩, hw.2⟩).1
+  obtain ⟨a, ha, hmass⟩ := countable_stopped_mass_ae_window_middle_regions P κ hκ hκ8
+    (fun ω => f ω) hSLE Fₙ F hFn hinj hF hne hconv Q hQ hQH pₙ p U hU hp hpconv hQR J hJ E D
+    hE hD hnull hretain v hv hlim η (fun _ => σ)
+    (fun n ω => frontTime (fun t => pₙ n (η n ω t)) 1)
+    Rₙ hregion (fun n => G n ∩ S n) (q / 2) (by positivity) hprob hwin
+  refine ⟨a, ha, ?_⟩
+  filter_upwards [hmass] with n hn
+  have heq : q / 2 / 4 = q / 8 := by ring
+  rw [heq] at hn
+  exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1.2, hω.2⟩))
+
+
+end SLEExactGauge.PassageSuccess
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedAdaptedBatch.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedAdaptedBatch.lean
index 27c76f2f34995e1955b58341950cf39133610db3..c4a08a81df04e87eb755fefc6bae36e7d88cdec4 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedAdaptedBatch.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedAdaptedBatch.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Gauge.PrescribedTestPast
-import OAI.Probability.SLE.Gauge.PrescribedCanonicalTests
-import OAI.Probability.SLE.Gauge.CanonicalBatchFailure
-import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular
-import OAI.Probability.SLE.Mass.DenseCoverMassTests
-
-namespace OAI
-
-/-! The exact chronological batch is adapted to its preceding endpoints.
-  The intervening gap remains available for finite-centered angle mixing;
-  the half-probability bound is conditioned before that gap. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (M : CanonicalMass P κ B hBm γ G)
-
-theorem prescribed_adapted_batch_failure
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    {n J : ℕ} (hn : 3≤n) (e : Fin J → ℤ×ℕ) (hi : Function.Injective e)
-    (he : ∀ kj, kj∈range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
-    (horder : StrictMono (fun i => Target.entryPosition κ W (n,some (e i))))
-    (C c A : ℝ) (z : ℂ) (hz : z∈upperHalfPlane)
-    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) A z)
-    (V : ℝ) (hV : 0 ≤ V)
-    (hstep : ∀ (i : Fin J) (E : Set Ω),
-      MeasurableSet[(actual_radiusLevelTime_stopping hB
-        (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
-        κ hκ hκ8 γ G F hw z hz (Target.prescribedTestPastLevel κ W n e A z i.val)
-        (Target.prescribedTestPastLevel_nonneg hκ W n e A z hl i.val)).measurableSpace] E →
-      ENNReal.ofReal
-        ((centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
-          {p | MomentBatches.dyadic (e i).1 ≤
-            M.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}/2)*
-        finiteCenteredLaw P κ B γ G z V E ≤
-      finiteCenteredLaw P κ B γ G z V (E∩M.prescribedBallTest
-        (fun ω => radiusLevelTime (γ ω) (G ω) z
-          (Target.entryPhysicalLevel κ W (n,some (e i)) A z+2*Target.a κ*(Target.batchLevel κ W n:ℝ)))
-        z (C*Target.entryRadius κ W (n,some (e i)))
-        (ENNReal.ofReal (c*Target.entryValue (n,some (e i))*
-          (Target.entryRadius κ W (n,some (e i)))^Target.d κ)))) :
-    finiteCenteredLaw P κ B γ G z V
-      (M.prescribedTestFailures W n e C c z
-        (fun i ω => radiusLevelTime (γ ω) (G ω) z
-          (Target.entryPhysicalLevel κ W (n,some (e i)) A z+2*Target.a κ*(Target.batchLevel κ W n:ℝ)))) ≤
-      ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
-  classical
-  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let u := Target.prescribedTestPastLevel κ W n e A z
-  have hu (j : ℕ) : 0 ≤ u j :=
-    Target.prescribedTestPastLevel_nonneg hκ W n e A z hl j
-  let S (j : ℕ) (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z (u j)
-  have hS (j : ℕ) : IsStoppingTime ℱ (S j) :=
-    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz (u j) (hu j)
-  have hm1 := M.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1
-  have hJn := M.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
-  have hei (i) : (e i).2 < Target.multiplicity κ W n (e i).1 := (he _).mp ⟨i,rfl⟩
-  have hum : Monotone u := Target.prescribedTestPastLevel_monotone hκ W (by omega) hm1 hJn e hei horder A z hl
-  have hSm (ω) : Monotone (fun j => S j ω) := by
-    intro i j hij
-    exact radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z (hu i) (hum hij)
-  let ℱc := clockStopFiltration ℱ S hS hSm
-  let T (i : Fin J) (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z
-    (Target.entryPhysicalLevel κ W (n,some (e i)) A z+2*Target.a κ*(Target.batchLevel κ W n:ℝ))
-  have hTlevel (i : Fin J) : 0 ≤ Target.entryPhysicalLevel κ W (n,some (e i)) A z+
-      2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by
-    have ha := Target.internal_a_pos hκ
-    exact add_nonneg (hl _) (by positivity)
-  have hT (i : Fin J) : IsStoppingTime ℱ (T i) :=
-    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz _ (hTlevel i)
-  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw) z hz V hV
-  apply M.ordered_batch_failure_bound hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
-    (finiteCenteredLaw P κ B γ G z V) ℱc hn e hi he
-    (fun i => M.prescribedBallTest (T i) z (C*Target.entryRadius κ W (n,some (e i)))
-      (ENNReal.ofReal (c*Target.entryValue (n,some (e i))*
-        (Target.entryRadius κ W (n,some (e i)))^Target.d κ)))
-  · intro i
-    apply (hT i).measurableSpace_mono (hS (i.val+1)) _ _
-      (M.prescribedBallTest_measurable (T i) (hT i) _ _ _)
-    intro ω
-    change radiusLevelTime (γ ω) (G ω) z _ ≤ radiusLevelTime (γ ω) (G ω) z _
-    dsimp only [u]
-    rw [Target.prescribedTestPastLevel_succ W n e A z i]
-  · intro i E hE
-    exact hstep i E hE
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedTestPast
+import OAI.Probability.SLE.Gauge.PrescribedCanonicalTests
+import OAI.Probability.SLE.Gauge.CanonicalBatchFailure
+import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular
+import OAI.Probability.SLE.Mass.DenseCoverMassTests
+
+namespace OAI
+
+/-! The exact chronological batch is adapted to its preceding endpoints.
+  The intervening gap remains available for finite-centered angle mixing;
+  the half-probability bound is conditioned before that gap. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (M : CanonicalMass P κ B hBm γ G)
+
+theorem prescribed_adapted_batch_failure
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (hgrowth : PublishedDiskUnivalentGrowth) (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    {n J : ℕ} (hn : 3≤n) (e : Fin J → ℤ×ℕ) (hi : Function.Injective e)
+    (he : ∀ kj, kj∈range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
+    (horder : StrictMono (fun i => Target.entryPosition κ W (n,some (e i))))
+    (C c A : ℝ) (z : ℂ) (hz : z∈upperHalfPlane)
+    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) A z)
+    (V : ℝ) (hV : 0 ≤ V)
+    (hstep : ∀ (i : Fin J) (E : Set Ω),
+      MeasurableSet[(actual_radiusLevelTime_stopping hB
+        (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
+        κ hκ hκ8 γ G F hw z hz (Target.prescribedTestPastLevel κ W n e A z i.val)
+        (Target.prescribedTestPastLevel_nonneg hκ W n e A z hl i.val)).measurableSpace] E →
+      ENNReal.ofReal
+        ((centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
+          {p | MomentBatches.dyadic (e i).1 ≤
+            M.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}/2)*
+        finiteCenteredLaw P κ B γ G z V E ≤
+      finiteCenteredLaw P κ B γ G z V (E∩M.prescribedBallTest
+        (fun ω => radiusLevelTime (γ ω) (G ω) z
+          (Target.entryPhysicalLevel κ W (n,some (e i)) A z+2*Target.a κ*(Target.batchLevel κ W n:ℝ)))
+        z (C*Target.entryRadius κ W (n,some (e i)))
+        (ENNReal.ofReal (c*Target.entryValue (n,some (e i))*
+          (Target.entryRadius κ W (n,some (e i)))^Target.d κ)))) :
+    finiteCenteredLaw P κ B γ G z V
+      (M.prescribedTestFailures W n e C c z
+        (fun i ω => radiusLevelTime (γ ω) (G ω) z
+          (Target.entryPhysicalLevel κ W (n,some (e i)) A z+2*Target.a κ*(Target.batchLevel κ W n:ℝ)))) ≤
+      ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
+  classical
+  let hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let u := Target.prescribedTestPastLevel κ W n e A z
+  have hu (j : ℕ) : 0 ≤ u j :=
+    Target.prescribedTestPastLevel_nonneg hκ W n e A z hl j
+  let S (j : ℕ) (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z (u j)
+  have hS (j : ℕ) : IsStoppingTime ℱ (S j) :=
+    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz (u j) (hu j)
+  have hm1 := M.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1
+  have hJn := M.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
+  have hei (i) : (e i).2 < Target.multiplicity κ W n (e i).1 := (he _).mp ⟨i,rfl⟩
+  have hum : Monotone u := Target.prescribedTestPastLevel_monotone hκ W (by omega) hm1 hJn e hei horder A z hl
+  have hSm (ω) : Monotone (fun j => S j ω) := by
+    intro i j hij
+    exact radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z (hu i) (hum hij)
+  let ℱc := clockStopFiltration ℱ S hS hSm
+  let T (i : Fin J) (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z
+    (Target.entryPhysicalLevel κ W (n,some (e i)) A z+2*Target.a κ*(Target.batchLevel κ W n:ℝ))
+  have hTlevel (i : Fin J) : 0 ≤ Target.entryPhysicalLevel κ W (n,some (e i)) A z+
+      2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by
+    have ha := Target.internal_a_pos hκ
+    exact add_nonneg (hl _) (by positivity)
+  have hT (i : Fin J) : IsStoppingTime ℱ (T i) :=
+    actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz _ (hTlevel i)
+  let := finiteCenteredLaw_probability hB κ hκ hκ8 γ G F (Eventually.of_forall hw) z hz V hV
+  apply M.ordered_batch_failure_bound hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
+    (finiteCenteredLaw P κ B γ G z V) ℱc hn e hi he
+    (fun i => M.prescribedBallTest (T i) z (C*Target.entryRadius κ W (n,some (e i)))
+      (ENNReal.ofReal (c*Target.entryValue (n,some (e i))*
+        (Target.entryRadius κ W (n,some (e i)))^Target.d κ)))
+  · intro i
+    apply (hT i).measurableSpace_mono (hS (i.val+1)) _ _
+      (M.prescribedBallTest_measurable (T i) (hT i) _ _ _)
+    intro ω
+    change radiusLevelTime (γ ω) (G ω) z _ ≤ radiusLevelTime (γ ω) (G ω) z _
+    dsimp only [u]
+    rw [Target.prescribedTestPastLevel_succ W n e A z i]
+  · intro i E hE
+    exact hstep i E hE
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedAngleMain.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedAngleMain.lean
index f0d7cd689ea8d330aa9166b67cc61ba6a5337bab..e1b21909813d5047e91045898502f403d90b1a9f 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedAngleMain.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedAngleMain.lean
@@ -1,77 +1,77 @@
-import OAI.Probability.SLE.Gauge.PrescribedLowerAngle
-import OAI.Probability.SLE.Gauge.PrescribedUpperAngle
-import OAI.Probability.SLE.Mass.RawScaledMassPaths
-import OAI.Probability.SLE.MainReduction
-
-namespace OAI
-
-/-! A uniform canonical angle estimate implies the exact prescribed-gauge
-Hausdorff conclusion. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance prescribedMainMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance prescribedMainBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-theorem CanonicalMass.prescribed_main_of_canonical_angle_bounds
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hγ : ∀ t, Measurable (fun ω => γ ω t))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (δ : ℝ) (hδ : 0<δ)
-    (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f →
-      ∀ θ∈Ioo (0:ℝ) Real.pi,
-      (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤
-          canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ∧
-      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤
-          2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) :
-    MainConclusion P (Target.gauge κ W) γ := by
-  obtain ⟨R,he,D,hD,hDf,hDp⟩ := C.exists_raw_positive_paths hc
-  have hg := C.prescribed_gauge_admissible hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
-  have hg' : IsGauge (Target.gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
-  have hSLE : IsOrdinaryChordalSLE κ γ P :=
-    ⟨hγ,B,hB,Eventually.of_forall (fun ω => ⟨G ω,F ω,hw ω⟩)⟩
-  have hp := C.positive_prescribed_all_segments_of_canonical_angle_bound hB hc h0 hκ hκ8
-    F hw R he D hD hDf hDp hgrowth hpublished W hW δ hδ
-    (fun v hv f hf θ hθ => (hangle v hv f hf θ hθ).2)
-  have hfinite : FiniteExpectedWholeTrace P (Target.gauge κ W) γ := by
-    intro L hL
-    let K : Set ℂ := closedBall 0 L ∩ {z : ℂ | 0≤z.im}
-    have hK : IsClosed K := isClosed_closedBall.inter
-      (isClosed_le continuous_const Complex.continuous_im)
-    have hb := C.prescribed_expectation_of_canonical_angle_bound hB hc h0 hκ hκ8
-      F hw R he D hD hDf hDp hgrowth hpublished W hW hγ δ hδ
-      (fun v hv f hf θ hθ => (hangle v hv f hf θ hθ).1)
-      L hL.le K hK inter_subset_left (fun _ h => h.2)
-    have heq (ω : Ω) : range (γ ω)∩K=wholeTrace (γ ω)∩closedBall 0 L := by
-      change range (γ ω)∩K=range (γ ω)∩closedBall 0 L
-      ext z
-      constructor
-      · exact fun h => ⟨h.1,h.2.1⟩
-      · rintro ⟨⟨t,rfl⟩,hz⟩
-        exact ⟨mem_range_self t,hz,(hw ω).2.2.1 t⟩
-    simpa only [Target.hausdorff_eq_hausdorffGauge,heq,Measure.toOuterMeasure_apply] using hb
-  apply mainConclusion_of_lower_and_finiteExpectation P (Target.gauge κ W) γ
-    (Eventually.of_forall (fun ω => (hw ω).1)) _
-    (fun _ _ => source_sle_whole_trace_aemeasurable P κ γ hSLE _ hg' _ isClosed_closedBall)
-    hfinite
-  filter_upwards [hp] with ω hω
-  intro s t _hs hst
-  simpa only [Target.hausdorff_eq_hausdorffGauge,Measure.toOuterMeasure_apply,segment]
-    using hω s t hst
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedLowerAngle
+import OAI.Probability.SLE.Gauge.PrescribedUpperAngle
+import OAI.Probability.SLE.Mass.RawScaledMassPaths
+import OAI.Probability.SLE.MainReduction
+
+namespace OAI
+
+/-! A uniform canonical angle estimate implies the exact prescribed-gauge
+Hausdorff conclusion. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance prescribedMainMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance prescribedMainBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+theorem CanonicalMass.prescribed_main_of_canonical_angle_bounds
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hγ : ∀ t, Measurable (fun ω => γ ω t))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (δ : ℝ) (hδ : 0<δ)
+    (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f →
+      ∀ θ∈Ioo (0:ℝ) Real.pi,
+      (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤
+          canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ∧
+      canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤
+          2*(∫⁻ φ, f φ ∂Target.angleLaw κ)) :
+    MainConclusion P (Target.gauge κ W) γ := by
+  obtain ⟨R,he,D,hD,hDf,hDp⟩ := C.exists_raw_positive_paths hc
+  have hg := C.prescribed_gauge_admissible hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
+  have hg' : IsGauge (Target.gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
+  have hSLE : IsOrdinaryChordalSLE κ γ P :=
+    ⟨hγ,B,hB,Eventually.of_forall (fun ω => ⟨G ω,F ω,hw ω⟩)⟩
+  have hp := C.positive_prescribed_all_segments_of_canonical_angle_bound hB hc h0 hκ hκ8
+    F hw R he D hD hDf hDp hgrowth hpublished W hW δ hδ
+    (fun v hv f hf θ hθ => (hangle v hv f hf θ hθ).2)
+  have hfinite : FiniteExpectedWholeTrace P (Target.gauge κ W) γ := by
+    intro L hL
+    let K : Set ℂ := closedBall 0 L ∩ {z : ℂ | 0≤z.im}
+    have hK : IsClosed K := isClosed_closedBall.inter
+      (isClosed_le continuous_const Complex.continuous_im)
+    have hb := C.prescribed_expectation_of_canonical_angle_bound hB hc h0 hκ hκ8
+      F hw R he D hD hDf hDp hgrowth hpublished W hW hγ δ hδ
+      (fun v hv f hf θ hθ => (hangle v hv f hf θ hθ).1)
+      L hL.le K hK inter_subset_left (fun _ h => h.2)
+    have heq (ω : Ω) : range (γ ω)∩K=wholeTrace (γ ω)∩closedBall 0 L := by
+      change range (γ ω)∩K=range (γ ω)∩closedBall 0 L
+      ext z
+      constructor
+      · exact fun h => ⟨h.1,h.2.1⟩
+      · rintro ⟨⟨t,rfl⟩,hz⟩
+        exact ⟨mem_range_self t,hz,(hw ω).2.2.1 t⟩
+    simpa only [Target.hausdorff_eq_hausdorffGauge,heq,Measure.toOuterMeasure_apply] using hb
+  apply mainConclusion_of_lower_and_finiteExpectation P (Target.gauge κ W) γ
+    (Eventually.of_forall (fun ω => (hw ω).1)) _
+    (fun _ _ => source_sle_whole_trace_aemeasurable P κ γ hSLE _ hg' _ isClosed_closedBall)
+    hfinite
+  filter_upwards [hp] with ω hω
+  intro s t _hs hst
+  simpa only [Target.hausdorff_eq_hausdorffGauge,Measure.toOuterMeasure_apply,segment]
+    using hω s t hst
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedApproachTransfer.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedApproachTransfer.lean
index 6f4a279e4d341a9aff003302f2fb211413bc49fe..3a4341949cb3844216f9a1ea2549f64a14d44c48 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedApproachTransfer.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedApproachTransfer.lean
@@ -1,97 +1,97 @@
-import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
-import OAI.Probability.SLE.Gauge.PrescribedVisitBound
-import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
-import OAI.Probability.SLE.Gauge.ActualMissedCell
-
-namespace OAI
-
-/-! Passing an adapted failure event through the final radius stop. The
-  terminal candidate event is used only for set inclusion, outside the
-  finite change of law. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem prescribed_missed_cell_bound_of_adapted_failure
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hBm : Measurable (Function.uncurry B))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (ν : Ω → Measure ℂ) (W : Target.WienerLaw) (n : ℕ)
-    {J : ℕ} (e : Fin J → ℤ×ℕ) (C c R : ℝ) (K : Set ℂ)
-    (ij : ℤ×ℤ) (hij : ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K)
-    (u : ℝ) (hu : 0 ≤ u)
-    (hgap : u+1 ≤ Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
-      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
-    (hsmall : 4*Target.entryRadius κ W (n,none) ≤
-      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im)
-    (A : Set Ω)
-    (hA : MeasurableSet[(actual_radiusLevelTime_stopping hB
-      (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc h0
-      κ hκ hκ8 γ G F hw _ (prescribedGrid_center_height (Real.exp_pos _) hij).2 u hu).measurableSpace] A)
-    (hfailure : ∀ᵐ ω ∂P,
-      ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K) (ν ω) → ω∈A) :
-    P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K) (ν ω)} ≤
-      ENNReal.ofReal (uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ*
-        initialGreen κ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)*
-        (Target.entryRadius κ W (n,none))^codimension κ)*
-      finiteCenteredLaw P κ B γ G (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
-        (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
-          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)) A := by
-  classical
-  let r := Target.entryRadius κ W (n,none)
-  let z := prescribedGridCenter r ij
-  let V := Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z
-  have hr : 0<r := Real.exp_pos _
-  have hz : 0<z.im := (prescribedGrid_center_height hr hij).2
-  have hgapV : u+1 ≤ V := hgap
-  have hV : 0 ≤ V := by linarith
-  let v : ℝ≥0 := ⟨V-u,by linarith⟩
-  have hv : 1 ≤ (v:ℝ) := by change 1 ≤ V-u; linarith
-  have huv : u+(v:ℝ)=V := by change u+(V-u)=V; ring
-  have hrad : (2*z.im)*Real.exp (-V)=(2048*Real.pi)*r :=
-    Target.entryPhysicalLevel_radius κ W _ (by positivity) hz
-  have hbound := actual_conditional_approach_at_radius_stop P κ hκ hκ8 B hB hBm hc h0
-    γ G F hw htrace z hz u hu v hv A hA
-  rw [huv] at hbound
-  have he : uniformInverseAngleConstant κ 1*Real.exp (-codimension κ*(v:ℝ))*
-      ((2*z.im)*Real.exp (-u))^codimension κ*initialGreen κ z =
-      uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ*
-        initialGreen κ z*r^codimension κ := by
-    calc
-      _ = uniformInverseAngleConstant κ 1*
-          (Real.exp (-codimension κ*(v:ℝ))*((2*z.im)*Real.exp (-u))^codimension κ)*
-          initialGreen κ z := by ring
-      _ = uniformInverseAngleConstant κ 1*((2*z.im)*Real.exp (-V))^codimension κ*
-          initialGreen κ z := by
-        rw [radius_gap_power _ u (v:ℝ) _ (by positivity),huv]
-      _ = _ := by
-        rw [hrad,Real.mul_rpow (by positivity) hr.le]
-        ring
-  rw [he] at hbound
-  refine (measure_mono_ae ?_).trans hbound
-  filter_upwards [hfailure] with ω hf hω
-  refine ⟨hf hω,?_⟩
-  obtain ⟨x,⟨⟨t,rfl⟩,_⟩,ht⟩ := (Finset.mem_filter.mp hω).2.1
-  have ht' : dist (γ ω t) z < 4*r := by
-    change dist (γ ω t) z ≤ 3*r at ht
-    linarith
-  obtain ⟨s,_,hsl,hsR⟩ := (hw ω).radius_approach_of_trace_approach
-    ((hc ω).const_mul _) z (4*r) (by positivity) hsmall t ht'
-  apply radiusLevelTime_finite_of_live_radius_lt ((hc ω).const_mul _) (hw ω) z V hV s hsl
-  apply hsR.trans_le
-  rw [hrad]
-  have hp := Real.pi_pos
-  nlinarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
+import OAI.Probability.SLE.Gauge.PrescribedVisitBound
+import OAI.Probability.SLE.Loewner.ActualApproachAtRadius
+import OAI.Probability.SLE.Gauge.ActualMissedCell
+
+namespace OAI
+
+/-! Passing an adapted failure event through the final radius stop. The
+  terminal candidate event is used only for set inclusion, outside the
+  finite change of law. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem prescribed_missed_cell_bound_of_adapted_failure
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hBm : Measurable (Function.uncurry B))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (htrace : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (ν : Ω → Measure ℂ) (W : Target.WienerLaw) (n : ℕ)
+    {J : ℕ} (e : Fin J → ℤ×ℕ) (C c R : ℝ) (K : Set ℂ)
+    (ij : ℤ×ℤ) (hij : ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K)
+    (u : ℝ) (hu : 0 ≤ u)
+    (hgap : u+1 ≤ Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
+      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
+    (hsmall : 4*Target.entryRadius κ W (n,none) ≤
+      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im)
+    (A : Set Ω)
+    (hA : MeasurableSet[(actual_radiusLevelTime_stopping hB
+      (fun _t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hc h0
+      κ hκ hκ8 γ G F hw _ (prescribedGrid_center_height (Real.exp_pos _) hij).2 u hu).measurableSpace] A)
+    (hfailure : ∀ᵐ ω ∂P,
+      ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K) (ν ω) → ω∈A) :
+    P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K) (ν ω)} ≤
+      ENNReal.ofReal (uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ*
+        initialGreen κ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)*
+        (Target.entryRadius κ W (n,none))^codimension κ)*
+      finiteCenteredLaw P κ B γ G (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
+        (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
+          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)) A := by
+  classical
+  let r := Target.entryRadius κ W (n,none)
+  let z := prescribedGridCenter r ij
+  let V := Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z
+  have hr : 0<r := Real.exp_pos _
+  have hz : 0<z.im := (prescribedGrid_center_height hr hij).2
+  have hgapV : u+1 ≤ V := hgap
+  have hV : 0 ≤ V := by linarith
+  let v : ℝ≥0 := ⟨V-u,by linarith⟩
+  have hv : 1 ≤ (v:ℝ) := by change 1 ≤ V-u; linarith
+  have huv : u+(v:ℝ)=V := by change u+(V-u)=V; ring
+  have hrad : (2*z.im)*Real.exp (-V)=(2048*Real.pi)*r :=
+    Target.entryPhysicalLevel_radius κ W _ (by positivity) hz
+  have hbound := actual_conditional_approach_at_radius_stop P κ hκ hκ8 B hB hBm hc h0
+    γ G F hw htrace z hz u hu v hv A hA
+  rw [huv] at hbound
+  have he : uniformInverseAngleConstant κ 1*Real.exp (-codimension κ*(v:ℝ))*
+      ((2*z.im)*Real.exp (-u))^codimension κ*initialGreen κ z =
+      uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ*
+        initialGreen κ z*r^codimension κ := by
+    calc
+      _ = uniformInverseAngleConstant κ 1*
+          (Real.exp (-codimension κ*(v:ℝ))*((2*z.im)*Real.exp (-u))^codimension κ)*
+          initialGreen κ z := by ring
+      _ = uniformInverseAngleConstant κ 1*((2*z.im)*Real.exp (-V))^codimension κ*
+          initialGreen κ z := by
+        rw [radius_gap_power _ u (v:ℝ) _ (by positivity),huv]
+      _ = _ := by
+        rw [hrad,Real.mul_rpow (by positivity) hr.le]
+        ring
+  rw [he] at hbound
+  refine (measure_mono_ae ?_).trans hbound
+  filter_upwards [hfailure] with ω hf hω
+  refine ⟨hf hω,?_⟩
+  obtain ⟨x,⟨⟨t,rfl⟩,_⟩,ht⟩ := (Finset.mem_filter.mp hω).2.1
+  have ht' : dist (γ ω t) z < 4*r := by
+    change dist (γ ω t) z ≤ 3*r at ht
+    linarith
+  obtain ⟨s,_,hsl,hsR⟩ := (hw ω).radius_approach_of_trace_approach
+    ((hc ω).const_mul _) z (4*r) (by positivity) hsmall t ht'
+  apply radiusLevelTime_finite_of_live_radius_lt ((hc ω).const_mul _) (hw ω) z V hV s hsl
+  apply hsR.trans_le
+  rw [hrad]
+  have hp := Real.pi_pos
+  nlinarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedBatchClock.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedBatchClock.lean
index 60ffbd573642c48ec1e9a41b9910798745e77e8e..0724b57717e6f9621cec36c0f8e15ad64bd0d02e 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedBatchClock.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedBatchClock.lean
@@ -1,80 +1,80 @@
-import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
-
-namespace OAI
-
-/-! The chronological repeated entries, followed by their cleanup level,
-  give a nondecreasing discrete clock for the adapted test calculation. -/
-noncomputable section
-open Set
-
-namespace SLEExactGauge.Target
-
-def prescribedBatchLevel {J : ℕ} (κ : ℝ) (W : WienerLaw) (n : ℕ)
-    (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) (i : ℕ) : ℝ :=
-  if h : i<J then entryPhysicalLevel κ W (n,some (e ⟨i,h⟩)) A z
-  else entryPhysicalLevel κ W (n,none) A z
-
-private lemma entryPhysicalLevel_mono_position {κ : ℝ} (hκ : 0<κ)
-    (W : WienerLaw) (n : ℕ) (A : ℝ) (z : ℂ)
-    (o o' : Option (ℤ×ℕ))
-    (hpos : entryPosition κ W (n,o) ≤ entryPosition κ W (n,o')) :
-    entryPhysicalLevel κ W (n,o) A z ≤ entryPhysicalLevel κ W (n,o') A z := by
-  have hp : (entryPosition κ W (n,o):ℝ) ≤ entryPosition κ W (n,o') := by exact_mod_cast hpos
-  have hc : 0 ≤ 2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ)) := by
-    have ha := internal_a_pos hκ
-    positivity
-  have hh := mul_le_mul_of_nonneg_left hp hc
-  dsimp only [entryPhysicalLevel]
-  nlinarith
-
-lemma prescribedBatchLevel_monotone {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
-    {n J : ℕ} (hn : 1≤n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ)
-    (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1)
-    (horder : StrictMono (fun i => entryPosition κ W (n,some (e i))))
-    (A : ℝ) (z : ℂ) : Monotone (prescribedBatchLevel κ W n e A z) := by
-  intro i j hij
-  unfold prescribedBatchLevel
-  by_cases hj : j<J
-  · have hi : i<J := hij.trans_lt hj
-    rw [dite_eq_left hi,dite_eq_left hj]
-    exact entryPhysicalLevel_mono_position hκ W n A z _ _
-      (horder.monotone (show (⟨i,hi⟩:Fin J) ≤ ⟨j,hj⟩ from hij))
-  · rw [dite_eq_right hj]
-    split_ifs with hi
-    · exact entryPhysicalLevel_mono_position hκ W n A z _ _
-        (entryPosition_lt_cleanup κ W hn hm1 hJ (he ⟨i,hi⟩)).le
-    · exact le_rfl
-
-lemma prescribedBatchLevel_test_end_le_next {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
-    {n J : ℕ} (hn : 1≤n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ)
-    (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1)
-    (horder : StrictMono (fun i => entryPosition κ W (n,some (e i))))
-    (A : ℝ) (z : ℂ) (i : Fin J) :
-    entryPhysicalLevel κ W (n,some (e i)) A z+2*a κ*(batchLevel κ W n:ℝ) ≤
-      prescribedBatchLevel κ W n e A z (i.val+1) := by
-  have ha := internal_a_pos hκ
-  have hnon : 0 ≤ 2*a κ*(n:ℝ) := by positivity
-  unfold prescribedBatchLevel
-  split_ifs with hi
-  · have hp := horder (show i < (⟨i.val+1,hi⟩:Fin J) from Nat.lt_succ_self _)
-    have hh := entryPhysicalLevel_gap hκ W hn hp A z
-    linarith
-  · have hh := entryPhysicalLevel_cleanup_gap hκ W hn hm1 hJ (he i) A z
-    linarith
-
-lemma prescribedBatchLevel_test_end_le_cleanup {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
-    {n J : ℕ} (hn : 1≤n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ)
-    (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1)
-    (A : ℝ) (z : ℂ) (i : Fin J) :
-    entryPhysicalLevel κ W (n,some (e i)) A z+2*a κ*(batchLevel κ W n:ℝ)+2*a κ*(n:ℝ) ≤
-      entryPhysicalLevel κ W (n,none) A z :=
-  entryPhysicalLevel_cleanup_gap hκ W hn hm1 hJ (he i) A z
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
+
+namespace OAI
+
+/-! The chronological repeated entries, followed by their cleanup level,
+  give a nondecreasing discrete clock for the adapted test calculation. -/
+noncomputable section
+open Set
+
+namespace SLEExactGauge.Target
+
+def prescribedBatchLevel {J : ℕ} (κ : ℝ) (W : WienerLaw) (n : ℕ)
+    (e : Fin J → ℤ×ℕ) (A : ℝ) (z : ℂ) (i : ℕ) : ℝ :=
+  if h : i<J then entryPhysicalLevel κ W (n,some (e ⟨i,h⟩)) A z
+  else entryPhysicalLevel κ W (n,none) A z
+
+private lemma entryPhysicalLevel_mono_position {κ : ℝ} (hκ : 0<κ)
+    (W : WienerLaw) (n : ℕ) (A : ℝ) (z : ℂ)
+    (o o' : Option (ℤ×ℕ))
+    (hpos : entryPosition κ W (n,o) ≤ entryPosition κ W (n,o')) :
+    entryPhysicalLevel κ W (n,o) A z ≤ entryPhysicalLevel κ W (n,o') A z := by
+  have hp : (entryPosition κ W (n,o):ℝ) ≤ entryPosition κ W (n,o') := by exact_mod_cast hpos
+  have hc : 0 ≤ 2*a κ*((batchLevel κ W n:ℝ)+(n:ℝ)) := by
+    have ha := internal_a_pos hκ
+    positivity
+  have hh := mul_le_mul_of_nonneg_left hp hc
+  dsimp only [entryPhysicalLevel]
+  nlinarith
+
+lemma prescribedBatchLevel_monotone {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
+    {n J : ℕ} (hn : 1≤n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ)
+    (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1)
+    (horder : StrictMono (fun i => entryPosition κ W (n,some (e i))))
+    (A : ℝ) (z : ℂ) : Monotone (prescribedBatchLevel κ W n e A z) := by
+  intro i j hij
+  unfold prescribedBatchLevel
+  by_cases hj : j<J
+  · have hi : i<J := hij.trans_lt hj
+    rw [dite_eq_left hi,dite_eq_left hj]
+    exact entryPhysicalLevel_mono_position hκ W n A z _ _
+      (horder.monotone (show (⟨i,hi⟩:Fin J) ≤ ⟨j,hj⟩ from hij))
+  · rw [dite_eq_right hj]
+    split_ifs with hi
+    · exact entryPhysicalLevel_mono_position hκ W n A z _ _
+        (entryPosition_lt_cleanup κ W hn hm1 hJ (he ⟨i,hi⟩)).le
+    · exact le_rfl
+
+lemma prescribedBatchLevel_test_end_le_next {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
+    {n J : ℕ} (hn : 1≤n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ)
+    (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1)
+    (horder : StrictMono (fun i => entryPosition κ W (n,some (e i))))
+    (A : ℝ) (z : ℂ) (i : Fin J) :
+    entryPhysicalLevel κ W (n,some (e i)) A z+2*a κ*(batchLevel κ W n:ℝ) ≤
+      prescribedBatchLevel κ W n e A z (i.val+1) := by
+  have ha := internal_a_pos hκ
+  have hnon : 0 ≤ 2*a κ*(n:ℝ) := by positivity
+  unfold prescribedBatchLevel
+  split_ifs with hi
+  · have hp := horder (show i < (⟨i.val+1,hi⟩:Fin J) from Nat.lt_succ_self _)
+    have hh := entryPhysicalLevel_gap hκ W hn hp A z
+    linarith
+  · have hh := entryPhysicalLevel_cleanup_gap hκ W hn hm1 hJ (he i) A z
+    linarith
+
+lemma prescribedBatchLevel_test_end_le_cleanup {κ : ℝ} (hκ : 0<κ) (W : WienerLaw)
+    {n J : ℕ} (hn : 1≤n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1^n ≤ moment κ W n) (e : Fin J → ℤ×ℕ)
+    (he : ∀ i, (e i).2 < multiplicity κ W n (e i).1)
+    (A : ℝ) (z : ℂ) (i : Fin J) :
+    entryPhysicalLevel κ W (n,some (e i)) A z+2*a κ*(batchLevel κ W n:ℝ)+2*a κ*(n:ℝ) ≤
+      entryPhysicalLevel κ W (n,none) A z :=
+  entryPhysicalLevel_cleanup_gap hκ W hn hm1 hJ (he i) A z
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedBatchMixing.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedBatchMixing.lean
index 73172a66673286493456eb85748ebc09b2a8bb0b..b6d1878447e193a4bb10494779363d66b7fb6151 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedBatchMixing.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedBatchMixing.lean
@@ -1,110 +1,110 @@
-import OAI.Probability.SLE.Gauge.PrescribedEntryKernel
-import OAI.Probability.SLE.Gauge.PrescribedAdaptedBatch
-import OAI.Probability.SLE.Geometry.CanonicalInteriorBound
-
-namespace OAI
-
-/-! Angular mixing on the gaps preceding the tests combines with the exact
-normalized disk kernel to give the prescribed batch estimate. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance mixingBatchMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance mixingBatchBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-theorem CanonicalMass.prescribed_batch_failure_of_angle_mixing
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
-    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
-    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    {n J : ℕ} (hn : 3≤n) (e : Fin J → ℤ×ℕ) (hi : Function.Injective e)
-    (he' : ∀ kj, kj∈range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
-    (horder : StrictMono (fun i => Target.entryPosition κ W (n,some (e i))))
-    (z : ℂ) (hz : z∈upperHalfPlane)
-    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) (2048*Real.pi) z)
-    (hmix : ∀ (i : Fin J) (E : Set Ω),
-      MeasurableSet[(actual_radiusLevelTime_stopping hB
-        (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
-        κ hκ hκ8 γ G F hw z hz (Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val)
-        (Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val)).measurableSpace] E →
-      ((∫⁻ θ, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some (e i)))
-          (2*Target.a κ*(Target.batchLevel κ W n:ℝ)) θ ∂Target.angleLaw κ)/2)*
-        finiteCenteredLaw P κ B γ G z (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z) E ≤
-      ∫⁻ ω in E, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some (e i)))
-        (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
-        (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z
-          (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z)) z ω)
-        ∂finiteCenteredLaw P κ B γ G z (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z)) :
-    C.prescribedCenteredFailure W n e (4096*Real.pi) ((128*Real.pi)^dimension κ) z ≤
-      ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) :=
-    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
-  have hm1 := C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1
-  have hJn := C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
-  have hei (i : Fin J) : (e i).2 < Target.multiplicity κ W n (e i).1 := (he' _).mp ⟨i,rfl⟩
-  have ha := Target.internal_a_pos hκ
-  have hd : 2*Target.a κ*(Target.batchLevel κ W n:ℝ)=4*(Target.batchLevel κ W n:ℝ)/κ := by
-    unfold Target.a
-    ring
-  have hv : 0 ≤ 4*(Target.batchLevel κ W n:ℝ)/κ := by positivity
-  let : IsProbabilityMeasure (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)) :=
-    centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw _ hv
-  apply C.prescribed_adapted_batch_failure hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
-    hn e hi he' horder (4096*Real.pi) ((128*Real.pi)^dimension κ) (2048*Real.pi) z hz hl
-    _ (hl none)
-  intro i E hE
-  have hgap := Target.prescribedTestPastLevel_gap hκ W (by omega) e horder
-    (2048*Real.pi) z 0 (fun j => hl (some (e j))) (by positivity) i
-  simp only [add_zero] at hgap
-  have hS := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz
-    (Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val)
-    (Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val)
-  have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz
-    (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z) (hl _)
-  have hET : MeasurableSet[hT.measurableSpace] E :=
-    hS.measurableSpace_mono hT (fun ω => radiusLevelTime_mono ((hc ω).const_mul _)
-      (hw ω) z (Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val) hgap) E hE
-  have hend := Target.prescribedBatchLevel_test_end_le_cleanup hκ W (by omega) hm1 hJn e hei
-    (2048*Real.pi) z i
-  have hV : Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+
-      2*Target.a κ*(Target.batchLevel κ W n:ℝ) ≤
-        Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z := by
-    have hn0 : 0 ≤ 2*Target.a κ*(n:ℝ) := by positivity
-    linarith
-  have htest := C.prescribed_entry_success_kernel hB hc h0 hκ hκ8 F hw R he D hD hDf hDp
-    hgrowth hpublished W n (e i) z hz (hl _) _ hV E hET
-  have hav := R.angleDiskTail_average (P.map (sourceDriverPath B hc)) κ hκ
-    (Target.entryValue (n,some (e i))) (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
-  rw [C.raw_centeredDiskTail_eq hc hκ F hw R he _ _ (by positivity),hd] at hav
-  have hhalf : ENNReal.ofReal
-      ((centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
-        {p | MomentBatches.dyadic (e i).1 ≤ C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}/2) =
-      (∫⁻ θ, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some (e i)))
-          (2*Target.a κ*(Target.batchLevel κ W n:ℝ)) θ ∂Target.angleLaw κ)/2 := by
-    rw [ENNReal.ofReal_div_of_pos (by norm_num),ofReal_measureReal,hd,hav]
-    rcases e i with ⟨k,j⟩
-    norm_num [CanonicalMass.centeredDiskTail,Target.entryValue,MomentBatches.dyadic]
-  rw [hhalf]
-  exact (hmix i E hE).trans htest
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedEntryKernel
+import OAI.Probability.SLE.Gauge.PrescribedAdaptedBatch
+import OAI.Probability.SLE.Geometry.CanonicalInteriorBound
+
+namespace OAI
+
+/-! Angular mixing on the gaps preceding the tests combines with the exact
+normalized disk kernel to give the prescribed batch estimate. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance mixingBatchMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance mixingBatchBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+theorem CanonicalMass.prescribed_batch_failure_of_angle_mixing
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
+    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
+    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    {n J : ℕ} (hn : 3≤n) (e : Fin J → ℤ×ℕ) (hi : Function.Injective e)
+    (he' : ∀ kj, kj∈range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
+    (horder : StrictMono (fun i => Target.entryPosition κ W (n,some (e i))))
+    (z : ℂ) (hz : z∈upperHalfPlane)
+    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) (2048*Real.pi) z)
+    (hmix : ∀ (i : Fin J) (E : Set Ω),
+      MeasurableSet[(actual_radiusLevelTime_stopping hB
+        (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
+        κ hκ hκ8 γ G F hw z hz (Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val)
+        (Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val)).measurableSpace] E →
+      ((∫⁻ θ, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some (e i)))
+          (2*Target.a κ*(Target.batchLevel κ W n:ℝ)) θ ∂Target.angleLaw κ)/2)*
+        finiteCenteredLaw P κ B γ G z (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z) E ≤
+      ∫⁻ ω in E, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some (e i)))
+        (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
+        (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z
+          (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z)) z ω)
+        ∂finiteCenteredLaw P κ B γ G z (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z)) :
+    C.prescribedCenteredFailure W n e (4096*Real.pi) ((128*Real.pi)^dimension κ) z ≤
+      ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) :=
+    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
+  have hm1 := C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1
+  have hJn := C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
+  have hei (i : Fin J) : (e i).2 < Target.multiplicity κ W n (e i).1 := (he' _).mp ⟨i,rfl⟩
+  have ha := Target.internal_a_pos hκ
+  have hd : 2*Target.a κ*(Target.batchLevel κ W n:ℝ)=4*(Target.batchLevel κ W n:ℝ)/κ := by
+    unfold Target.a
+    ring
+  have hv : 0 ≤ 4*(Target.batchLevel κ W n:ℝ)/κ := by positivity
+  let : IsProbabilityMeasure (centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)) :=
+    centeredDiskLaw_isProbability hB hBm hc hκ hκ8 γ G F hw _ hv
+  apply C.prescribed_adapted_batch_failure hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW
+    hn e hi he' horder (4096*Real.pi) ((128*Real.pi)^dimension κ) (2048*Real.pi) z hz hl
+    _ (hl none)
+  intro i E hE
+  have hgap := Target.prescribedTestPastLevel_gap hκ W (by omega) e horder
+    (2048*Real.pi) z 0 (fun j => hl (some (e j))) (by positivity) i
+  simp only [add_zero] at hgap
+  have hS := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz
+    (Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val)
+    (Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val)
+  have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz
+    (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z) (hl _)
+  have hET : MeasurableSet[hT.measurableSpace] E :=
+    hS.measurableSpace_mono hT (fun ω => radiusLevelTime_mono ((hc ω).const_mul _)
+      (hw ω) z (Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val) hgap) E hE
+  have hend := Target.prescribedBatchLevel_test_end_le_cleanup hκ W (by omega) hm1 hJn e hei
+    (2048*Real.pi) z i
+  have hV : Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+
+      2*Target.a κ*(Target.batchLevel κ W n:ℝ) ≤
+        Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z := by
+    have hn0 : 0 ≤ 2*Target.a κ*(n:ℝ) := by positivity
+    linarith
+  have htest := C.prescribed_entry_success_kernel hB hc h0 hκ hκ8 F hw R he D hD hDf hDp
+    hgrowth hpublished W n (e i) z hz (hl _) _ hV E hET
+  have hav := R.angleDiskTail_average (P.map (sourceDriverPath B hc)) κ hκ
+    (Target.entryValue (n,some (e i))) (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
+  rw [C.raw_centeredDiskTail_eq hc hκ F hw R he _ _ (by positivity),hd] at hav
+  have hhalf : ENNReal.ofReal
+      ((centeredDiskLaw P κ B hc (4*(Target.batchLevel κ W n:ℝ)/κ)).real
+        {p | MomentBatches.dyadic (e i).1 ≤ C.diskAtRadius p.1 (4*(Target.batchLevel κ W n:ℝ)/κ) p.2}/2) =
+      (∫⁻ θ, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some (e i)))
+          (2*Target.a κ*(Target.batchLevel κ W n:ℝ)) θ ∂Target.angleLaw κ)/2 := by
+    rw [ENNReal.ofReal_div_of_pos (by norm_num),ofReal_measureReal,hd,hav]
+    rcases e i with ⟨k,j⟩
+    norm_num [CanonicalMass.centeredDiskTail,Target.entryValue,MomentBatches.dyadic]
+  rw [hhalf]
+  exact (hmix i E hE).trans htest
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCandidateCover.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCandidateCover.lean
index ed7b6111852ff8c77f7e3ade6ce706a22fb03d21..925da3e8d30adc81c4ba5b6a2d1fcf9e59cf443c 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCandidateCover.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCandidateCover.lean
@@ -1,153 +1,153 @@
-import OAI.Probability.SLE.Gauge.PrescribedCoverGrid
-import OAI.Probability.SLE.Gauge.PrescribedGaugeCost
-import OAI.Probability.SLE.Gauge.BatchRadiusGaps
-import OAI.Probability.SLE.Gauge.GaugeEntrySmallness
-import OAI.Probability.SLE.Gauge.FiniteCoverAssembly
-
-namespace OAI
-
-/-! Actual finite candidate and missed-cell covers for a prescribed batch.
-Candidate selection uses the terminal spatial measure. No terminal event
-is inserted into the adapted finite change of law. -/
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def prescribedCandidateIndices {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ)
-    (e : Fin J → ℤ × ℕ) (C c R : ℝ) (K : Set ℂ) (ν : Measure ℂ) :
-    Finset ((ℤ × ℤ) × Fin J) := by
-  classical
-  let r := Target.entryRadius κ W (n,none)
-  exact ((prescribedGridIndex R r K) ×ˢ Finset.univ).filter (fun ij =>
-    ENNReal.ofReal (c*Target.entryValue (n,some (e ij.2))*
-      (Target.entryRadius κ W (n,some (e ij.2)))^Target.d κ) ≤
-    ν (closedBall (prescribedGridCenter r ij.1)
-      (C*Target.entryRadius κ W (n,some (e ij.2)))))
-
-def prescribedMissedIndices {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ)
-    (e : Fin J → ℤ × ℕ) (C c R : ℝ) (K E : Set ℂ) (ν : Measure ℂ) : Finset (ℤ × ℤ) := by
-  classical
-  let r := Target.entryRadius κ W (n,none)
-  exact (prescribedGridIndex R r K).filter (fun ij =>
-    (E ∩ prescribedFineCell r ij).Nonempty ∧ ∀ j,
-      ν (closedBall (prescribedGridCenter r ij) (C*Target.entryRadius κ W (n,some (e j)))) <
-      ENNReal.ofReal (c*Target.entryValue (n,some (e j))*
-        (Target.entryRadius κ W (n,some (e j)))^Target.d κ))
-
-theorem prescribed_candidates_finite_cover
-    {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw)
-    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
-    {n J : ℕ} (hn : 1 ≤ n) (hm1 : 0<Target.moment κ W 1)
-    (hJ : Target.moment κ W 1^n ≤ Target.moment κ W n)
-    (e : Fin J → ℤ × ℕ) (he : ∀ j, (e j).2<Target.multiplicity κ W n (e j).1)
-    (C c R : ℝ) (hC : 3 ≤ C) (hc : 0<c) (hR : 0 ≤ R)
-    (K E : Set ℂ) (hK : K ⊆ closedBall 0 R) (hH : ∀ z∈K, 0 ≤ z.im)
-    (hE : E ⊆ K) (ν : Measure ℂ) :
-    ∃ (N : ℕ) (U : Fin N → Set ℂ), E ⊆ ⋃ i, U i ∧
-      (∀ i, ediam (U i) ≤ ENNReal.ofReal (8*C*Real.exp (-(n:ℝ)^2))) ∧
-      (∑ i, extendedGauge (Target.gauge κ W) (ediam (U i))) ≤
-        ENNReal.ofReal ((8*C)^2/c)*ν (closedBall 0 (R+3+C)) +
-        (prescribedMissedIndices κ W n e C c R K E ν).card *
-          ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ) := by
-  classical
-  let r := Target.entryRadius κ W (n,none)
-  let s (j : Fin J) := Target.entryRadius κ W (n,some (e j))
-  let I := prescribedCandidateIndices κ W n e C c R K ν
-  let M := prescribedMissedIndices κ W n e C c R K E ν
-  let x (ij : (ℤ×ℤ)×Fin J) := prescribedGridCenter r ij.1
-  let rad (ij : (ℤ×ℤ)×Fin J) := C*s ij.2
-  let δ := ENNReal.ofReal (8*C*Real.exp (-(n:ℝ)^2))
-  have hC0 : 0 ≤ C := by linarith
-  have hr : 0<r := Real.exp_pos _
-  have hs (j) : 0<s j := Real.exp_pos _
-  have hsmall (j) : s j ≤ Real.exp (-(n:ℝ)^2) := Target.entryRadius_le_batch_exp hκ W _
-  have hrsmall : r ≤ Real.exp (-(n:ℝ)^2) := Target.entryRadius_le_batch_exp hκ W _
-  have hunit : r ≤ 1 := hrsmall.trans (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
-  have hrs (j) : r ≤ s j := (Target.entryRadius_lt_of_position_lt hκ W hn
-    (Target.entryPosition_lt_cleanup κ W hn hm1 hJ (he j))).le
-  have hcontain (j) : 3*r ≤ C*s j := by
-    nlinarith [mul_le_mul_of_nonneg_left (hrs j) (by norm_num : (0:ℝ) ≤ 3)]
-  have himem {ij : (ℤ×ℤ)×Fin J} (hij : ij∈I) :
-      ij.1∈prescribedGridIndex R r K ∧
-      ENNReal.ofReal (c*Target.entryValue (n,some (e ij.2))*(s ij.2)^Target.d κ) ≤
-        ν (closedBall (x ij) (rad ij)) := by
-    exact ⟨(Finset.mem_product.mp (Finset.mem_filter.mp hij).1).1,
-      (Finset.mem_filter.mp hij).2⟩
-  have hcover : ∀ z∈E, (∃ ij∈I, z∈closedBall (x ij) (rad ij)) ∨
-      ∃ ij∈M, z∈prescribedFineCell r ij := by
-    intro z hz
-    obtain ⟨ij,hij,hzcell⟩ := mem_iUnion₂.mp (prescribedGrid_covers hR hr hK hH (hE hz))
-    by_cases hq : ∃ j, ENNReal.ofReal (c*Target.entryValue (n,some (e j))*(s j)^Target.d κ) ≤
-        ν (closedBall (prescribedGridCenter r ij) (C*s j))
-    · obtain ⟨j,hj⟩ := hq
-      refine Or.inl ⟨(ij,j),Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨hij,Finset.mem_univ _⟩,hj⟩,?_⟩
-      exact hzcell.trans (hcontain j)
-    · refine Or.inr ⟨ij,Finset.mem_filter.mpr ⟨hij,⟨z,hz,hzcell⟩,?_⟩,hzcell⟩
-      intro j
-      exact lt_of_not_ge (fun hj => hq ⟨j,hj⟩)
-  have hball : ∀ ij∈I, closedBall (x ij) (rad ij) ⊆ closedBall 0 (R+3+C) := by
-    intro ij hij z hz
-    have hx := prescribedGrid_center_bound hK (himem hij).1
-    have hs1 : s ij.2 ≤ 1 := (hsmall _).trans (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
-    have hrad : rad ij ≤ C := by dsimp only [rad]; nlinarith
-    have hz' : dist z (x ij) ≤ rad ij := hz
-    have ht := dist_triangle z (x ij) 0
-    simp only [dist_zero_right] at ht
-    change dist z 0 ≤ R+3+C
-    rw [dist_zero_right]
-    dsimp only [x] at ht
-    nlinarith
-  have hdiam : ∀ ij∈I, ediam (closedBall (x ij) (4*rad ij)) ≤ δ := by
-    intro ij _
-    apply ediam_le_of_forall_dist_le
-    intro z hz w hw
-    have hz' : dist z (x ij) ≤ 4*rad ij := hz
-    have hw' : dist (x ij) w ≤ 4*rad ij := by simpa only [mem_closedBall,dist_comm] using hw
-    apply (dist_triangle z (x ij) w).trans
-    have hh := mul_le_mul_of_nonneg_left (hsmall ij.2) (show 0 ≤ 8*C by positivity)
-    dsimp only [rad] at hz' hw'
-    nlinarith
-  have hmiss : ∀ ij∈M, ediam (prescribedFineCell r ij) ≤ δ := by
-    intro ij _
-    apply ediam_le_of_forall_dist_le
-    intro z hz w hw
-    have hz' : dist z (prescribedGridCenter r ij) ≤ 3*r := hz
-    have hw' : dist (prescribedGridCenter r ij) w ≤ 3*r := by
-      simpa only [prescribedFineCell,mem_closedBall,dist_comm] using hw
-    apply (dist_triangle z (prescribedGridCenter r ij) w).trans
-    have hh := mul_le_mul_of_nonneg_left hrsmall (show (0:ℝ) ≤ 6 by norm_num)
-    have hexp := Real.exp_pos (-(n:ℝ)^2)
-    nlinarith
-  obtain ⟨N,U,hcov,hU,hcost⟩ := exists_finite_selected_missed_cover I M x rad
-    (prescribedFineCell r) E (closedBall 0 (R+3+C)) ν (Target.gauge κ W)
-    (C*Real.exp (-(n:ℝ)^2)) (fun ij _ => mul_le_mul_of_nonneg_left (hsmall _) hC0)
-    δ (ENNReal.ofReal ((8*C)^2/c)) hball
-    (fun ij hij => by
-      simpa only [rad,mul_assoc] using Target.enlarged_candidate_gauge_cost W hg
-        (n,some (e ij.2)) ⟨hn,he ij.2⟩ (x ij) (by linarith : 1 ≤ C) hc ν (himem hij).2)
-    hdiam hmiss hcover
-  refine ⟨N,U,hcov,hU,hcost.trans ?_⟩
-  apply add_le_add le_rfl
-  calc
-    (∑ ij∈M, extendedGauge (Target.gauge κ W) (ediam (prescribedFineCell r ij))) ≤
-        ∑ _ij∈M, ENNReal.ofReal (36*(n:ℝ)*r^Target.d κ) := by
-      apply Finset.sum_le_sum
-      intro ij _
-      have hh : IsGauge (Target.gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
-      refine (extendedGauge_closedBall_le hh (prescribedGridCenter r ij) (by positivity)).trans ?_
-      apply ENNReal.ofReal_le_ofReal
-      have hb := Target.gauge_scaled_entry_le κ W (n,none) ⟨hn,trivial⟩ 6
-      norm_num [Target.entryValue] at hb
-      convert hb using 1
-      dsimp only [r]
-      congr 1
-      ring
-    _ = _ := by rw [Finset.sum_const,nsmul_eq_mul]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedCoverGrid
+import OAI.Probability.SLE.Gauge.PrescribedGaugeCost
+import OAI.Probability.SLE.Gauge.BatchRadiusGaps
+import OAI.Probability.SLE.Gauge.GaugeEntrySmallness
+import OAI.Probability.SLE.Gauge.FiniteCoverAssembly
+
+namespace OAI
+
+/-! Actual finite candidate and missed-cell covers for a prescribed batch.
+Candidate selection uses the terminal spatial measure. No terminal event
+is inserted into the adapted finite change of law. -/
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def prescribedCandidateIndices {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ)
+    (e : Fin J → ℤ × ℕ) (C c R : ℝ) (K : Set ℂ) (ν : Measure ℂ) :
+    Finset ((ℤ × ℤ) × Fin J) := by
+  classical
+  let r := Target.entryRadius κ W (n,none)
+  exact ((prescribedGridIndex R r K) ×ˢ Finset.univ).filter (fun ij =>
+    ENNReal.ofReal (c*Target.entryValue (n,some (e ij.2))*
+      (Target.entryRadius κ W (n,some (e ij.2)))^Target.d κ) ≤
+    ν (closedBall (prescribedGridCenter r ij.1)
+      (C*Target.entryRadius κ W (n,some (e ij.2)))))
+
+def prescribedMissedIndices {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ)
+    (e : Fin J → ℤ × ℕ) (C c R : ℝ) (K E : Set ℂ) (ν : Measure ℂ) : Finset (ℤ × ℤ) := by
+  classical
+  let r := Target.entryRadius κ W (n,none)
+  exact (prescribedGridIndex R r K).filter (fun ij =>
+    (E ∩ prescribedFineCell r ij).Nonempty ∧ ∀ j,
+      ν (closedBall (prescribedGridCenter r ij) (C*Target.entryRadius κ W (n,some (e j)))) <
+      ENNReal.ofReal (c*Target.entryValue (n,some (e j))*
+        (Target.entryRadius κ W (n,some (e j)))^Target.d κ))
+
+theorem prescribed_candidates_finite_cover
+    {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw)
+    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
+    {n J : ℕ} (hn : 1 ≤ n) (hm1 : 0<Target.moment κ W 1)
+    (hJ : Target.moment κ W 1^n ≤ Target.moment κ W n)
+    (e : Fin J → ℤ × ℕ) (he : ∀ j, (e j).2<Target.multiplicity κ W n (e j).1)
+    (C c R : ℝ) (hC : 3 ≤ C) (hc : 0<c) (hR : 0 ≤ R)
+    (K E : Set ℂ) (hK : K ⊆ closedBall 0 R) (hH : ∀ z∈K, 0 ≤ z.im)
+    (hE : E ⊆ K) (ν : Measure ℂ) :
+    ∃ (N : ℕ) (U : Fin N → Set ℂ), E ⊆ ⋃ i, U i ∧
+      (∀ i, ediam (U i) ≤ ENNReal.ofReal (8*C*Real.exp (-(n:ℝ)^2))) ∧
+      (∑ i, extendedGauge (Target.gauge κ W) (ediam (U i))) ≤
+        ENNReal.ofReal ((8*C)^2/c)*ν (closedBall 0 (R+3+C)) +
+        (prescribedMissedIndices κ W n e C c R K E ν).card *
+          ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ) := by
+  classical
+  let r := Target.entryRadius κ W (n,none)
+  let s (j : Fin J) := Target.entryRadius κ W (n,some (e j))
+  let I := prescribedCandidateIndices κ W n e C c R K ν
+  let M := prescribedMissedIndices κ W n e C c R K E ν
+  let x (ij : (ℤ×ℤ)×Fin J) := prescribedGridCenter r ij.1
+  let rad (ij : (ℤ×ℤ)×Fin J) := C*s ij.2
+  let δ := ENNReal.ofReal (8*C*Real.exp (-(n:ℝ)^2))
+  have hC0 : 0 ≤ C := by linarith
+  have hr : 0<r := Real.exp_pos _
+  have hs (j) : 0<s j := Real.exp_pos _
+  have hsmall (j) : s j ≤ Real.exp (-(n:ℝ)^2) := Target.entryRadius_le_batch_exp hκ W _
+  have hrsmall : r ≤ Real.exp (-(n:ℝ)^2) := Target.entryRadius_le_batch_exp hκ W _
+  have hunit : r ≤ 1 := hrsmall.trans (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
+  have hrs (j) : r ≤ s j := (Target.entryRadius_lt_of_position_lt hκ W hn
+    (Target.entryPosition_lt_cleanup κ W hn hm1 hJ (he j))).le
+  have hcontain (j) : 3*r ≤ C*s j := by
+    nlinarith [mul_le_mul_of_nonneg_left (hrs j) (by norm_num : (0:ℝ) ≤ 3)]
+  have himem {ij : (ℤ×ℤ)×Fin J} (hij : ij∈I) :
+      ij.1∈prescribedGridIndex R r K ∧
+      ENNReal.ofReal (c*Target.entryValue (n,some (e ij.2))*(s ij.2)^Target.d κ) ≤
+        ν (closedBall (x ij) (rad ij)) := by
+    exact ⟨(Finset.mem_product.mp (Finset.mem_filter.mp hij).1).1,
+      (Finset.mem_filter.mp hij).2⟩
+  have hcover : ∀ z∈E, (∃ ij∈I, z∈closedBall (x ij) (rad ij)) ∨
+      ∃ ij∈M, z∈prescribedFineCell r ij := by
+    intro z hz
+    obtain ⟨ij,hij,hzcell⟩ := mem_iUnion₂.mp (prescribedGrid_covers hR hr hK hH (hE hz))
+    by_cases hq : ∃ j, ENNReal.ofReal (c*Target.entryValue (n,some (e j))*(s j)^Target.d κ) ≤
+        ν (closedBall (prescribedGridCenter r ij) (C*s j))
+    · obtain ⟨j,hj⟩ := hq
+      refine Or.inl ⟨(ij,j),Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨hij,Finset.mem_univ _⟩,hj⟩,?_⟩
+      exact hzcell.trans (hcontain j)
+    · refine Or.inr ⟨ij,Finset.mem_filter.mpr ⟨hij,⟨z,hz,hzcell⟩,?_⟩,hzcell⟩
+      intro j
+      exact lt_of_not_ge (fun hj => hq ⟨j,hj⟩)
+  have hball : ∀ ij∈I, closedBall (x ij) (rad ij) ⊆ closedBall 0 (R+3+C) := by
+    intro ij hij z hz
+    have hx := prescribedGrid_center_bound hK (himem hij).1
+    have hs1 : s ij.2 ≤ 1 := (hsmall _).trans (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
+    have hrad : rad ij ≤ C := by dsimp only [rad]; nlinarith
+    have hz' : dist z (x ij) ≤ rad ij := hz
+    have ht := dist_triangle z (x ij) 0
+    simp only [dist_zero_right] at ht
+    change dist z 0 ≤ R+3+C
+    rw [dist_zero_right]
+    dsimp only [x] at ht
+    nlinarith
+  have hdiam : ∀ ij∈I, ediam (closedBall (x ij) (4*rad ij)) ≤ δ := by
+    intro ij _
+    apply ediam_le_of_forall_dist_le
+    intro z hz w hw
+    have hz' : dist z (x ij) ≤ 4*rad ij := hz
+    have hw' : dist (x ij) w ≤ 4*rad ij := by simpa only [mem_closedBall,dist_comm] using hw
+    apply (dist_triangle z (x ij) w).trans
+    have hh := mul_le_mul_of_nonneg_left (hsmall ij.2) (show 0 ≤ 8*C by positivity)
+    dsimp only [rad] at hz' hw'
+    nlinarith
+  have hmiss : ∀ ij∈M, ediam (prescribedFineCell r ij) ≤ δ := by
+    intro ij _
+    apply ediam_le_of_forall_dist_le
+    intro z hz w hw
+    have hz' : dist z (prescribedGridCenter r ij) ≤ 3*r := hz
+    have hw' : dist (prescribedGridCenter r ij) w ≤ 3*r := by
+      simpa only [prescribedFineCell,mem_closedBall,dist_comm] using hw
+    apply (dist_triangle z (prescribedGridCenter r ij) w).trans
+    have hh := mul_le_mul_of_nonneg_left hrsmall (show (0:ℝ) ≤ 6 by norm_num)
+    have hexp := Real.exp_pos (-(n:ℝ)^2)
+    nlinarith
+  obtain ⟨N,U,hcov,hU,hcost⟩ := exists_finite_selected_missed_cover I M x rad
+    (prescribedFineCell r) E (closedBall 0 (R+3+C)) ν (Target.gauge κ W)
+    (C*Real.exp (-(n:ℝ)^2)) (fun ij _ => mul_le_mul_of_nonneg_left (hsmall _) hC0)
+    δ (ENNReal.ofReal ((8*C)^2/c)) hball
+    (fun ij hij => by
+      simpa only [rad,mul_assoc] using Target.enlarged_candidate_gauge_cost W hg
+        (n,some (e ij.2)) ⟨hn,he ij.2⟩ (x ij) (by linarith : 1 ≤ C) hc ν (himem hij).2)
+    hdiam hmiss hcover
+  refine ⟨N,U,hcov,hU,hcost.trans ?_⟩
+  apply add_le_add le_rfl
+  calc
+    (∑ ij∈M, extendedGauge (Target.gauge κ W) (ediam (prescribedFineCell r ij))) ≤
+        ∑ _ij∈M, ENNReal.ofReal (36*(n:ℝ)*r^Target.d κ) := by
+      apply Finset.sum_le_sum
+      intro ij _
+      have hh : IsGauge (Target.gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
+      refine (extendedGauge_closedBall_le hh (prescribedGridCenter r ij) (by positivity)).trans ?_
+      apply ENNReal.ofReal_le_ofReal
+      have hb := Target.gauge_scaled_entry_le κ W (n,none) ⟨hn,trivial⟩ 6
+      norm_num [Target.entryValue] at hb
+      convert hb using 1
+      dsimp only [r]
+      congr 1
+      ring
+    _ = _ := by rw [Finset.sum_const,nsmul_eq_mul]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalMixing.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalMixing.lean
index 7a6c8f07cdccfa062072bf6be81fac5d19a4a34f..a7bb2c0f382408246de0d0497272c5bcf1a61724 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalMixing.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalMixing.lean
@@ -1,84 +1,84 @@
-import OAI.Probability.SLE.Gauge.PrescribedBatchMixing
-import OAI.Probability.SLE.Angle.RadiusAngleBounds
-
-namespace OAI
-
-/-! A canonical angle transition bound across the reserved preceding gaps
-implies the exact prescribed batch estimate. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance canonicalBatchMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance canonicalBatchBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-theorem CanonicalMass.prescribed_batch_failure_of_canonical_angle_bound
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
-    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
-    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    {n J : ℕ} (hn : 3≤n) (e : Fin J → ℤ×ℕ) (hi : Function.Injective e)
-    (he' : ∀ kj, kj∈range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
-    (horder : StrictMono (fun i => Target.entryPosition κ W (n,some (e i))))
-    (z : ℂ) (hz : z∈upperHalfPlane)
-    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) (2048*Real.pi) z)
-    (δ : ℝ) (hδ : 0<δ)
-    (hgap : ∀ i : Fin J, Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val+δ ≤
-      Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z)
-    (hangle : ∀ (v : ℝ), δ≤v → ∀ (f : ℝ → ℝ≥0∞), Measurable f →
-      ∀ θ∈Ioo (0:ℝ) Real.pi, (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤
-        canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ) :
-    C.prescribedCenteredFailure W n e (4096*Real.pi) ((128*Real.pi)^dimension κ) z ≤
-      ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let ν := P.map (sourceDriverPath B hc)
-  let : IsProbabilityMeasure ν :=
-    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
-  have hm1 := C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1
-  have hJn := C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
-  have hei (i : Fin J) : (e i).2 < Target.multiplicity κ W n (e i).1 := (he' _).mp ⟨i,rfl⟩
-  apply C.prescribed_batch_failure_of_angle_mixing hB hc h0 hκ hκ8 F hw R he D hD hDf hDp
-    hgrowth hpublished W hW hn e hi he' horder z hz hl
-  intro i E hE
-  let s := Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val
-  let u := Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z
-  let V := Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z
-  let f := R.angleDiskTail ν κ (Target.entryValue (n,some (e i)))
-    (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
-  have hs : 0 ≤ s := Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val
-  have hsu : δ ≤ u-s := by have hh := hgap i; dsimp only [s,u]; linarith
-  have huV : u≤V := by
-    have hend := Target.prescribedBatchLevel_test_end_le_cleanup hκ W (by omega) hm1 hJn e hei
-      (2048*Real.pi) z i
-    have ha := Target.internal_a_pos hκ
-    have hnon1 : 0≤2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by positivity
-    have hnon2 : 0≤2*Target.a κ*(n:ℝ) := by positivity
-    dsimp only [u,V]
-    linarith
-  have hf : Measurable f := R.angleDiskTail_measurable ν κ _ _
-  have hb := (radius_angle_average_bounds hB hBm hc h0 hκ hκ8 γ G F hw hpublished
-    z hz s (u-s) V hs (hδ.trans_le hsu) (by linarith) f hf
-    ((∫⁻ φ, f φ ∂Target.angleLaw κ)/2) ∞
-    (fun θ hθ => ⟨hangle (u-s) hsu f hf θ hθ,le_top⟩) E hE).1
-  have heq : s+(u-s)=u := by ring
-  simpa only [heq] using hb
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedBatchMixing
+import OAI.Probability.SLE.Angle.RadiusAngleBounds
+
+namespace OAI
+
+/-! A canonical angle transition bound across the reserved preceding gaps
+implies the exact prescribed batch estimate. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance canonicalBatchMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance canonicalBatchBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+theorem CanonicalMass.prescribed_batch_failure_of_canonical_angle_bound
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
+    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
+    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    {n J : ℕ} (hn : 3≤n) (e : Fin J → ℤ×ℕ) (hi : Function.Injective e)
+    (he' : ∀ kj, kj∈range e ↔ kj.2 < Target.multiplicity κ W n kj.1)
+    (horder : StrictMono (fun i => Target.entryPosition κ W (n,some (e i))))
+    (z : ℂ) (hz : z∈upperHalfPlane)
+    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) (2048*Real.pi) z)
+    (δ : ℝ) (hδ : 0<δ)
+    (hgap : ∀ i : Fin J, Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val+δ ≤
+      Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z)
+    (hangle : ∀ (v : ℝ), δ≤v → ∀ (f : ℝ → ℝ≥0∞), Measurable f →
+      ∀ θ∈Ioo (0:ℝ) Real.pi, (∫⁻ φ, f φ ∂Target.angleLaw κ)/2 ≤
+        canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ) :
+    C.prescribedCenteredFailure W n e (4096*Real.pi) ((128*Real.pi)^dimension κ) z ≤
+      ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let ν := P.map (sourceDriverPath B hc)
+  let : IsProbabilityMeasure ν :=
+    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
+  have hm1 := C.moment_positive hB hc h0 hκ hκ8 F hw hpublished hgrowth W hW 1
+  have hJn := C.moment_jensen hB hc h0 hκ hκ8 F hw hgrowth W hW n
+  have hei (i : Fin J) : (e i).2 < Target.multiplicity κ W n (e i).1 := (he' _).mp ⟨i,rfl⟩
+  apply C.prescribed_batch_failure_of_angle_mixing hB hc h0 hκ hκ8 F hw R he D hD hDf hDp
+    hgrowth hpublished W hW hn e hi he' horder z hz hl
+  intro i E hE
+  let s := Target.prescribedTestPastLevel κ W n e (2048*Real.pi) z i.val
+  let u := Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z
+  let V := Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi) z
+  let f := R.angleDiskTail ν κ (Target.entryValue (n,some (e i)))
+    (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
+  have hs : 0 ≤ s := Target.prescribedTestPastLevel_nonneg hκ W n e (2048*Real.pi) z hl i.val
+  have hsu : δ ≤ u-s := by have hh := hgap i; dsimp only [s,u]; linarith
+  have huV : u≤V := by
+    have hend := Target.prescribedBatchLevel_test_end_le_cleanup hκ W (by omega) hm1 hJn e hei
+      (2048*Real.pi) z i
+    have ha := Target.internal_a_pos hκ
+    have hnon1 : 0≤2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by positivity
+    have hnon2 : 0≤2*Target.a κ*(n:ℝ) := by positivity
+    dsimp only [u,V]
+    linarith
+  have hf : Measurable f := R.angleDiskTail_measurable ν κ _ _
+  have hb := (radius_angle_average_bounds hB hBm hc h0 hκ hκ8 γ G F hw hpublished
+    z hz s (u-s) V hs (hδ.trans_le hsu) (by linarith) f hf
+    ((∫⁻ φ, f φ ∂Target.angleLaw κ)/2) ∞
+    (fun θ hθ => ⟨hangle (u-s) hsu f hf θ hθ,le_top⟩) E hE).1
+  have heq : s+(u-s)=u := by ring
+  simpa only [heq] using hb
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTests.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTests.lean
index 4ba7b81bf964d673a24bef4fc4e15842378ae4a5..55cac75ae7bb82d798811a61f5b14bddb43247cb 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTests.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTests.lean
@@ -1,82 +1,82 @@
-import OAI.Probability.SLE.Gauge.FiniteStoppedThreshold
-import OAI.Probability.SLE.Mass.CanonicalMassRemaining
-import OAI.Probability.SLE.Mass.CanonicalPhysicalMass
-import OAI.Probability.SLE.Gauge.PrescribedCandidateCover
-
-namespace OAI
-
-/-! Actual adapted tests use canonical mass at the finite test endpoint.
-  Terminal candidate failure implies failure of all these earlier tests. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (M : CanonicalMass P κ B hBm γ G)
-
-def prescribedBallTest (S : Ω → WithTop ℝ≥0) (z : ℂ) (r : ℝ) (b : ℝ≥0∞) : Set Ω :=
-  finiteStoppedThreshold
-    (fun t ω => M.process t ω (Subtype.val ⁻¹' closedBall z r)) S b
-
-lemma prescribedBallTest_measurable
-    (S : Ω → WithTop ℝ≥0)
-    (hS : IsStoppingTime (Filtration.natural B (fun r =>
-      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S)
-    (z : ℂ) (r : ℝ) (b : ℝ≥0∞) :
-    MeasurableSet[hS.measurableSpace] (M.prescribedBallTest S z r b) := by
-  apply finiteStoppedThreshold_measurable _ _ S hS b
-  intro t
-  exact (Measure.measurable_coe
-    (isClosed_closedBall.measurableSet.preimage measurable_subtype_coe)).comp (M.progressive t)
-
-def prescribedTestFailures {J : ℕ} (W : Target.WienerLaw) (n : ℕ)
-    (e : Fin J → ℤ×ℕ) (C c : ℝ) (z : ℂ) (S : Fin J → Ω → WithTop ℝ≥0) : Set Ω :=
-  ⋂ j, (M.prescribedBallTest (S j) z (C*Target.entryRadius κ W (n,some (e j)))
-    (ENNReal.ofReal (c*Target.entryValue (n,some (e j))*
-      (Target.entryRadius κ W (n,some (e j)))^Target.d κ)))ᶜ
-
-lemma prescribedTestFailures_measurable_at_end
-    {J : ℕ} (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ) (C c : ℝ) (z : ℂ)
-    (S : Fin J → Ω → WithTop ℝ≥0)
-    (hS : ∀ j, IsStoppingTime (Filtration.natural B (fun r =>
-      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) (S j))
-    (T : Ω → WithTop ℝ≥0)
-    (hT : IsStoppingTime (Filtration.natural B (fun r =>
-      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) T)
-    (hST : ∀ j ω, S j ω≤T ω) :
-    MeasurableSet[hT.measurableSpace] (M.prescribedTestFailures W n e C c z S) := by
-  apply MeasurableSet.iInter
-  intro j
-  exact ((hS j).measurableSpace_mono hT (hST j)) _
-    (M.prescribedBallTest_measurable (S j) (hS j) _ _ _).compl
-
-lemma prescribed_missed_implies_test_failures
-    {J : ℕ} (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ) (C c R : ℝ)
-    (K : Set ℂ) (ij : ℤ×ℤ) (S : Fin J → Ω → WithTop ℝ≥0) :
-    ∀ᵐ ω ∂P,
-      ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K)
-        (M.rational.physicalTerminal ω) →
-      ω∈M.prescribedTestFailures W n e C c
-        (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij) S := by
-  classical
-  filter_upwards [M.paths] with ω hω hm
-  apply mem_iInter.mpr
-  intro j hj
-  have htest := hj.2
-  have hdom := hω.dominated ((S j ω).untopD 0)
-  have hball := hdom (Subtype.val ⁻¹' closedBall
-    (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
-    (C*Target.entryRadius κ W (n,some (e j))))
-  have hterm := (Finset.mem_filter.mp hm).2.2 j
-  rw [M.rational.physicalTerminal_apply ω isClosed_closedBall.measurableSet] at hterm
-  exact (not_le_of_gt hterm) (htest.trans hball)
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.FiniteStoppedThreshold
+import OAI.Probability.SLE.Mass.CanonicalMassRemaining
+import OAI.Probability.SLE.Mass.CanonicalPhysicalMass
+import OAI.Probability.SLE.Gauge.PrescribedCandidateCover
+
+namespace OAI
+
+/-! Actual adapted tests use canonical mass at the finite test endpoint.
+  Terminal candidate failure implies failure of all these earlier tests. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (M : CanonicalMass P κ B hBm γ G)
+
+def prescribedBallTest (S : Ω → WithTop ℝ≥0) (z : ℂ) (r : ℝ) (b : ℝ≥0∞) : Set Ω :=
+  finiteStoppedThreshold
+    (fun t ω => M.process t ω (Subtype.val ⁻¹' closedBall z r)) S b
+
+lemma prescribedBallTest_measurable
+    (S : Ω → WithTop ℝ≥0)
+    (hS : IsStoppingTime (Filtration.natural B (fun r =>
+      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S)
+    (z : ℂ) (r : ℝ) (b : ℝ≥0∞) :
+    MeasurableSet[hS.measurableSpace] (M.prescribedBallTest S z r b) := by
+  apply finiteStoppedThreshold_measurable _ _ S hS b
+  intro t
+  exact (Measure.measurable_coe
+    (isClosed_closedBall.measurableSet.preimage measurable_subtype_coe)).comp (M.progressive t)
+
+def prescribedTestFailures {J : ℕ} (W : Target.WienerLaw) (n : ℕ)
+    (e : Fin J → ℤ×ℕ) (C c : ℝ) (z : ℂ) (S : Fin J → Ω → WithTop ℝ≥0) : Set Ω :=
+  ⋂ j, (M.prescribedBallTest (S j) z (C*Target.entryRadius κ W (n,some (e j)))
+    (ENNReal.ofReal (c*Target.entryValue (n,some (e j))*
+      (Target.entryRadius κ W (n,some (e j)))^Target.d κ)))ᶜ
+
+lemma prescribedTestFailures_measurable_at_end
+    {J : ℕ} (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ) (C c : ℝ) (z : ℂ)
+    (S : Fin J → Ω → WithTop ℝ≥0)
+    (hS : ∀ j, IsStoppingTime (Filtration.natural B (fun r =>
+      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) (S j))
+    (T : Ω → WithTop ℝ≥0)
+    (hT : IsStoppingTime (Filtration.natural B (fun r =>
+      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) T)
+    (hST : ∀ j ω, S j ω≤T ω) :
+    MeasurableSet[hT.measurableSpace] (M.prescribedTestFailures W n e C c z S) := by
+  apply MeasurableSet.iInter
+  intro j
+  exact ((hS j).measurableSpace_mono hT (hST j)) _
+    (M.prescribedBallTest_measurable (S j) (hS j) _ _ _).compl
+
+lemma prescribed_missed_implies_test_failures
+    {J : ℕ} (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ) (C c R : ℝ)
+    (K : Set ℂ) (ij : ℤ×ℤ) (S : Fin J → Ω → WithTop ℝ≥0) :
+    ∀ᵐ ω ∂P,
+      ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K)
+        (M.rational.physicalTerminal ω) →
+      ω∈M.prescribedTestFailures W n e C c
+        (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij) S := by
+  classical
+  filter_upwards [M.paths] with ω hω hm
+  apply mem_iInter.mpr
+  intro j hj
+  have htest := hj.2
+  have hdom := hω.dominated ((S j ω).untopD 0)
+  have hball := hdom (Subtype.val ⁻¹' closedBall
+    (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
+    (C*Target.entryRadius κ W (n,some (e j))))
+  have hterm := (Finset.mem_filter.mp hm).2.2 j
+  rw [M.rational.physicalTerminal_apply ω isClosed_closedBall.measurableSet] at hterm
+  exact (not_le_of_gt hterm) (htest.trans hball)
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTransfer.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTransfer.lean
index a34d807b60fb16be6e35d7ab15b19e454772a29b..13bd0f92d9eb63284312a3d8d1ebe1c315be8fd0 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTransfer.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCanonicalTransfer.lean
@@ -1,98 +1,98 @@
-import OAI.Probability.SLE.Gauge.PrescribedApproachTransfer
-import OAI.Probability.SLE.Gauge.PrescribedCanonicalTests
-
-namespace OAI
-
-/-! The original-law missed-cell estimate in terms of the finite centered
-failure probability for the canonical tests. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (M : CanonicalMass P κ B hBm γ G)
-
-theorem prescribed_missed_from_finite_failure
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (W : Target.WienerLaw) (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ)
-    (C c R : ℝ) (K : Set ℂ) (ij : ℤ×ℤ)
-    (hij : ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K)
-    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) (2048*Real.pi)
-      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
-    (u : ℝ) (hu : 0 ≤ u)
-    (hgap : u+1 ≤ Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
-      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
-    (hend : ∀ i, Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi)
-      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)+
-        2*Target.a κ*(Target.batchLevel κ W n:ℝ) ≤ u)
-    (hsmall : 4*Target.entryRadius κ W (n,none) ≤
-      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im)
-    (ε : ℝ) (hε : 0 ≤ ε) (hfailure :
-      finiteCenteredLaw P κ B γ G (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
-        (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
-          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
-        (M.prescribedTestFailures W n e C c
-          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
-          (fun i ω => radiusLevelTime (γ ω) (G ω)
-            (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
-            (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi)
-              (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)+
-              2*Target.a κ*(Target.batchLevel κ W n:ℝ)))) ≤ ENNReal.ofReal ε) :
-    P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K)
-      (M.rational.physicalTerminal ω)} ≤
-      ENNReal.ofReal ((uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ*
-        (2:ℝ)^(-codimension κ))*
-        (Target.entryRadius κ W (n,none))^codimension κ*
-        (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*ε) := by
-  let z := prescribedGridCenter (Target.entryRadius κ W (n,none)) ij
-  have hz : 0<z.im := (prescribedGrid_center_height (Real.exp_pos _) hij).2
-  let ℱ := Filtration.natural B (fun t =>
-    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)
-  let T (i : Fin J) (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z
-    (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+
-      2*Target.a κ*(Target.batchLevel κ W n:ℝ))
-  have ht0 (i : Fin J) : 0 ≤ Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+
-      2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by
-    have ha := Target.internal_a_pos hκ
-    exact add_nonneg (hl _) (by positivity)
-  have hT (i : Fin J) : IsStoppingTime ℱ (T i) :=
-    actual_radiusLevelTime_stopping hB
-      (fun t => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t))) hBm hc h0 κ hκ hκ8 γ G F hw z hz _ (ht0 i)
-  let S (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z u
-  have hS : IsStoppingTime ℱ S :=
-    actual_radiusLevelTime_stopping hB
-      (fun t => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t))) hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu
-  have hTS (i : Fin J) (ω : Ω) : T i ω≤S ω :=
-    radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z (ht0 i) (hend i)
-  have hA := M.prescribedTestFailures_measurable_at_end W n e C c z T hT S hS hTS
-  have hm := prescribed_missed_cell_bound_of_adapted_failure P κ hκ hκ8 B hB hBm hc h0
-    γ G F hw hpublished M.rational.physicalTerminal W n e C c R K ij hij u hu hgap hsmall
-    (M.prescribedTestFailures W n e C c z T) hA
-    (M.prescribed_missed_implies_test_failures W n e C c R K ij T)
-  have hr : 0<Target.entryRadius κ W (n,none) := Real.exp_pos _
-  have hC := uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
-  have hG := initialGreen_pos κ z hz
-  refine (hm.trans (mul_le_mul' le_rfl hfailure)).trans ?_
-  rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ uniformInverseAngleConstant κ 1*
-    (2048*Real.pi)^codimension κ*initialGreen κ z*(Target.entryRadius κ W (n,none))^codimension κ)]
-  apply ENNReal.ofReal_le_ofReal
-  have hh := mul_le_mul_of_nonneg_left (initialGreen_le_height κ hκ hκ8 z hz)
-    (by positivity : 0 ≤ uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ)
-  have hh' := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh
-    (Real.rpow_nonneg hr.le (codimension κ))) hε
-  dsimp only [z] at hh'
-  convert hh' using 1
-  ring
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedApproachTransfer
+import OAI.Probability.SLE.Gauge.PrescribedCanonicalTests
+
+namespace OAI
+
+/-! The original-law missed-cell estimate in terms of the finite centered
+failure probability for the canonical tests. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (M : CanonicalMass P κ B hBm γ G)
+
+theorem prescribed_missed_from_finite_failure
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (W : Target.WienerLaw) (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ)
+    (C c R : ℝ) (K : Set ℂ) (ij : ℤ×ℤ)
+    (hij : ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K)
+    (hl : ∀ o, 0 ≤ Target.entryPhysicalLevel κ W (n,o) (2048*Real.pi)
+      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
+    (u : ℝ) (hu : 0 ≤ u)
+    (hgap : u+1 ≤ Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
+      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
+    (hend : ∀ i, Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi)
+      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)+
+        2*Target.a κ*(Target.batchLevel κ W n:ℝ) ≤ u)
+    (hsmall : 4*Target.entryRadius κ W (n,none) ≤
+      (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im)
+    (ε : ℝ) (hε : 0 ≤ ε) (hfailure :
+      finiteCenteredLaw P κ B γ G (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
+        (Target.entryPhysicalLevel κ W (n,none) (2048*Real.pi)
+          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij))
+        (M.prescribedTestFailures W n e C c
+          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
+          (fun i ω => radiusLevelTime (γ ω) (G ω)
+            (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)
+            (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi)
+              (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij)+
+              2*Target.a κ*(Target.batchLevel κ W n:ℝ)))) ≤ ENNReal.ofReal ε) :
+    P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (γ ω)∩K)
+      (M.rational.physicalTerminal ω)} ≤
+      ENNReal.ofReal ((uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ*
+        (2:ℝ)^(-codimension κ))*
+        (Target.entryRadius κ W (n,none))^codimension κ*
+        (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*ε) := by
+  let z := prescribedGridCenter (Target.entryRadius κ W (n,none)) ij
+  have hz : 0<z.im := (prescribedGrid_center_height (Real.exp_pos _) hij).2
+  let ℱ := Filtration.natural B (fun t =>
+    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)
+  let T (i : Fin J) (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z
+    (Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+
+      2*Target.a κ*(Target.batchLevel κ W n:ℝ))
+  have ht0 (i : Fin J) : 0 ≤ Target.entryPhysicalLevel κ W (n,some (e i)) (2048*Real.pi) z+
+      2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by
+    have ha := Target.internal_a_pos hκ
+    exact add_nonneg (hl _) (by positivity)
+  have hT (i : Fin J) : IsStoppingTime ℱ (T i) :=
+    actual_radiusLevelTime_stopping hB
+      (fun t => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t))) hBm hc h0 κ hκ hκ8 γ G F hw z hz _ (ht0 i)
+  let S (ω : Ω) : WithTop ℝ≥0 := radiusLevelTime (γ ω) (G ω) z u
+  have hS : IsStoppingTime ℱ S :=
+    actual_radiusLevelTime_stopping hB
+      (fun t => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t))) hBm hc h0 κ hκ hκ8 γ G F hw z hz u hu
+  have hTS (i : Fin J) (ω : Ω) : T i ω≤S ω :=
+    radiusLevelTime_mono ((hc ω).const_mul _) (hw ω) z (ht0 i) (hend i)
+  have hA := M.prescribedTestFailures_measurable_at_end W n e C c z T hT S hS hTS
+  have hm := prescribed_missed_cell_bound_of_adapted_failure P κ hκ hκ8 B hB hBm hc h0
+    γ G F hw hpublished M.rational.physicalTerminal W n e C c R K ij hij u hu hgap hsmall
+    (M.prescribedTestFailures W n e C c z T) hA
+    (M.prescribed_missed_implies_test_failures W n e C c R K ij T)
+  have hr : 0<Target.entryRadius κ W (n,none) := Real.exp_pos _
+  have hC := uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
+  have hG := initialGreen_pos κ z hz
+  refine (hm.trans (mul_le_mul' le_rfl hfailure)).trans ?_
+  rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ uniformInverseAngleConstant κ 1*
+    (2048*Real.pi)^codimension κ*initialGreen κ z*(Target.entryRadius κ W (n,none))^codimension κ)]
+  apply ENNReal.ofReal_le_ofReal
+  have hh := mul_le_mul_of_nonneg_left (initialGreen_le_height κ hκ hκ8 z hz)
+    (by positivity : 0 ≤ uniformInverseAngleConstant κ 1*(2048*Real.pi)^codimension κ)
+  have hh' := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh
+    (Real.rpow_nonneg hr.le (codimension κ))) hε
+  dsimp only [z] at hh'
+  convert hh' using 1
+  ring
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCellNormalization.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCellNormalization.lean
index fc31b1aab80c5b85d6052b9eae7508b15572c821..99763bb10619a21dd0fae494a15777946abdae6c 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCellNormalization.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCellNormalization.lean
@@ -1,45 +1,45 @@
-import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
-import OAI.Probability.SLE.Geometry.ArbitraryDiskTail
-import OAI.Probability.SLE.Geometry.CenteredCellTail
-
-namespace OAI
-
-/-! The lower-bound grid uses doubled cell radii around each sampled
-point. These identities match its threshold with the exact gauge weights. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma cell_subset_ball_at_member {K : Set ℂ} {c x : ℂ} {r : ℝ}
-    (hK : K⊆closedBall c (2*r)) (hx : x∈K) : K⊆closedBall x (2*(2*r)) := by
-  intro y hy
-  have hxc := mem_closedBall.mp (hK hx)
-  have hyc := mem_closedBall.mp (hK hy)
-  apply mem_closedBall.mpr
-  calc
-    dist y x ≤ dist y c+dist c x := dist_triangle _ _ _
-    _ ≤ 2*r+2*r := add_le_add hyc (by simpa only [dist_comm] using hxc)
-    _ = 2*(2*r) := by ring
-
-lemma prescribed_lower_threshold (κ r w : ℝ) (hr : 0<r) :
-    ((32768*Real.pi)*(2*r))^dimension κ*(16*w) =
-      (16*(65536*Real.pi)^dimension κ)*w*r^Target.d κ := by
-  have hd : Target.d κ=dimension κ := by rw [Target.dimension_eq]; rfl
-  rw [show (32768*Real.pi)*(2*r)=(65536*Real.pi)*r by ring,
-    Real.mul_rpow (by positivity : 0≤65536*Real.pi) hr.le,hd]
-  ring
-
-lemma prescribed_lower_radius (κ : ℝ) (W : Target.WienerLaw) (e : Target.Entry)
-    {z : ℂ} (hz : 0<z.im) :
-    (2*z.im)*Real.exp (-Target.entryPhysicalLevel κ W e (4096*Real.pi) z)=
-      (2048*Real.pi)*(2*Target.entryRadius κ W e) := by
-  rw [Target.entryPhysicalLevel_radius κ W e (by positivity) hz]
-  ring
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
+import OAI.Probability.SLE.Geometry.ArbitraryDiskTail
+import OAI.Probability.SLE.Geometry.CenteredCellTail
+
+namespace OAI
+
+/-! The lower-bound grid uses doubled cell radii around each sampled
+point. These identities match its threshold with the exact gauge weights. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma cell_subset_ball_at_member {K : Set ℂ} {c x : ℂ} {r : ℝ}
+    (hK : K⊆closedBall c (2*r)) (hx : x∈K) : K⊆closedBall x (2*(2*r)) := by
+  intro y hy
+  have hxc := mem_closedBall.mp (hK hx)
+  have hyc := mem_closedBall.mp (hK hy)
+  apply mem_closedBall.mpr
+  calc
+    dist y x ≤ dist y c+dist c x := dist_triangle _ _ _
+    _ ≤ 2*r+2*r := add_le_add hyc (by simpa only [dist_comm] using hxc)
+    _ = 2*(2*r) := by ring
+
+lemma prescribed_lower_threshold (κ r w : ℝ) (hr : 0<r) :
+    ((32768*Real.pi)*(2*r))^dimension κ*(16*w) =
+      (16*(65536*Real.pi)^dimension κ)*w*r^Target.d κ := by
+  have hd : Target.d κ=dimension κ := by rw [Target.dimension_eq]; rfl
+  rw [show (32768*Real.pi)*(2*r)=(65536*Real.pi)*r by ring,
+    Real.mul_rpow (by positivity : 0≤65536*Real.pi) hr.le,hd]
+  ring
+
+lemma prescribed_lower_radius (κ : ℝ) (W : Target.WienerLaw) (e : Target.Entry)
+    {z : ℂ} (hz : 0<z.im) :
+    (2*z.im)*Real.exp (-Target.entryPhysicalLevel κ W e (4096*Real.pi) z)=
+      (2048*Real.pi)*(2*Target.entryRadius κ W e) := by
+  rw [Target.entryPhysicalLevel_radius κ W e (by positivity) hz]
+  ring
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCostConvergence.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCostConvergence.lean
index d323e234b0c2dc627fc451cf8261faad17470899..e8ee1cb4e252f949ab82e65ebf459773fec371cf 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCostConvergence.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCostConvergence.lean
@@ -1,87 +1,87 @@
-import OAI.Probability.SLE.Gauge.PrescribedGridError
-
-namespace OAI
-
-/-! The source's pointwise missed-cell bounds imply vanishing expected
-  cleanup cost for the exact prescribed gauge. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance prescribedCostPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance prescribedCostPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-theorem expected_prescribed_missed_cost_le
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    (ν : Ω → Measure ℂ) (hν : Measurable ν)
-    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) (W : Target.WienerLaw)
-    (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ)
-    (C c R : ℝ) (hR : 0 ≤ R) (K : Set ℂ) (hK : IsClosed K)
-    (η ε A : ℝ) (hη : 0 ≤ η) (hε : 0 ≤ ε) (hA : 0 ≤ A)
-    (hr1 : Target.entryRadius κ W (n,none) ≤ 1)
-    (hb : ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
-      P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)} ≤
-        ENNReal.ofReal (A*(Target.entryRadius κ W (n,none))^codimension κ*
-          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*
-          (ε + if (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im<η then 1 else 0))) :
-    (∫⁻ ω,prescribedMissedCost f ν κ W n e C c R K ω ∂P) ≤
-      ENNReal.ofReal ((36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ))))*
-        ((R+3)^(1-codimension κ)*((n:ℝ)*ε)+(n:ℝ)*(η+Target.entryRadius κ W (n,none))^(1-codimension κ))) := by
-  let r := Target.entryRadius κ W (n,none)
-  let b := fun ij => A*r^codimension κ*(prescribedGridCenter r ij).im^(-codimension κ)*
-    (ε + if (prescribedGridCenter r ij).im<η then 1 else 0)
-  have hr : 0<r := Real.exp_pos _
-  have hb0 (ij) (hij : ij∈prescribedGridIndex R r K) : 0 ≤ b ij := by
-    dsimp only [b]
-    have hy := (prescribedGrid_center_height hr hij).2
-    split_ifs <;> positivity
-  calc
-    _ ≤ ENNReal.ofReal (36*(n:ℝ)*r^Target.d κ)*
-        ∑ ij∈prescribedGridIndex R r K, ENNReal.ofReal (b ij) :=
-      expected_prescribed_missed_cost_bound P f hf ν hν κ W n e C c R K hK _ hb
-    _ = ENNReal.ofReal ((36*(n:ℝ)*r^Target.d κ)*∑ ij∈prescribedGridIndex R r K,b ij) := by
-      rw [← ENNReal.ofReal_sum_of_nonneg hb0,← ENNReal.ofReal_mul (by positivity : 0 ≤ 36*(n:ℝ)*r^Target.d κ)]
-    _ ≤ _ := ENNReal.ofReal_le_ofReal
-      (prescribed_cleanup_error_bound κ hκ hκ8 hR hr hr1 hη hε hA K n)
-
-theorem prescribed_missed_cost_tendsto_zero
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    (ν : Ω → Measure ℂ) (hν : Measurable ν)
-    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) (W : Target.WienerLaw)
-    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
-    (C c R : ℝ) (hR : 0 ≤ R) (K : Set ℂ) (hK : IsClosed K)
-    (A : ℝ) (hA : 0 ≤ A)
-    (hb : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
-      P {ω | ij∈prescribedMissedIndices κ W n (e n) C c R K (range (f ω)∩K) (ν ω)} ≤
-        ENNReal.ofReal (A*(Target.entryRadius κ W (n,none))^codimension κ*
-          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*
-          (Real.exp (-((2:ℝ)^n/16)) +
-            if (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im<Real.exp (-(n:ℝ)) then 1 else 0))) :
-    Tendsto (fun n => ∫⁻ ω,prescribedMissedCost f ν κ W n (e n) C c R K ω ∂P)
-      atTop (𝓝 0) := by
-  let D := 36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ)))
-  let F := fun n : ℕ => D*((R+3)^(1-codimension κ)*((n:ℝ)*Real.exp (-((2:ℝ)^n/16)))+
-    (n:ℝ)*(Real.exp (-(n:ℝ))+Target.entryRadius κ W (n,none))^(1-codimension κ))
-  have ht : Tendsto F atTop (𝓝 0) := by
-    have h := (prescribed_batch_failure_cost_tendsto_zero.const_mul ((R+3)^(1-codimension κ))).add
-      (prescribed_boundary_cost_tendsto_zero hκ W)
-    simpa only [mul_zero,add_zero,F] using h.const_mul D
-  have he : Tendsto (fun n => ENNReal.ofReal (F n)) atTop (𝓝 0) := by
-    simpa only [ENNReal.ofReal_zero,Function.comp_def] using (ENNReal.continuous_ofReal.tendsto 0).comp ht
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds he
-  · exact Eventually.of_forall (fun _ => bot_le)
-  · filter_upwards [hb,eventually_ge_atTop 1] with n hn hn1
-    have hr1 : Target.entryRadius κ W (n,none) ≤ 1 :=
-      (Target.entryRadius_le_batch_exp hκ W _).trans
-        (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
-    exact expected_prescribed_missed_cost_le P f hf ν hν hκ hκ8 W n (e n) C c R hR K hK
-      _ _ A (Real.exp_pos _).le (Real.exp_pos _).le hA hr1 hn
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedGridError
+
+namespace OAI
+
+/-! The source's pointwise missed-cell bounds imply vanishing expected
+  cleanup cost for the exact prescribed gauge. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance prescribedCostPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance prescribedCostPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+theorem expected_prescribed_missed_cost_le
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    (ν : Ω → Measure ℂ) (hν : Measurable ν)
+    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) (W : Target.WienerLaw)
+    (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ)
+    (C c R : ℝ) (hR : 0 ≤ R) (K : Set ℂ) (hK : IsClosed K)
+    (η ε A : ℝ) (hη : 0 ≤ η) (hε : 0 ≤ ε) (hA : 0 ≤ A)
+    (hr1 : Target.entryRadius κ W (n,none) ≤ 1)
+    (hb : ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
+      P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)} ≤
+        ENNReal.ofReal (A*(Target.entryRadius κ W (n,none))^codimension κ*
+          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*
+          (ε + if (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im<η then 1 else 0))) :
+    (∫⁻ ω,prescribedMissedCost f ν κ W n e C c R K ω ∂P) ≤
+      ENNReal.ofReal ((36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ))))*
+        ((R+3)^(1-codimension κ)*((n:ℝ)*ε)+(n:ℝ)*(η+Target.entryRadius κ W (n,none))^(1-codimension κ))) := by
+  let r := Target.entryRadius κ W (n,none)
+  let b := fun ij => A*r^codimension κ*(prescribedGridCenter r ij).im^(-codimension κ)*
+    (ε + if (prescribedGridCenter r ij).im<η then 1 else 0)
+  have hr : 0<r := Real.exp_pos _
+  have hb0 (ij) (hij : ij∈prescribedGridIndex R r K) : 0 ≤ b ij := by
+    dsimp only [b]
+    have hy := (prescribedGrid_center_height hr hij).2
+    split_ifs <;> positivity
+  calc
+    _ ≤ ENNReal.ofReal (36*(n:ℝ)*r^Target.d κ)*
+        ∑ ij∈prescribedGridIndex R r K, ENNReal.ofReal (b ij) :=
+      expected_prescribed_missed_cost_bound P f hf ν hν κ W n e C c R K hK _ hb
+    _ = ENNReal.ofReal ((36*(n:ℝ)*r^Target.d κ)*∑ ij∈prescribedGridIndex R r K,b ij) := by
+      rw [← ENNReal.ofReal_sum_of_nonneg hb0,← ENNReal.ofReal_mul (by positivity : 0 ≤ 36*(n:ℝ)*r^Target.d κ)]
+    _ ≤ _ := ENNReal.ofReal_le_ofReal
+      (prescribed_cleanup_error_bound κ hκ hκ8 hR hr hr1 hη hε hA K n)
+
+theorem prescribed_missed_cost_tendsto_zero
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    (ν : Ω → Measure ℂ) (hν : Measurable ν)
+    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) (W : Target.WienerLaw)
+    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
+    (C c R : ℝ) (hR : 0 ≤ R) (K : Set ℂ) (hK : IsClosed K)
+    (A : ℝ) (hA : 0 ≤ A)
+    (hb : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
+      P {ω | ij∈prescribedMissedIndices κ W n (e n) C c R K (range (f ω)∩K) (ν ω)} ≤
+        ENNReal.ofReal (A*(Target.entryRadius κ W (n,none))^codimension κ*
+          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*
+          (Real.exp (-((2:ℝ)^n/16)) +
+            if (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im<Real.exp (-(n:ℝ)) then 1 else 0))) :
+    Tendsto (fun n => ∫⁻ ω,prescribedMissedCost f ν κ W n (e n) C c R K ω ∂P)
+      atTop (𝓝 0) := by
+  let D := 36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ)))
+  let F := fun n : ℕ => D*((R+3)^(1-codimension κ)*((n:ℝ)*Real.exp (-((2:ℝ)^n/16)))+
+    (n:ℝ)*(Real.exp (-(n:ℝ))+Target.entryRadius κ W (n,none))^(1-codimension κ))
+  have ht : Tendsto F atTop (𝓝 0) := by
+    have h := (prescribed_batch_failure_cost_tendsto_zero.const_mul ((R+3)^(1-codimension κ))).add
+      (prescribed_boundary_cost_tendsto_zero hκ W)
+    simpa only [mul_zero,add_zero,F] using h.const_mul D
+  have he : Tendsto (fun n => ENNReal.ofReal (F n)) atTop (𝓝 0) := by
+    simpa only [ENNReal.ofReal_zero,Function.comp_def] using (ENNReal.continuous_ofReal.tendsto 0).comp ht
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds he
+  · exact Eventually.of_forall (fun _ => bot_le)
+  · filter_upwards [hb,eventually_ge_atTop 1] with n hn hn1
+    have hr1 : Target.entryRadius κ W (n,none) ≤ 1 :=
+      (Target.entryRadius_le_batch_exp hκ W _).trans
+        (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
+    exact expected_prescribed_missed_cost_le P f hf ν hν hκ hκ8 W n (e n) C c R hR K hK
+      _ _ A (Real.exp_pos _).le (Real.exp_pos _).le hA hr1 hn
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverEvents.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverEvents.lean
index 69d0bc7c062ea1923382c5ad97b43a596d72e9dd..41f789a3d08c9186412c89b90bc42a2a338d88c7 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverEvents.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverEvents.lean
@@ -1,88 +1,88 @@
-import OAI.Probability.SLE.Gauge.PrescribedCandidateCover
-
-namespace OAI
-
-/-! The finite terminal candidate family has measurable missed-cell cost.
-Only this cost envelope is integrated; the spatial selection itself need
-not be chosen measurably. -/
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance prescribedCoverPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance prescribedCoverPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-private lemma trace_inter_nonempty (f : C(ℝ≥0,ℂ)) (K A : Set ℂ) :
-    ((Set.range f ∩ K) ∩ A).Nonempty ↔ ∃ t, f t∈K∩A := by
-  constructor
-  · rintro ⟨z,⟨⟨t,rfl⟩,hzK⟩,hzA⟩
-    exact ⟨t,hzK,hzA⟩
-  · rintro ⟨t,htK,htA⟩
-    exact ⟨f t,⟨⟨t,rfl⟩,htK⟩,htA⟩
-
-theorem prescribedMissedIndices_measurable
-    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    (ν : Ω → Measure ℂ) (hν : Measurable ν)
-    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
-    (C c R : ℝ) (K : Set ℂ) (hK : IsClosed K) (ij : ℤ×ℤ) :
-    MeasurableSet {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)} := by
-  classical
-  let r := Target.entryRadius κ W (n,none)
-  have hv : MeasurableSet {ω | ((range (f ω)∩K)∩prescribedFineCell r ij).Nonempty} := by
-    simp_rw [trace_inter_nonempty]
-    exact measurableSet_trace_hits_closed f hf (hK.inter isClosed_closedBall)
-  have hm (j : Fin J) : MeasurableSet {ω |
-      ν ω (closedBall (prescribedGridCenter r ij) (C*Target.entryRadius κ W (n,some (e j)))) <
-      ENNReal.ofReal (c*Target.entryValue (n,some (e j))*
-        (Target.entryRadius κ W (n,some (e j)))^Target.d κ)} :=
-    measurableSet_lt ((Measure.measurable_coe isClosed_closedBall.measurableSet).comp hν)
-      measurable_const
-  have hall := MeasurableSet.iInter hm
-  have hpair := hv.inter hall
-  by_cases hij : ij∈prescribedGridIndex R r K
-  · convert hpair using 1
-    ext ω
-    simp only [prescribedMissedIndices,Finset.mem_filter,hij,true_and,mem_inter_iff,
-      mem_iInter,mem_ofPred_eq,r]
-  · have heq : {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}=∅ := by
-      ext ω
-      have hij' : ij∉prescribedGridIndex R (Target.entryRadius κ W (n,none)) K := hij
-      simp only [prescribedMissedIndices,Finset.mem_filter,hij',false_and,mem_ofPred_eq,mem_empty_iff_false]
-    rw [heq]
-    exact MeasurableSet.empty
-
-theorem prescribed_missed_cost_measurable
-    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    (ν : Ω → Measure ℂ) (hν : Measurable ν)
-    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
-    (C c R : ℝ) (K : Set ℂ) (hK : IsClosed K) (a : ℝ≥0∞) :
-    Measurable (fun ω =>
-      (prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)).card*a) := by
-  classical
-  let I := prescribedGridIndex R (Target.entryRadius κ W (n,none)) K
-  let E (ij : ℤ×ℤ) := {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}
-  have hE (ij) : MeasurableSet (E ij) := prescribedMissedIndices_measurable f hf ν hν κ W n e C c R K hK ij
-  have heq (ω) :
-      (prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)).card*a =
-      ∑ ij∈I, (E ij).indicator (fun _ => a) ω := by
-    rw [← nsmul_eq_mul,← Finset.sum_const]
-    rw [prescribedMissedIndices,Finset.sum_filter]
-    apply Finset.sum_congr rfl
-    intro ij hij
-    by_cases h : ω∈E ij
-    · rw [indicator_of_mem h]
-      rw [ite_eq_left]
-      exact (Finset.mem_filter.mp h).2
-    · rw [indicator_of_notMem h]
-      rw [ite_eq_right]
-      intro hp
-      exact h (Finset.mem_filter.mpr ⟨hij,hp⟩)
-  simp_rw [heq]
-  exact Finset.measurable_sum I (fun ij _ => measurable_const.indicator (hE ij))
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedCandidateCover
+
+namespace OAI
+
+/-! The finite terminal candidate family has measurable missed-cell cost.
+Only this cost envelope is integrated; the spatial selection itself need
+not be chosen measurably. -/
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance prescribedCoverPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance prescribedCoverPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+private lemma trace_inter_nonempty (f : C(ℝ≥0,ℂ)) (K A : Set ℂ) :
+    ((Set.range f ∩ K) ∩ A).Nonempty ↔ ∃ t, f t∈K∩A := by
+  constructor
+  · rintro ⟨z,⟨⟨t,rfl⟩,hzK⟩,hzA⟩
+    exact ⟨t,hzK,hzA⟩
+  · rintro ⟨t,htK,htA⟩
+    exact ⟨f t,⟨⟨t,rfl⟩,htK⟩,htA⟩
+
+theorem prescribedMissedIndices_measurable
+    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    (ν : Ω → Measure ℂ) (hν : Measurable ν)
+    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
+    (C c R : ℝ) (K : Set ℂ) (hK : IsClosed K) (ij : ℤ×ℤ) :
+    MeasurableSet {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)} := by
+  classical
+  let r := Target.entryRadius κ W (n,none)
+  have hv : MeasurableSet {ω | ((range (f ω)∩K)∩prescribedFineCell r ij).Nonempty} := by
+    simp_rw [trace_inter_nonempty]
+    exact measurableSet_trace_hits_closed f hf (hK.inter isClosed_closedBall)
+  have hm (j : Fin J) : MeasurableSet {ω |
+      ν ω (closedBall (prescribedGridCenter r ij) (C*Target.entryRadius κ W (n,some (e j)))) <
+      ENNReal.ofReal (c*Target.entryValue (n,some (e j))*
+        (Target.entryRadius κ W (n,some (e j)))^Target.d κ)} :=
+    measurableSet_lt ((Measure.measurable_coe isClosed_closedBall.measurableSet).comp hν)
+      measurable_const
+  have hall := MeasurableSet.iInter hm
+  have hpair := hv.inter hall
+  by_cases hij : ij∈prescribedGridIndex R r K
+  · convert hpair using 1
+    ext ω
+    simp only [prescribedMissedIndices,Finset.mem_filter,hij,true_and,mem_inter_iff,
+      mem_iInter,mem_ofPred_eq,r]
+  · have heq : {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}=∅ := by
+      ext ω
+      have hij' : ij∉prescribedGridIndex R (Target.entryRadius κ W (n,none)) K := hij
+      simp only [prescribedMissedIndices,Finset.mem_filter,hij',false_and,mem_ofPred_eq,mem_empty_iff_false]
+    rw [heq]
+    exact MeasurableSet.empty
+
+theorem prescribed_missed_cost_measurable
+    {Ω : Type*} [MeasurableSpace Ω] (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    (ν : Ω → Measure ℂ) (hν : Measurable ν)
+    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
+    (C c R : ℝ) (K : Set ℂ) (hK : IsClosed K) (a : ℝ≥0∞) :
+    Measurable (fun ω =>
+      (prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)).card*a) := by
+  classical
+  let I := prescribedGridIndex R (Target.entryRadius κ W (n,none)) K
+  let E (ij : ℤ×ℤ) := {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}
+  have hE (ij) : MeasurableSet (E ij) := prescribedMissedIndices_measurable f hf ν hν κ W n e C c R K hK ij
+  have heq (ω) :
+      (prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)).card*a =
+      ∑ ij∈I, (E ij).indicator (fun _ => a) ω := by
+    rw [← nsmul_eq_mul,← Finset.sum_const]
+    rw [prescribedMissedIndices,Finset.sum_filter]
+    apply Finset.sum_congr rfl
+    intro ij hij
+    by_cases h : ω∈E ij
+    · rw [indicator_of_mem h]
+      rw [ite_eq_left]
+      exact (Finset.mem_filter.mp h).2
+    · rw [indicator_of_notMem h]
+      rw [ite_eq_right]
+      intro hp
+      exact h (Finset.mem_filter.mpr ⟨hij,hp⟩)
+  simp_rw [heq]
+  exact Finset.measurable_sum I (fun ij _ => measurable_const.indicator (hE ij))
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverExpectation.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverExpectation.lean
index 2aeec0d340445d8d8e26e1b2c40570d0115e1aac..30ed36bcb92976bd579b40bad137857cf5384079 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverExpectation.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverExpectation.lean
@@ -1,76 +1,76 @@
-import OAI.Probability.SLE.Gauge.PrescribedCoverEvents
-import OAI.Probability.SLE.Gauge.PrescribedCoverLimits
-import OAI.Probability.SLE.Geometry.BoundedTerminalMean
-
-namespace OAI
-
-/-! The actual prescribed finite covers reduce the Hausdorff expectation
-to the fixed terminal-mass charge and vanishing expected missed-cell cost.
-The stochastic missed-cell estimate is kept explicit in this reduction. -/
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance prescribedExpectationPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance prescribedExpectationPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-def prescribedMissedCost {Ω : Type*} (f : Ω → C(ℝ≥0,ℂ)) (ν : Ω → Measure ℂ)
-    (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ)
-    (C c R : ℝ) (K : Set ℂ) (ω : Ω) : ℝ≥0∞ :=
-  (prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)).card *
-    ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ)
-
-theorem expected_prescribed_hausdorff_le_of_missed_cost
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw)
-    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
-    (hm1 : 0<Target.moment κ W 1)
-    (hJ : ∀ n, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
-    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    (ν : Ω → Measure ℂ) (hν : Measurable ν)
-    (C c R : ℝ) (hC : 3 ≤ C) (hc : 0<c) (hR : 0 ≤ R)
-    (K : Set ℂ) (hK : IsClosed K) (hKR : K ⊆ closedBall 0 R)
-    (hH : ∀ z∈K, 0 ≤ z.im)
-    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
-    (he : ∀ n j, (e n j).2<Target.multiplicity κ W (n+1) (e n j).1)
-    (herr : Tendsto (fun n => ∫⁻ ω, prescribedMissedCost f ν κ W (n+1) (e n) C c R K ω ∂P)
-      atTop (𝓝 0)) :
-    (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (f ω)∩K) ∂P) ≤
-      ENNReal.ofReal ((8*C)^2/c)*(∫⁻ ω, ν ω (closedBall 0 (R+3+C)) ∂P) := by
-  let A := ENNReal.ofReal ((8*C)^2/c)
-  let Q := closedBall (0:ℂ) (R+3+C)
-  let error (n : ℕ) := prescribedMissedCost f ν κ W (n+1) (e n) C c R K
-  let Z (n : ℕ) (ω : Ω) := A*ν ω Q + error n ω
-  have hm (n : ℕ) : Measurable (error n) :=
-    prescribed_missed_cost_measurable f hf ν hν κ W (n+1) (e n) C c R K hK _
-  have hZ (n : ℕ) : AEMeasurable (Z n) P :=
-    ((measurable_const.mul ((Measure.measurable_coe isClosed_closedBall.measurableSet).comp hν)).add (hm n)).aemeasurable
-  let δ (n : ℕ) := ENNReal.ofReal (8*C*Real.exp (-((n+1:ℕ):ℝ)^2))
-  have hδ : Tendsto δ atTop (𝓝 0) := by
-    have h := (ENNReal.continuous_ofReal.tendsto 0).comp
-      ((prescribed_cover_diameter_tendsto_zero C).comp (tendsto_add_atTop_nat 1))
-    simpa only [ENNReal.ofReal_zero,Function.comp_def] using h
-  have hcover : ∀ᵐ ω ∂P, ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
-      range (f ω)∩K ⊆ ⋃ i,U i ∧ (∀ i,ediam (U i) ≤ δ n) ∧
-        ∑ i, extendedGauge (Target.gauge κ W) (ediam (U i)) ≤ Z n ω := by
-    apply Eventually.of_forall
-    intro ω n
-    exact prescribed_candidates_finite_cover hκ W hg (by omega) hm1 (hJ (n+1) (by omega))
-      (e n) (he n) C c R hC hc hR K (range (f ω)∩K) hKR hH inter_subset_right (ν ω)
-  have hcost : Tendsto (fun n => ∫⁻ ω, Z n ω ∂P) atTop
-      (𝓝 (A*(∫⁻ ω,ν ω Q ∂P))) := by
-    have heq (n) : (∫⁻ ω,Z n ω ∂P) = A*(∫⁻ ω,ν ω Q ∂P) + ∫⁻ ω,error n ω ∂P := by
-      dsimp only [Z]
-      rw [lintegral_add_right' _ (hm n).aemeasurable,lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
-    simp_rw [heq]
-    simpa only [add_zero] using tendsto_const_nhds.add herr
-  simp_rw [Target.hausdorff_eq_hausdorffGauge]
-  exact expected_hausdorff_le_of_finite_cover_envelopes P (Target.gauge κ W)
-    (fun ω => range (f ω)∩K) δ hδ Z hZ hcover _ (le_of_eq hcost.liminf_eq)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedCoverEvents
+import OAI.Probability.SLE.Gauge.PrescribedCoverLimits
+import OAI.Probability.SLE.Geometry.BoundedTerminalMean
+
+namespace OAI
+
+/-! The actual prescribed finite covers reduce the Hausdorff expectation
+to the fixed terminal-mass charge and vanishing expected missed-cell cost.
+The stochastic missed-cell estimate is kept explicit in this reduction. -/
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance prescribedExpectationPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance prescribedExpectationPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+def prescribedMissedCost {Ω : Type*} (f : Ω → C(ℝ≥0,ℂ)) (ν : Ω → Measure ℂ)
+    (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) {J : ℕ} (e : Fin J → ℤ×ℕ)
+    (C c R : ℝ) (K : Set ℂ) (ω : Ω) : ℝ≥0∞ :=
+  (prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)).card *
+    ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ)
+
+theorem expected_prescribed_hausdorff_le_of_missed_cost
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw)
+    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
+    (hm1 : 0<Target.moment κ W 1)
+    (hJ : ∀ n, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
+    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    (ν : Ω → Measure ℂ) (hν : Measurable ν)
+    (C c R : ℝ) (hC : 3 ≤ C) (hc : 0<c) (hR : 0 ≤ R)
+    (K : Set ℂ) (hK : IsClosed K) (hKR : K ⊆ closedBall 0 R)
+    (hH : ∀ z∈K, 0 ≤ z.im)
+    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
+    (he : ∀ n j, (e n j).2<Target.multiplicity κ W (n+1) (e n j).1)
+    (herr : Tendsto (fun n => ∫⁻ ω, prescribedMissedCost f ν κ W (n+1) (e n) C c R K ω ∂P)
+      atTop (𝓝 0)) :
+    (∫⁻ ω, Target.hausdorff (Target.gauge κ W) (range (f ω)∩K) ∂P) ≤
+      ENNReal.ofReal ((8*C)^2/c)*(∫⁻ ω, ν ω (closedBall 0 (R+3+C)) ∂P) := by
+  let A := ENNReal.ofReal ((8*C)^2/c)
+  let Q := closedBall (0:ℂ) (R+3+C)
+  let error (n : ℕ) := prescribedMissedCost f ν κ W (n+1) (e n) C c R K
+  let Z (n : ℕ) (ω : Ω) := A*ν ω Q + error n ω
+  have hm (n : ℕ) : Measurable (error n) :=
+    prescribed_missed_cost_measurable f hf ν hν κ W (n+1) (e n) C c R K hK _
+  have hZ (n : ℕ) : AEMeasurable (Z n) P :=
+    ((measurable_const.mul ((Measure.measurable_coe isClosed_closedBall.measurableSet).comp hν)).add (hm n)).aemeasurable
+  let δ (n : ℕ) := ENNReal.ofReal (8*C*Real.exp (-((n+1:ℕ):ℝ)^2))
+  have hδ : Tendsto δ atTop (𝓝 0) := by
+    have h := (ENNReal.continuous_ofReal.tendsto 0).comp
+      ((prescribed_cover_diameter_tendsto_zero C).comp (tendsto_add_atTop_nat 1))
+    simpa only [ENNReal.ofReal_zero,Function.comp_def] using h
+  have hcover : ∀ᵐ ω ∂P, ∀ n, ∃ (N : ℕ) (U : Fin N → Set ℂ),
+      range (f ω)∩K ⊆ ⋃ i,U i ∧ (∀ i,ediam (U i) ≤ δ n) ∧
+        ∑ i, extendedGauge (Target.gauge κ W) (ediam (U i)) ≤ Z n ω := by
+    apply Eventually.of_forall
+    intro ω n
+    exact prescribed_candidates_finite_cover hκ W hg (by omega) hm1 (hJ (n+1) (by omega))
+      (e n) (he n) C c R hC hc hR K (range (f ω)∩K) hKR hH inter_subset_right (ν ω)
+  have hcost : Tendsto (fun n => ∫⁻ ω, Z n ω ∂P) atTop
+      (𝓝 (A*(∫⁻ ω,ν ω Q ∂P))) := by
+    have heq (n) : (∫⁻ ω,Z n ω ∂P) = A*(∫⁻ ω,ν ω Q ∂P) + ∫⁻ ω,error n ω ∂P := by
+      dsimp only [Z]
+      rw [lintegral_add_right' _ (hm n).aemeasurable,lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+    simp_rw [heq]
+    simpa only [add_zero] using tendsto_const_nhds.add herr
+  simp_rw [Target.hausdorff_eq_hausdorffGauge]
+  exact expected_hausdorff_le_of_finite_cover_envelopes P (Target.gauge κ W)
+    (fun ω => range (f ω)∩K) δ hδ Z hZ hcover _ (le_of_eq hcost.liminf_eq)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverGrid.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverGrid.lean
index cdfea04e61be219597ba4362ad1bc3e53efa9abf..393220ae7762684ebf80eb6dc53bb3f20b01d33d 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverGrid.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverGrid.lean
@@ -1,84 +1,84 @@
-import OAI.Probability.SLE.Gauge.CoverGrid
-
-namespace OAI
-
-/-! The final mesh is shifted by half a row. Every retained center is in
-the open half-plane, while its closed cell can cover the real boundary. -/
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def prescribedGridCenter (r : ℝ) (ij : ℤ × ℤ) : ℂ :=
-  coverGridCenter r ij + (r/2:ℝ)*Complex.I
-
-def prescribedFineCell (r : ℝ) (ij : ℤ × ℤ) : Set ℂ :=
-  closedBall (prescribedGridCenter r ij) (3*r)
-
-def prescribedGridIndex (R r : ℝ) (K : Set ℂ) : Finset (ℤ × ℤ) := by
-  classical
-  exact (coverGridIndex R r).filter (fun ij =>
-    0 ≤ ij.2 ∧ (K ∩ prescribedFineCell r ij).Nonempty)
-
-lemma prescribedGrid_floor_distance {r : ℝ} (hr : 0<r) (z : ℂ) :
-    dist z (prescribedGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋))  ≤  3*r := by
-  apply (dist_triangle z (coverGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋)) _).trans
-  have hdist : dist (coverGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋))
-      (prescribedGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋)) = r/2 := by
-    rw [prescribedGridCenter,dist_self_add_right,norm_mul,Complex.norm_real,
-      Complex.norm_I,mul_one,Real.norm_eq_abs,abs_of_pos (by positivity)]
-  rw [hdist]
-  linarith [coverGrid_floor_distance hr z]
-
-lemma prescribedGrid_covers {R r : ℝ} (hR : 0 ≤ R) (hr : 0<r)
-    {K : Set ℂ} (hK : K ⊆ closedBall 0 R) (hH : ∀ z∈K, 0 ≤ z.im) :
-    K ⊆ ⋃ ij∈prescribedGridIndex R r K, prescribedFineCell r ij := by
-  classical
-  intro z hz
-  let ij : ℤ × ℤ := (⌊z.re/r⌋,⌊z.im/r⌋)
-  have hnorm : ‖z‖ ≤ R := by simpa only [mem_closedBall,dist_zero_right] using hK hz
-  have hre := coverGrid_coordinate hR hr ((Complex.abs_re_le_norm z).trans hnorm)
-  have him := coverGrid_coordinate hR hr ((Complex.abs_im_le_norm z).trans hnorm)
-  have hmem : ij∈coverGridIndex R r := by
-    simpa only [ij,coverGridIndex,Finset.mem_product,Finset.mem_Icc] using And.intro hre him
-  have hrow : 0 ≤ ij.2 := Int.floor_nonneg.mpr (div_nonneg (hH z hz) hr.le)
-  have hcell : z∈prescribedFineCell r ij := prescribedGrid_floor_distance hr z
-  exact mem_iUnion₂.mpr ⟨ij,Finset.mem_filter.mpr ⟨hmem,hrow,⟨z,hz,hcell⟩⟩,hcell⟩
-
-lemma prescribedGrid_center_height {R r : ℝ} (hr : 0<r) {K : Set ℂ}
-    {ij : ℤ × ℤ} (hij : ij∈prescribedGridIndex R r K) :
-    (prescribedGridCenter r ij).im = ((ij.2:ℝ)+1/2)*r ∧
-      0<(prescribedGridCenter r ij).im := by
-  classical
-  have hrow : (0:ℝ) ≤ ij.2 := by exact_mod_cast (Finset.mem_filter.mp hij).2.1
-  have he : (prescribedGridCenter r ij).im = ((ij.2:ℝ)+1/2)*r := by
-    simp only [prescribedGridCenter,coverGridCenter,Complex.add_im,Complex.ofReal_im,
-      Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero,
-      add_zero,zero_add]
-    ring
-  exact ⟨he,by rw [he]; positivity⟩
-
-lemma prescribedGrid_center_bound {R r : ℝ} {K : Set ℂ}
-    (hK : K ⊆ closedBall 0 R) {ij : ℤ × ℤ} (hij : ij∈prescribedGridIndex R r K) :
-    ‖prescribedGridCenter r ij‖  ≤  R+3*r := by
-  classical
-  obtain ⟨z,hz,hcell⟩ := (Finset.mem_filter.mp hij).2.2
-  have hnorm : ‖z‖ ≤ R := by simpa only [mem_closedBall,dist_zero_right] using hK hz
-  have hdist : dist (prescribedGridCenter r ij) z ≤ 3*r := by
-    simpa only [prescribedFineCell,mem_closedBall,dist_comm] using hcell
-  have ht := dist_triangle (prescribedGridCenter r ij) z 0
-  simp only [dist_zero_right] at ht
-  linarith
-
-lemma prescribedGrid_card_bound {R r : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1)
-    (K : Set ℂ) :
-    ((prescribedGridIndex R r K).card:ℝ)  ≤  (2*R+5)^2*r^(-2:ℝ) := by
-  classical
-  exact (Nat.cast_le.mpr (Finset.card_filter_le _ _)).trans (coverGrid_card_bound hR hr hr1)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverGrid
+
+namespace OAI
+
+/-! The final mesh is shifted by half a row. Every retained center is in
+the open half-plane, while its closed cell can cover the real boundary. -/
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def prescribedGridCenter (r : ℝ) (ij : ℤ × ℤ) : ℂ :=
+  coverGridCenter r ij + (r/2:ℝ)*Complex.I
+
+def prescribedFineCell (r : ℝ) (ij : ℤ × ℤ) : Set ℂ :=
+  closedBall (prescribedGridCenter r ij) (3*r)
+
+def prescribedGridIndex (R r : ℝ) (K : Set ℂ) : Finset (ℤ × ℤ) := by
+  classical
+  exact (coverGridIndex R r).filter (fun ij =>
+    0 ≤ ij.2 ∧ (K ∩ prescribedFineCell r ij).Nonempty)
+
+lemma prescribedGrid_floor_distance {r : ℝ} (hr : 0<r) (z : ℂ) :
+    dist z (prescribedGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋))  ≤  3*r := by
+  apply (dist_triangle z (coverGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋)) _).trans
+  have hdist : dist (coverGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋))
+      (prescribedGridCenter r (⌊z.re/r⌋,⌊z.im/r⌋)) = r/2 := by
+    rw [prescribedGridCenter,dist_self_add_right,norm_mul,Complex.norm_real,
+      Complex.norm_I,mul_one,Real.norm_eq_abs,abs_of_pos (by positivity)]
+  rw [hdist]
+  linarith [coverGrid_floor_distance hr z]
+
+lemma prescribedGrid_covers {R r : ℝ} (hR : 0 ≤ R) (hr : 0<r)
+    {K : Set ℂ} (hK : K ⊆ closedBall 0 R) (hH : ∀ z∈K, 0 ≤ z.im) :
+    K ⊆ ⋃ ij∈prescribedGridIndex R r K, prescribedFineCell r ij := by
+  classical
+  intro z hz
+  let ij : ℤ × ℤ := (⌊z.re/r⌋,⌊z.im/r⌋)
+  have hnorm : ‖z‖ ≤ R := by simpa only [mem_closedBall,dist_zero_right] using hK hz
+  have hre := coverGrid_coordinate hR hr ((Complex.abs_re_le_norm z).trans hnorm)
+  have him := coverGrid_coordinate hR hr ((Complex.abs_im_le_norm z).trans hnorm)
+  have hmem : ij∈coverGridIndex R r := by
+    simpa only [ij,coverGridIndex,Finset.mem_product,Finset.mem_Icc] using And.intro hre him
+  have hrow : 0 ≤ ij.2 := Int.floor_nonneg.mpr (div_nonneg (hH z hz) hr.le)
+  have hcell : z∈prescribedFineCell r ij := prescribedGrid_floor_distance hr z
+  exact mem_iUnion₂.mpr ⟨ij,Finset.mem_filter.mpr ⟨hmem,hrow,⟨z,hz,hcell⟩⟩,hcell⟩
+
+lemma prescribedGrid_center_height {R r : ℝ} (hr : 0<r) {K : Set ℂ}
+    {ij : ℤ × ℤ} (hij : ij∈prescribedGridIndex R r K) :
+    (prescribedGridCenter r ij).im = ((ij.2:ℝ)+1/2)*r ∧
+      0<(prescribedGridCenter r ij).im := by
+  classical
+  have hrow : (0:ℝ) ≤ ij.2 := by exact_mod_cast (Finset.mem_filter.mp hij).2.1
+  have he : (prescribedGridCenter r ij).im = ((ij.2:ℝ)+1/2)*r := by
+    simp only [prescribedGridCenter,coverGridCenter,Complex.add_im,Complex.ofReal_im,
+      Complex.mul_im,Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero,
+      add_zero,zero_add]
+    ring
+  exact ⟨he,by rw [he]; positivity⟩
+
+lemma prescribedGrid_center_bound {R r : ℝ} {K : Set ℂ}
+    (hK : K ⊆ closedBall 0 R) {ij : ℤ × ℤ} (hij : ij∈prescribedGridIndex R r K) :
+    ‖prescribedGridCenter r ij‖  ≤  R+3*r := by
+  classical
+  obtain ⟨z,hz,hcell⟩ := (Finset.mem_filter.mp hij).2.2
+  have hnorm : ‖z‖ ≤ R := by simpa only [mem_closedBall,dist_zero_right] using hK hz
+  have hdist : dist (prescribedGridCenter r ij) z ≤ 3*r := by
+    simpa only [prescribedFineCell,mem_closedBall,dist_comm] using hcell
+  have ht := dist_triangle (prescribedGridCenter r ij) z 0
+  simp only [dist_zero_right] at ht
+  linarith
+
+lemma prescribedGrid_card_bound {R r : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1)
+    (K : Set ℂ) :
+    ((prescribedGridIndex R r K).card:ℝ)  ≤  (2*R+5)^2*r^(-2:ℝ) := by
+  classical
+  exact (Nat.cast_le.mpr (Finset.card_filter_le _ _)).trans (coverGrid_card_bound hR hr hr1)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverLimits.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverLimits.lean
index 9f7faaa8ee6464ac960e5a5d9afa11a545c5479d..e12a2b7d12e869d6e02fec424d8464c4d308bb98 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedCoverLimits.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedCoverLimits.lean
@@ -1,91 +1,91 @@
-import OAI.Probability.SLE.Gauge.GaugeEntrySmallness
-import OAI.Probability.SLE.Gauge.CoverNumerics
-
-namespace OAI
-
-/-! The missed-cell error in the prescribed cover vanishes, including
-the bottom rows whose centers approach the real boundary. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma prescribed_cleanup_radius_le_exp {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw)
-    {n : ℕ} (hn : 1 ≤ n) :
-    Target.entryRadius κ W (n,none) ≤ Real.exp (-(n:ℝ)) := by
-  refine (Target.entryRadius_le_batch_exp hκ W _).trans ?_
-  apply Real.exp_le_exp.mpr
-  have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn
-  nlinarith
-
-lemma prescribed_cover_diameter_tendsto_zero (C : ℝ) :
-    Tendsto (fun n : ℕ => 8*C*Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
-  have he : Tendsto (fun n : ℕ => Real.exp (-(n:ℝ))) atTop (𝓝 0) := by
-    simpa only [Real.rpow_zero,one_mul,neg_mul,Function.comp_def] using
-      (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 0 1 (by norm_num)).comp
-        tendsto_natCast_atTop_atTop
-  have hsq : Tendsto (fun n : ℕ => Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
-    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds he
-    · exact Eventually.of_forall (fun _ => (Real.exp_pos _).le)
-    · filter_upwards [eventually_ge_atTop 1] with n hn
-      have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn
-      exact Real.exp_le_exp.mpr (by nlinarith)
-  simpa only [mul_zero] using hsq.const_mul (8*C)
-
-lemma prescribed_batch_failure_cost_tendsto_zero :
-    Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-((2:ℝ)^n/16))) atTop (𝓝 0) := by
-  have ht : Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-(1/16:ℝ)*(n:ℝ))) atTop (𝓝 0) := by
-    simpa only [Real.rpow_one,Function.comp_def] using
-      (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 (1/16) (by norm_num)).comp
-        tendsto_natCast_atTop_atTop
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds ht
-  · exact Eventually.of_forall (fun n => mul_nonneg (Nat.cast_nonneg n) (Real.exp_pos _).le)
-  · apply Eventually.of_forall
-    intro n
-    apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
-    apply Real.exp_le_exp.mpr
-    have hn : (n:ℝ) ≤ (2:ℝ)^n := by exact_mod_cast (Nat.lt_two_pow_self (n := n)).le
-    linarith
-
-lemma prescribed_boundary_cost_tendsto_zero {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw) :
-    Tendsto (fun n : ℕ => (n:ℝ)*(Real.exp (-(n:ℝ))+
-      Target.entryRadius κ W (n,none))^(1-codimension κ)) atTop (𝓝 0) := by
-  let q := 1-codimension κ
-  have hq : 0<q := by dsimp only [q,codimension,dimension]; linarith
-  have ht : Tendsto (fun n : ℕ => (2:ℝ)^q*((n:ℝ)*Real.exp (-q*(n:ℝ)))) atTop (𝓝 0) := by
-    have h := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 q hq).comp
-      tendsto_natCast_atTop_atTop
-    simpa only [Real.rpow_one,Function.comp_def,mul_zero] using h.const_mul ((2:ℝ)^q)
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds ht
-  · apply Eventually.of_forall
-    intro n
-    exact mul_nonneg (Nat.cast_nonneg n) (Real.rpow_nonneg
-      (add_nonneg (Real.exp_pos _).le (show 0 ≤ Target.entryRadius κ W (n,none) from (Real.exp_pos _).le)) _)
-  · filter_upwards [eventually_ge_atTop 1] with n hn
-    have hr := prescribed_cleanup_radius_le_exp hκ W hn
-    have hrad : 0 < Target.entryRadius κ W (n,none) := Real.exp_pos _
-    have hp : (Real.exp (-(n:ℝ))+Target.entryRadius κ W (n,none))^q ≤
-        (2*Real.exp (-(n:ℝ)))^q := by
-      apply Real.rpow_le_rpow (by positivity) (by linarith) hq.le
-    have he : (2*Real.exp (-(n:ℝ)))^q = (2:ℝ)^q*Real.exp (-q*(n:ℝ)) := by
-      rw [Real.mul_rpow (by norm_num) (Real.exp_pos _).le,← Real.exp_mul]
-      congr 2
-      ring
-    rw [he] at hp
-    calc
-      _ ≤ (n:ℝ)*((2:ℝ)^q*Real.exp (-q*(n:ℝ))) := mul_le_mul_of_nonneg_left hp (Nat.cast_nonneg n)
-      _ = _ := by ring
-
-lemma prescribed_cleanup_error_tendsto_zero {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw) :
-    Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-((2:ℝ)^n/16)) +
-      (n:ℝ)*(Real.exp (-(n:ℝ))+Target.entryRadius κ W (n,none))^(1-codimension κ))
-        atTop (𝓝 0) := by
-  simpa only [add_zero] using prescribed_batch_failure_cost_tendsto_zero.add
-    (prescribed_boundary_cost_tendsto_zero hκ W)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.GaugeEntrySmallness
+import OAI.Probability.SLE.Gauge.CoverNumerics
+
+namespace OAI
+
+/-! The missed-cell error in the prescribed cover vanishes, including
+the bottom rows whose centers approach the real boundary. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma prescribed_cleanup_radius_le_exp {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw)
+    {n : ℕ} (hn : 1 ≤ n) :
+    Target.entryRadius κ W (n,none) ≤ Real.exp (-(n:ℝ)) := by
+  refine (Target.entryRadius_le_batch_exp hκ W _).trans ?_
+  apply Real.exp_le_exp.mpr
+  have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn
+  nlinarith
+
+lemma prescribed_cover_diameter_tendsto_zero (C : ℝ) :
+    Tendsto (fun n : ℕ => 8*C*Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
+  have he : Tendsto (fun n : ℕ => Real.exp (-(n:ℝ))) atTop (𝓝 0) := by
+    simpa only [Real.rpow_zero,one_mul,neg_mul,Function.comp_def] using
+      (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 0 1 (by norm_num)).comp
+        tendsto_natCast_atTop_atTop
+  have hsq : Tendsto (fun n : ℕ => Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
+    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds he
+    · exact Eventually.of_forall (fun _ => (Real.exp_pos _).le)
+    · filter_upwards [eventually_ge_atTop 1] with n hn
+      have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn
+      exact Real.exp_le_exp.mpr (by nlinarith)
+  simpa only [mul_zero] using hsq.const_mul (8*C)
+
+lemma prescribed_batch_failure_cost_tendsto_zero :
+    Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-((2:ℝ)^n/16))) atTop (𝓝 0) := by
+  have ht : Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-(1/16:ℝ)*(n:ℝ))) atTop (𝓝 0) := by
+    simpa only [Real.rpow_one,Function.comp_def] using
+      (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 (1/16) (by norm_num)).comp
+        tendsto_natCast_atTop_atTop
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds ht
+  · exact Eventually.of_forall (fun n => mul_nonneg (Nat.cast_nonneg n) (Real.exp_pos _).le)
+  · apply Eventually.of_forall
+    intro n
+    apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
+    apply Real.exp_le_exp.mpr
+    have hn : (n:ℝ) ≤ (2:ℝ)^n := by exact_mod_cast (Nat.lt_two_pow_self (n := n)).le
+    linarith
+
+lemma prescribed_boundary_cost_tendsto_zero {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw) :
+    Tendsto (fun n : ℕ => (n:ℝ)*(Real.exp (-(n:ℝ))+
+      Target.entryRadius κ W (n,none))^(1-codimension κ)) atTop (𝓝 0) := by
+  let q := 1-codimension κ
+  have hq : 0<q := by dsimp only [q,codimension,dimension]; linarith
+  have ht : Tendsto (fun n : ℕ => (2:ℝ)^q*((n:ℝ)*Real.exp (-q*(n:ℝ)))) atTop (𝓝 0) := by
+    have h := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 q hq).comp
+      tendsto_natCast_atTop_atTop
+    simpa only [Real.rpow_one,Function.comp_def,mul_zero] using h.const_mul ((2:ℝ)^q)
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds ht
+  · apply Eventually.of_forall
+    intro n
+    exact mul_nonneg (Nat.cast_nonneg n) (Real.rpow_nonneg
+      (add_nonneg (Real.exp_pos _).le (show 0 ≤ Target.entryRadius κ W (n,none) from (Real.exp_pos _).le)) _)
+  · filter_upwards [eventually_ge_atTop 1] with n hn
+    have hr := prescribed_cleanup_radius_le_exp hκ W hn
+    have hrad : 0 < Target.entryRadius κ W (n,none) := Real.exp_pos _
+    have hp : (Real.exp (-(n:ℝ))+Target.entryRadius κ W (n,none))^q ≤
+        (2*Real.exp (-(n:ℝ)))^q := by
+      apply Real.rpow_le_rpow (by positivity) (by linarith) hq.le
+    have he : (2*Real.exp (-(n:ℝ)))^q = (2:ℝ)^q*Real.exp (-q*(n:ℝ)) := by
+      rw [Real.mul_rpow (by norm_num) (Real.exp_pos _).le,← Real.exp_mul]
+      congr 2
+      ring
+    rw [he] at hp
+    calc
+      _ ≤ (n:ℝ)*((2:ℝ)^q*Real.exp (-q*(n:ℝ))) := mul_le_mul_of_nonneg_left hp (Nat.cast_nonneg n)
+      _ = _ := by ring
+
+lemma prescribed_cleanup_error_tendsto_zero {κ : ℝ} (hκ : 0<κ) (W : Target.WienerLaw) :
+    Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-((2:ℝ)^n/16)) +
+      (n:ℝ)*(Real.exp (-(n:ℝ))+Target.entryRadius κ W (n,none))^(1-codimension κ))
+        atTop (𝓝 0) := by
+  simpa only [add_zero] using prescribed_batch_failure_cost_tendsto_zero.add
+    (prescribed_boundary_cost_tendsto_zero hκ W)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedDiskConditional.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedDiskConditional.lean
index 77555adab93ff73919bd40b0cd58a7f766f858e1..7c6b00ed1618ecbcefd3de3387e70a9e06cb0a5a 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedDiskConditional.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedDiskConditional.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Gauge.PrescribedDiskTestKernel
-import OAI.Probability.SLE.Geometry.PastDensityIntegral
-
-namespace OAI
-
-/-! The normalized disk success probability is a genuine conditional lower
-bound under the actual finite centered law, tested against its full past. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance diskConditionalMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance diskConditionalBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-theorem CanonicalMass.prescribed_disk_conditional_bound
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
-    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
-    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t =>
-      (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable))
-      (fun ω => (τ ω:WithTop ℝ≥0)))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b : ℝ)
-    (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r)
-    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
-    (hlive : ∀ ω∈E, z ∈ survivingDomain (γ ω) (τ ω))
-    (hbase : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u))
-    (hR : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2048*Real.pi)*r) :
-    ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
-      (∫⁻ ω in A∩E, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v
-        (stoppedMarkedCenter κ B γ G τ z ω).arg
-        ∂finiteCenteredLaw P κ B γ G z (u+v)) ≤
-      finiteCenteredLaw P κ B γ G z (u+v)
-        (A∩E∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v))
-          z ((4096*Real.pi)*r)
-          (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by
-  obtain ⟨M,hM,_hMf,hMp,hbound⟩ := C.prescribed_disk_test_kernel hB hc h0 hκ hκ8
-    F hw R he D hD hDf hDp hgrowth hpublished τ hτ z hz u v r b hu hv hr E hE hlive hbase hR
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  have hq : Measurable[hτ.measurableSpace] (fun ω =>
-      R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v
-        (stoppedMarkedCenter κ B γ G τ z ω).arg) := by
-    let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) :=
-      (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
-    exact (R.angleDiskTail_measurable _ κ b v).comp
-      (source_stoppedMarkedCenter_measurable B hm hc κ γ G F hw τ hτ z).carg
-  intro A hA
-  rw [← past_density_test_integral P (finiteCenteredLaw P κ B γ G z (u+v))
-    hτ.measurableSpace hτ.measurableSpace_le M hM E hE hMp _ hq A hA]
-  exact hbound A hA
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedDiskTestKernel
+import OAI.Probability.SLE.Geometry.PastDensityIntegral
+
+namespace OAI
+
+/-! The normalized disk success probability is a genuine conditional lower
+bound under the actual finite centered law, tested against its full past. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance diskConditionalMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance diskConditionalBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+theorem CanonicalMass.prescribed_disk_conditional_bound
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
+    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
+    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t =>
+      (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable))
+      (fun ω => (τ ω:WithTop ℝ≥0)))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b : ℝ)
+    (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r)
+    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
+    (hlive : ∀ ω∈E, z ∈ survivingDomain (γ ω) (τ ω))
+    (hbase : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u))
+    (hR : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2048*Real.pi)*r) :
+    ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
+      (∫⁻ ω in A∩E, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v
+        (stoppedMarkedCenter κ B γ G τ z ω).arg
+        ∂finiteCenteredLaw P κ B γ G z (u+v)) ≤
+      finiteCenteredLaw P κ B γ G z (u+v)
+        (A∩E∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v))
+          z ((4096*Real.pi)*r)
+          (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by
+  obtain ⟨M,hM,_hMf,hMp,hbound⟩ := C.prescribed_disk_test_kernel hB hc h0 hκ hκ8
+    F hw R he D hD hDf hDp hgrowth hpublished τ hτ z hz u v r b hu hv hr E hE hlive hbase hR
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  have hq : Measurable[hτ.measurableSpace] (fun ω =>
+      R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v
+        (stoppedMarkedCenter κ B γ G τ z ω).arg) := by
+    let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) :=
+      (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
+    exact (R.angleDiskTail_measurable _ κ b v).comp
+      (source_stoppedMarkedCenter_measurable B hm hc κ γ G F hw τ hτ z).carg
+  intro A hA
+  rw [← past_density_test_integral P (finiteCenteredLaw P κ B γ G z (u+v))
+    hτ.measurableSpace hτ.measurableSpace_le M hM E hE hMp _ hq A hA]
+  exact hbound A hA
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedDiskTestKernel.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedDiskTestKernel.lean
index ac8f650de9d079c1605a36613847dfb71049ebe7..1f5ce2e80070001c829bdaac8b292ed8dfc35bc8 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedDiskTestKernel.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedDiskTestKernel.lean
@@ -1,180 +1,180 @@
-import OAI.Probability.SLE.Geometry.NormalizedRadiusCell
-import OAI.Probability.SLE.Geometry.NormalizedDiskDensity
-import OAI.Probability.SLE.Loewner.RawDiskAngleTail
-import OAI.Probability.SLE.Geometry.StoppedMarkedStateMeasurability
-import OAI.Probability.SLE.Geometry.FiniteDensityAtRadiusState
-import OAI.Probability.SLE.Gauge.PrescribedCanonicalTests
-import OAI.Probability.SLE.Geometry.RestartWeightedKernel
-
-namespace OAI
-
-/-! The actual disk test has its finite centered success kernel at the
-starting angle. Only finite stopped events enter the change of law. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance diskTestMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance diskTestBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-theorem CanonicalMass.prescribed_disk_test_kernel
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
-    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
-    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t =>
-      (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable))
-      (fun ω => (τ ω:WithTop ℝ≥0)))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b : ℝ)
-    (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r)
-    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
-    (hlive : ∀ ω∈E, z ∈ survivingDomain (γ ω) (τ ω))
-    (hbase : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u))
-    (hR : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2048*Real.pi)*r) :
-    ∃ M : Ω → ℝ≥0∞, Measurable[hτ.measurableSpace] M ∧
-      (∀ ω, M ω ≠ ∞) ∧
-      (∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
-        (∫⁻ ω in A∩E, M ω ∂P)=finiteCenteredLaw P κ B γ G z (u+v) (A∩E)) ∧
-      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
-        (∫⁻ ω in A∩E, M ω*R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v
-          (stoppedMarkedCenter κ B γ G τ z ω).arg ∂P) ≤
-        finiteCenteredLaw P κ B γ G z (u+v)
-          (A∩E∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v))
-            z ((4096*Real.pi)*r)
-            (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by
-  classical
-  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
-  let ν := P.map (sourceDriverPath B hc)
-  let : IsProbabilityMeasure ν :=
-    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
-  let a := stoppedMarkedScale κ B γ G τ z
-  let θ := fun ω => (stoppedMarkedCenter κ B γ G τ z ω).arg
-  let Y := normalizedRestartDriverPath B hc τ a
-  have ham : Measurable[hτ.measurableSpace] a :=
-    source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z
-  have ha : ∀ ω, 0 < a ω := stoppedMarkedScale_pos κ B γ G τ z
-  have hθm : Measurable[hτ.measurableSpace] θ :=
-    (source_stoppedMarkedCenter_measurable B hm hc κ γ G F hw τ hτ z).carg
-  have hY : Measurable Y := measurable_scaleDriverPath.comp
-    ((ham.mono hτ.measurableSpace_le le_rfl).inv.prodMk
-      (sourceRestartDriverPath_measurable B hm hc τ hτ))
-  have hYlaw (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
-      (P.restrict A).map Y=P A • ν :=
-    source_restart_random_scale_driver_law B hB hm hc τ hτ
-      (fun ω => (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) A hA
-  have hpolar (ω : Ω) (hω : ω∈E) :
-      θ ω∈Ioo (0:ℝ) Real.pi ∧ G ω (τ ω) z=
-        (Real.sqrt κ*B (τ ω) ω:ℝ)+(a ω:ℂ)*Complex.exp (θ ω*Complex.I) := by
-    have hh := (hw ω).live_marked_polar (τ ω) z (hlive ω hω)
-    simpa only [θ,a,stoppedMarkedCenter_eq κ B γ G τ z ω (hlive ω hω),
-      stoppedMarkedScale_eq_on_live B κ γ G F hw τ z ω (hlive ω hω)] using hh.2
-  obtain ⟨M,hM,hMf,hMactual,hMpast⟩ := exists_finiteDensity_at_radius_state
-    hB hBm hc h0 κ hκ hκ8 γ G F hw τ hτ z hz u v hu hv E hE hlive hbase
-  refine ⟨M,hM,hMf,hMpast,?_⟩
-  intro A hA
-  let T := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)
-  let q := fun ω => R.angleDiskTail ν κ b v (θ ω)
-  let test := C.prescribedBallTest T z ((4096*Real.pi)*r)
-    (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)
-  let S := E∩test
-  let Z := fun ω => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-    (γ ω) (G ω) z (u+v)/initialGreen κ z)
-  let Q : Set (Ω × C(ℝ≥0,ℝ)) := {x | x.1∈E ∧ b ≤ R.diskAtRadius κ (θ x.1) v x.2}
-  let K : Ω × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun x =>
-    M x.1*canonicalFiniteDensity κ (v,Target.diskMapInv (θ x.1) 0,x.2)
-  have hQ : @MeasurableSet _ (hτ.measurableSpace.prod (borel _)) Q :=
-    (hE.preimage measurable_fst).inter (measurableSet_le measurable_const
-      (@RawCanonicalMass.diskAtRadius_measurable (Ω × C(ℝ≥0,ℝ))
-        (hτ.measurableSpace.prod (borel _)) R κ _ _ (hθm.comp measurable_fst)
-        measurable_const Prod.snd measurable_snd))
-  have hzm : Measurable[hτ.measurableSpace] (fun ω => Target.diskMapInv (θ ω) 0) := by
-    unfold Target.diskMapInv
-    fun_prop
-  have hK : @Measurable _ _ (hτ.measurableSpace.prod (borel _)) _ K :=
-    (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp
-      (measurable_const.prodMk ((hzm.comp measurable_fst).prodMk measurable_snd)))
-  have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz
-    (u+v) (by linarith)
-  have htest : MeasurableSet test := hT.measurableSpace_le _
-    (C.prescribedBallTest_measurable T hT _ _ _)
-  have hS : MeasurableSet S := (hτ.measurableSpace_le _ hE).inter htest
-  have hsec : ∀ᵐ ω ∂P, (1:ℝ≥0∞)*E.indicator (fun ω => M ω*q ω) ω ≤
-      ∫⁻ p, Q.indicator K (ω,p) ∂ν := by
-    apply Eventually.of_forall
-    intro ω
-    by_cases hω : ω∈E
-    · let H := {p | b ≤ R.diskAtRadius κ (θ ω) v p}
-      have hH : MeasurableSet H := measurableSet_le measurable_const
-        (R.diskAtRadius_measurable κ (fun _ => θ ω) (fun _ => v)
-          measurable_const measurable_const id measurable_id)
-      have hid : (fun p => Q.indicator K (ω,p)) =
-          (fun p => M ω*H.indicator (fun p => canonicalFiniteDensity κ
-            (v,Target.diskMapInv (θ ω) 0,p)) p) := by
-        funext p
-        by_cases hp : p∈H
-        · rw [indicator_of_mem (show (ω,p)∈Q from ⟨hω,hp⟩),indicator_of_mem hp]
-        · rw [indicator_of_notMem (show (ω,p)∉Q from fun h => hp h.2),
-            indicator_of_notMem hp,mul_zero]
-      rw [hid,lintegral_const_mul' _ _ (hMf ω),lintegral_indicator hH,
-        ← withDensity_apply _ hH,indicator_of_mem hω,one_mul]
-      exact le_rfl
-    · rw [indicator_of_notMem hω,mul_zero]
-      exact bot_le
-  have hden := normalized_restart_disk_density hB hm hc γ G F hw hpublished τ hτ a ham ha
-  have hcell := C.normalized_cell_bounds_at_radius hB hc hκ F hw R he D hD hDf hDp
-    hgrowth hpublished τ hτ a ham ha
-  have hactual : ∀ᵐ ω ∂P, Q.indicator K (ω,Y ω) ≤ S.indicator Z ω := by
-    filter_upwards [hden,hcell,hMactual] with ω hd hb hMω
-    by_cases hQω : (ω,Y ω)∈Q
-    · obtain ⟨hωE,hbval⟩ := hQω
-      have hpol := hpolar ω hωE
-      have hKZ : K (ω,Y ω)=Z ω := by
-        change M ω*canonicalFiniteDensity κ (v,Target.diskMapInv (θ ω) 0,Y ω)=Z ω
-        rw [hMω hωE]
-        exact (hd _ hpol.1 z (hlive ω hωE) hz hpol.2 u v hu hv (hbase ω hωE)).symm
-      rw [indicator_of_mem (show (ω,Y ω)∈Q from ⟨hωE,hbval⟩),hKZ]
-      by_cases hf : T ω<⊤
-      · have hmass := (hb (θ ω) hpol.1 z (hlive ω hωE) r hr (hR ω hωE)
-          hpol.2 u hu (hbase ω hωE) v hv hf).1
-        have hvb := (mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal hbval)).trans hmass
-        have heval : extendedCanonicalMass C.process C.rational.terminal (T ω) ω =
-            C.process ((T ω).untopD 0) ω := by
-          dsimp only [extendedCanonicalMass]
-          rw [ite_eq_left hf]
-          rfl
-        rw [heval] at hvb
-        have ht : ω∈test := ⟨hf,hvb⟩
-        rw [indicator_of_mem (show ω∈S from ⟨hωE,ht⟩)]
-      · have hz0 : Z ω=0 := by
-          simp only [Z,stoppedGreen,show ¬radiusLevelTime (γ ω) (G ω) z (u+v)<∞ from hf,
-            ite_false,zero_div,ENNReal.ofReal_zero]
-        rw [hz0]
-        exact bot_le
-    · rw [indicator_of_notMem hQω]
-      exact bot_le
-  have hh := @restart_weighted_event_lower_bound Ω C(ℝ≥0,ℝ) mΩ (borel _)
-    P inferInstance ν inferInstance hτ.measurableSpace hτ.measurableSpace_le
-    Y hY hYlaw K hK Q hQ (E.indicator (fun ω => M ω*q ω)) Z 1 (by simp) S hS
-    hsec hactual A hA
-  rw [one_mul,setLIntegral_indicator (hτ.measurableSpace_le _ hE),inter_comm E A] at hh
-  simpa only [ν,q,θ,Z,S,test,finiteCenteredLaw,← inter_assoc] using hh
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.NormalizedRadiusCell
+import OAI.Probability.SLE.Geometry.NormalizedDiskDensity
+import OAI.Probability.SLE.Loewner.RawDiskAngleTail
+import OAI.Probability.SLE.Geometry.StoppedMarkedStateMeasurability
+import OAI.Probability.SLE.Geometry.FiniteDensityAtRadiusState
+import OAI.Probability.SLE.Gauge.PrescribedCanonicalTests
+import OAI.Probability.SLE.Geometry.RestartWeightedKernel
+
+namespace OAI
+
+/-! The actual disk test has its finite centered success kernel at the
+starting angle. Only finite stopped events enter the change of law. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance diskTestMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance diskTestBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+theorem CanonicalMass.prescribed_disk_test_kernel
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
+    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
+    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t =>
+      (hBm.of_uncurry_left : Measurable (B t)).stronglyMeasurable))
+      (fun ω => (τ ω:WithTop ℝ≥0)))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u v r b : ℝ)
+    (hu : 0 ≤ u) (hv : 0 < v) (hr : 0 < r)
+    (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E)
+    (hlive : ∀ ω∈E, z ∈ survivingDomain (γ ω) (τ ω))
+    (hbase : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2*z.im)*Real.exp (-u))
+    (hR : ∀ ω∈E, fullRadius (G ω) (τ ω) z=(2048*Real.pi)*r) :
+    ∃ M : Ω → ℝ≥0∞, Measurable[hτ.measurableSpace] M ∧
+      (∀ ω, M ω ≠ ∞) ∧
+      (∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
+        (∫⁻ ω in A∩E, M ω ∂P)=finiteCenteredLaw P κ B γ G z (u+v) (A∩E)) ∧
+      ∀ A : Set Ω, MeasurableSet[hτ.measurableSpace] A →
+        (∫⁻ ω in A∩E, M ω*R.angleDiskTail (P.map (sourceDriverPath B hc)) κ b v
+          (stoppedMarkedCenter κ B γ G τ z ω).arg ∂P) ≤
+        finiteCenteredLaw P κ B γ G z (u+v)
+          (A∩E∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z (u+v))
+            z ((4096*Real.pi)*r)
+            (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)) := by
+  classical
+  have hm (t : ℝ≥0) : Measurable (B t) := hBm.of_uncurry_left
+  let ν := P.map (sourceDriverPath B hc)
+  let : IsProbabilityMeasure ν :=
+    (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance
+  let a := stoppedMarkedScale κ B γ G τ z
+  let θ := fun ω => (stoppedMarkedCenter κ B γ G τ z ω).arg
+  let Y := normalizedRestartDriverPath B hc τ a
+  have ham : Measurable[hτ.measurableSpace] a :=
+    source_stoppedMarkedScale_measurable B hm hc κ γ G F hw τ hτ z
+  have ha : ∀ ω, 0 < a ω := stoppedMarkedScale_pos κ B γ G τ z
+  have hθm : Measurable[hτ.measurableSpace] θ :=
+    (source_stoppedMarkedCenter_measurable B hm hc κ γ G F hw τ hτ z).carg
+  have hY : Measurable Y := measurable_scaleDriverPath.comp
+    ((ham.mono hτ.measurableSpace_le le_rfl).inv.prodMk
+      (sourceRestartDriverPath_measurable B hm hc τ hτ))
+  have hYlaw (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
+      (P.restrict A).map Y=P A • ν :=
+    source_restart_random_scale_driver_law B hB hm hc τ hτ
+      (fun ω => (a ω)⁻¹) ham.inv (fun ω => inv_pos.mpr (ha ω)) A hA
+  have hpolar (ω : Ω) (hω : ω∈E) :
+      θ ω∈Ioo (0:ℝ) Real.pi ∧ G ω (τ ω) z=
+        (Real.sqrt κ*B (τ ω) ω:ℝ)+(a ω:ℂ)*Complex.exp (θ ω*Complex.I) := by
+    have hh := (hw ω).live_marked_polar (τ ω) z (hlive ω hω)
+    simpa only [θ,a,stoppedMarkedCenter_eq κ B γ G τ z ω (hlive ω hω),
+      stoppedMarkedScale_eq_on_live B κ γ G F hw τ z ω (hlive ω hω)] using hh.2
+  obtain ⟨M,hM,hMf,hMactual,hMpast⟩ := exists_finiteDensity_at_radius_state
+    hB hBm hc h0 κ hκ hκ8 γ G F hw τ hτ z hz u v hu hv E hE hlive hbase
+  refine ⟨M,hM,hMf,hMpast,?_⟩
+  intro A hA
+  let T := fun ω => radiusLevelTime (γ ω) (G ω) z (u+v)
+  let q := fun ω => R.angleDiskTail ν κ b v (θ ω)
+  let test := C.prescribedBallTest T z ((4096*Real.pi)*r)
+    (ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b)
+  let S := E∩test
+  let Z := fun ω => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+    (γ ω) (G ω) z (u+v)/initialGreen κ z)
+  let Q : Set (Ω × C(ℝ≥0,ℝ)) := {x | x.1∈E ∧ b ≤ R.diskAtRadius κ (θ x.1) v x.2}
+  let K : Ω × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun x =>
+    M x.1*canonicalFiniteDensity κ (v,Target.diskMapInv (θ x.1) 0,x.2)
+  have hQ : @MeasurableSet _ (hτ.measurableSpace.prod (borel _)) Q :=
+    (hE.preimage measurable_fst).inter (measurableSet_le measurable_const
+      (@RawCanonicalMass.diskAtRadius_measurable (Ω × C(ℝ≥0,ℝ))
+        (hτ.measurableSpace.prod (borel _)) R κ _ _ (hθm.comp measurable_fst)
+        measurable_const Prod.snd measurable_snd))
+  have hzm : Measurable[hτ.measurableSpace] (fun ω => Target.diskMapInv (θ ω) 0) := by
+    unfold Target.diskMapInv
+    fun_prop
+  have hK : @Measurable _ _ (hτ.measurableSpace.prod (borel _)) _ K :=
+    (hM.comp measurable_fst).mul ((canonicalFiniteDensity_measurable κ).comp
+      (measurable_const.prodMk ((hzm.comp measurable_fst).prodMk measurable_snd)))
+  have hT := actual_radiusLevelTime_stopping hB hm hBm hc h0 κ hκ hκ8 γ G F hw z hz
+    (u+v) (by linarith)
+  have htest : MeasurableSet test := hT.measurableSpace_le _
+    (C.prescribedBallTest_measurable T hT _ _ _)
+  have hS : MeasurableSet S := (hτ.measurableSpace_le _ hE).inter htest
+  have hsec : ∀ᵐ ω ∂P, (1:ℝ≥0∞)*E.indicator (fun ω => M ω*q ω) ω ≤
+      ∫⁻ p, Q.indicator K (ω,p) ∂ν := by
+    apply Eventually.of_forall
+    intro ω
+    by_cases hω : ω∈E
+    · let H := {p | b ≤ R.diskAtRadius κ (θ ω) v p}
+      have hH : MeasurableSet H := measurableSet_le measurable_const
+        (R.diskAtRadius_measurable κ (fun _ => θ ω) (fun _ => v)
+          measurable_const measurable_const id measurable_id)
+      have hid : (fun p => Q.indicator K (ω,p)) =
+          (fun p => M ω*H.indicator (fun p => canonicalFiniteDensity κ
+            (v,Target.diskMapInv (θ ω) 0,p)) p) := by
+        funext p
+        by_cases hp : p∈H
+        · rw [indicator_of_mem (show (ω,p)∈Q from ⟨hω,hp⟩),indicator_of_mem hp]
+        · rw [indicator_of_notMem (show (ω,p)∉Q from fun h => hp h.2),
+            indicator_of_notMem hp,mul_zero]
+      rw [hid,lintegral_const_mul' _ _ (hMf ω),lintegral_indicator hH,
+        ← withDensity_apply _ hH,indicator_of_mem hω,one_mul]
+      exact le_rfl
+    · rw [indicator_of_notMem hω,mul_zero]
+      exact bot_le
+  have hden := normalized_restart_disk_density hB hm hc γ G F hw hpublished τ hτ a ham ha
+  have hcell := C.normalized_cell_bounds_at_radius hB hc hκ F hw R he D hD hDf hDp
+    hgrowth hpublished τ hτ a ham ha
+  have hactual : ∀ᵐ ω ∂P, Q.indicator K (ω,Y ω) ≤ S.indicator Z ω := by
+    filter_upwards [hden,hcell,hMactual] with ω hd hb hMω
+    by_cases hQω : (ω,Y ω)∈Q
+    · obtain ⟨hωE,hbval⟩ := hQω
+      have hpol := hpolar ω hωE
+      have hKZ : K (ω,Y ω)=Z ω := by
+        change M ω*canonicalFiniteDensity κ (v,Target.diskMapInv (θ ω) 0,Y ω)=Z ω
+        rw [hMω hωE]
+        exact (hd _ hpol.1 z (hlive ω hωE) hz hpol.2 u v hu hv (hbase ω hωE)).symm
+      rw [indicator_of_mem (show (ω,Y ω)∈Q from ⟨hωE,hbval⟩),hKZ]
+      by_cases hf : T ω<⊤
+      · have hmass := (hb (θ ω) hpol.1 z (hlive ω hωE) r hr (hR ω hωE)
+          hpol.2 u hu (hbase ω hωE) v hv hf).1
+        have hvb := (mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal hbval)).trans hmass
+        have heval : extendedCanonicalMass C.process C.rational.terminal (T ω) ω =
+            C.process ((T ω).untopD 0) ω := by
+          dsimp only [extendedCanonicalMass]
+          rw [ite_eq_left hf]
+          rfl
+        rw [heval] at hvb
+        have ht : ω∈test := ⟨hf,hvb⟩
+        rw [indicator_of_mem (show ω∈S from ⟨hωE,ht⟩)]
+      · have hz0 : Z ω=0 := by
+          simp only [Z,stoppedGreen,show ¬radiusLevelTime (γ ω) (G ω) z (u+v)<∞ from hf,
+            ite_false,zero_div,ENNReal.ofReal_zero]
+        rw [hz0]
+        exact bot_le
+    · rw [indicator_of_notMem hQω]
+      exact bot_le
+  have hh := @restart_weighted_event_lower_bound Ω C(ℝ≥0,ℝ) mΩ (borel _)
+    P inferInstance ν inferInstance hτ.measurableSpace hτ.measurableSpace_le
+    Y hY hYlaw K hK Q hQ (E.indicator (fun ω => M ω*q ω)) Z 1 (by simp) S hS
+    hsec hactual A hA
+  rw [one_mul,setLIntegral_indicator (hτ.measurableSpace_le _ hE),inter_comm E A] at hh
+  simpa only [ν,q,θ,Z,S,test,finiteCenteredLaw,← inter_assoc] using hh
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedEntryKernel.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedEntryKernel.lean
index 902d4bee7a34ebac4698afe468f94ce148349dbf..c83cdf2b65f5a0d44d11717f4306a3f03487574c 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedEntryKernel.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedEntryKernel.lean
@@ -1,78 +1,78 @@
-import OAI.Probability.SLE.Gauge.PrescribedRadiusFinalLaw
-import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
-
-namespace OAI
-
-/-! The physical radius, exact prescribed inner-clock duration, threshold,
-and normalized angle kernel of one entry all agree with the batch test. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance entryKernelMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance entryKernelBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-lemma prescribed_ball_threshold (κ r b : ℝ) (hr : 0 < r) :
-    ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b =
-      ENNReal.ofReal ((128*Real.pi)^dimension κ*b*r^Target.d κ) := by
-  have hd : Target.d κ=dimension κ := by rw [Target.dimension_eq]; rfl
-  rw [← ENNReal.ofReal_mul (Real.rpow_nonneg (by positivity) _),
-    Real.mul_rpow (by positivity : 0 ≤ 128*Real.pi) hr.le,hd]
-  congr 1
-  ring
-
-theorem CanonicalMass.prescribed_entry_success_kernel
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
-    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
-    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (W : Target.WienerLaw) (n : ℕ) (e : ℤ×ℕ) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (hu : 0 ≤ Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z)
-    (V : ℝ) (hV : Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z+
-      2*Target.a κ*(Target.batchLevel κ W n:ℝ) ≤ V) :
-    ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB
-      (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
-      κ hκ hκ8 γ G F hw z hz _ hu).measurableSpace] A →
-      (∫⁻ ω in A, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some e))
-        (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
-        (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z
-          (Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z)) z ω)
-        ∂finiteCenteredLaw P κ B γ G z V) ≤
-      finiteCenteredLaw P κ B γ G z V
-        (A∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z
-          (Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z+
-            2*Target.a κ*(Target.batchLevel κ W n:ℝ)))
-          z ((4096*Real.pi)*Target.entryRadius κ W (n,some e))
-          (ENNReal.ofReal ((128*Real.pi)^dimension κ*Target.entryValue (n,some e)*
-            (Target.entryRadius κ W (n,some e))^Target.d κ))) := by
-  have hr : 0 < Target.entryRadius κ W (n,some e) := Real.exp_pos _
-  have hℓ : (0:ℝ)<Target.batchLevel κ W n := by
-    exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one (Target.batchLevel_ge_one κ W n))
-  have hv : 0 < 2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by
-    exact mul_pos (mul_pos (by norm_num) (Target.internal_a_pos hκ)) hℓ
-  intro A hA
-  have h := C.prescribed_disk_test_under_final_law hB hc h0 hκ hκ8 F hw R he D hD hDf hDp
-    hgrowth hpublished z hz (Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z)
-    (2*Target.a κ*(Target.batchLevel κ W n:ℝ)) (Target.entryRadius κ W (n,some e))
-    (Target.entryValue (n,some e)) V hu hv hr
-    (Target.entryPhysicalLevel_radius κ W (n,some e) (by positivity) hz) hV A hA
-  rwa [prescribed_ball_threshold κ _ _ hr] at h
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedRadiusFinalLaw
+import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels
+
+namespace OAI
+
+/-! The physical radius, exact prescribed inner-clock duration, threshold,
+and normalized angle kernel of one entry all agree with the batch test. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance entryKernelMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance entryKernelBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+lemma prescribed_ball_threshold (κ r b : ℝ) (hr : 0 < r) :
+    ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*ENNReal.ofReal b =
+      ENNReal.ofReal ((128*Real.pi)^dimension κ*b*r^Target.d κ) := by
+  have hd : Target.d κ=dimension κ := by rw [Target.dimension_eq]; rfl
+  rw [← ENNReal.ofReal_mul (Real.rpow_nonneg (by positivity) _),
+    Real.mul_rpow (by positivity : 0 ≤ 128*Real.pi) hr.le,hd]
+  congr 1
+  ring
+
+theorem CanonicalMass.prescribed_entry_success_kernel
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
+    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
+    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (W : Target.WienerLaw) (n : ℕ) (e : ℤ×ℕ) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (hu : 0 ≤ Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z)
+    (V : ℝ) (hV : Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z+
+      2*Target.a κ*(Target.batchLevel κ W n:ℝ) ≤ V) :
+    ∀ A : Set Ω, MeasurableSet[(actual_radiusLevelTime_stopping hB
+      (fun t => (hBm.of_uncurry_left : Measurable (B t))) hBm hc h0
+      κ hκ hκ8 γ G F hw z hz _ hu).measurableSpace] A →
+      (∫⁻ ω in A, R.angleDiskTail (P.map (sourceDriverPath B hc)) κ (Target.entryValue (n,some e))
+        (2*Target.a κ*(Target.batchLevel κ W n:ℝ))
+        (extendedMarkedAngle κ B γ G (fun ω => radiusLevelTime (γ ω) (G ω) z
+          (Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z)) z ω)
+        ∂finiteCenteredLaw P κ B γ G z V) ≤
+      finiteCenteredLaw P κ B γ G z V
+        (A∩C.prescribedBallTest (fun ω => radiusLevelTime (γ ω) (G ω) z
+          (Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z+
+            2*Target.a κ*(Target.batchLevel κ W n:ℝ)))
+          z ((4096*Real.pi)*Target.entryRadius κ W (n,some e))
+          (ENNReal.ofReal ((128*Real.pi)^dimension κ*Target.entryValue (n,some e)*
+            (Target.entryRadius κ W (n,some e))^Target.d κ))) := by
+  have hr : 0 < Target.entryRadius κ W (n,some e) := Real.exp_pos _
+  have hℓ : (0:ℝ)<Target.batchLevel κ W n := by
+    exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one (Target.batchLevel_ge_one κ W n))
+  have hv : 0 < 2*Target.a κ*(Target.batchLevel κ W n:ℝ) := by
+    exact mul_pos (mul_pos (by norm_num) (Target.internal_a_pos hκ)) hℓ
+  intro A hA
+  have h := C.prescribed_disk_test_under_final_law hB hc h0 hκ hκ8 F hw R he D hD hDf hDp
+    hgrowth hpublished z hz (Target.entryPhysicalLevel κ W (n,some e) (2048*Real.pi) z)
+    (2*Target.a κ*(Target.batchLevel κ W n:ℝ)) (Target.entryRadius κ W (n,some e))
+    (Target.entryValue (n,some e)) V hu hv hr
+    (Target.entryPhysicalLevel_radius κ W (n,some e) (by positivity) hz) hV A hA
+  rwa [prescribed_ball_threshold κ _ _ hr] at h
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedGaugeCost.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedGaugeCost.lean
index 22abd6e50e98c8d5a50542c28453686df36af9ff..a71f84a032c00c9d659399e7ee6d2d94969957cc 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedGaugeCost.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedGaugeCost.lean
@@ -1,91 +1,91 @@
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLE.Gauge.DenseCandidateCover
-
-namespace OAI
-
-/-! The prescribed quadratic envelope directly pays for enlarged candidate
-balls and for final cells. Its infinite-diameter convention gives the same
-Hausdorff outer measure, since the defining limit uses small covers. -/
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma hausdorff_eq_hausdorffGauge (h : ℝ → ℝ) :
-    Target.hausdorff h = (hausdorffGauge h).toOuterMeasure := by
-  rw [hausdorffGauge,Measure.mkMetric_toOuterMeasure]
-  unfold Target.hausdorff
-  apply le_antisymm <;> apply OuterMeasure.mkMetric_mono
-  · filter_upwards [gt_mem_nhds (show (0:ℝ≥0∞)<∞ from ENNReal.zero_lt_top)] with r hr
-    simp only [ne_of_lt hr,ite_false,extendedGauge,le_refl]
-  · filter_upwards [] with r
-    unfold extendedGauge
-    split_ifs <;> simp
-
-lemma gauge_le_entry (κ : ℝ) (W : WienerLaw) (e : Entry)
-    (he : ValidEntry κ W e) (r : ℝ) : gauge κ W r ≤ entryCost κ W e r := by
-  rw [gauge_eq_full]
-  exact QuadraticEnvelope.full_le_profile
-    (fun i => (weights_pos κ W i).le) (radii_pos κ W) ⟨e,he⟩ r
-
-lemma gauge_scaled_entry_le (κ : ℝ) (W : WienerLaw) (e : Entry)
-    (he : ValidEntry κ W e) (C : ℝ) :
-    gauge κ W (C*entryRadius κ W e) ≤
-      max 1 (C^2)*entryValue e*(entryRadius κ W e)^d κ := by
-  have hr : entryRadius κ W e ≠ 0 := (Real.exp_pos _).ne'
-  convert gauge_le_entry κ W e he (C*entryRadius κ W e) using 1
-  unfold entryCost
-  rw [mul_div_cancel_right₀ _ hr]
-  ring
-
-lemma enlarged_candidate_gauge_cost {κ : ℝ} (W : WienerLaw)
-    (hg : IsHausdorffGauge (gauge κ W)) (e : Entry) (he : ValidEntry κ W e)
-    (z : ℂ) {C c : ℝ} (hC : 1≤C) (hc : 0<c) (ν : Measure ℂ)
-    (hqualified : ENNReal.ofReal (c*entryValue e*(entryRadius κ W e)^d κ) ≤
-      ν (closedBall z (C*entryRadius κ W e))) :
-    extendedGauge (gauge κ W) (ediam (closedBall z (4*C*entryRadius κ W e))) ≤
-      ENNReal.ofReal ((8*C)^2/c)*ν (closedBall z (C*entryRadius κ W e)) := by
-  have hh : IsGauge (gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
-  have hC0 : 0≤C := by linarith
-  have hr := radii_pos κ W ⟨e,he⟩
-  have hm : max 1 ((8*C)^2)=(8*C)^2 := by
-    apply max_eq_right
-    nlinarith
-  have hbound := gauge_scaled_entry_le κ W e he (8*C)
-  rw [hm] at hbound
-  have hconst : 0≤(8*C)^2/c := div_nonneg (sq_nonneg _) hc.le
-  calc
-    _ ≤ ENNReal.ofReal (gauge κ W (2*(4*C*entryRadius κ W e))) :=
-      extendedGauge_closedBall_le hh z (by positivity)
-    _ = ENNReal.ofReal (gauge κ W ((8*C)*entryRadius κ W e)) := by congr 2; ring
-    _ ≤ ENNReal.ofReal ((8*C)^2*entryValue e*(entryRadius κ W e)^d κ) :=
-      ENNReal.ofReal_le_ofReal hbound
-    _ = ENNReal.ofReal ((8*C)^2/c)*
-        ENNReal.ofReal (c*entryValue e*(entryRadius κ W e)^d κ) := by
-      rw [← ENNReal.ofReal_mul hconst]
-      congr 1
-      field_simp
-    _ ≤ _ := mul_le_mul' le_rfl hqualified
-
-lemma cleanup_cell_gauge_cost {κ : ℝ} (W : WienerLaw)
-    (hg : IsHausdorffGauge (gauge κ W)) (n : ℕ) (hn : 1≤n) (z : ℂ) :
-    extendedGauge (gauge κ W)
-      (ediam (closedBall z (2*entryRadius κ W (n,none)))) ≤
-      ENNReal.ofReal (16*(n:ℝ)*(entryRadius κ W (n,none))^d κ) := by
-  have hh : IsGauge (gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
-  have hr : 0<entryRadius κ W (n,none) := Real.exp_pos _
-  calc
-    _ ≤ ENNReal.ofReal (gauge κ W (2*(2*entryRadius κ W (n,none)))) :=
-      extendedGauge_closedBall_le hh z (by positivity)
-    _ = ENNReal.ofReal (gauge κ W (4*entryRadius κ W (n,none))) := by congr 2; ring
-    _ ≤ _ := by
-      apply ENNReal.ofReal_le_ofReal
-      simpa only [show (max (1:ℝ) (4^2))=16 by norm_num,entryValue] using
-        gauge_scaled_entry_le κ W (n,none) ⟨hn,trivial⟩ 4
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLE.Gauge.DenseCandidateCover
+
+namespace OAI
+
+/-! The prescribed quadratic envelope directly pays for enlarged candidate
+balls and for final cells. Its infinite-diameter convention gives the same
+Hausdorff outer measure, since the defining limit uses small covers. -/
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma hausdorff_eq_hausdorffGauge (h : ℝ → ℝ) :
+    Target.hausdorff h = (hausdorffGauge h).toOuterMeasure := by
+  rw [hausdorffGauge,Measure.mkMetric_toOuterMeasure]
+  unfold Target.hausdorff
+  apply le_antisymm <;> apply OuterMeasure.mkMetric_mono
+  · filter_upwards [gt_mem_nhds (show (0:ℝ≥0∞)<∞ from ENNReal.zero_lt_top)] with r hr
+    simp only [ne_of_lt hr,ite_false,extendedGauge,le_refl]
+  · filter_upwards [] with r
+    unfold extendedGauge
+    split_ifs <;> simp
+
+lemma gauge_le_entry (κ : ℝ) (W : WienerLaw) (e : Entry)
+    (he : ValidEntry κ W e) (r : ℝ) : gauge κ W r ≤ entryCost κ W e r := by
+  rw [gauge_eq_full]
+  exact QuadraticEnvelope.full_le_profile
+    (fun i => (weights_pos κ W i).le) (radii_pos κ W) ⟨e,he⟩ r
+
+lemma gauge_scaled_entry_le (κ : ℝ) (W : WienerLaw) (e : Entry)
+    (he : ValidEntry κ W e) (C : ℝ) :
+    gauge κ W (C*entryRadius κ W e) ≤
+      max 1 (C^2)*entryValue e*(entryRadius κ W e)^d κ := by
+  have hr : entryRadius κ W e ≠ 0 := (Real.exp_pos _).ne'
+  convert gauge_le_entry κ W e he (C*entryRadius κ W e) using 1
+  unfold entryCost
+  rw [mul_div_cancel_right₀ _ hr]
+  ring
+
+lemma enlarged_candidate_gauge_cost {κ : ℝ} (W : WienerLaw)
+    (hg : IsHausdorffGauge (gauge κ W)) (e : Entry) (he : ValidEntry κ W e)
+    (z : ℂ) {C c : ℝ} (hC : 1≤C) (hc : 0<c) (ν : Measure ℂ)
+    (hqualified : ENNReal.ofReal (c*entryValue e*(entryRadius κ W e)^d κ) ≤
+      ν (closedBall z (C*entryRadius κ W e))) :
+    extendedGauge (gauge κ W) (ediam (closedBall z (4*C*entryRadius κ W e))) ≤
+      ENNReal.ofReal ((8*C)^2/c)*ν (closedBall z (C*entryRadius κ W e)) := by
+  have hh : IsGauge (gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
+  have hC0 : 0≤C := by linarith
+  have hr := radii_pos κ W ⟨e,he⟩
+  have hm : max 1 ((8*C)^2)=(8*C)^2 := by
+    apply max_eq_right
+    nlinarith
+  have hbound := gauge_scaled_entry_le κ W e he (8*C)
+  rw [hm] at hbound
+  have hconst : 0≤(8*C)^2/c := div_nonneg (sq_nonneg _) hc.le
+  calc
+    _ ≤ ENNReal.ofReal (gauge κ W (2*(4*C*entryRadius κ W e))) :=
+      extendedGauge_closedBall_le hh z (by positivity)
+    _ = ENNReal.ofReal (gauge κ W ((8*C)*entryRadius κ W e)) := by congr 2; ring
+    _ ≤ ENNReal.ofReal ((8*C)^2*entryValue e*(entryRadius κ W e)^d κ) :=
+      ENNReal.ofReal_le_ofReal hbound
+    _ = ENNReal.ofReal ((8*C)^2/c)*
+        ENNReal.ofReal (c*entryValue e*(entryRadius κ W e)^d κ) := by
+      rw [← ENNReal.ofReal_mul hconst]
+      congr 1
+      field_simp
+    _ ≤ _ := mul_le_mul' le_rfl hqualified
+
+lemma cleanup_cell_gauge_cost {κ : ℝ} (W : WienerLaw)
+    (hg : IsHausdorffGauge (gauge κ W)) (n : ℕ) (hn : 1≤n) (z : ℂ) :
+    extendedGauge (gauge κ W)
+      (ediam (closedBall z (2*entryRadius κ W (n,none)))) ≤
+      ENNReal.ofReal (16*(n:ℝ)*(entryRadius κ W (n,none))^d κ) := by
+  have hh : IsGauge (gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
+  have hr : 0<entryRadius κ W (n,none) := Real.exp_pos _
+  calc
+    _ ≤ ENNReal.ofReal (gauge κ W (2*(2*entryRadius κ W (n,none)))) :=
+      extendedGauge_closedBall_le hh z (by positivity)
+    _ = ENNReal.ofReal (gauge κ W (4*entryRadius κ W (n,none))) := by congr 2; ring
+    _ ≤ _ := by
+      apply ENNReal.ofReal_le_ofReal
+      simpa only [show (max (1:ℝ) (4^2))=16 by norm_num,entryValue] using
+        gauge_scaled_entry_le κ W (n,none) ⟨hn,trivial⟩ 4
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedGridError.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedGridError.lean
index 68cac9a0962458648d7563bf6aee4f08c0f7a224..5c11598a0e2626f1f246fbcf39b7b7fd64532f7b 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedGridError.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedGridError.lean
@@ -1,74 +1,74 @@
-import OAI.Probability.SLE.Gauge.PrescribedGridStripSum
-import OAI.Probability.SLE.Gauge.PrescribedMissedExpectation
-
-namespace OAI
-
-/-! Summing the interior failure bound and the boundary visit bound over
-  the actual shifted mesh cancels the three powers of its scale. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma prescribed_grid_error_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
-    {R r η ε A : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1)
-    (hη : 0 ≤ η) (hε : 0 ≤ ε) (hA : 0 ≤ A) (K : Set ℂ) :
-    (∑ ij∈prescribedGridIndex R r K,
-      A*r^α*(prescribedGridCenter r ij).im^(-α)*
-        (ε + if (prescribedGridCenter r ij).im<η then 1 else 0)) ≤
-      A*r^α*(((2*R+5)*((2:ℝ)^α/(1-α)))*r^(-2:ℝ)*
-        ((R+3)^(1-α)*ε+(η+r)^(1-α))) := by
-  classical
-  let I := prescribedGridIndex R r K
-  let w := fun ij => (prescribedGridCenter r ij).im^(-α)
-  have heq : (∑ ij∈I, A*r^α*w ij*(ε + if (prescribedGridCenter r ij).im<η then 1 else 0)) =
-      A*r^α*(ε*(∑ ij∈I,w ij)+
-        ∑ ij∈I.filter (fun ij => (prescribedGridCenter r ij).im<η),w ij) := by
-    rw [Finset.sum_filter,Finset.mul_sum,← Finset.sum_add_distrib,Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro ij _
-    split_ifs <;> ring
-  rw [heq]
-  have hfull := prescribed_grid_height_sum_bound α hα0 hα1 hR hr hr1 K
-  have hstrip := prescribed_grid_lower_strip_sum α hα0 hα1 hR hr hr1 hη K
-  have hh := add_le_add (mul_le_mul_of_nonneg_left hfull hε) hstrip
-  apply (mul_le_mul_of_nonneg_left hh (mul_nonneg hA (Real.rpow_nonneg hr.le _))).trans
-  apply le_of_eq
-  ring
-
-lemma prescribed_cleanup_power_cancellation (κ : ℝ) {r : ℝ} (hr : 0<r) :
-    r^Target.d κ*r^codimension κ*r^(-2:ℝ)=1 := by
-  have he : Target.d κ=dimension κ := by
-    rw [Target.dimension_eq]
-    rfl
-  rw [he]
-  simpa only [mul_comm,mul_left_comm,mul_assoc] using dense_cover_power_cancellation κ hr
-
-lemma prescribed_cleanup_error_bound (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    {R r η ε A : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1)
-    (hη : 0 ≤ η) (hε : 0 ≤ ε) (hA : 0 ≤ A) (K : Set ℂ) (n : ℕ) :
-    (36*(n:ℝ)*r^Target.d κ) *
-      (∑ ij∈prescribedGridIndex R r K,
-        A*r^codimension κ*(prescribedGridCenter r ij).im^(-codimension κ)*
-          (ε + if (prescribedGridCenter r ij).im<η then 1 else 0)) ≤
-      (36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ))))*
-        ((R+3)^(1-codimension κ)*((n:ℝ)*ε)+(n:ℝ)*(η+r)^(1-codimension κ)) := by
-  have hα0 : 0 ≤ codimension κ := (codimension_pos κ hκ8).le
-  have hα1 : codimension κ<1 := codimension_lt_one κ hκ
-  apply (mul_le_mul_of_nonneg_left
-    (prescribed_grid_error_sum (codimension κ) hα0 hα1 hR hr hr1 hη hε hA K)
-      (by positivity : 0 ≤ 36*(n:ℝ)*r^Target.d κ)).trans
-  apply le_of_eq
-  have hp := prescribed_cleanup_power_cancellation κ hr
-  calc
-    _ = (36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ))))*
-        (r^Target.d κ*r^codimension κ*r^(-2:ℝ))*
-        ((R+3)^(1-codimension κ)*((n:ℝ)*ε)+(n:ℝ)*(η+r)^(1-codimension κ)) := by ring
-    _ = _ := by rw [hp,mul_one]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedGridStripSum
+import OAI.Probability.SLE.Gauge.PrescribedMissedExpectation
+
+namespace OAI
+
+/-! Summing the interior failure bound and the boundary visit bound over
+  the actual shifted mesh cancels the three powers of its scale. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma prescribed_grid_error_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
+    {R r η ε A : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1)
+    (hη : 0 ≤ η) (hε : 0 ≤ ε) (hA : 0 ≤ A) (K : Set ℂ) :
+    (∑ ij∈prescribedGridIndex R r K,
+      A*r^α*(prescribedGridCenter r ij).im^(-α)*
+        (ε + if (prescribedGridCenter r ij).im<η then 1 else 0)) ≤
+      A*r^α*(((2*R+5)*((2:ℝ)^α/(1-α)))*r^(-2:ℝ)*
+        ((R+3)^(1-α)*ε+(η+r)^(1-α))) := by
+  classical
+  let I := prescribedGridIndex R r K
+  let w := fun ij => (prescribedGridCenter r ij).im^(-α)
+  have heq : (∑ ij∈I, A*r^α*w ij*(ε + if (prescribedGridCenter r ij).im<η then 1 else 0)) =
+      A*r^α*(ε*(∑ ij∈I,w ij)+
+        ∑ ij∈I.filter (fun ij => (prescribedGridCenter r ij).im<η),w ij) := by
+    rw [Finset.sum_filter,Finset.mul_sum,← Finset.sum_add_distrib,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro ij _
+    split_ifs <;> ring
+  rw [heq]
+  have hfull := prescribed_grid_height_sum_bound α hα0 hα1 hR hr hr1 K
+  have hstrip := prescribed_grid_lower_strip_sum α hα0 hα1 hR hr hr1 hη K
+  have hh := add_le_add (mul_le_mul_of_nonneg_left hfull hε) hstrip
+  apply (mul_le_mul_of_nonneg_left hh (mul_nonneg hA (Real.rpow_nonneg hr.le _))).trans
+  apply le_of_eq
+  ring
+
+lemma prescribed_cleanup_power_cancellation (κ : ℝ) {r : ℝ} (hr : 0<r) :
+    r^Target.d κ*r^codimension κ*r^(-2:ℝ)=1 := by
+  have he : Target.d κ=dimension κ := by
+    rw [Target.dimension_eq]
+    rfl
+  rw [he]
+  simpa only [mul_comm,mul_left_comm,mul_assoc] using dense_cover_power_cancellation κ hr
+
+lemma prescribed_cleanup_error_bound (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    {R r η ε A : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1)
+    (hη : 0 ≤ η) (hε : 0 ≤ ε) (hA : 0 ≤ A) (K : Set ℂ) (n : ℕ) :
+    (36*(n:ℝ)*r^Target.d κ) *
+      (∑ ij∈prescribedGridIndex R r K,
+        A*r^codimension κ*(prescribedGridCenter r ij).im^(-codimension κ)*
+          (ε + if (prescribedGridCenter r ij).im<η then 1 else 0)) ≤
+      (36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ))))*
+        ((R+3)^(1-codimension κ)*((n:ℝ)*ε)+(n:ℝ)*(η+r)^(1-codimension κ)) := by
+  have hα0 : 0 ≤ codimension κ := (codimension_pos κ hκ8).le
+  have hα1 : codimension κ<1 := codimension_lt_one κ hκ
+  apply (mul_le_mul_of_nonneg_left
+    (prescribed_grid_error_sum (codimension κ) hα0 hα1 hR hr hr1 hη hε hA K)
+      (by positivity : 0 ≤ 36*(n:ℝ)*r^Target.d κ)).trans
+  apply le_of_eq
+  have hp := prescribed_cleanup_power_cancellation κ hr
+  calc
+    _ = (36*A*((2*R+5)*((2:ℝ)^codimension κ/(1-codimension κ))))*
+        (r^Target.d κ*r^codimension κ*r^(-2:ℝ))*
+        ((R+3)^(1-codimension κ)*((n:ℝ)*ε)+(n:ℝ)*(η+r)^(1-codimension κ)) := by ring
+    _ = _ := by rw [hp,mul_one]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedGridHeightSum.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedGridHeightSum.lean
index c05a6ae81e333e7c9cf79ce2f14840355b46b8d5..cc649a8e1cf6f9299db04e946f9312aac261bd36 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedGridHeightSum.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedGridHeightSum.lean
@@ -1,94 +1,94 @@
-import OAI.Probability.SLE.Gauge.PrescribedCoverGrid
-import OAI.Probability.SLE.Gauge.PrescribedHeightSum
-
-namespace OAI
-
-/-! Summing the one-point height factor over the actual finite shifted
-grid. The constant depends on the bounded region, not on the mesh. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-private lemma int_row_sum (α r : ℝ) (N : ℤ) :
-    (∑ j∈Finset.Icc 0 N, (((j:ℝ)+1/2)*r)^(-α)) =
-      ∑ j∈Finset.range (N+1).toNat, (((j:ℝ)+1/2)*r)^(-α) := by
-  rw [Int.Icc_eq_finset_map,Finset.sum_map]
-  simp only [sub_zero,Function.Embedding.trans_apply,addLeftEmbedding_apply,zero_add,
-    Nat.castEmbedding_apply,Int.cast_natCast]
-
-theorem prescribed_grid_height_sum_bound (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
-    {R r : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1) (K : Set ℂ) :
-    (∑ ij∈prescribedGridIndex R r K, (prescribedGridCenter r ij).im^(-α)) ≤
-      ((2*R+5)*((2:ℝ)^α/(1-α))*(R+3)^(1-α))*r^(-2:ℝ) := by
-  classical
-  let N : ℤ := ⌈R/r⌉+1
-  have hN0 : 0 ≤ N := by
-    have hceil := Int.le_ceil (R/r)
-    have hquot : 0 ≤ R/r := div_nonneg hR hr.le
-    have hNreal : (0:ℝ) ≤ N := by dsimp only [N]; push_cast; linarith
-    exact_mod_cast hNreal
-  have hN : (N:ℝ) ≤ R/r+2 := by
-    have hceil := Int.ceil_lt_add_one (R/r)
-    dsimp only [N]
-    push_cast
-    linarith
-  let M := (N+1).toNat
-  have hM : (M:ℝ)=(N:ℝ)+1 := by
-    have h := Int.toNat_of_nonneg (show 0 ≤ N+1 by omega)
-    exact_mod_cast h
-  have hwidth : (M:ℝ)*r ≤ R+3 := by
-    rw [hM]
-    have h := mul_le_mul_of_nonneg_right hN hr.le
-    rw [add_mul,div_mul_cancel₀ _ hr.ne'] at h
-    nlinarith
-  have hcol : (2*(N:ℝ)+1) ≤ (2*R+5)/r := by
-    apply (le_div_iff₀ hr).mpr
-    have h := mul_le_mul_of_nonneg_right hN hr.le
-    rw [add_mul,div_mul_cancel₀ _ hr.ne'] at h
-    nlinarith
-  have hsub : prescribedGridIndex R r K ⊆ (Finset.Icc (-N) N) ×ˢ (Finset.Icc 0 N) := by
-    intro ij hij
-    have hmem := Finset.mem_filter.mp hij
-    have hi := Finset.mem_product.mp hmem.1
-    exact Finset.mem_product.mpr ⟨hi.1,Finset.mem_Icc.mpr
-      ⟨hmem.2.1,(Finset.mem_Icc.mp hi.2).2⟩⟩
-  have hrows : (∑ j∈Finset.Icc 0 N, (((j:ℝ)+1/2)*r)^(-α)) ≤
-      ((2:ℝ)^α/(1-α))*r⁻¹*(R+3)^(1-α) := by
-    rw [int_row_sum]
-    refine (scaled_half_integer_row_sum α hα0 hα1 hr M).trans ?_
-    exact mul_le_mul_of_nonneg_left
-      (Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg M) hr.le) hwidth (by linarith)) (by positivity)
-  calc
-    _ = ∑ ij∈prescribedGridIndex R r K, (((ij.2:ℝ)+1/2)*r)^(-α) := by
-      apply Finset.sum_congr rfl
-      intro ij hij
-      rw [(prescribedGrid_center_height hr hij).1]
-    _ ≤ ∑ ij∈(Finset.Icc (-N) N) ×ˢ (Finset.Icc 0 N), (((ij.2:ℝ)+1/2)*r)^(-α) :=
-      Finset.sum_le_sum_of_subset_of_nonneg hsub (fun ij hij _ =>
-        Real.rpow_nonneg (mul_nonneg (by
-          have hj := (Finset.mem_Icc.mp (Finset.mem_product.mp hij).2).1
-          have hjR : (0:ℝ) ≤ ij.2 := by exact_mod_cast hj
-          linarith) hr.le) _)
-    _ = (2*(N:ℝ)+1)*∑ j∈Finset.Icc 0 N, (((j:ℝ)+1/2)*r)^(-α) := by
-      have hcard : ((Finset.Icc (-N) N).card:ℝ)=2*(N:ℝ)+1 := by
-        have hh := Int.card_Icc_of_le (a := -N) (b := N) (by omega)
-        have hh' : ((Finset.Icc (-N) N).card:ℝ)=(N:ℝ)+1-(-(N:ℝ)) := by exact_mod_cast hh
-        linarith
-      rw [Finset.sum_product]
-      dsimp only [Prod.snd]
-      rw [Finset.sum_const,nsmul_eq_mul,hcard]
-    _ ≤ ((2*R+5)/r)*(((2:ℝ)^α/(1-α))*r⁻¹*(R+3)^(1-α)) :=
-      mul_le_mul hcol hrows (Finset.sum_nonneg (fun j hj => Real.rpow_nonneg (by
-        have hjR : (0:ℝ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
-        positivity) _)) (by positivity)
-    _ = _ := by
-      rw [Real.rpow_neg hr.le,Real.rpow_two]
-      field_simp
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedCoverGrid
+import OAI.Probability.SLE.Gauge.PrescribedHeightSum
+
+namespace OAI
+
+/-! Summing the one-point height factor over the actual finite shifted
+grid. The constant depends on the bounded region, not on the mesh. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+private lemma int_row_sum (α r : ℝ) (N : ℤ) :
+    (∑ j∈Finset.Icc 0 N, (((j:ℝ)+1/2)*r)^(-α)) =
+      ∑ j∈Finset.range (N+1).toNat, (((j:ℝ)+1/2)*r)^(-α) := by
+  rw [Int.Icc_eq_finset_map,Finset.sum_map]
+  simp only [sub_zero,Function.Embedding.trans_apply,addLeftEmbedding_apply,zero_add,
+    Nat.castEmbedding_apply,Int.cast_natCast]
+
+theorem prescribed_grid_height_sum_bound (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
+    {R r : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1) (K : Set ℂ) :
+    (∑ ij∈prescribedGridIndex R r K, (prescribedGridCenter r ij).im^(-α)) ≤
+      ((2*R+5)*((2:ℝ)^α/(1-α))*(R+3)^(1-α))*r^(-2:ℝ) := by
+  classical
+  let N : ℤ := ⌈R/r⌉+1
+  have hN0 : 0 ≤ N := by
+    have hceil := Int.le_ceil (R/r)
+    have hquot : 0 ≤ R/r := div_nonneg hR hr.le
+    have hNreal : (0:ℝ) ≤ N := by dsimp only [N]; push_cast; linarith
+    exact_mod_cast hNreal
+  have hN : (N:ℝ) ≤ R/r+2 := by
+    have hceil := Int.ceil_lt_add_one (R/r)
+    dsimp only [N]
+    push_cast
+    linarith
+  let M := (N+1).toNat
+  have hM : (M:ℝ)=(N:ℝ)+1 := by
+    have h := Int.toNat_of_nonneg (show 0 ≤ N+1 by omega)
+    exact_mod_cast h
+  have hwidth : (M:ℝ)*r ≤ R+3 := by
+    rw [hM]
+    have h := mul_le_mul_of_nonneg_right hN hr.le
+    rw [add_mul,div_mul_cancel₀ _ hr.ne'] at h
+    nlinarith
+  have hcol : (2*(N:ℝ)+1) ≤ (2*R+5)/r := by
+    apply (le_div_iff₀ hr).mpr
+    have h := mul_le_mul_of_nonneg_right hN hr.le
+    rw [add_mul,div_mul_cancel₀ _ hr.ne'] at h
+    nlinarith
+  have hsub : prescribedGridIndex R r K ⊆ (Finset.Icc (-N) N) ×ˢ (Finset.Icc 0 N) := by
+    intro ij hij
+    have hmem := Finset.mem_filter.mp hij
+    have hi := Finset.mem_product.mp hmem.1
+    exact Finset.mem_product.mpr ⟨hi.1,Finset.mem_Icc.mpr
+      ⟨hmem.2.1,(Finset.mem_Icc.mp hi.2).2⟩⟩
+  have hrows : (∑ j∈Finset.Icc 0 N, (((j:ℝ)+1/2)*r)^(-α)) ≤
+      ((2:ℝ)^α/(1-α))*r⁻¹*(R+3)^(1-α) := by
+    rw [int_row_sum]
+    refine (scaled_half_integer_row_sum α hα0 hα1 hr M).trans ?_
+    exact mul_le_mul_of_nonneg_left
+      (Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg M) hr.le) hwidth (by linarith)) (by positivity)
+  calc
+    _ = ∑ ij∈prescribedGridIndex R r K, (((ij.2:ℝ)+1/2)*r)^(-α) := by
+      apply Finset.sum_congr rfl
+      intro ij hij
+      rw [(prescribedGrid_center_height hr hij).1]
+    _ ≤ ∑ ij∈(Finset.Icc (-N) N) ×ˢ (Finset.Icc 0 N), (((ij.2:ℝ)+1/2)*r)^(-α) :=
+      Finset.sum_le_sum_of_subset_of_nonneg hsub (fun ij hij _ =>
+        Real.rpow_nonneg (mul_nonneg (by
+          have hj := (Finset.mem_Icc.mp (Finset.mem_product.mp hij).2).1
+          have hjR : (0:ℝ) ≤ ij.2 := by exact_mod_cast hj
+          linarith) hr.le) _)
+    _ = (2*(N:ℝ)+1)*∑ j∈Finset.Icc 0 N, (((j:ℝ)+1/2)*r)^(-α) := by
+      have hcard : ((Finset.Icc (-N) N).card:ℝ)=2*(N:ℝ)+1 := by
+        have hh := Int.card_Icc_of_le (a := -N) (b := N) (by omega)
+        have hh' : ((Finset.Icc (-N) N).card:ℝ)=(N:ℝ)+1-(-(N:ℝ)) := by exact_mod_cast hh
+        linarith
+      rw [Finset.sum_product]
+      dsimp only [Prod.snd]
+      rw [Finset.sum_const,nsmul_eq_mul,hcard]
+    _ ≤ ((2*R+5)/r)*(((2:ℝ)^α/(1-α))*r⁻¹*(R+3)^(1-α)) :=
+      mul_le_mul hcol hrows (Finset.sum_nonneg (fun j hj => Real.rpow_nonneg (by
+        have hjR : (0:ℝ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
+        positivity) _)) (by positivity)
+    _ = _ := by
+      rw [Real.rpow_neg hr.le,Real.rpow_two]
+      field_simp
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedGridLower.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedGridLower.lean
index 629096e53f12eb14201899cc27b404b1dfdc5508..d6522e349e7add6402a9c1a620d4e8ca16521e85 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedGridLower.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedGridLower.lean
@@ -1,110 +1,110 @@
-import OAI.Probability.SLE.Mass.LocalGridMass
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLE.HausdorffLimits
-
-namespace OAI
-
-/-! Eventual bounds on the exact prescribed grid imply an arbitrary-cover
-Hausdorff lower bound. Finitely many early batches are restored uniformly. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma prescribed_grid_ball_bound
-    (μ : Measure ℂ) [IsFiniteMeasure μ] (κ : ℝ) (W : Target.WienerLaw)
-    (hm1 : 0 < Target.moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
-    (n₀ : ℕ) (D : ℝ) (hD : 0 ≤ D)
-    (hcell : ∀ e : Target.Entry, Target.ValidEntry κ W e → n₀ ≤ e.1 →
-      ∀ i : ℤ×ℤ, μ (closedBall (coverGridCenter (Target.entryRadius κ W e) i)
-        (2*Target.entryRadius κ W e)) ≤ ENNReal.ofReal
-          (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ)) :
-    ∃ C : ℝ, 0 < C ∧ ∀ x : ℂ, ∀ s : ℝ, 0 < s →
-      μ (ball x s) ≤ ENNReal.ofReal (C*Target.gauge κ W s) := by
-  let F : ℝ → ℝ := fun s => ⨆ x : ℂ, (μ (ball x s)).toReal
-  have hμ (x : ℂ) (s : ℝ) : (μ (ball x s)).toReal ≤ (μ univ).toReal :=
-    ENNReal.toReal_mono (measure_ne_top μ univ) (measure_mono (subset_univ _))
-  have hbd (s : ℝ) : BddAbove (range (fun x : ℂ => (μ (ball x s)).toReal)) :=
-    ⟨(μ univ).toReal,by rintro _ ⟨x,rfl⟩; exact hμ x s⟩
-  have hbound (s : ℝ) (_hs : 0 < s) : F s ≤ (μ univ).toReal := ciSup_le (fun x => hμ x s)
-  have htail (e : Target.Entry) (he : Target.ValidEntry κ W e) (hen : n₀ ≤ e.1)
-      (s : ℝ) (hs : 0 < s) : F s ≤ (324*D)*Target.entryCost κ W e s := by
-    apply ciSup_le
-    intro x
-    have hr : 0 < Target.entryRadius κ W e := Target.radii_pos κ W ⟨e,he⟩
-    have hv : 0 < Target.entryValue e := Target.weights_pos κ W ⟨e,he⟩
-    let H := D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ
-    have hH : 0 ≤ H := mul_nonneg (mul_nonneg hD hv.le) (Real.rpow_nonneg hr.le _)
-    have hg := localGrid_mass_bound μ hr (show 0 ≤ 2*s by positivity) hH
-      (hcell e he hen) (ball x s) (fun y hy z hz =>
-        (dist_triangle_right y z x).trans (by
-          have hy' : dist y x < s := hy
-          have hz' : dist z x < s := hz
-          linarith))
-    have ht := ENNReal.toReal_mono ENNReal.ofReal_ne_top hg
-    rw [ENNReal.toReal_ofReal (by positivity)] at ht
-    have hmax : max 1 ((2*s/Target.entryRadius κ W e)^2) ≤
-        4*max 1 ((s/Target.entryRadius κ W e)^2) := by
-      apply max_le
-      · nlinarith [le_max_left (1:ℝ) ((s/Target.entryRadius κ W e)^2)]
-      · have heq : (2*s/Target.entryRadius κ W e)^2=4*(s/Target.entryRadius κ W e)^2 := by ring
-        rw [heq]
-        exact mul_le_mul_of_nonneg_left (le_max_right _ _) (by norm_num)
-    calc
-      (μ (ball x s)).toReal ≤ 81*H*max 1 ((2*s/Target.entryRadius κ W e)^2) := ht
-      _ ≤ 81*H*(4*max 1 ((s/Target.entryRadius κ W e)^2)) :=
-        mul_le_mul_of_nonneg_left hmax (mul_nonneg (by norm_num) hH)
-      _ = (324*D)*Target.entryCost κ W e s := by unfold H Target.entryCost; ring
-  obtain ⟨C,hC,hCf⟩ := Target.gauge_restore_early_batches κ W hm1 hJ n₀ F
-    ENNReal.toReal_nonneg hbound htail
-  refine ⟨C,hC,?_⟩
-  intro x s hs
-  have ht : (μ (ball x s)).toReal ≤ C*Target.gauge κ W s :=
-    (le_ciSup (hbd s) x).trans (hCf s hs)
-  rw [← ENNReal.ofReal_toReal (measure_ne_top μ (ball x s))]
-  exact ENNReal.ofReal_le_ofReal ht
-
-lemma Target.hausdorffGauge_le_hausdorff (h : ℝ → ℝ) :
-    (hausdorffGauge h).toOuterMeasure ≤ Target.hausdorff h := by
-  rw [hausdorffGauge,Measure.mkMetric_toOuterMeasure]
-  apply OuterMeasure.mkMetric_mono
-  filter_upwards [] with r
-  dsimp only [extendedGauge]
-  by_cases hr : r=∞
-  · simp only [hr,ite_true,le_top]
-  · simp only [hr,ite_false,le_refl]
-
-lemma prescribed_grid_mass_distribution
-    (μ : Measure ℂ) [IsFiniteMeasure μ] (κ : ℝ) (W : Target.WienerLaw)
-    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
-    (hm1 : 0 < Target.moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
-    (n₀ : ℕ) (D : ℝ) (hD : 0 ≤ D)
-    (hcell : ∀ e : Target.Entry, Target.ValidEntry κ W e → n₀ ≤ e.1 →
-      ∀ i : ℤ×ℤ, μ (closedBall (coverGridCenter (Target.entryRadius κ W e) i)
-        (2*Target.entryRadius κ W e)) ≤ ENNReal.ofReal
-          (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ)) :
-    ∃ C : ℝ, 0 < C ∧ μ.toOuterMeasure ≤ ENNReal.ofReal C • Target.hausdorff (Target.gauge κ W) := by
-  obtain ⟨C,hC,hball⟩ := prescribed_grid_ball_bound μ κ W hm1 hJ n₀ D hD hcell
-  let C' : ℝ≥0 := ⟨C,hC.le⟩
-  have hC' : 0 < C' := hC
-  have hg' : IsGauge (Target.gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
-  have hc : (C':ℝ≥0∞)=ENNReal.ofReal C := ENNReal.ofReal_coe_nnreal.symm
-  have hm := gauge_mass_distribution hg' μ C' hC' 1 (by norm_num)
-    (fun x r hr _ => by simpa only [ENNReal.ofReal_mul hC.le,← hc] using hball x r hr)
-  refine ⟨C,hC,?_⟩
-  intro A
-  have hm' := hm A
-  change μ A ≤ (C':ℝ≥0∞)*hausdorffGauge (Target.gauge κ W) A at hm'
-  change μ A ≤ ENNReal.ofReal C*Target.hausdorff (Target.gauge κ W) A
-  rw [hc] at hm'
-  exact hm'.trans (mul_le_mul' le_rfl (Target.hausdorffGauge_le_hausdorff _ A))
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.LocalGridMass
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLE.HausdorffLimits
+
+namespace OAI
+
+/-! Eventual bounds on the exact prescribed grid imply an arbitrary-cover
+Hausdorff lower bound. Finitely many early batches are restored uniformly. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma prescribed_grid_ball_bound
+    (μ : Measure ℂ) [IsFiniteMeasure μ] (κ : ℝ) (W : Target.WienerLaw)
+    (hm1 : 0 < Target.moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
+    (n₀ : ℕ) (D : ℝ) (hD : 0 ≤ D)
+    (hcell : ∀ e : Target.Entry, Target.ValidEntry κ W e → n₀ ≤ e.1 →
+      ∀ i : ℤ×ℤ, μ (closedBall (coverGridCenter (Target.entryRadius κ W e) i)
+        (2*Target.entryRadius κ W e)) ≤ ENNReal.ofReal
+          (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ)) :
+    ∃ C : ℝ, 0 < C ∧ ∀ x : ℂ, ∀ s : ℝ, 0 < s →
+      μ (ball x s) ≤ ENNReal.ofReal (C*Target.gauge κ W s) := by
+  let F : ℝ → ℝ := fun s => ⨆ x : ℂ, (μ (ball x s)).toReal
+  have hμ (x : ℂ) (s : ℝ) : (μ (ball x s)).toReal ≤ (μ univ).toReal :=
+    ENNReal.toReal_mono (measure_ne_top μ univ) (measure_mono (subset_univ _))
+  have hbd (s : ℝ) : BddAbove (range (fun x : ℂ => (μ (ball x s)).toReal)) :=
+    ⟨(μ univ).toReal,by rintro _ ⟨x,rfl⟩; exact hμ x s⟩
+  have hbound (s : ℝ) (_hs : 0 < s) : F s ≤ (μ univ).toReal := ciSup_le (fun x => hμ x s)
+  have htail (e : Target.Entry) (he : Target.ValidEntry κ W e) (hen : n₀ ≤ e.1)
+      (s : ℝ) (hs : 0 < s) : F s ≤ (324*D)*Target.entryCost κ W e s := by
+    apply ciSup_le
+    intro x
+    have hr : 0 < Target.entryRadius κ W e := Target.radii_pos κ W ⟨e,he⟩
+    have hv : 0 < Target.entryValue e := Target.weights_pos κ W ⟨e,he⟩
+    let H := D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ
+    have hH : 0 ≤ H := mul_nonneg (mul_nonneg hD hv.le) (Real.rpow_nonneg hr.le _)
+    have hg := localGrid_mass_bound μ hr (show 0 ≤ 2*s by positivity) hH
+      (hcell e he hen) (ball x s) (fun y hy z hz =>
+        (dist_triangle_right y z x).trans (by
+          have hy' : dist y x < s := hy
+          have hz' : dist z x < s := hz
+          linarith))
+    have ht := ENNReal.toReal_mono ENNReal.ofReal_ne_top hg
+    rw [ENNReal.toReal_ofReal (by positivity)] at ht
+    have hmax : max 1 ((2*s/Target.entryRadius κ W e)^2) ≤
+        4*max 1 ((s/Target.entryRadius κ W e)^2) := by
+      apply max_le
+      · nlinarith [le_max_left (1:ℝ) ((s/Target.entryRadius κ W e)^2)]
+      · have heq : (2*s/Target.entryRadius κ W e)^2=4*(s/Target.entryRadius κ W e)^2 := by ring
+        rw [heq]
+        exact mul_le_mul_of_nonneg_left (le_max_right _ _) (by norm_num)
+    calc
+      (μ (ball x s)).toReal ≤ 81*H*max 1 ((2*s/Target.entryRadius κ W e)^2) := ht
+      _ ≤ 81*H*(4*max 1 ((s/Target.entryRadius κ W e)^2)) :=
+        mul_le_mul_of_nonneg_left hmax (mul_nonneg (by norm_num) hH)
+      _ = (324*D)*Target.entryCost κ W e s := by unfold H Target.entryCost; ring
+  obtain ⟨C,hC,hCf⟩ := Target.gauge_restore_early_batches κ W hm1 hJ n₀ F
+    ENNReal.toReal_nonneg hbound htail
+  refine ⟨C,hC,?_⟩
+  intro x s hs
+  have ht : (μ (ball x s)).toReal ≤ C*Target.gauge κ W s :=
+    (le_ciSup (hbd s) x).trans (hCf s hs)
+  rw [← ENNReal.ofReal_toReal (measure_ne_top μ (ball x s))]
+  exact ENNReal.ofReal_le_ofReal ht
+
+lemma Target.hausdorffGauge_le_hausdorff (h : ℝ → ℝ) :
+    (hausdorffGauge h).toOuterMeasure ≤ Target.hausdorff h := by
+  rw [hausdorffGauge,Measure.mkMetric_toOuterMeasure]
+  apply OuterMeasure.mkMetric_mono
+  filter_upwards [] with r
+  dsimp only [extendedGauge]
+  by_cases hr : r=∞
+  · simp only [hr,ite_true,le_top]
+  · simp only [hr,ite_false,le_refl]
+
+lemma prescribed_grid_mass_distribution
+    (μ : Measure ℂ) [IsFiniteMeasure μ] (κ : ℝ) (W : Target.WienerLaw)
+    (hg : Target.IsHausdorffGauge (Target.gauge κ W))
+    (hm1 : 0 < Target.moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → Target.moment κ W 1^n ≤ Target.moment κ W n)
+    (n₀ : ℕ) (D : ℝ) (hD : 0 ≤ D)
+    (hcell : ∀ e : Target.Entry, Target.ValidEntry κ W e → n₀ ≤ e.1 →
+      ∀ i : ℤ×ℤ, μ (closedBall (coverGridCenter (Target.entryRadius κ W e) i)
+        (2*Target.entryRadius κ W e)) ≤ ENNReal.ofReal
+          (D*Target.entryValue e*(Target.entryRadius κ W e)^Target.d κ)) :
+    ∃ C : ℝ, 0 < C ∧ μ.toOuterMeasure ≤ ENNReal.ofReal C • Target.hausdorff (Target.gauge κ W) := by
+  obtain ⟨C,hC,hball⟩ := prescribed_grid_ball_bound μ κ W hm1 hJ n₀ D hD hcell
+  let C' : ℝ≥0 := ⟨C,hC.le⟩
+  have hC' : 0 < C' := hC
+  have hg' : IsGauge (Target.gauge κ W) := ⟨hg.1,hg.2.1,hg.2.2.1,hg.2.2.2⟩
+  have hc : (C':ℝ≥0∞)=ENNReal.ofReal C := ENNReal.ofReal_coe_nnreal.symm
+  have hm := gauge_mass_distribution hg' μ C' hC' 1 (by norm_num)
+    (fun x r hr _ => by simpa only [ENNReal.ofReal_mul hC.le,← hc] using hball x r hr)
+  refine ⟨C,hC,?_⟩
+  intro A
+  have hm' := hm A
+  change μ A ≤ (C':ℝ≥0∞)*hausdorffGauge (Target.gauge κ W) A at hm'
+  change μ A ≤ ENNReal.ofReal C*Target.hausdorff (Target.gauge κ W) A
+  rw [hc] at hm'
+  exact hm'.trans (mul_le_mul' le_rfl (Target.hausdorffGauge_le_hausdorff _ A))
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedGridStripSum.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedGridStripSum.lean
index 6a4f71507ef78b0a7ed50474622f1acc21ad6657..0bac94d88b09bbd2e6c8ce8b16e54451232bcf01 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedGridStripSum.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedGridStripSum.lean
@@ -1,97 +1,97 @@
-import OAI.Probability.SLE.Gauge.PrescribedGridHeightSum
-
-namespace OAI
-
-/-! The height weight of cells near the real boundary. The bound tends
-  to zero with the strip height, uniformly in the cleanup mesh. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-private lemma int_filtered_row_sum (α r η : ℝ) (N : ℤ) :
-    (∑ j∈(Finset.Icc 0 N).filter (fun j : ℤ => ((j:ℝ)+1/2)*r<η),
-      (((j:ℝ)+1/2)*r)^(-α)) =
-      ∑ j∈(Finset.range (N+1).toNat).filter (fun j : ℕ => ((j:ℝ)+1/2)*r<η),
-        (((j:ℝ)+1/2)*r)^(-α) := by
-  rw [Finset.sum_filter, Int.Icc_eq_finset_map,Finset.sum_map,Finset.sum_filter]
-  simp only [sub_zero,Function.Embedding.trans_apply,addLeftEmbedding_apply,zero_add,
-    Nat.castEmbedding_apply,Int.cast_natCast]
-
-theorem prescribed_grid_lower_strip_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
-    {R r η : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1) (hη : 0 ≤ η) (K : Set ℂ) :
-    (∑ ij∈(prescribedGridIndex R r K).filter
-      (fun ij => (prescribedGridCenter r ij).im<η), (prescribedGridCenter r ij).im^(-α)) ≤
-      ((2*R+5)*((2:ℝ)^α/(1-α)))*r^(-2:ℝ)*(η+r)^(1-α) := by
-  classical
-  let N : ℤ := ⌈R/r⌉+1
-  have hN0 : 0 ≤ N := by
-    have hceil := Int.le_ceil (R/r)
-    have hquot : 0 ≤ R/r := div_nonneg hR hr.le
-    have hNreal : (0:ℝ) ≤ N := by dsimp only [N]; push_cast; linarith
-    exact_mod_cast hNreal
-  have hN : (N:ℝ) ≤ R/r+2 := by
-    have hceil := Int.ceil_lt_add_one (R/r)
-    dsimp only [N]
-    push_cast
-    linarith
-  have hcol : (2*(N:ℝ)+1) ≤ (2*R+5)/r := by
-    apply (le_div_iff₀ hr).mpr
-    have h := mul_le_mul_of_nonneg_right hN hr.le
-    rw [add_mul,div_mul_cancel₀ _ hr.ne'] at h
-    nlinarith
-  have hsub : (prescribedGridIndex R r K).filter
-      (fun ij => (prescribedGridCenter r ij).im<η) ⊆
-      (Finset.Icc (-N) N) ×ˢ ((Finset.Icc 0 N).filter
-        (fun j : ℤ => ((j:ℝ)+1/2)*r<η)) := by
-    intro ij hij
-    obtain ⟨hgrid,hstrip⟩ := Finset.mem_filter.mp hij
-    have hmem := Finset.mem_filter.mp hgrid
-    have hi := Finset.mem_product.mp hmem.1
-    apply Finset.mem_product.mpr
-    refine ⟨hi.1,Finset.mem_filter.mpr ⟨Finset.mem_Icc.mpr
-      ⟨hmem.2.1,(Finset.mem_Icc.mp hi.2).2⟩,?_⟩⟩
-    rwa [← (prescribedGrid_center_height hr hgrid).1]
-  have hrows : (∑ j∈(Finset.Icc 0 N).filter
-      (fun j : ℤ => ((j:ℝ)+1/2)*r<η), (((j:ℝ)+1/2)*r)^(-α)) ≤
-      ((2:ℝ)^α/(1-α))*r⁻¹*(η+r)^(1-α) := by
-    rw [int_filtered_row_sum]
-    exact half_integer_lower_strip_sum α hα0 hα1 hr hη _
-  calc
-    _ = ∑ ij∈(prescribedGridIndex R r K).filter
-        (fun ij => (prescribedGridCenter r ij).im<η), (((ij.2:ℝ)+1/2)*r)^(-α) := by
-      apply Finset.sum_congr rfl
-      intro ij hij
-      rw [(prescribedGrid_center_height hr (Finset.mem_filter.mp hij).1).1]
-    _ ≤ ∑ ij∈(Finset.Icc (-N) N) ×ˢ ((Finset.Icc 0 N).filter
-        (fun j : ℤ => ((j:ℝ)+1/2)*r<η)), (((ij.2:ℝ)+1/2)*r)^(-α) :=
-      Finset.sum_le_sum_of_subset_of_nonneg hsub (fun ij hij _ =>
-        Real.rpow_nonneg (mul_nonneg (by
-          have hj := (Finset.mem_Icc.mp
-            (Finset.mem_filter.mp (Finset.mem_product.mp hij).2).1).1
-          have hjR : (0:ℝ) ≤ ij.2 := by exact_mod_cast hj
-          linarith) hr.le) _)
-    _ = (2*(N:ℝ)+1)*∑ j∈(Finset.Icc 0 N).filter
-        (fun j : ℤ => ((j:ℝ)+1/2)*r<η), (((j:ℝ)+1/2)*r)^(-α) := by
-      have hcard : ((Finset.Icc (-N) N).card:ℝ)=2*(N:ℝ)+1 := by
-        have hh := Int.card_Icc_of_le (a := -N) (b := N) (by omega)
-        have hh' : ((Finset.Icc (-N) N).card:ℝ)=(N:ℝ)+1-(-(N:ℝ)) := by exact_mod_cast hh
-        linarith
-      rw [Finset.sum_product]
-      dsimp only [Prod.snd]
-      rw [Finset.sum_const,nsmul_eq_mul,hcard]
-    _ ≤ ((2*R+5)/r)*(((2:ℝ)^α/(1-α))*r⁻¹*(η+r)^(1-α)) :=
-      mul_le_mul hcol hrows (Finset.sum_nonneg (fun j hj => Real.rpow_nonneg (by
-        have hjR : (0:ℝ) ≤ j := by
-          exact_mod_cast (Finset.mem_Icc.mp (Finset.mem_filter.mp hj).1).1
-        positivity) _)) (by positivity)
-    _ = _ := by
-      rw [Real.rpow_neg hr.le,Real.rpow_two]
-      field_simp
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedGridHeightSum
+
+namespace OAI
+
+/-! The height weight of cells near the real boundary. The bound tends
+  to zero with the strip height, uniformly in the cleanup mesh. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+private lemma int_filtered_row_sum (α r η : ℝ) (N : ℤ) :
+    (∑ j∈(Finset.Icc 0 N).filter (fun j : ℤ => ((j:ℝ)+1/2)*r<η),
+      (((j:ℝ)+1/2)*r)^(-α)) =
+      ∑ j∈(Finset.range (N+1).toNat).filter (fun j : ℕ => ((j:ℝ)+1/2)*r<η),
+        (((j:ℝ)+1/2)*r)^(-α) := by
+  rw [Finset.sum_filter, Int.Icc_eq_finset_map,Finset.sum_map,Finset.sum_filter]
+  simp only [sub_zero,Function.Embedding.trans_apply,addLeftEmbedding_apply,zero_add,
+    Nat.castEmbedding_apply,Int.cast_natCast]
+
+theorem prescribed_grid_lower_strip_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
+    {R r η : ℝ} (hR : 0 ≤ R) (hr : 0<r) (hr1 : r ≤ 1) (hη : 0 ≤ η) (K : Set ℂ) :
+    (∑ ij∈(prescribedGridIndex R r K).filter
+      (fun ij => (prescribedGridCenter r ij).im<η), (prescribedGridCenter r ij).im^(-α)) ≤
+      ((2*R+5)*((2:ℝ)^α/(1-α)))*r^(-2:ℝ)*(η+r)^(1-α) := by
+  classical
+  let N : ℤ := ⌈R/r⌉+1
+  have hN0 : 0 ≤ N := by
+    have hceil := Int.le_ceil (R/r)
+    have hquot : 0 ≤ R/r := div_nonneg hR hr.le
+    have hNreal : (0:ℝ) ≤ N := by dsimp only [N]; push_cast; linarith
+    exact_mod_cast hNreal
+  have hN : (N:ℝ) ≤ R/r+2 := by
+    have hceil := Int.ceil_lt_add_one (R/r)
+    dsimp only [N]
+    push_cast
+    linarith
+  have hcol : (2*(N:ℝ)+1) ≤ (2*R+5)/r := by
+    apply (le_div_iff₀ hr).mpr
+    have h := mul_le_mul_of_nonneg_right hN hr.le
+    rw [add_mul,div_mul_cancel₀ _ hr.ne'] at h
+    nlinarith
+  have hsub : (prescribedGridIndex R r K).filter
+      (fun ij => (prescribedGridCenter r ij).im<η) ⊆
+      (Finset.Icc (-N) N) ×ˢ ((Finset.Icc 0 N).filter
+        (fun j : ℤ => ((j:ℝ)+1/2)*r<η)) := by
+    intro ij hij
+    obtain ⟨hgrid,hstrip⟩ := Finset.mem_filter.mp hij
+    have hmem := Finset.mem_filter.mp hgrid
+    have hi := Finset.mem_product.mp hmem.1
+    apply Finset.mem_product.mpr
+    refine ⟨hi.1,Finset.mem_filter.mpr ⟨Finset.mem_Icc.mpr
+      ⟨hmem.2.1,(Finset.mem_Icc.mp hi.2).2⟩,?_⟩⟩
+    rwa [← (prescribedGrid_center_height hr hgrid).1]
+  have hrows : (∑ j∈(Finset.Icc 0 N).filter
+      (fun j : ℤ => ((j:ℝ)+1/2)*r<η), (((j:ℝ)+1/2)*r)^(-α)) ≤
+      ((2:ℝ)^α/(1-α))*r⁻¹*(η+r)^(1-α) := by
+    rw [int_filtered_row_sum]
+    exact half_integer_lower_strip_sum α hα0 hα1 hr hη _
+  calc
+    _ = ∑ ij∈(prescribedGridIndex R r K).filter
+        (fun ij => (prescribedGridCenter r ij).im<η), (((ij.2:ℝ)+1/2)*r)^(-α) := by
+      apply Finset.sum_congr rfl
+      intro ij hij
+      rw [(prescribedGrid_center_height hr (Finset.mem_filter.mp hij).1).1]
+    _ ≤ ∑ ij∈(Finset.Icc (-N) N) ×ˢ ((Finset.Icc 0 N).filter
+        (fun j : ℤ => ((j:ℝ)+1/2)*r<η)), (((ij.2:ℝ)+1/2)*r)^(-α) :=
+      Finset.sum_le_sum_of_subset_of_nonneg hsub (fun ij hij _ =>
+        Real.rpow_nonneg (mul_nonneg (by
+          have hj := (Finset.mem_Icc.mp
+            (Finset.mem_filter.mp (Finset.mem_product.mp hij).2).1).1
+          have hjR : (0:ℝ) ≤ ij.2 := by exact_mod_cast hj
+          linarith) hr.le) _)
+    _ = (2*(N:ℝ)+1)*∑ j∈(Finset.Icc 0 N).filter
+        (fun j : ℤ => ((j:ℝ)+1/2)*r<η), (((j:ℝ)+1/2)*r)^(-α) := by
+      have hcard : ((Finset.Icc (-N) N).card:ℝ)=2*(N:ℝ)+1 := by
+        have hh := Int.card_Icc_of_le (a := -N) (b := N) (by omega)
+        have hh' : ((Finset.Icc (-N) N).card:ℝ)=(N:ℝ)+1-(-(N:ℝ)) := by exact_mod_cast hh
+        linarith
+      rw [Finset.sum_product]
+      dsimp only [Prod.snd]
+      rw [Finset.sum_const,nsmul_eq_mul,hcard]
+    _ ≤ ((2*R+5)/r)*(((2:ℝ)^α/(1-α))*r⁻¹*(η+r)^(1-α)) :=
+      mul_le_mul hcol hrows (Finset.sum_nonneg (fun j hj => Real.rpow_nonneg (by
+        have hjR : (0:ℝ) ≤ j := by
+          exact_mod_cast (Finset.mem_Icc.mp (Finset.mem_filter.mp hj).1).1
+        positivity) _)) (by positivity)
+    _ = _ := by
+      rw [Real.rpow_neg hr.le,Real.rpow_two]
+      field_simp
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedHeightSum.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedHeightSum.lean
index f9d45b054b7a1ade60737bb583e86032fb8f6d29..4909d5808094a22785a99ba058bb050b4b5744cb 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedHeightSum.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedHeightSum.lean
@@ -1,68 +1,68 @@
-import OAI.Probability.SLE.Gauge.PrescribedRowSum
-
-namespace OAI
-
-/-! Uniform sums of the actual half-row heights, including a lower strip
-of arbitrary height. The bound is independent of the final mesh size. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma scaled_half_integer_row_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
-    {r : ℝ} (hr : 0<r) (N : ℕ) :
-    (∑ j∈Finset.range N, (((j:ℝ)+1/2)*r)^(-α)) ≤
-      ((2:ℝ)^α/(1-α))*r⁻¹*((N:ℝ)*r)^(1-α) := by
-  have he : r^(-α)=r⁻¹*r^(1-α) := by
-    rw [← Real.rpow_neg_one,← Real.rpow_add hr]
-    congr 1
-    ring
-  calc
-    _ = r^(-α)*∑ j∈Finset.range N, ((j:ℝ)+1/2)^(-α) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro j _
-      rw [Real.mul_rpow (by positivity) hr.le,mul_comm]
-    _ ≤ r^(-α)*(((2:ℝ)^α/(1-α))*(N:ℝ)^(1-α)) :=
-      mul_le_mul_of_nonneg_left (half_integer_row_sum_bound α hα0 hα1 N)
-        (Real.rpow_nonneg hr.le _)
-    _ = _ := by
-      rw [Real.mul_rpow (Nat.cast_nonneg N) hr.le,he]
-      ring
-
-lemma half_integer_lower_strip_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
-    {r η : ℝ} (hr : 0<r) (hη : 0 ≤ η) (N : ℕ) :
-    (∑ j∈(Finset.range N).filter (fun j : ℕ => ((j:ℝ)+1/2)*r<η),
-      (((j:ℝ)+1/2)*r)^(-α)) ≤
-      ((2:ℝ)^α/(1-α))*r⁻¹*(η+r)^(1-α) := by
-  classical
-  let M := ⌊η/r⌋₊+1
-  have hsub : (Finset.range N).filter (fun j : ℕ => ((j:ℝ)+1/2)*r<η) ⊆ Finset.range M := by
-    intro j hj
-    have hy := (Finset.mem_filter.mp hj).2
-    have hjR : (j:ℝ) ≤ η/r := by
-      apply (le_div_iff₀ hr).mpr
-      nlinarith
-    have hjM : j ≤ ⌊η/r⌋₊ := Nat.le_floor hjR
-    exact Finset.mem_range.mpr (Nat.lt_succ_of_le hjM)
-  have hMr : (M:ℝ)*r ≤ η+r := by
-    have hfloor := Nat.floor_le (div_nonneg hη hr.le)
-    have h := mul_le_mul_of_nonneg_right hfloor hr.le
-    rw [div_mul_cancel₀ _ hr.ne'] at h
-    dsimp only [M]
-    push_cast
-    nlinarith
-  calc
-    _ ≤ ∑ j∈Finset.range M, (((j:ℝ)+1/2)*r)^(-α) :=
-      Finset.sum_le_sum_of_subset_of_nonneg hsub (fun j _ _ => Real.rpow_nonneg (by positivity) _)
-    _ ≤ ((2:ℝ)^α/(1-α))*r⁻¹*((M:ℝ)*r)^(1-α) := scaled_half_integer_row_sum α hα0 hα1 hr M
-    _ ≤ _ := mul_le_mul_of_nonneg_left
-      (Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg M) hr.le) hMr (by linarith))
-      (by positivity)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedRowSum
+
+namespace OAI
+
+/-! Uniform sums of the actual half-row heights, including a lower strip
+of arbitrary height. The bound is independent of the final mesh size. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma scaled_half_integer_row_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
+    {r : ℝ} (hr : 0<r) (N : ℕ) :
+    (∑ j∈Finset.range N, (((j:ℝ)+1/2)*r)^(-α)) ≤
+      ((2:ℝ)^α/(1-α))*r⁻¹*((N:ℝ)*r)^(1-α) := by
+  have he : r^(-α)=r⁻¹*r^(1-α) := by
+    rw [← Real.rpow_neg_one,← Real.rpow_add hr]
+    congr 1
+    ring
+  calc
+    _ = r^(-α)*∑ j∈Finset.range N, ((j:ℝ)+1/2)^(-α) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro j _
+      rw [Real.mul_rpow (by positivity) hr.le,mul_comm]
+    _ ≤ r^(-α)*(((2:ℝ)^α/(1-α))*(N:ℝ)^(1-α)) :=
+      mul_le_mul_of_nonneg_left (half_integer_row_sum_bound α hα0 hα1 N)
+        (Real.rpow_nonneg hr.le _)
+    _ = _ := by
+      rw [Real.mul_rpow (Nat.cast_nonneg N) hr.le,he]
+      ring
+
+lemma half_integer_lower_strip_sum (α : ℝ) (hα0 : 0 ≤ α) (hα1 : α<1)
+    {r η : ℝ} (hr : 0<r) (hη : 0 ≤ η) (N : ℕ) :
+    (∑ j∈(Finset.range N).filter (fun j : ℕ => ((j:ℝ)+1/2)*r<η),
+      (((j:ℝ)+1/2)*r)^(-α)) ≤
+      ((2:ℝ)^α/(1-α))*r⁻¹*(η+r)^(1-α) := by
+  classical
+  let M := ⌊η/r⌋₊+1
+  have hsub : (Finset.range N).filter (fun j : ℕ => ((j:ℝ)+1/2)*r<η) ⊆ Finset.range M := by
+    intro j hj
+    have hy := (Finset.mem_filter.mp hj).2
+    have hjR : (j:ℝ) ≤ η/r := by
+      apply (le_div_iff₀ hr).mpr
+      nlinarith
+    have hjM : j ≤ ⌊η/r⌋₊ := Nat.le_floor hjR
+    exact Finset.mem_range.mpr (Nat.lt_succ_of_le hjM)
+  have hMr : (M:ℝ)*r ≤ η+r := by
+    have hfloor := Nat.floor_le (div_nonneg hη hr.le)
+    have h := mul_le_mul_of_nonneg_right hfloor hr.le
+    rw [div_mul_cancel₀ _ hr.ne'] at h
+    dsimp only [M]
+    push_cast
+    nlinarith
+  calc
+    _ ≤ ∑ j∈Finset.range M, (((j:ℝ)+1/2)*r)^(-α) :=
+      Finset.sum_le_sum_of_subset_of_nonneg hsub (fun j _ _ => Real.rpow_nonneg (by positivity) _)
+    _ ≤ ((2:ℝ)^α/(1-α))*r⁻¹*((M:ℝ)*r)^(1-α) := scaled_half_integer_row_sum α hα0 hα1 hr M
+    _ ≤ _ := mul_le_mul_of_nonneg_left
+      (Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg M) hr.le) hMr (by linarith))
+      (by positivity)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedInteriorCost.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedInteriorCost.lean
index a8c3f776b5ac17f0bf7998a32be376e6bdaebfc3..6522bb0fefcbcdc0d019371e46a04dfab15c936a 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedInteriorCost.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedInteriorCost.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Gauge.PrescribedCostConvergence
-import OAI.Probability.SLE.Gauge.PrescribedVisitBound
-
-namespace OAI
-
-/-! The interior adapted-failure estimate bounds the cleanup cost, while
-the one-point theorem pays for the boundary rows. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance prescribedInteriorPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance prescribedInteriorPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-theorem prescribed_missed_cost_tendsto_zero_of_interior
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (f ω) (G ω) (F ω))
-    (ν : Ω → Measure ℂ) (hν : Measurable ν) (W : Target.WienerLaw)
-    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
-    (C c R : ℝ) (hR : 0 ≤ R) (K : Set ℂ) (hK : IsClosed K)
-    (A : ℝ)
-    (hinterior : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
-      Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im →
-      P {ω | ij∈prescribedMissedIndices κ W n (e n) C c R K (range (f ω)∩K) (ν ω)} ≤
-        ENNReal.ofReal (A*(Target.entryRadius κ W (n,none))^codimension κ*
-          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*
-          Real.exp (-((2:ℝ)^n/16)))) :
-    Tendsto (fun n => ∫⁻ ω,prescribedMissedCost f ν κ W n (e n) C c R K ω ∂P)
-      atTop (𝓝 0) := by
-  let A₀ := sausagePointConstant κ*(4:ℝ)^codimension κ
-  have hA₀ : 0<A₀ := by
-    have hC := sausagePointConstant_pos κ hκ hκ8
-    dsimp only [A₀]
-    positivity
-  apply prescribed_missed_cost_tendsto_zero P f hf ν hν hκ hκ8 W J e C c R hR K hK
-    (max A₀ A) (hA₀.le.trans (le_max_left _ _))
-  filter_upwards [hinterior] with n hn
-  intro ij hij
-  let r := Target.entryRadius κ W (n,none)
-  let z := prescribedGridCenter r ij
-  have hr : 0<r := Real.exp_pos _
-  have hz : 0<z.im := (prescribedGrid_center_height hr hij).2
-  have hnon : 0 ≤ max A₀ A*r^codimension κ*z.im^(-codimension κ) := by
-    have hh : 0 < max A₀ A := hA₀.trans_le (le_max_left _ _)
-    positivity
-  by_cases hlow : z.im<Real.exp (-(n:ℝ))
-  · have hv := prescribed_missed_cell_visit_bound hB hκ hκ8 (fun ω => f ω) G F hw ν W n (e n) C c R K ij hij
-    refine hv.trans (ENNReal.ofReal_le_ofReal ?_)
-    rw [ite_eq_left hlow]
-    calc
-      _ ≤ max A₀ A*r^codimension κ*z.im^(-codimension κ) := by
-        apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg hz.le _)
-        exact mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.rpow_nonneg hr.le _)
-      _ ≤ _ := le_mul_of_one_le_right hnon (by linarith [Real.exp_pos (-((2:ℝ)^n/16))])
-  · refine (hn ij hij (le_of_not_gt hlow)).trans (ENNReal.ofReal_le_ofReal ?_)
-    rw [ite_eq_right hlow,add_zero]
-    exact mul_le_mul_of_nonneg_right
-      (mul_le_mul_of_nonneg_right
-        (mul_le_mul_of_nonneg_right (le_max_right A₀ A) (Real.rpow_nonneg hr.le _))
-        (Real.rpow_nonneg hz.le _)) (Real.exp_pos _).le
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedCostConvergence
+import OAI.Probability.SLE.Gauge.PrescribedVisitBound
+
+namespace OAI
+
+/-! The interior adapted-failure estimate bounds the cleanup cost, while
+the one-point theorem pays for the boundary rows. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance prescribedInteriorPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance prescribedInteriorPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+theorem prescribed_missed_cost_tendsto_zero_of_interior
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (f ω) (G ω) (F ω))
+    (ν : Ω → Measure ℂ) (hν : Measurable ν) (W : Target.WienerLaw)
+    (J : ℕ → ℕ) (e : ∀ n, Fin (J n) → ℤ×ℕ)
+    (C c R : ℝ) (hR : 0 ≤ R) (K : Set ℂ) (hK : IsClosed K)
+    (A : ℝ)
+    (hinterior : ∀ᶠ n in atTop, ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
+      Real.exp (-(n:ℝ)) ≤ (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im →
+      P {ω | ij∈prescribedMissedIndices κ W n (e n) C c R K (range (f ω)∩K) (ν ω)} ≤
+        ENNReal.ofReal (A*(Target.entryRadius κ W (n,none))^codimension κ*
+          (prescribedGridCenter (Target.entryRadius κ W (n,none)) ij).im^(-codimension κ)*
+          Real.exp (-((2:ℝ)^n/16)))) :
+    Tendsto (fun n => ∫⁻ ω,prescribedMissedCost f ν κ W n (e n) C c R K ω ∂P)
+      atTop (𝓝 0) := by
+  let A₀ := sausagePointConstant κ*(4:ℝ)^codimension κ
+  have hA₀ : 0<A₀ := by
+    have hC := sausagePointConstant_pos κ hκ hκ8
+    dsimp only [A₀]
+    positivity
+  apply prescribed_missed_cost_tendsto_zero P f hf ν hν hκ hκ8 W J e C c R hR K hK
+    (max A₀ A) (hA₀.le.trans (le_max_left _ _))
+  filter_upwards [hinterior] with n hn
+  intro ij hij
+  let r := Target.entryRadius κ W (n,none)
+  let z := prescribedGridCenter r ij
+  have hr : 0<r := Real.exp_pos _
+  have hz : 0<z.im := (prescribedGrid_center_height hr hij).2
+  have hnon : 0 ≤ max A₀ A*r^codimension κ*z.im^(-codimension κ) := by
+    have hh : 0 < max A₀ A := hA₀.trans_le (le_max_left _ _)
+    positivity
+  by_cases hlow : z.im<Real.exp (-(n:ℝ))
+  · have hv := prescribed_missed_cell_visit_bound hB hκ hκ8 (fun ω => f ω) G F hw ν W n (e n) C c R K ij hij
+    refine hv.trans (ENNReal.ofReal_le_ofReal ?_)
+    rw [ite_eq_left hlow]
+    calc
+      _ ≤ max A₀ A*r^codimension κ*z.im^(-codimension κ) := by
+        apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg hz.le _)
+        exact mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.rpow_nonneg hr.le _)
+      _ ≤ _ := le_mul_of_one_le_right hnon (by linarith [Real.exp_pos (-((2:ℝ)^n/16))])
+  · refine (hn ij hij (le_of_not_gt hlow)).trans (ENNReal.ofReal_le_ofReal ?_)
+    rw [ite_eq_right hlow,add_zero]
+    exact mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_right
+        (mul_le_mul_of_nonneg_right (le_max_right A₀ A) (Real.rpow_nonneg hr.le _))
+        (Real.rpow_nonneg hz.le _)) (Real.exp_pos _).le
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedLowerAngle.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedLowerAngle.lean
index 65b7ad9fafbce944919cee7eb0e3101c3c902cf7..bd21807417f311d5e4a87653b58523e90d00742b 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedLowerAngle.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedLowerAngle.lean
@@ -1,102 +1,102 @@
-import OAI.Probability.SLE.Geometry.CompactCellTail
-import OAI.Probability.SLE.Geometry.CanonicalLowerSegments
-
-namespace OAI
-
-/-! A canonical angle mixing inequality implies the prescribed-gauge
-Hausdorff lower bound. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-local instance lowerAngleMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance lowerAngleBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance lowerAngleLocallyCompact : LocallyCompactSpace upperHalfPlane :=
-  isOpen_upperHalfPlane.locallyCompactSpace
-
-theorem CanonicalMass.compact_positive_prescribed_of_canonical_angle_bound
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
-    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
-    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (δ : ℝ) (hδ : 0<δ)
-    (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f →
-      ∀ θ∈Ioo (0:ℝ) Real.pi, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤
-        2*(∫⁻ φ, f φ ∂Target.angleLaw κ))
-    (K : Set upperHalfPlane) (hK : IsCompact K) :
-    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E →
-      0<C.rational.terminal ω (K∩Subtype.val ⁻¹' E) →
-      0<Target.hausdorff (Target.gauge κ W) E := by
-  obtain ⟨N,hN⟩ := Target.eventually_entryPhysicalLevel_large_on_compact hκ W
-    (by positivity : 0<4096*Real.pi) δ K hK
-  apply C.ae_positive_prescribed_of_late_disk_tails hB hc h0 hκ hκ8 F hw hpublished
-    hgrowth W hW (fun n : ℕ => (n:ℝ)) (fun n => Nat.cast_nonneg n)
-    (fun _ _ h => Nat.cast_le.mpr h) tendsto_natCast_atTop_atTop
-    (TopologicalSpace.denseSeq ℝ≥0) (TopologicalSpace.denseRange_denseSeq ℝ≥0)
-    K hK (16*(65536*Real.pi)^dimension κ) (by positivity) N
-  intro e heN i j
-  obtain ⟨M,hM⟩ := Target.entryPhysicalLevel_bounded_on_compact κ W e.val
-    (by positivity : 0<4096*Real.pi) K hK
-  filter_upwards [tendsto_natCast_atTop_atTop.eventually (eventually_gt_atTop M)] with n hn
-  apply C.compact_cell_tail_of_angle_bound hB hc h0 hκ hκ8 F hw R he
-    D hD hDf hDp hgrowth hpublished W e δ hδ hangle
-    (prescribedCompactGridCell κ W K e i)
-    (compactSpatialMass_compact_cell K hK _ _)
-    (coverGridCenter (Target.radii κ W e) i)
-  · rintro _ ⟨z,hz,rfl⟩
-    exact hz.2
-  · intro z hz
-    exact (hN e heN z hz.1).le
-  · intro z hz
-    exact (hM z hz.1).trans_lt hn
-
-theorem CanonicalMass.positive_prescribed_all_segments_of_canonical_angle_bound
-    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
-    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
-    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (δ : ℝ) (hδ : 0<δ)
-    (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f →
-      ∀ θ∈Ioo (0:ℝ) Real.pi, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤
-        2*(∫⁻ φ, f φ ∂Target.angleLaw κ))
-    : ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, s<t →
-      0<Target.hausdorff (Target.gauge κ W) (γ ω '' Icc s t) := by
-  apply C.positive_all_segments_of_compact_lower hB hc h0 hκ F hw hpublished
-    (Target.gauge κ W)
-  intro n
-  exact C.compact_positive_prescribed_of_canonical_angle_bound hB hc h0 hκ hκ8
-    F hw R he D hD hDf hDp hgrowth hpublished W hW δ hδ hangle
-    (compactCovering upperHalfPlane n) (isCompact_compactCovering _ _)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CompactCellTail
+import OAI.Probability.SLE.Geometry.CanonicalLowerSegments
+
+namespace OAI
+
+/-! A canonical angle mixing inequality implies the prescribed-gauge
+Hausdorff lower bound. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+local instance lowerAngleMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance lowerAngleBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance lowerAngleLocallyCompact : LocallyCompactSpace upperHalfPlane :=
+  isOpen_upperHalfPlane.locallyCompactSpace
+
+theorem CanonicalMass.compact_positive_prescribed_of_canonical_angle_bound
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
+    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
+    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (δ : ℝ) (hδ : 0<δ)
+    (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f →
+      ∀ θ∈Ioo (0:ℝ) Real.pi, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤
+        2*(∫⁻ φ, f φ ∂Target.angleLaw κ))
+    (K : Set upperHalfPlane) (hK : IsCompact K) :
+    ∀ᵐ ω ∂P, ∀ E : Set ℂ, MeasurableSet E →
+      0<C.rational.terminal ω (K∩Subtype.val ⁻¹' E) →
+      0<Target.hausdorff (Target.gauge κ W) E := by
+  obtain ⟨N,hN⟩ := Target.eventually_entryPhysicalLevel_large_on_compact hκ W
+    (by positivity : 0<4096*Real.pi) δ K hK
+  apply C.ae_positive_prescribed_of_late_disk_tails hB hc h0 hκ hκ8 F hw hpublished
+    hgrowth W hW (fun n : ℕ => (n:ℝ)) (fun n => Nat.cast_nonneg n)
+    (fun _ _ h => Nat.cast_le.mpr h) tendsto_natCast_atTop_atTop
+    (TopologicalSpace.denseSeq ℝ≥0) (TopologicalSpace.denseRange_denseSeq ℝ≥0)
+    K hK (16*(65536*Real.pi)^dimension κ) (by positivity) N
+  intro e heN i j
+  obtain ⟨M,hM⟩ := Target.entryPhysicalLevel_bounded_on_compact κ W e.val
+    (by positivity : 0<4096*Real.pi) K hK
+  filter_upwards [tendsto_natCast_atTop_atTop.eventually (eventually_gt_atTop M)] with n hn
+  apply C.compact_cell_tail_of_angle_bound hB hc h0 hκ hκ8 F hw R he
+    D hD hDf hDp hgrowth hpublished W e δ hδ hangle
+    (prescribedCompactGridCell κ W K e i)
+    (compactSpatialMass_compact_cell K hK _ _)
+    (coverGridCenter (Target.radii κ W e) i)
+  · rintro _ ⟨z,hz,rfl⟩
+    exact hz.2
+  · intro z hz
+    exact (hN e heN z hz.1).le
+  · intro z hz
+    exact (hM z hz.1).trans_lt hn
+
+theorem CanonicalMass.positive_prescribed_all_segments_of_canonical_angle_bound
+    {Ω : Type} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+      R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+    (D : Set C(ℝ≥0,ℝ)) (hD : MeasurableSet D)
+    (hDf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈D)
+    (hDp : ∀ p∈D, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (δ : ℝ) (hδ : 0<δ)
+    (hangle : ∀ v : ℝ, δ≤v → ∀ f : ℝ → ℝ≥0∞, Measurable f →
+      ∀ θ∈Ioo (0:ℝ) Real.pi, canonicalAngleAverage (P.map (sourceDriverPath B hc)) κ v f θ ≤
+        2*(∫⁻ φ, f φ ∂Target.angleLaw κ))
+    : ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, s<t →
+      0<Target.hausdorff (Target.gauge κ W) (γ ω '' Icc s t) := by
+  apply C.positive_all_segments_of_compact_lower hB hc h0 hκ F hw hpublished
+    (Target.gauge κ W)
+  intro n
+  exact C.compact_positive_prescribed_of_canonical_angle_bound hB hc h0 hκ hκ8
+    F hw R he D hD hDf hDp hgrowth hpublished W hW δ hδ hangle
+    (compactCovering upperHalfPlane n) (isCompact_compactCovering _ _)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Gauge/PrescribedMissedExpectation.lean b/lean/OAI/Probability/SLE/Gauge/PrescribedMissedExpectation.lean
index 068d8d081cccb6147b28b8e1f3cf6aecc7dce6a6..a0885dae74fda5670cb2f2b0565f7c5ac7b9ab4b 100644
--- a/lean/OAI/Probability/SLE/Gauge/PrescribedMissedExpectation.lean
+++ b/lean/OAI/Probability/SLE/Gauge/PrescribedMissedExpectation.lean
@@ -1,61 +1,61 @@
-import OAI.Probability.SLE.Gauge.PrescribedCoverExpectation
-import OAI.Probability.SLE.Gauge.CoverNumerics
-
-namespace OAI
-
-/-! Expected cleanup cost is the finite sum of the actual missed-cell
-probabilities. Terminal candidate events remain outside the stopped-law
-calculation that will bound these probabilities. -/
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-local instance prescribedMissedPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance prescribedMissedPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-lemma prescribed_missed_cost_eq_sum
-    {Ω : Type*} (f : Ω → C(ℝ≥0,ℂ)) (ν : Ω → Measure ℂ)
-    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
-    (C c R : ℝ) (K : Set ℂ) (ω : Ω) :
-    prescribedMissedCost f ν κ W n e C c R K ω =
-      ∑ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
-        {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}.indicator
-          (fun _ => ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ)) ω := by
-  classical
-  unfold prescribedMissedCost
-  rw [← nsmul_eq_mul,← Finset.sum_const]
-  rw [prescribedMissedIndices,Finset.sum_filter]
-  apply Finset.sum_congr rfl
-  intro ij hij
-  by_cases h : ω∈{ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}
-  · rw [indicator_of_mem h]
-    rw [ite_eq_left]
-    exact (Finset.mem_filter.mp h).2
-  · rw [indicator_of_notMem h]
-    rw [ite_eq_right]
-    intro hp
-    exact h (Finset.mem_filter.mpr ⟨hij,hp⟩)
-
-theorem expected_prescribed_missed_cost_bound
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
-    (ν : Ω → Measure ℂ) (hν : Measurable ν)
-    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
-    (C c R : ℝ) (K : Set ℂ) (hK : IsClosed K) (bound : ℤ×ℤ → ℝ≥0∞)
-    (hb : ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
-      P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)} ≤ bound ij) :
-    (∫⁻ ω,prescribedMissedCost f ν κ W n e C c R K ω ∂P) ≤
-      ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ) *
-        ∑ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K, bound ij := by
-  simp_rw [prescribed_missed_cost_eq_sum f ν κ W n e C c R K]
-  rw [Finset.mul_sum]
-  exact expected_missed_cell_cost P _ _
-    (fun ij _ => prescribedMissedIndices_measurable f hf ν hν κ W n e C c R K hK ij)
-    (fun _ => _) bound hb
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.PrescribedCoverExpectation
+import OAI.Probability.SLE.Gauge.CoverNumerics
+
+namespace OAI
+
+/-! Expected cleanup cost is the finite sum of the actual missed-cell
+probabilities. Terminal candidate events remain outside the stopped-law
+calculation that will bound these probabilities. -/
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+local instance prescribedMissedPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance prescribedMissedPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+lemma prescribed_missed_cost_eq_sum
+    {Ω : Type*} (f : Ω → C(ℝ≥0,ℂ)) (ν : Ω → Measure ℂ)
+    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
+    (C c R : ℝ) (K : Set ℂ) (ω : Ω) :
+    prescribedMissedCost f ν κ W n e C c R K ω =
+      ∑ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
+        {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}.indicator
+          (fun _ => ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ)) ω := by
+  classical
+  unfold prescribedMissedCost
+  rw [← nsmul_eq_mul,← Finset.sum_const]
+  rw [prescribedMissedIndices,Finset.sum_filter]
+  apply Finset.sum_congr rfl
+  intro ij hij
+  by_cases h : ω∈{ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)}
+  · rw [indicator_of_mem h]
+    rw [ite_eq_left]
+    exact (Finset.mem_filter.mp h).2
+  · rw [indicator_of_notMem h]
+    rw [ite_eq_right]
+    intro hp
+    exact h (Finset.mem_filter.mpr ⟨hij,hp⟩)
+
+theorem expected_prescribed_missed_cost_bound
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (f : Ω → C(ℝ≥0,ℂ)) (hf : Measurable f)
+    (ν : Ω → Measure ℂ) (hν : Measurable ν)
+    {J : ℕ} (κ : ℝ) (W : Target.WienerLaw) (n : ℕ) (e : Fin J → ℤ×ℕ)
+    (C c R : ℝ) (K : Set ℂ) (hK : IsClosed K) (bound : ℤ×ℤ → ℝ≥0∞)
+    (hb : ∀ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K,
+      P {ω | ij∈prescribedMissedIndices κ W n e C c R K (range (f ω)∩K) (ν ω)} ≤ bound ij) :
+    (∫⁻ ω,prescribedMissedCost f ν κ W n e C c R K ω ∂P) ≤
+      ENNReal.ofReal (36*(n:ℝ)*(Target.entryRadius κ W (n,none))^Target.d κ) *
+        ∑ ij∈prescribedGridIndex R (Target.entryRadius κ W (n,none)) K, bound ij := by
+  simp_rw [prescribed_missed_cost_eq_sum f ν κ W n e C c R K]
+  rw [Finset.mul_sum]
+  exact expected_missed_cell_cost P _ _
+    (fun ij _ => prescribedMissedIndices_measurable f hf ν hν κ W n e C c R K hK ij)
+    (fun _ => _) bound hb
+
+end SLEExactGauge
+
+end
+
+end OAI